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V O L . I I
LECTURES PRESENTED AT ANIN TER N ATIO N AL SEMINAR COURSE TRIESTE, 11 SEPTEMBER - 29 NOVEMBER 1974 ORGANIZED BY THE INTER N ATIO N AL CENTRE FOR THEORETICAL PHYSICS TRIESTE
OJ- . 0 + -
INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1 976• r r *
s
CONTROL THEORY AND
TOPICS IN FUNCTIONAL ANALYSIS
Vol.II
INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, TRIESTE
CONTROL THEORY AND
TOPICS IN FUNCTIONAL ANALYSIS
LECTURES PRESENTED AT AN INTERNATIONAL SEMINAR COURSE
AT TRIESTE FROM 11 SEPTEMBER TO 29 NOVEMBER 1974 ORGANIZED BY THE
INTERNATIONAL. CENTRE FOR THEORETICAL PHYSICS, TRIESTE
In three volumes
VOL. II
INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA, 1976
T H E IN T E R N A T IO N A L C E N T R E FO R T H E O R E T IC A L P H Y S IC S (IC TP) in Trieste was established by the International Atomic Energy Agency (IA E A ) in 1964 under an agreement with the Italian Government, and with the assistance of the C ity and University of Trieste.
The IA E A and the United Nations Educational, Scientific and Cultural Organization (U N ESCO ) subsequently agreed to operate the Centre jo intly from 1 January 1970.
Member States of both organizations participate in the work of the Centre, the main purpose of which is to foster, through training and research, the advancement of theoretical physics, with special regard to the needs of developing countries.
CONTROL THEORY AND TOPICS IN FUNCTIONAL ANALYSIS IAEA, VIENNA, 1976
STI / P U B /415 ISBN 9 2 -0 - 1 3 0 1 7 6 - 6Printed by the IAEA in Austria
May 1976
FOREWORD
The International Centre for Theoretical Physics has maintained an interdisciplinary character in its research and training programmes in different branches of theoretical physics and related applied mathematics. In pursuance of this objective, the Centre has - since 1964 — organized extended research courses in various disciplines; most of the Proceedings of these courses have been published by the International Atomic Energy Agency.
The present three volumes record the Proceedings of the 1974 Autumn Course on Control Theory and Topics in Functional Analysis held from 11 September to 29 November 1974. The first volume consists of fundamental courses on differential systems, functional analysis and optimization in theory and applications; the second contains lectures on control theory and optimal control of ordinary differential systems; the third volume deals with infinitedimensional (hereditary, stochastic and partial differential) systems. The programme of lectures was organized by Professors R. Conti (Florence, Italy), L. Markus (Warwick, United Kingdom) and C. Olech (Warsaw, Poland).
A bdus Salam
CONTENTS OF VOL.II
Time-optimal feedback control for linear systems (IA E A -S M R -17 /4 6 ) ................................... 1S. MiricS
Applications of functional analysis to optimal control problems ( IA E A -S M R -1 7 /4 7 ) ........ 39K. M izukami
Non-linear functional analysis and applications to optimal control theory( IA E A -S M R -1 7 /3 0 ) ............................................................................................................................ 75C. Vârsan
Some topics in the mathematical theory of optimal control ( IA E A -S M R -17 /4 6 )................. 157T. Zolezzi
An application of Pontrjagin’s principle to the study of the optimal growthof population ( IA E A -S M R -17 /4 9 ).................................................................................................. 189Vera de Spinadel
Functional analysis in the study of differential and integral equations(IA E A -S M R -17 /5 0 ).............................................................................................................................. 201G.R. Sell
Spectral theories for linear differential equations ( 1A E A -S M R -1 7 /5 1) ..................................... 2 19G.R. Sell
Some examples of dynamical systems (IA E A -S M R -17/66) ......................................................... 227S. Shahshahani
Realization theory of linear dynamical systems (IA E A -S M R -17 / 5 3 ) ........................................ 235R E. Kalman
Basic equation of input-output models and some related topics (IA E A -S M R -17 / 8 1 ) ........... 257M. Ribaric
Support functions and the integration of set-valued mappings (IA E A -S M R -17 /5 9 ) ............. 281G.S. Goodman
Topics in set-valued integration (IA E A -S M R -17/60) ..................................................................... 297R. Pallu de la Barrière
IAEA-SMR-17/46
TIM E-O PTIM AL FEEDBACK CONTROL FOR LINEAR SYSTEMS
S. MIRICA
Department of Mathematics,
University of Bucharest,
Bucharest, Romania
Abstract
TIME-OPTIMAL FEEDBACK CONTROL FOR LINEAR SYSTEMS.The paper dèals with the results of qualitative investigations of the time-optimal feedback control for
linear systems with constant coefficients.In the first section, after some definitions and notations, two examples are given and it is shown that
even the time-optimal control problem for linear systems with constant coefficients which looked like "completely solved" requires a further qualitative investigation of the stability to "permanent perturbations" of optimal feedback control.
In the second section some basic results of the linear time-optimal control problem are reviewed.The third section deals with the definition of Boltyanskii's "regular synthesis" and its connection to
Filippov’s theory of right-hand side discontinuous differential equations.In the fourth section a theorem is proved concerning the stability to perturbations of time-optimal
feedback control for linear systems with scalar control.In the last two sections it is proved that, if the matrix which defines the system has only real eigenvalues
or is three-dimensional, the time-optimal feedback control defines a regular synthesis and therefore is stable to perturbations.
1. THE LINEAR TIM E-O PTIM AL CONTROL PROBLEM . NOTATIONS,DEFINITIONS AND EXAMPLES
We con sider an n X n rea l m atrix A € L(Rn, R n), an n X p rea l m atrix В e L (R P, Rn) and a c losed , convex, p -d im ensional polyhedron U С Rp such that 0 £ U, but 0 is not a vertex of U.
The elem ents A, B, U define the linear param etrized differential system
^ = Ax + Bu, x G Rn, u e U (1.1)dt
Rn is ca lled the phase space and U the con trol space .An adm issible con trol with resp ect to the state x' 6 R n is a p iecew ise
continuous map u :[t0, t j -» U such that the solution cpu(.; t0, XgJ^tg, t 1J - Rn o f the "con tro lled " d ifferential system
H v— = Ax + Bu(t), x(t0) = x 0 (1.2)
reaches the orig in 0 G Rn fo r the fir s t tim e at the moment t j> t Q (that is , <pu(t ;t0, x 0) € R rt\ { 0 } fo r t e [t0, tx) and 9 u(t1; t0, x 0) = 0). We say that the control u(.) steers the state x 0 to the orig in in the tim e ^ - tQ;. the solution cpu(.; t0, x 0) : [t0, t jj -► Rn is called an adm issible tra jectory and (u(.), cpu(.; t0, tj)) an adm issible pair with resp ect to the state x 0 € Rn.
1
2 MIRICA
We reca ll that a map u(.): [t0, tj] -*• U is said to be p iecew ise continuous if there exist f о = To < Ti < ••• < 7k < Tk + l = * 1 suc^ the restriction maps u| (tí , t 1 + 1 ) , i = 0, 1 , ... к are continuous and on e-sided lim its exist at the points t 0, Tj, .. . тк + 1 . Since the values o f u(.) at the discontinuity points
... тк do not count in the differential equation ( 1 . 2 ), we may take u(.) to be continuous to the right-hand at each point: lim u(t) = u(s) as s - t and t < s, and continuous at the ends of the interval [t0, tj].
If we denote b y ® ' the set of all adm issible con trols with resp ect to the state x € Rn, then the linear tim e-optim al control problem may be stated as fo llow s: given A G L (R n, R n), B G L(RP, Rn) U С RP a p-dim ensional c losed , convex polyhedron (such that 0 G U, but 0 is not a vertex of U) which define the system ( 1 . 1 ); find an adm issible control ux(.) G<% fo r each point x e Rn fo r which is not empty, such that flY(.) steers the point x to the~ orig in 0 G Rn in m inim al tim e, among all the adm issible con trols u„(.) G í / X. We say that <ïx(.) is an optim al control with resp ect to x and the c o r r e s ponding solution cpïï(.; t0, x0) : lt0, txJ -► R n of (1.2) is called an optimal tra jectory of x.
T h ere fore , the linear tim e-optim al con trol problem is a standard optim al control problem where the "term inal m anifold" or the "target" is
the orig in {0 } and the optim ality cr iterion is min( j dt). So we can apply
the general resu lts of optim al control theory to this problem , namely the theorem s of existence and maximum princip le.
Let us denote by G С Rn the set of those point x G R n fo r which the set ^ of the adm issible con trols with resp ect to x is not empty. We ca ll G the controllability set of the system .
T heorem 1.1 (existence of optim al con trols [2,6,12,20]. F or any state x e G there exists an optim al control ux : [0,T(x)]-> U which steers the point x to the orig in 0 within the m inimal tim e T (x).
Definition 1.2 (the Maximum P rincip le [2])
L et H : Rn X Rn X U -> R be a map defined by
where A* is the transposed of the m atrix A.We say that the adm issible control u(.) G <8^ , u (.):[t0, t j -* U satisfies
the maximum principle; if there exists a non-triv ia l solution X :[t0, txJ -* Rn of (1.4) such that:
't,
H(X, x, u) = <X, Ax + Bu> and let us con sider the differential system
(1.3)
(1.4)
H(Mt), q>u(t; t 0, x), u(t)) = m ax H(X(t), q>u(t; t 0, x ), u) (1.5)u e U
HfXitj), 9 u(tj; tQ, x), u(tj)) ë 0 (1.6)
where фи(.; t 0, x ^ : ^ , ^ ] — Rn is the solution through x of (1.2).
IAEA-SMR-17/46 3
Theorem 1.3 (N ecessary condition fo r optim ality [2,12,20]). If u(.) E<$/xis an optim al control with resp ect to x e G, then u(.) sa tisfies the maximum princip le.
In addition to these general resu lts , also particular resu lts w ere obtained to the tim e-optim al control problem fo r linear system s which are dealt with in the next section ; owing to these results the problem was considered as "so lv ed ".
T here are problem s we are interested in when studying optim al con trols only with resp ect to som e initial states of the system . But even in this ca se , fo r som e "p ra ctica l" reasons, we must find the optim al con trols for each state in a neighbourhood of the required optim al tra jectory since, if owing to natural perturbations (or computation erro rs ) the state deviates from the optim al tra jectory , we must know how to control the system such that the state is steered to the desired final state in the shortest possib le tim e.
T h ere fore , the problem of "synthesizing" the optimal control with resp ect to all states in a whole region or in the whole controllability set is natural from the point of view of further use of the m athem atical results of this problem .
Let us finally give two o f the sim plest exam ples o f tim e-optim al con trol problem s fo r linear system s in the plane R 2.
Exam ple 1 ([2 ,12 ,20])
A particle whose position is described by x 6 R is moving under the law x = u and the external fo r ce u is constrained to take values in the interval [-1 , +1]. The problem is to find a control u(.): [t0, tj] -» U = [-1 , +1] which steers a given initial state (x0, x0) of the particle to the final state (0 , 0 ) in the m inim al tim e.
With the notation x = x 1, x = x 2 the above problem becom es the follow ing tim e-optim al control problem in R2:
^ x2 d - 7 )
dx2 _ dt U
2 2 2 where the m atrices A e L(R , R ), В € L(R , R ) are given by
In ord er to so lve this problem we apply the maximum princip le : the function H:R 2 X R2 X U -» R defined by (1.3) is given by
H(A j, X2, x 1, x 2, u) = A.j x 2 + X2u (1.8)
4 MI MCA
and the system (1.4) becom es
S1-j\ (1.9)« 2 . 4Ж-- ’ X l
i.e . the solutions of (1.9) are o f the form (X^t), X2(t ) = (c i. "C it + c 2) where c i, C2 G R. If üx(.) :[ t0, ti] - U is an optim al con trol with respect to the point x = (x1, x 2) E R2 then from the maximum condition (1.4) we obtain: ux(t) = s ig n (-c it + c2).
It fo llow s that any optimal con trol is p iecew ise constant, takes values in the set { -1 , +1} and has at m ost one switching point. Hence any optimal tra jectory is a continuous, p iecew ise sm ooth curve obtained by "join ing" at m ost two p ieces o f solutions of the follow ing differential system s:
dx 1
dt
dt
x 2
(1 . 1 0 )±
±1
Its solutions are
(<pi(t; x0), cp2 (t; Xq)) = (± ^ — + tx2 + x¿, ±t + x 2
i.e . the parabolas
„ 1 - ^ (x2)2
Using the fact that, as we shall see in the next section, the maximum princip le is also a sufficient condition fo r optim ality in linear tim e-optim al
FIG. 1. Optim al trajectories, Case 1.
IA E A -SM R -17/46 5
control prob lem s, one can easily prove that the global p icture of the optim al tra jectories fo r the considered problem is that shown in F ig .l .
With v(x) = ux (t0) for any x = (x1, x 2) e R 2 it follow s that the optimal tra jector ies of the problem defined by the system (1.7) are the unique solutions o f the differential system
dx 2
d T = x -
% r= v(x1’ x2)
( 1 . 1 1 )
and the map v : R2 -»■ [-1 , +1 ] is given by
+ 1 if x 1 - f (x 2) 2 è 0 and x 1 + 1 (x2)2 > 0
or x 1 - \ (x2 )2 < 0 and x 2> 0 , x 1 + | (x 2)2 > 0
0 if (x1, x2) = (0 , 0 )v tx1, x2) :
- 1 if x ^ K x 2 )2 < 0 , xi + K x 2)2 0
x 2< 0 , x x + i (x2)2> 0
,1. A I v2 \2(x2)2 < 0 ,
The map v: R2 - [-1 , +1] so defined is called the tim e-optim al feedback (or c lo se d -lo o p ) control and has the follow ing property: for any x e R 2
there exists a uniqu-e solution through x of the system (1.11), cp(.;x):[0,T(x)] -» R n, such that ф (Т(х),х) = 0 and u x(t) = v(cp(t; x)). The tim e-optim al feedback control v (.), together with the global picture of the optimal tra jectories of F ig . l , is ca lled the optim al synthesis of the problem defined by (1.7).
FIG. 2. Optim al trajectories, Case 2.
6 M iracÂ
We con sider now the linear tim e-optim al con trol problem defined by the system
E x a m p l e 2 ( [ 3 ] )
where u = (u1, u2) G U = Co {u (1), u(2), u ,3) } , Co M being the convex hull o f the set M, u (1) = ( 1 , 0 ), u(2) = ( - 1 / 2 , 1 ), u (3) = ( - 1 , - l / 2 ).
F rom the maximum principle it follow s easily that the optimal tra jectories of this problem are obtained by joining "p ie ce s " of the solutions of the differential system s:
By d irect computation we find that the tim e-optim al feedback control v (.):R 2 -* U of this problem is given by
If fo r a given tim e-optim al control problem we find an optim al fe e d back control then the problem may be considered com pletely solved from the m athem atical point of view . Such a "g loba l" solution of the optimal con trol problem has the advantage that it is no longer n ecessary to compute
whenever an instantaneous perturbation deflects the state from its optimal tra jectory , an optim al control with resp ect to the new state is available.
H ow ever, the follow ing problem is still open: what happens to the solutions o f the system defined by the tim e-optim al feedback con trol if som e permanent perturbations occu r, as e .g . in a p rocess of num erical computation o r m odelling on a "technical d ev ice ".
One can see that if som e permanent perturbations occu r in the feedback control o f the system o f exam ple 2 , then the points in a neighbourhood of the segm ent ( - 1 , 0 ) on the x x-axis do not reach the orig in within a finite tim e. T h ere fore , som e optim al feedback con trols are unstable to som e sort o f perturbations.
( 1 . 1 2 )
(i) (1.13)
i = 1, 2, 3, U(i) = (U(i) , ufi) )
u(1) if x 2 = 0 , x 1 < 0
v fx 1, x 2) = - u(2) if x 1 < 0 , x 1 +| - I e - *2 s 0
u (3) if x2 > 0 , x 1 + 1 - \ e~x s 0 o r if x 1 + j - f e ”x > 0
(1.14)
and the global p icture of the optimal tra jectories is that shown in F ig .2.
the optim al solution with resp ect to each point of the phase space and b esides,
IA E A -SM R -17/46 7
2. THE BASIC RESULTS FOR THE LINEAR TIM E-O PTIM AL CONTROL PROBLEM
Definition 2.1 (the general position [2]). We say that the m atrices A 6 L (R n, R n), В e L (R P, R n) and the polyhedron U С Rp are in the general position if fo r any edge w of U the vectors Bw, ABw, .. . , An-iBw are linearly independent.
It can be easily proved that (A, B, U) are in the general position if and only if fo r any edge w of U the vector Bw does not belong to any proper invariant subspace with resp ect to the linear map defined by the m atrix A.
Rem ark 2.2
If (A, B , U) are in the general position then the rank of the m atrix (В, AB, .. . An B) is n and the pair (A,B) is said to be com pletely con trollab le ; in this ca se , if the control space U coincides with the whole space Rp, the controllability set G of the system (1.1) is the whole space R n.
If (A, B, U) are in the general position and the polyhedron U С Rp contains the origin in its in terior then the system ( 1 . 1 ) is said to be norm al o r non- degenerate^ [2 , 1 0 ,1 2 , 2 0 ].
If p = 1, U = [-1 , +1] С R and В = b 6 Rn = L (R ,R n) then (A,b,U) is in a general position if and only if the pair (A,b) is com pletely controllable.
F or proofs of these statem ents as w ell as fo r other results and com m ents the reader is re ferred to [2 , 1 0 , 1 2 , 2 0 , 2 1 ].
T heorem 2.3 (sufficient condition fo r optimality [2]). Let A 6 L (R n, Rn),В £ L(RP, Rn) and U С RP such that (A, B, U) are in the general position.Let x 6 G and u (.) :[t0, t tJ -> U be an adm issible control with resp ect to x that sa tisfies the maximum princip le. Then u(.) is an optim al control with resp ect to the point x.
P roo f
We assume that u(.) is not optim al so that there exists an adm issible con trol u(.):[to, t2] -*• U, t2 < ti, with respect to x. If X (.):[t0, t j -► Rn is the non-triv ia l solution of (1.4) which satisfies (1.5) and (1.6), then d /d 7 <X(T),<Pu( t ; t0,x)> = <X(also < ( X ( t ) , B u ( t ) > and therefore , since 9 u(tx; t0, x) = cpjr(t2; t0, x) = 0 6 R n it follow s that<X(t2), cpu(t2; t0, x) У = (t^), cpu(t2, t0, x)> - ^Xft^,
4cpù(t2; t0, x )> -< X (tQ), cpu(t0; t0, x )> +<X(t0), cp~(t0; t0, x) > = / [<X (t), Bu(t)> • '(X(t), Bu(r)^]dT s 0. 10
On the other hand, from (1.5) and from the fact that 0 6 U it follow s that <X(t), Bu(t) У è 0 fo r any t e [t0, t jj hence <X(t2), cpu(t2; tQ, x)>
4= - / <X(t), B u ( t ) > dr s 0 so <X(t2), <T>u(t2 ; to, x)> = 0 and <Х(т), сри(т; t0, x)> = 0
l 2
fo r any г e [t2 , tx] .F rom the fact that 0 € U is not a vertex o f U it follow s that there exists
an edge w of U such that <X(t), B w ) = 0 fo r any t e [t2, t j . Taking the derivatives with resp ect to t o f the above identity and, from the fact that X(.) is not triv ia l, it follow s that the v ectors Bw, ABw, .. An"1bw are linearly dependent; this is a contradiction.
8 MIRICA
T heorem 2.4 (the range of tim e-optim al con trols [2,20]). F or any non- triv ia l solution X (.):[t0, tj] -► Rn of the system (1.4), condition (1.5) defines uniquely a p iecew ise constant adm issible control u(.):[to, t j -*■ U with respect to the point x = cpuftp; tj, 0) and u(.) takes values only in the set V of sill v ertices of U.
P roo f
Since the map u O - ( t ) , B u ) is linear, its m axim al values on theconvex closed polyhedron U are reached on the set V of ve rtices of U. Itfollow s that there exists either a single vertex v of U such that <A(t), B v )= max <(X(t), B u ) or an edge w of U such that O-(t), B w ) = 0. Using the
u e Ufact that (A, B, U) are in the general position and that the map tn> <(X(t), B u ) is analytic, one can prove easily that there exists only a finite number of points t0 = t 0 < 7i < . . . < т к < т к + 1 = ti such that, fo r t = t 1# . . . т к, the vertex v at which u -» <(X(t), B u ) is not uniquely determ ined. It follow s im m ediately that the map u(.) defined by (1.5) is constant on any interval ( t j , T j + 1 ) , , j = 1 , 2 , . . . к and the theorem is proved.
T heorem 2.5 (the number of switching points [2,20]). Let U С Rp be the parallelepiped U = {u - (u1, .. .u p), a1 s u1 s b1, i = 1, 2, . .p } , a1, b 1 € R and suppose that the m atrix A has only rea l eigenvalues. Then for tim e-optim al control u(.) = (u1(.), .. uP(.)):[t0, tiJ -» U with resp ect to a point x € G , any com ponent u‘ (.) of u(.) is a p iecew ise constant map taking only the values a ! o r b 1 and it has at m ost n - 1 switching points (i.e . d iscontinuities).
P roo f
If the m atrix В € L(RP, Rn) has the elem ents (b ¡j), i = 1, 2, .. u, j = 1, 2, .. . p and if X(.) = (X1^), ..,X n(.)):[t0, tx] -* Rn is the non-triv ia l solution
p rnof (1.4) which satisfies (1.5) then max <X(t), B u ) = max (£ } X¡(t) b*)u' ;
u s a i = 1 а> <=и* s b* i = lso fo r any i = 1 , 2 , .. p, u ‘ (t) is either a1 o r b 1 and is p iecew ise constant accord ing to theorem 2.4. Further on, from the general theory of linear d ifferential system s with constant coefficien ts it follow s that, if the m atrix A has the rea l eigenvalues аг, . . . , ak è R with the orders of m ultiplicity
кm j, . . . , m k respective ly , X / m s = n< then there exist the polynom ials
e = i nï^ (t), j = 1, 2, .. . p, i = 1, .. к such that degPjj(t) s т { -1 and Yj ^-i(t)bj =
i = lк
23 (t)-fa£t. j = 1,2, .. . L The theorem results from the follow ing lem m ai = iwhich can be easily proved by induction
Lem m a 2.6 ([2 0 ]) . Let a-¡, a? , .. . g R be p a ir-w ise distinct rea ls and let Pj (t), ... Pk (t) be polynom ials with rea l coefficien ts such that
к кdeg P„ (t) s m , - 1 ; then the map t>-» Yj Р» (t )ia<!t has at m ost X / m f - 1 zeros .
i=i i=i
T heorem 2.7 (uniqueness o f the tim e-optim al con trols [2,20].) Letux(.) :Lt0, tj] ->■ U, u2 (.):[tQ, t2] -* U be optim al con trols with resp ect to x € G;t h e n U i ( . ) a n d u 2( . ) a r e d e f i n e d o n t h e s a m e i n t e r v a l ( i . e . t j = t ^ a n d U j(t )= u2 (t) fo r any t e [t0, t j ] .
P roo f
F rom the optim ality o f u-^.) and u2(.),it follow s that % = t2. Further on, using the varia tion s-of-constants form ula we have
0 = eAt>[x0+ J' е"Ат В u ^ ^ d -r] = eAt‘ [x0+ J' е ‘ Ат Bu2 (r)dT]t0 о
Henceti 4
J' е"АтВ Uj(t ) dT = J e"ArB u2 (-r)dTto to
and if we take the sca la r product of both sides o f the last equality to A^to) where t j -» Rn is the non-triv ia l solution o f (1.4) correspondingto the optim al control иг(.), then we have
t i 4
J <X1 (t), В Ul(t) > dt = J <Xj(t), В u2 (t)> dt4 to
using (1.5) we obtain «(Xj^t), В UjCt) = ^Xj(t), B u 2 (t)) whence, with theorem 2.5, it follow s that ux(t) = u2 (t) fo r any t e [ t 0, t-J and the theorem is proved.
Rem ark 2.8
Since the linear d ifferential system (1.1) has constant coe ffic ien ts , we can always make a translation of the time variable t fo r the tim e-optim al con trol ui.^ttg, tx] — U such that we obtain fo r each point x e G a unique optim al con trol ux:[0 ,T (x)] -*■ U with resp ect to the point x € G, T(x) being the m inim al tim e in which x can be steered to the origin by adm issible con tro ls. The functionT :G -► R defined in this way is ca lled the m inim al-tim e function; in R e f.[8 ] it was proved that, if the system (1.1) is non-degenerate, T (.) is continuous.
Rem ark 2.9
In Refs [8 ] and [20] it was proved that the map v: G-*U defined by: v(x) = ux(0), x 6 G being w ell defined (v(0) = 0), takes values only in the set V of the v ertices of U and has the property that any optim al tra jectory is a solution to the d ifferential system
1 0 MIRICA
that is , if ф(.; x ):[0 ,T (x )] ->• G is the solution through x(cp(0;x) = x) of (2.1), then ux(t) = v(<p(t; x)). The map v: G -»■ U with this property is ca lled the tim e-optim al feedback control.
3. THE REGULAR SYNTHESIS AND RIGHT-HAND-SIDE DISCONTINUOUSDIFFERENTIAL EQUATIONS
The sim plest exam ples o f tim e-optim al control problem s show that the tim e-optim al feedback con trol v: G ->■ U (Rem ark 2.9) is a discontinuous map so that the system (2 . 1 ) is a right-hand-side discontinuous d ifferential system .
It is w ell known that fo r such differential system s the concept of a solution, even in C arathéodory 's sense, is no longer useful since one cannot obtain theorem s o f existence, dependence on the initial data, etc. At present it is generally accepted that the best notion o f a solution fo r right-hand-side discontinuous d ifferential equations is that given by F ilippov in R ef.[7],
On the other hand, the tim e-optim al tra jectories are Carathéodory solutions of the discontinuous differential system (2 . 1 ) and so we cannot use forthwith F ilippov 's resu lts fo r the study of tim e-optim al tra jector ies .
As the tim e-optim al feedback control is not yet very w ell known in general, we must make som e assumptions about the behaviour o f optimal tra je cto r ie s . We have a fa irly good description of this ob ject if we assume that it defines a regu lar synthesis.
Definition 3.1 (regular synthesis [2,14,15])
We say that the tim e-optim al feedback control v: G -» U defines a regu lar synthesis if there exists a partition of G into a fam ily E = и г*1) и ... u E(n) of disjoint subsets of G such that the properties (A), (B), .. (H) are satisfied :
(A) F o r any к = 0, 1, .. n, the fam ily E(k) is a set o f connected differentiable subm anifolds o f G, of dim ension k, ca lled the к -c e lls of the synthesis;
(B) {0 } G E(0) and E(0) as a set o f points in G does not have lim it points in G;
(C) F o r any ce ll g G E, a f {0 } , the restr iction map vn = v|a o f v to a is a constant map talcing values in the set V of the v ertices of U;
(D) E very ce ll cr G E of the synthesis is either of type I or of type II; the 0 -c e lls (i.e . the points in E<°)) are of type II and the n -ce lls are of type I; if we denote by EMTIthe sets of к -ce lls of type I and type II, respective ly , then we may write
I = uк = 1
n I (3.1)
(к)(E) F or any к = 1,2, .. n and fo r any к - ce ll o f type I, g £ ET , the
IA E A -SM R -17/46 1 1
vector fie ld x -» f0 (x) defined by
f0 ( x )= A x + bv0 (3.2)
is tangent to g and fo r any point x € g there exists a unique Carathéodory solution ф(.;х) through x of the system (2 . 1 ) which coincides on an interval ( - 6 , +e) with the solution cp0 (.;x) through x of the system
= Ax +-bv0 (= f0 (x)) (3.3)
i.e . it rem ains on a on the interval ( -e , +e), e > 0 .M oreover, there exists a (k - l ) - c e l l П (g) £ such that any solution
of the system (2 . 1 ) starting at a point in g reaches n (g ) transversally (fo(x) ^ TYg), in a finite tim e;
(F) F or any к - 0, 1, .. n-1 and fo r any к -c e ll of type II,g g E^] g f { 0 } , there exists a (k+1)-cell o f type I, E (g) g E*k+1) such that from any point ofg a unique solution of the system (2.1) w ill start which enters E(g) and in tersects with a only at the starting point. M oreover, v0 = V£(a) ;
(G) F or any point x € G, the unique solution <p(.;x) o f (2.1) through x, ca lled the optim al tra jectory (which reaches the orig in 0 e Rn at the m inim al tim e T (x)) in tersects with only a finite number of ce lls ;
(H) The fam ily E o f all ce lls of the synthesis is lo ca lly finite, that is , for any point x £ G there exists a neighbourhood W o f x which in tersects with only a finite number of ce lls .
Rem ark 3.2.
In this definition we dropped the condition of R ef.[2] that the sets P°, P 1, .. Pndefined by
P ° = U {g|g G E(0) } , P k = U {g|g € E (k>} U Рк'\ к = 1, 2, .. . n (3.4)
are p iecew ise smooth sets of dim ension 0 , 1 , .. n, respectively , since we do not need it; this condition was used in R ef.[2] only to prove that the unique solutions of the discontinuous differential system (2 . 1 ) are the corresponding optim al tra jector ies while in our ca se this fact follow s from other reasons.
Rem ark 3.3
F rom definition 3.1 it fo llow s that for any ce ll g e E, a f {0 } there exists a finite number o f ce lls of type I, g j, g 2, .. . g{ € Ej such that any optim al tra jectory starting at a point o f g passes through g j, g2, ... gj b e fore it reaches the orig in . T h ere fore we have
П (gj ) = { 0 } , g j = и if cr 6 E j, g j = E(a) if g G En , gj+ j = П (gj) if
n (g j) G Ej, gj + 1 - E (n (g j)) if П (gj) e Ejj fo r j = 1, 2, . . . i - 1
1 2 MI MCA
Further on, if we denote by 0 < Tj(x) < .. . < r 5(x) = t(x) the switching points of the optim al control 1Ц,: [0, T (x)] ->• V with resp ect to the point x, that is , if we have
ux(t) = vQi fo r t e [0jT l(x)) and ux(t) = v fo r t e It^ jIx), т ;(х )) (3.5)
and j = 2, 3, .. jI, then we know from R e f.[l4 ] that the maps x 1-*- t ¡(x ), x « X;(x) = ф(^ (x);x), j = 1 , 2 , . . . jÍ, are differentiable.
We re ca ll also the definition of the F ilippov solution:
Definition 3.4
If f : G С Rn « Rn is a m easurable map which is bounded on any com pact subset of G then the map ф: [tn, t-j -» G is ca lled a F ilippov solution of the system
§ ■ f (x ) (3 .6 )
i f ф is an absolutely continuous map, and fo r a lm ost all t [tn, t-J (with resp ect to the Lebesgue m easure) we have
| £(t) e F(0(t)) (3.7)
where the set-valued map F : G -> ^ ( R n) is given by
F(x) = П П co f(B 6 (x )\ E ) (3.8)6 > 0 |i (E )=0
(со M denotes the convex c losu re of the set M, ц denotes the Lebesguem easure, B¿(x) denotes the open ball o f radius ó centred at x and (Rn)denotes the fam ily of all the subsets of Rn).
Rem ark 3.5
If the tim e-optim al feedback control v : G -*• U defines a regular synthesis then it follow s im m ediately that the map f : G -* Rn given by
f(x) = Ax + Bv(x) (3.9)
is m easurable and bounded on any com pact subset of G (in fact it is p ie ce - w ise differentiable) and the set-valued map F : G ->■ (Rn) defined by (3.8) is given in this case by
F(x) = co { A x + B v&|ct 6 £ (n), x G a} (3.10)
In this ca se ,a s proved in [16], any optim al tra jectory is a F ilippov solution to the system (2 . 1 ) if an only if fo r any ce ll of type I, <r 6 E j, and fo r any point x G a, there exists an n -ce ll G E(n) such that x 6 ? ¡ and
IA E A -SM R -17/46 1 3
L et us suppose now that the right-hand side of (2.1) is "slightly" perturbed by a map g : R X G -*■ Rn, that is , the "re a l" tra jectories o f (2.1) are solutions of the differential system
•jji = f(x) + g(t,x) (3.11)
and there exists an e > 0 such that 11 g(t,x) 11 s e fo r any (t,x) £ R X G.The only way of studying the system (3.11) is to use F ilippov 's
solutions fo r right-hand-side-discontinuous differential equations.With the above concept of solution Filippov has proved in R ef.[7 ] that
analogous resu lts are obtained fo r alm ost all fundamental theorem s in the general theory of ordinary differential equations with a continuous (in x) right-hand side.
In particular., he proved the follow ing resu lt concerning the continuous dependence of solutions on the right-hand side of the equation.
T heorem 3.6 ([7] )
Let f: G С R n-> Rn be a m easurable, bounded map such that the differentia l- system
= f(x) (3.12)
has a unique cp(.;x) to the right solution in the sense of F ilippov, cp(.;x):[0, t : ]-♦ G,cp(0;x) = x, through each point x £ G.
Then fo r any x £ G, and any e > 0 there exists a 6 > 0 such that fo r any m easurable mapi g^O.tj] X G -> R n such that || g(t,x) || s <//(t) a.e. on [0, t: ]_ _
where j (//(t)dt < 6 , the F ilippov solution cp(.;x) through x of the differential о
system
~ = f(x) + g(t,x) (3.13)
is defined on the interval [0 , t-,] and || <p(t;x) - <p(t;x) || < e a.e. on [0 ,t1].It follow s that, if all optim al tra jectories are F ilippov solutions o f the
system (2.1), we may derive from F ilippov 's resu lts the properties needed fo r the tim e-optim al feedback con trol, in particular a good behaviour of the optim al tra jectories (as solutions of (2 . 1 )) with respect to perturbations o f the right-hand side o f (2 . 1 ).
On the other hand, according to a result o f H erm es (Ref. [9]), if there exists an optim al tra jectory which is not a F ilippov solution of (2.1), then the-system (2 . 1 ) is not stable to e r ro rs of m easurem ents o f the state.
Definition 3.7 (stability to m easurem ents [9])
Let f: G С Rn -* Rn be a m easurable and bounded map such that through each point x £ G passes a C arathéodory solution cp(.;x):L0, t 1 J -*■ G of the system (3.12). Then f(.) is ca lled stable to m easurem ent if fo r any x e G any T > 0 and any e > 0 there exists a 6 > 0 such that for any m easurable
map g:[Q,T] -> Rn such that Ц g Ц = ess sup { || g(t) ||, t 6 [0 ,t ]} < ó for which there exists a F ilippov solution cp(.;x):[0,T] •* G of the differential system
g = f ( x + g(t)) (3.14)
then
II ф(.;х) - ф(.;х) ü = ess sup { ü ç(t;x) - <p(t;x) ü, t e [ 0 ,T ] } < e
T heorem 3.8 (instability to m easurem ents [9])
If there exists a Carathéodory solution of (3.12) which is not a F ilippov solution then f(.) is not stable to m easurem ents.
One situation in which it is intuitively obvious that system (2.1) is not stable to perturbations of the tim e-optim al feedback control is that in which the regular synthesis contains a so -ca lle d "slid ing reg im e" or "universal su rface" (R ef.[lO ]).
Definition 3.9 (universal surfaces [16])
An (n - l ) - c e l l a o f type I of the regular synthesis, rea lized by the vertex v o f U is said to be a universal surface of the synthesis if there exist two n -ce lls Oj and g2 such that cr = П(а-[) = П(сг2) and ГНст-,) О П (стя) f ф, П(а; ) being the set of points in a reached by optimal tra jectories com ing from ai, i = 1 , 2 .
Theorem 3.10 ([16 ])
If the regu lar synthesis of the problem defined by system (1.1) contains a universal su rface then there exist optimal tra jectories which are not F ilippov solutions fo r the system (2.1) and so the system (2.1) is not stable to m easurem ents.
If system (1.1) is 2 -dim ensional, that is if n = p = dim U = 2 then, using the behaviour of the solutions o f a 2 -dim ensional linear differential system with constant coe fficien ts , we can find easily the cases in which the synthesis contains universal su rfaces. If we denote by V j,.. vk the vertices of the polygon U and fo r any vertex v¡ we define the cone A¡ = {X e Rn|■( X, B (vj - Vj))>ë 0} fo r any vertex v¡ fo r which v¡-Vj is an edge o f U, then it is easy to see that if there exist tra jectories of (1.4) entering through either side o f it, then the vertex v¡ defines a universal surface. This situation may occu r only if the m atrix -A *, which defines the system (1.1), has rea l distinct eigenvalues and the cone Ai in tersects only that eigenspace o f -A * which correspon ds to its greatest eigenvalue.
In R efs [3] and [4] Brunovsky has proved much m ore detailed results to this problem :
T heorem 3.11 ([3])
L et n = p = 2 and le t the system (1.1) be norm al (0 £ int U). Then the optim al tra jector ies coincide with the corresponding F ilippov solutions
14 MIRICA
o f the system (2.1) if and only if no vertex v¡ of U exists such that the
corresponding cone A¡ in tersects only that eigenspace of -A * which correspon ds to its greatest eigenvalue and does not in tersect the other eigenspace.
The stability resu lts proved in R ef.[4] to the sam e problem are s tr icter than those which can be obtained from F ilippov 's theorem 3.6.
IA E A -SM R -17/46 15
4. A CASE OF STABILITY UNDER PERTURBATIONS OF TIM E-OPTIM AL FEEDBACK CONTROL
The aim of this section is to prove the follow ing:
T heorem 4.1
If the tim e-optim al feedback control of the problem (1.1) holds, where В = b € R n, and U = [-1 , +1] defines a regular synthesis (definition 3.1), then any optim al tra jectory is a F ilippov solution to the system (2 .1).
P roo f
A ccord in g to Rem ark 3.5, it is sufficient to prove that fo r any ce ll o f type I, a G E j, and fo r any point x G a there exists a n -ce ll UjG such that x E cti and v0 = v„. This statement follow s from the next four lem m as:
Lem m a 4.2
Let x n G G and let { x p} be a sequence of points from G such that lim x p = xn. Denote up(.) = Uxp(*) and extend Up(.) as w ell as <p(.;xp) by
uD(t) = 0 and q>(t;Xp) = 0 fo r t a T (xD).Then lim <p(t;x_) =cp(t;xn) uniform ly on the interval [0 ,T (xn)] and
P —> oo
lim u (.) = un(.) weakly in the space L 2(0,T(x J).p —* «s “
M oreover, there exists a subsequence {u p ^ .)} o f {up(.)} and a finite sequence o f points Тч, t, , .. . Tm G [о , T(Xn)] such that lim u (t)= uQ(t)
Pk kfo r any t G Т2, . . . , тт } .
P ro o f
We firs t prove that u p(.) -»■ u„(.) weakly in L 2 (0,T(xq)). A ssum e the con trary . Then, there exists a neighbourhood of u0(.) in L 2 (0 ,T (x0))(in the weak topology) and a subsequence (upk(.)} of {u p(.)} such that Upk ( . ) fo r any p, . F rom R e f .[ l2 ] , (Chapter 2, Appendix, Lem m a 1A) it fo llow s that there exists a subsequence {u k (.)} of {u ^ f.) } which is weakly convergent to a control u*(.). W ithout^oss of generality we may assum e that the sequence {u Pk{.)} itse lf converges weakly to u*(.). Let us denote by q>*(.) the solution of
-тг = A x + bu^ft) (4.1)
1 6 MIRICA
which satisfies ф*(0) = x 0. Then, fo r every t e [0 ,T (x0)] we have
t
<p(t; x p) - cp*(t) = e tA(xp- x q ) + J b(up(s) - u*(s))ds (4.2)0
t
Since the map u(.) - J bu(s)ds is a linear functional on L2 (0, Tfxg)) о
and Up f.) - u(.) -*• 0 weakly, it follow s from (4.2) that
lim <p(t; xD) = ф*(t) for any t e [0, T (x 0)] (4.3)P CO
Further on, we have
II Ф*(Т(х0)) II S ||ф*(Т(х0)) -tp(T(x0); xp) К + II ф(Т(х0); з )|| (4.4)
F rom (4.3) it follow s that the firs t term of the right-hand side of (4.4) tends to zero as p -* As to the second term , since ф (Т(хр); x p) = 0, we have
II ф(Т(х0); x p) II s ||ф(Т(х0); x¡) -ф (Т (хр); xp)||s (L + 2 |] b || ) |t(x„) -T(Xp)|
where (4.5)
L = a max{|| AetA|| , t e [0, ¡3 ]} , a = sup ||xj( < °o, /3 = sup T (xJ < °oII It p II p n p p
(note that the sequences {x p} , (T (x p)} are bounded, being convergent). Thus, the second term of the right-hand side o f (4.4) also tends to zero and so Ф * ( Т ( х 0)) = 0 which means that the control u(.) is adm issible with respect to the point Xq. F rom the uniqueness of optim al con trols it follow su*(.) = u0(.) and this contradicts the fact that uPk(.) $ . The firs t statement of the lem m a follow s now from (4.3), the uniform convergence of { ф ( . ; х р) } being the consequence of (4.3) and of the uniform L ipschitzian continuity o f the functions ф ( . ; х р) .
T o prove the second part o f the lem m a, we note that, owing to the lo ca l finiteness of the fam ily E o f the ce lls of the synthesis, we may assume without lo s s o f generality that all points x p, p ï 1 belong to the sam e ce ll, u. T h ere fore , every control up(.) has the same number of switching points,0 < t1 (Xp) < ... < .. <Tm(xp). Since (T j(xp)} is bounded we can choose a subsequence {x Pr} of {x p} such that all sequences {T j(xp ) } , j = 1 , .. m are convergent. Denote т, = lim r, (x ) j = 1, 2, .. m. The fact that
D —► oo *r rUp (t) -» u0(t) as p r -*■ °° for any t e [0, Т (х 0) ] \ { т , , . . . , тт } is an im mediate consequence of the weak convergence of {u p (.)} and the fact that fo r any e > 0 there exists a pe such that uPr(.) is constant on the intervals [tj + e, rj + 1- e], j = 0 , 1 , .. m - 1 , fo r any pr > e.
C orollary 4.3
Let a € £ , be a ce ll o f type I and xn 6 G such that x n 6 g. If vnf v fx j then xn e П(стп).
IA E A -SM R -17/46 17
P r o o f
S ince x0 G cf, th e re e x is ts a sequence { x p} С a such th a t x p-* x0 as
p -» °o; a c co rd in g to le m m a 4.2 we c a n choose the sequence {xp} such
th a t the c o r re s p o n d in g sequences {T j(x )} , j = 1 , 2 , .. m of the sw itc h in g
po in ts o f the c o n tro ls u .( .) conve rge : l im (xp) = t¡ and th e re fo re
u p(t) - u 0(t) fo r any t G [0 ,Т (х0) ]\ {т 1, т2Г Л ., }. S ince up(t) = v0 f v (x0)
fo r t G [0, TjfXp)) by hy p o th e s is , i t fo llo w s th a t т1 = 0 ( if Tj > 0 then
u p(t) = va -*■ u 0(t) f v (x 0) fo r any t G l O , ^ ) ) . Now , s in c e ф(т1 (хр); х р)еП (a) and
T i ( x p) - O x p -»x0 a s p - * «> , i t fo l lo ws th a t ф ^ Х р ) ; x p) = ф ^т ^Х р ); x p) - x Q
as p ->■ °o, w h ich p ro ves x 0 G П (a).
L e m m a 4.4
L e t k G {1 , 2, .. n-1}, le t a G E*k) be a c e ll o f type I and xq G ct. T hen
th e re e x is ts e ith e r an n- ce ll crq G E(tf) such th a t x 0 G <jq and v0£| - vn o r
a (k+ 1)-cell gn G E(1k + i) o f type I s uch th a t П (cr0) = a (and hence van = -v0)
and such th a t x0 G П (ст0) w here fl (ст0) deno tes the se t of po in ts in П (сто)
re ache d by t r a je c to r ie s c o m in g fr o m ct0.
S ince the f a m ily E o f a l l c e lls of the syn th e s is is lo c a l ly f in ite , th e re
ex is ts o n ly a f in ite n u m b e r of n - ce lls , a x, cr2, .. crq e E(ni such th a t
X q G c t i П . . . П CTq.C o n s id e r a c e ll cr G E¡ o f type I and fo r any p o in t x G u de fine :
w here фо (.;х) deno tes the so lu tio n s th ro ug h x of the d if fe r e n t ia l sy s tem
W e choose now a c e ll c^G E (n) of d im e n s io n n such th a t x 0G ci^. T he re
a re on ly two p o s s ib i l i t ie s : (a) e ith e r th e re e x is ts a sequence {x p} С
such th a t xp -» x 0 and т0(хр) -* 0 as p -* (b) o r th e re e x is ts 6 > 0 such
th a t T o ( x p) < -6 < 0 fo r any sequence {xp} С such th a t x p -» x 0 as p -* «>.
In case (a) th e re e x is ts a c e ll o f type I I , cr G E(" 1' such th a t
X0(Xp) = 4bj (т0(хр); xp) G ст1 and, th e re fo re , s uch th a t E(ct-{) = c^. F u r t h e r
m o r e , fr o m the d e f in it io n 3.1 o f the sy n th e s is i t fo llow s at once th a t th e re
e x is ts an n - ce ll, ct2 G E (n), such th a t П (cr2) = cr and such th a t x 0 G ст2.
M o re o v e r , i f fo r the c e ll cr2 c ase (a) take s p la ce then we f in d ag a in an
n - ce ll ct3 G E(n) such th a t x 0G ст3, ct2 = Е(П(сгд)) and obv iou s ly стд f ct2 f CTj.
S ince th e re is on ly a f in ite n u m b e r o f n - ce lls o^, .. o;q G E (n su ch th a t
x 0G CTj, j = 1, 2, .. q , f in a l ly we f in d an n - ce ll, say tjq G EM , such th a t
x 0 С crq and fo r crq case (b) take s p la c e . W e c o n s id e r now a sequence
{xp} e CTq such th a t xp -*x0 as p-> ® and ó > 0 such th a t т0(хр) < -6 < 0. In
th is c a se we have y p = Фач(-0 ; x p) -* y0 = Фач (-ó; xq) as p -» «> and s in ce
{yp} с crq it fo llo w s th a t y 0 G TTq. M o re o v e r , by v ir tu e of le m m a 4.2 we m ay
a s su m e th a t т^Ур) -*■ t i as p - > « and uy(t) = v0q fo r t G [0,6). S ince
P ro o f
T0 (x) = inf{T < 0 I 9 0(t; x) G ст fo r t G [т,0]} (4.6)
(4.7)
18 MIRICA
Фач(6 ; у0) = х 0 i t fo llow s th a t e ith e r y0 G cr (and th e re fo re v0q = va), o r , i f
y0 $ ст, th en y0 b e lo ngs to a (k+ l)- ce ll ct0 G E<k+1) such th a t П(ст0) = a and
the le m m a is p roved .
L e m m a 4.5
L e t ct G Е(;П be a c e ll of type I and le t xn G ст. T hen th e re ex is ts an
n- c e ll, cti G E(n) , such th a t x q G cti and v D = vD .
P ro o f
A c c o rd in g to le m m a 4 .4 th e re ex is ts e ith e r an n- ce ll G E (n such
th a t x 0G ô i and v0 = v0 o r an n- ce ll cr0 G E (n) such th a t П(ст0) = ст, vO0 = -v0 and x 0e П(ст0). In the f i r s t c ase the p ro o f is f in is h e d so le t us suppose
th a t we a re in the second c a se . W e c o n s id e r now a n e ighbou rhood W of
x 0 in G and a new co - o rd in a te sy s te m = a* (x1, .. x "), j = 1, 2 , .. n in
W such th a t а(хо) = 0 and a ( W ) = B 6(0) с Rn, 6 > 0, a {W П ct) = {§ ||n = 0}
and we deno te by g0o(f) = (g¿0 (€X, . . . . ? n), . . . , g ^ fë 1, . . . , €n)) the re p re s e n ta t iv e
o f the v e c to r f ie ld x « A x + bv0|) (= Ax -bv0) in the new co - o rd in a te s .
F r o m the fa c t th a t П(сг0) = ст i t fo llo w s th a t g o ^ f1. . . . . i " ’ 1» 0) / 0 say
g(cS} (S1. . . . C "'1,0) > 0, fo r any ( I 1, . . , f " " 1, 0) G B 6(0). W e c an choose the
n e ig hb ou rho o d W o f x 0 and 6 > 0 such th a t gSof?1. . . . €n) > 0 fo r any
( I 1, . . . , €") 6 B s(0). I f we denote by Ф (.'Л ) = (Фг. ' Л ) , •••, Фп( . ' Л ) the s o lu t io n
th ro ug h the p o in t f = (Ç1, . . . , Çn) o f the sy s te m
^ - = g j^ K 1. . . . , €n). j = 1. 2, .. n (4.8)
then , s in c e g ° ( f . . . , Çn) > 0 we m u s t have
Ф п (t; (€\ ..., i n )) > i n (4.9)
fo r t > 0 s u f f ic ie n t ly s m a l l and th e re fo re a (W П ct0) G { f G B 6(0) |fn < 0} .
S ince the se t { f G B 6(0 )| fn > 0} is open , th e re ex is t s om e c e lls , say
q q. . . crq G E (n' such th a t XqGO ct¡ and U a (W Ocrj) c {Ç G B6 (0) |fn > 0} .
P ro c e e d in g exac tly as in the p ro o f o f le m m a 4 .4 we p ro ve th a t fo r at
le a s t one o f the c e lls o \ , ct2, . . . crq, say fo r ctj, c ase (b) m e n tio n e d in the
p ro o f takes p la c e , th a t is th e re e x is ts a 6 > 0 s u ch th a t fo r any sequence
{xp} с ст i w ith the p ro p e r ty : xp -*x0 as p-> ® , we have т0(Хр) < -ó < 0 w here
T0(x p) is d e fin ed by (4 .6 ). T h e re fo re we c a n conc lude th a t e ith e r v0l = vD
o r , i f v„ f v0 th en ITÍo-j) = a w h ich c o n tra d ic ts (4 .9); so the le m m a is p ro ved .
L e m m a 4.6
L e t k G {1 , 2, .. (n-2)}, le t ct 6 be a к - ce ll o f type I and x 0 g c t .T hen th e re e x is ts an n- ce ll o'! G E (n s uch th a t XqGctj and vDl = v0.
IA E A -S M R -n /4 6 19
L e t us c o n s id e r an n- ce ll g j 6 £ (tf) such th a t x 0 G ci^. I f v0l = v0 then
the le m m a is p ro v ed, so le t us suppose th a t v0¡ f v0. F r o m c o ro l la r y 4.3
i t fo llo w s th a t x 0 G П (o^). I f П ^ ) G ' E ^" '1) th en we m u s t have v n( = -v
and hence v D(0l) = v0 . I f П(ст1) е ¿п ""1' then ст2 = E O lío ^ )) is an n- ce ll w ith
th e p ro p e r ty x 0 G g2. A g a in , i f v02 f v0 and hence v0z = va (th is is not
c o n tra d ic ted by the d e f in it io n of the r e g u la r syn th e s is !) th en we m u s t
have x 0 G П(<т2). E v e n tu a lly we f in d e ith e r an n - ce ll g j G E <n such th a t
x0 G cti and v0 = v0 o r a n (n - l)- c e ll g 0 G E(In’ 1) w ith th e s a m e p ro p e r t ie s
( i.e . such th a t x 0 G cf0 and v0o = va). In the f i r s t c ase the le m m a is p ro ved .
In the second case we c o n s id e r a sequence {xp} с g0 s uch th a t x p -> x 0 as p -*■00. F r o m le m m a 4.5 i t fo llow s th a t fo r any p o in t x p G ст0 th e re ex is ts
an n- ce ll CTP G E(ni such th a t x p G gp and va = v (x p) = vO0 = v0 . S ince the
f a m ily E o f the c e lls is lo c a l ly f in ite , i t fo llow s th a t th e re ex is ts an n- ce ll
a ' G E <n) and a subsequence {xpk} с {xp} such th a t x ^ G cr1 and v0, = v0.
T h e re fo re , s in ce x n = l im x„ we have xn G a ' and v , = v : the le m m a is Pk 00 °
p ro ved and so is th e o re m 3.1.
The F i l ip p o v s o lu tio n s a re o p t im a l tr a je c to r ie s
W e p rove now the fo llo w in g r e s u lt :
T h e o re m 4.7
I f the t im e - o p t im a l fe edback c o n tro l v : G -* 1-1, +1] de fines a r e g u la r
n-1 .sy n th es is (d e f in it io n 3.1) and i f fo r any c e ll a G U E T of type I and fo r --------------------------------------- k = i --------------
any po in t x G a we have Ax - bvn ^ Tyg w here Tx cr denotes the tangen t space
of g at the po in t x , then any F i l ip p o v so lu t io n of the sy s tem (2.1) is a
C a ra th è o d o ry s o lu t io n and thus an o p t im a l t r a je c to ry .
P ro o f
F o r any x G G and т > 0 s u ff ic ie n tly s m a l l we denote b y $ T(x) the set
o f F i l ip p o v so lu tio n s of the sys tem (2 .1 ) on the in te rv a l [0, t ] s ta r t in g at
x ( i.e . ^ G Фт (x) i f (//:l0,т]-» G is a b so lu te ly con tinuous and dÿ//dt(t) G F(íí<(t))
a .e . on Г0,т] w here the se t-va lued m ap F : G-* & (Rn) is de fined by (3 .8)).
A s in s e c t io n 3 we denote by <p(.;x) the o p t im a l tr a je c to r y th ro ug h x ( i.e .
<p(.;x):[0,T(x)] -* G is the un ique C a ra th èo do ry so lu t io n th rough x 'of the
sy s te m (2 . 1 )).
W e need the fo llow in g tw o le m m a s :
L e m m a 4.8
(a) L e t g G E[ be a c e ll of type I , x G g , т > 0, and ф G Фт (x) a F i l ip p o v
s o lu t io n th ro ug h x such th a t ф(t) G g fo r tG 10,т]. T hen ф(t) = cp(t;x) fo r any
t G 10, t ] ;
(b) L e t g G Ец be a c e ll of type I I , x G g , т > О and ф G Фт (х) such that
ф(t) G E (g) fo r t G [ 0 , t J . T hen ÿ/(t) = p(t;x) fo r t G [ 0 , t J .
P r o o f
20 MIRICA
(a) S ince u £ E j is a c e ll o f type I , f r o m the d e f in it io n o f the syn thes is
it fo llow s th a t A x+ bva € Tx cr fo r any x G cr. B y hy po thes is o f the th e o re m
we have Ax - bv Ç Txcr. O n the o the r Jiand, fr o m th e o re m 4.1 it fo llo w s th a t
A x+ bv 0 G F (x ) fo r any x G ст and s in ce i t fo llow s fr o m (3.10) th a t F (x ) is
e ith e r A x + b v 0 o r co {A x+ bv0, Ax -bv0} (th is la s t case happens when th e re
ex is ts an n- ce ll cti G E<n) such th a t x G ctj and v0l = -va), o bv iou s ly
F (x ) П Tx ct = {A x+ bv0} . T h e re fo re , if ф{t)G a fo r t G [0 ,т], we m u s t have
^ ( t ) G F ( 9 ( t ) ) n T m ct = {Aÿ/(t)+ bv0}
hence 0(t) = cp(t;x) fo r tG [0,т].
I f we re p la c e a by E(cr) in the p ro o f, we o b ta in s ta te m e n t (b) of o u r le m m
L e m m a 4.9
F o r any x G G we denote by y (x) the n u m b e r o f the c e lls ct G E such
th a t x G ct ( y ( x ) is f in ite s in ce the fa m ily of the c e lls is lo c a l ly f in ite ) . L e t
crn G E be a c e l l o f the sy n th e s is and x nG q n; th en th e re e x is ts a n e ig hb ou r -
hood W o f x n in G such th a t y (x) < y (x0) fo r any xG W\cr0.
P ro o f o f le m m a 4.9
S ince the s u b m a n ifo ld cr0 o f G is a lo c a l ly c lo se d se t in G , i t fo llow s
th a t th e re e x is ts a ne ig hbou rho o d W of x 0 in G such th a t W П ct0 is a c lo se d
sub se t o f W . I f W is ta k e n s u ff ic ie n t ly s m a l l , i t in te rs e c ts on ly the c e lls
c tG E fo r w h ich x 0G c t . T hen , o b v io u s ly , fo r any x GW\ct0 we have x Ç cr0,
hence y (x) < y ( x q ) .
P ro o f o f th e o re m 4.7
The s ta te m e n t o f the th e o re m is e qu iv a le n t to the fa c t th a t fo r any
x G G th e re ex is ts а т > 0 such th a t ф(t) = cp(t; x) fo r any ф G Ф т (x) and
tG [0 ,т ] .
W e s h a l l p ro ve th is s ta te m e n t by in d u c t io n on the n u m b e r y (x) d e fined
in le m m a 4.9.
I f y (x) = 1 th en th e re e x is ts an n - ce ll ct G E(n) such th a t xG cr, and s in ce
ct is an open se t o f G fo r w h ich F (x ) = {A x+ bv0 } , the above s ta te m e n t is
t r iv ia l ly v e r if ie d .
L e t us suppose th a t the s ta te m e n t is tr u e fo r any x G G fo r w h ich
у (x) < к and le t us c o n s id e r x0 G G such th a t y (x0) = k . L e t ct0 G E be the
c e ll o f the sy n th e s is c o n ta in in g Xq.
W e c la im the e x is te nce o f а т > 0 such th a t fo r any ф G Фг(х0) and any
t G (0,т] we have ф(t) G ст0 i f cr0 is o f type I and ф(t) G E (cr0) if ct0 is o f type II.
The th e o re m w i l l th en fo llo w fr o m le m m a 4 .8 .
S in ce cp(.;xo) deno tes the o p t im a l t r a je c to r y th ro ug h x 0Gcto It fo llow s
th a t th e re e x is ts a ? j > 0 s u ch th a t <p (t; xQ) G ст0 i f ст0 is o f type I and
Ф (t; x 0) G E(ctq) if CT0 is o f type I I , fo r any t G (0,т].
P r o o f o f le m m a 4.8
IA E A -SM R -17/46 21
W e s h a l l p rove th a t i f fo r any e > 0 and any ф е Ф т(х0) th e re e x is ts
t e G (0,e] such th a t ф(t e )Ç cr0 i f cr0 is o f type I and i¿/(te ) $ Е(ст0) i f cr0 is of
type I I th en we get a c o n tra d ic t io n to the fa c t th a t cp ( .;x 0) is the o p t im a l
t r a je c to r y th ro u g h x0 G cr0.
L e t us suppose th a t ct0 G E j is of type I; a c c o rd in g to le m m a 4.9 th e re
e x is ts a ne ig hb ou rho o d W of Xq such th a t W П ct0 is c lo se d in W and
7 (x) < y (x0) fo r any x G W \ ct0.
W e c o n s id e r e > 0, a F i l ip p o v s o lu t io n ф G i e(xo), and the c o r re s p o n d in g
t eG (0 ,e ] s u ch th a t ф(\) Ç ст0. S ince the fu n c tio n <//(.):[0,e] -» G is co n t in uo u s ,
th e re e x is ts a 6 > 0 such th a t ф(t) G W fo r any t G[0,ô] ; a c co rd in g to the
above a s su m p t io n , th e re e x is ts a t j G (0,6 ] such th a t х г = Ф ^ ) G W\ ст0; s in ce
W П <т0 is c lo se d in W and ф{.) is c o n t in uo u s , i t fo llo w s th a t th e re ex is ts
a t 2G [ОД]) such th a t x 2 = ф{'t^) € a 0 and ^ ( t )G W\a0 fo r t G (1^, t j . F r o m the
in d u c t io n h y po the s is (y (x ) < к fo r any x G W\ctq) it fo llo w s th a t fo r any
t G (t2, tj] th e re e x is ts a T(t) > 0 such th a t ф(s+ t) = q>(s;^(t)) fo r any
s G t0 ,r ( t)] . F r o m the u n iq ue ness o f the o p t im a l tr a je c to r y q>(.;x) fo r any
x G G i t fo llo w s th a t ф(s +t) = çp(s; cp(t)) fo r any sG [0 ,^- t ] and any
t G (t2, t j w hence cp(s; x 2) G W\ct0 fo r any sG (0 ,t2 - tj] and th is c o n tra d ic ts
the fa c t th a t qp(.;x2) is the o p t im a l t r a je c to r y th ro ug h the po in t x2e aQ. If
ct0 is o f type I I th en o bv io u s ly aQ U E(ctq) is a s u b m a n ifo ld w ith b ou nd a ry
(the b o u nd a ry be ing <r0), hence lo c a l ly c lo se d in G , and the s am e a rg u m e n t
as above app lie s to E (aQ).
5. T IM E - O P T IM A L F E E D B A C K C O N T R O L F O R T H E C A SE O F R E A L
E IG E N V A L U E S A N D S C A L A R C O N T R O L
W e c o n s id e r the t im e - o p t im a l c o n tro l p ro b le m de fined by
The a im o f th is s e c t io n is to p ro ve the fo llow in g :
T h e o re m 5.1 ([ 17] )
F o r the t im e - o p t im a l c o n tro l p ro b le m de fined by (5 .1 ), th e re e x is t
the se ts Р ° С P 1 С . . . P ^ C P n = G and N C G such th a t {PO, P 1, .. P °, N , v (.)}
is a r e g u la r sy n th e s is (d e f in it io n 3.1 ).
M o re o v e r , the sy n th es is has the fo llo w in g p ro p e r t ie s :
(1) T he re a re no c e lls o f type I I o th e r th an the f in a l p o in t , 0, and the
" in d if fe re n c e s e t" N is e m p ty ;
(2) E v e ry c e ll o f the sy n th e s is is r e a l iz e d e ith e r by u = +1 o r by
u = -1 ;
^ = A x + bu , x G R n, u G U (5.1)
w here U = [-1, + 1], the p a ir (A ,b) is c o m p le te ly c o n tro lla b le ( i.e .
det (b, A b , .. A n l b) f 0) and A has the r e a l e ig enva lue s a 1 , e 2, . . . , amG H of o rd e rsm
22 MIRICA
(3) I f a c e ll cr o f type I , o f d im e n s io n le s s th an n , is r e a l iz e d by u = 1
(re s p e c t iv e ly by u = -1) then fo r each x e a the v e c to r Ax - b (re sp e c tiv e ly
A x + b) is tr a n s v e r s e to cr a t x ( i.e . A x - b $ Tx a w here Tx a is the tangen t
space a t cr a t the p o in t x).
R e m a rk 5.2
S ta te m e n t (3) in th eo re m 5.1 is the t r a n s v e r s a l i ty c o nd it io n w h ich was
needed in s e c t io n 4 in o rd e r to p rove th a t each F i l ip p o v so lu t io n o f (2.1)
is a t im e - o p t im a l tr a je c to r y ; in th is p a r t ic u la r case i t c o in c id e s w ith the
t r a n s v e r s a l i ty c o nd it io n in the d e f in it io n o f the r e g u la r s y n th e s is .
In o rd e r to de fine the c e lls o f the syn th e s is and the se ts P °, .. P n, we
c o n s id e r the fo llo w in g d if fe r e n t ia l sy s te m s :
dx . , ,— = A x + b dt
(5 .2 )+
-rr - A x - b dt
(5 .2 ).
and we denote by
(5.3)+
о
(5 .3).
о
the so lu tio n s th ro ug h xq 6 R n of (5 .2)+ and (5.2)_, r e s p e c t iv e ly .
W e de fine now the c e lls o f the sy n th e s is :
The on ly c e ll o f d im e n s io n 0 o r o f type I I is
aj°> = a i°> = { 0 } (5.4)(0)
The c e lls o f d im e n s io n 1 (of type I) a re
The c e lls o f d im e n s io n k , 2 s к s n (of type I) , a re :
W e d e fin e the se ts P °, P 1, .. P n, N , as fo llow s :
P ° = cr(+0) = a i 0) = {0 } , I * = P ^ U a (+k) U ст(.к>, к = 1, 2 .........n , N = ф (5.5)
IA E A -SM R -17/46 23
and the t im e - o p t im a l feedback c o n tro l by
k = l
v ( x ) = • 0 i f x = 0
1 i f x e u cr<k)V = 1 +
(5.6)
- 1 i f x G U c i к = 1
P ro o f o f th e o re m 5.1
T h e o re m 5.1 fo llow s fr o m the next fo u r le m m a s :
L e m m a 5.3
L e t A € L (R n, R n) and b 6 R ° such th a t the p a ir (A , b) is c o m p le te ly
c o n tro lla b le and the m a t r ix A has the r e a l e ig enva lue s а х, a 2 , . . . a m e R
T hen fo r any X e R" th e re e x is ts th e se t o f p o ly n o m ia ls Pj (t) , . . . Pm (t)
such th a t
a n d j C o n v e r s e l y , f o r a n y s e t o f p o l y n o m i a l s { P i( t ) , . . . , ( t ) } w h i c h s a t i s f i e s(5 .7 ), t h e r e e x i s t s a v e c t o r X 6 R n s u c h t h a t (5.8) h o l d s .
F r o m the c a n o n ic a l fo rm of the p a ir (A , b) (Re f. [21]) and f r o m the
th e o ry o f h ig h e r- o rd e r l in e a r d if fe r e n t ia l equa tio ns w ith co ns tan t c o e ff ic ie n ts
it fo llo w s th a t the s ta te m e n t in the le m m a is e q u iv a le n t to the fo llow in g :
a fu n c t io n f : R -* R is a s o lu t io n o f the d if fe r e n t ia l e qua tio n
w here -yn - а-т"*1’ = (- l) n d e t(A - rE ), i f and o n ly i f th e re e x is ts a v e c to r X G R n
j = isu ch th a t f(t) = <(X(t), eAtb^> fo r any t e R .
I f X £ R n is g iv e n , th en the fu n c tio n f : R -*■ R de fined by f(t) = <X, e ^ b ^
is a n a ly tic and its d e r iv a t iv e s a re : fM ( t) = <(X, eAtAk b X к = 1, 2, . . . and
s in ce A n = 2 A (n (the H am ilto n - C ay le y th e o re m ), we have s u c ce ss iv e ly :
m
of o rd e r s k 1t - k m( £ k , = n ) , r e s p e c t iv e ly .
d e g P j( t ) s kj - 1, j = 1, 2, .. m ; (5.7)
m
(5.8)
i = i
P ro o f
n
(5.9)
n
n
24 MIRICA
Anb = Y j a.ja " i b th en e AtAnb = a. eAtA n_ib and f in a lly : <X, eAtAnb >
j=i i=l J= aj^X , eAtAn" ib / ’ so f( .) is a s o lu t io n o f (5 . 9).
C o n v e rs e ly , i f f : R R is a g iven so lu t io n o f (5.9) th en it is c o m p le te ly
d e te rm in e d by its v a lu e and its f i r s t (n- 1 ) d e r iv a t iv e s at t = 0 : f ( 0), f (1) ( 0 ), . . . ftn-i) (0).
S ince de t (b, A b , . . . An' 1b) f 0 by hy p o th e s is , th e re e x is ts a un ique
v e c to r X G Rn w h ich s a t is f ie s <X, Akb 5 = f (k) (0), к = 0, 1 , . . . n-1 and th e re fo re
the s o lu tio n s f(t) and t |-* <^X, eAtb)> co in c id e .
L e m m a 5.4
L e t (A ,b ) be de fin ed as in le m m a 5.3 . T hen , fo r any t , , . . . t n 6 R such
th a t t,- f tk fo r j f k , j ,k = 1 , 2 , . . . n, the v e c to rs eAt‘ b , e г b , . . . , e Atnb
a re l in e a r ly independen t.
P ro o f
Suppose th a t eAtib , eAt¡ b , . . . eAtnb a re l in e a r ly dependent, th en th e re
w i l l e x is t a non- ze ro v e c to r X e R n such th a t <^X, eAtkb)> = 0 fo r к = 1, 2, . . . n;
f r o m le m m a 5.3 i t fo llow s th a t f(t) = <(X, eAtb)> is a s o lu t io n of the
e qua t io n (5. 9) and s in ce the m a t r ix A h as on ly r e a l e ig e n v a lu e s , i t fo llow s
f r o m le m m a 2.6 th a t f(t) c an have a t m o s t n-1 d is t in c t z e ro s . T h e re fo re
X = 0 and the le m m a is p roved .
L e m m a 5.5
The se t P 11 de fin ed by (5.5) c o in c id e s w ith the r e a c h a b il i ty se t G of
the c o n tro l p ro b le m .
M o re o v e r , if x n = x(+k) (s p s2, . . . sk) e crjk ), к 6 {1 , 2, . „ , n } , then the
o p t im a l c o n tro l w ith r e s p e c t to x0, u X(i :[0 ,T (x0)] -* [-1, 1] has exac tly k-1
sw itc h in g p o in ts , tj = • S sk-í> j = ••• k "2 andi = o
k - i
T (x0) = - ^ s k_j
J2= 0
(5.10)
M o re p re c is e ly :
U x „W
1 fo r t e [o,-sk )
j ) +i( - l ) i+1 fo r t e I- E sk. t , - S sk-i)< j = 0, 1, .. k-2 (5.11)
4=0 fi=0
A s im i la r s ta te m e n t is v a l id fo r x0 € c r® .
IA E A -SM R -17/46 25
P r o o f
O b v io u s ly O e P " i l G s o le t x0 G P n, x0 f 0; f r o m the d e f in it io n (5.5)
o f P n it fo llow s th a t th e re e x is ts a k G { 1 ,2 , . . . , n} such th a t х 0€ ст ^ и a W .
L e t us suppose x 0 G ст® (fo r x 0 G o ik5 the p ro o f is exac tly the s am e ),
and i f it ' s a t is f ie s the m a x im u m p r in c ip le (tha t is , th e re e x is ts a v e c to r
X G R n such tha t:
th en f r o m th e o re m s 2.3 and 2.7 o f s e c t io n 2 i t fo llo w s th a t x 0 G G , k-1
T (x0) = - Y j s k-£ anc* u Xo(.) is the o p t im a l c o n tro l w ith re s p e c t to x 0, so£ = 0
P n C G . U s ing le m m a 2.6 one can e a s ily p ro v e th a t G С P n.
C o ro l la r y 5.6
If we denote R k_ = {(s1, s2, .. sk) | S j, s2, .. sk < 0 } th en the m ap s
x(+k) . х (‘ > : R - G g iv e n by (5 .4 )(k) a re in je c t iv e m a p s and the c e lls a ( l ) j CTU ) , . . . p-W, CT(_n) a re p a ir w j se d is jo in t .
L e m m a 5.7
F o r any k G {1 ,2 , . . . n } , the fo llo w in g s ta te m e n ts ho ld :
(1) <jM \ = P k_1 = ст-(к) \ ст-к w here ctW deno tes the c lo s u re of the
se t crffl in the r e la t iv e to po logy o f G ;
(2) The c e lls cxM and crM a re a n a ly t ic a l (em bedded ) k - d im e n s io n a l
s u b m a n ifo ld s o f G ;
(3) The se ts o ® U ст(к_1) and ст® U a (+k_1 a re a n a ly t ic a l k - d im e n s io n a l
s u b m a n ifo ld s w ith b o u n d a r ie s , o f G , and 9 ( jW U ctO*"1) ) = ст(_к_1) ; Э(ст1к> U crû'-1))
k-1L e t uXo : [0, - s k_{] -*■ [-1, l] be the m a p de fin ed by (5 .11 ). I f we
k-1show th a t th is c o n tro l is a d m is s ib le (that is , the s o lu t io n cp(.;x ):[0 ,- j } sk_£]
ТЭП Л-P +U Л .R o f the sy s te m :
= A x+ buXo (t), ф(0; Xq) = x0 (5.12)
s a t is f ie s
k-1
(5.13)
k-1
s ig n X , b ) = u (t) fo rx0
(5.14)
26 MIRICA
(4) F o r any x G cr<W (x G ст<_к> ), A x : b $ i ; crUO (Ax +b f Тхст<к) ) w here
T; cr|k) deno tes the ta ng en t space at o<+k) at the p o in t x.
P ro o f
S ta te m e n t (1) fo llo w s fr o m the fa c t th a t if x p = x ® (s j, . . . , s k ) G c r® ,
Xp->-x0 G ст« fo r p ~ l , 2, . . . the sequence {(sP, . . . , sP)} С R k has a l im it in g
po in t (s j, . . . , s k) G R k \ R k.
It is a lso e asy to p ro v e , by in d u c tio n on k", th a t the fo llo w in g fo rm u la s
ho ld :
X(k) (S j, . . . , sk) = £ ( - l ) H J exp A ^ ^ s j - s ) b ds (5.15)
j = 1
X<k> (S j, . . . , s j = -XW (Sj, . . . , sk), к = 1, 2, . . . , n , (s j, . . . sk)G R k. (5.16)
B y d ir e c t c o m p u ta tio n , it fo llow s th a t f r o m (5.15) and (5.16)
(к) кЭх
3s— (S j, . . . , sk) = (- l)k 1 exp A ^ s
c= l
(5.17)
3 x a )±_ (S j, . . . , s k) = (- l)k' 1exp A ^ s t |b + 2 ^ ( - l ) k_i exp A ^ b ,
j= 2 1 = J
P = 2 .........к (5.18)
S ince a c c o rd in g to le m m a 5 .4 the v e c to rs exp [Askb), exp[A (sk+ sk-i) ] b ,
exp (A I Si)b' к s n , a re l in e a r ly in dependen t fo r any (S j, . . . , sk) G Rk,
i t fo llo w s im m e d ia te ly th a t the v e c to rs9x(k)3i T (Sl’ -• Sk) sk>
a re a lso l in e a r ly independen t; so the m ap s x+k) < X-1 : Rk_ “*■ G a re a n a ly t ic a l
im m e r s io n s and s in c e , by c o r o l la r y 5.6 o f the le m m a 5.5 , they a re 1-1 m a p s ,
i t fo llo w s th a t o ^ 1 and ct® a re im m e rs e d s u b m a n ifo ld s o f G . I f we suppose
th a t (x(+k) Г 2 : crjk' -► Rk a re not c o n t in uo u s , i t is easy to d e r iv e a c o n tra d ic t io n
f r o m the fa c t th a t T : G -» R is c o n tin uo us ; hence crM and crOO a re a n a ly t ic a lly
em bedded s u b m a n ifo ld s o f G .
S ta te m en ts (3) and (4) o f the le m m a fo llo w e a s ily f r o m the d e f in it io n s
o f the m ap s x ® and x ® and f r o m (5 .15 )- (5 .18).
IA E A -SM R -17/46 27
F r o m d e f in it io n (5.5) o f the se ts P k, к = 0, 1, . . . , n and f r o m le m m a 5.7
i t fo llo w s th a t the connec ted com ponen ts o f the s e t P k\Pk_1 a re the c e lls
ct® and or® w h ich a re d if fe re n t ia b le m a n ifo ld s o f d im e n s io n k . F r o m the
d e f in it io n (5.6) o f the t im e - o p t im a l feedback c o n tro l v : G -► [- l,+ l] , it
fo llo w s th a t fo r any c e lls cr® , a ® , the r e s t r ic t io n m a p s v|tj№ and v|crik)
a re the co ns ta n t m ap s and so a d m it sm o o th (a c tu a lly cons tan t) ex tens ions
to som e n e ighbou rhoods o f the c e lls crjk) and cKJO.F r o m the above le m m a s it fo llo w s th a t fo r any p o in t x 6 aft) (x G cr® ),
k i 1 , th e re e x is ts a u n iq ue s o lu t io n cpx : [0 ,T (x )] -» R n of the d if fe r e n t ia l
s y s te m (2 . 1 ) w h ich r e m a in s an in te r v a l on (j® (re s p e c t iv e ly on a ik)) (the-
in te r v a l is [0,-sk) i f x = x ® (s1( . . . , sk) o r x = x ® (S j, . . . . sk), ( s 1# . . . , s k) G R k).
A ny such s o lu t io n in te rs e c ts o n ly a f in ite n u m b e r o f c e lls (a c tu a lly th e re
a re o n ly (2 n + l) d is t in c t c e lls ) and re a ch e s the o r ig in w ith in a f in ite t im e .
F o r the к - ce lls cr® and cr® , th e re e x is t (к-1 ) - ce lls = П(сг+(к)) and
c r® 1) = П(сг_®), r e s p e c t iv e ly , such th a t fo r any x G ст® (x G cr®) the
t r a je c to r y ф( . ; х ) o f (2.1) th ro u g h x (the "m a rk e d t r a je c to r y " ) re ache s
a ik' 1) (cr*+k_1)), t r a n s v e r s a l ly ( le m m a 5 .7 , 4)), w ith in a f in ite t im e .
A l l the o th e r ax io m s of the r e g u la r sy n th e s is (d e f in it io n 3.1) a re c le a r ly
v e r if ie d and the th e o re m is p ro ved .
P r o o f o f th e o re m 5.1
R e m a rk
W e have no t a c tu a lly p ro ved th a t the se ts P 1, P 2, . . . , P n of the r e g u la r
sy n th e s is a re p ie ce w ise sm o o th se ts . F i r s t , th is c o nd it io n is not re q u ir e d
fo r the p ro o fs in s e c t io n 4 and so i t does no t a ffe c t the s ta b il i t y p ro p e r ty .
Second , the fa c t th a t the se ts P k, к = 1, . . . , n , a re p ie ce w ise sm o o th is
needed o n ly to p ro ve th a t the m a x im u m p r in c ip le is a s u f f ic ie n t c o nd it io n
fo r the o p t im a lity o f the t r a je c to r ie s de fined by the s y n th e s is , b u t in o u r
ca se , th is fa c t fo llo w s fr o m o th e r re a s o n s . F in a l ly , the p ro o f of the fac t
th a t the se ts P 1, . . . , P n a re p ie ce w ise sm o o th m ay be v e ry lo n g .
F o r in s ta n c e , the se t P 1 is a p ie ce w ise sm o o th se t o f d im e n s io n one
s in ce it is th e im a g e o f R by the m a p f <4: R R n de fined by
f(1) (t ) =
(T)
X ® ( -
fo r t § 0
fo r t > 0
w h ich is o bv io u s ly an in je c t iv e im m e r s io n (a c tu a lly a h o m e o m o rp h is m ).
In o rd e r to p rove th a t P 2 is a p ie ce w ise sm o o th se t we m ay choose fo r
in s ta n ce f (2 : R2 -► R n as fo llow s :
f <2) (Ti , T„) =
*x(2) (■v/'T2, -7 -T2 - Tx ) fo r T2 < 0, Tx > \f-T2
w h ich is an in je c t iv e im m e r s io n o f R2 onto P 2.
28 MIRICA
6 . T IM E - O P T IM A L F E E D B A C K C O N T R O L F O R T H IR D - O R D E R L IN E A R
SYST E M S W IT H C O M P L E X E IG E N V A L U E S
T he a im of th is s e c t io n is to p rove the fo llow in g :
T h e o re m 6.1
The t im e - o p t im a l feedback c o n tro l v : G — [~1, +l] of p ro b le m (1.1)
d e fin es a r e g u la r syn th e s is (d e fin it io n 3.1) if n = 3.
M o re o v e r , the sy n th e s is has the fo llo w in g p ro p e r t ie s :
(1) The syn th e s is has on ly one c e ll o f d im e n s io n ze ro (of type II) ,
the f in a l p o in t , {0 } , it has tw o c e lls of d im e n s io n one of type I and coun tab ly
m an y c e lls o f each of the o th e r p o s s ib le k in d s (of d im e n s io n one and type I I ,
o f d im e n s io n two and the types I and I I and of d im e n s io n th re e (of type I);
(2) E v e ry c e ll q o f type I is gene ra ted ("sw e p t" ) by t r a je c to r ie s of
s y s te m ( 1 . 1 ) w here e ith e r u = + 1 o r u = -1 ; we say tha t a is r e a l iz e d by
u = + 1 o r , r e s p e c t iv e ly , by u = -1 ;
(3) I f a c e ll a o f type I of d im e n s io n le s s th an th re e is r e a l iz e d by
u = +1 (u = -1) th en fo r any x £ q we have A x - b $ T , g (re sp e c tiv e ly
A x + b $ T xq) w h ere Ty q denotes the tangen t space of a at the p o in t x.
T h e o re m 6.1 fo llo w s fr o m the next m o re te c h n ic a l r e s u lt :
T he o re m 6.2
If the m a t r ix A £ L ( R 3, R 3) has the e ig enva lue s a € R and /3 ± i-y € С
w here )3 € R and у > 0 then th e re e x is ts a t € L- 0), a C 1- function
тп : ( t , 0) -» (- °o, 0) and two sequences of u p pe r s e m ic o n tin u o u s functions.:
such th a t the fo llo w in g p ro p e r t ie s a re v e r if ie d :
(1) F o r any Si 6 ( t , 0) and any (S j, s^) e ^ the fo llo w in g in e q u a lit ie s
ho ld :
Tlk : (T ,0 ) - [ - » , 0), т2к : - [-» , 0), к = 1, 2, . . .
w here the se t С R 2 is d e fined by
(6 . 1)
= { (S j , s2)| s : e (t , o), s2 e (tqÍSj), Sj) } (6 . 2 )
s i > To(si) - Ti ( s i)
0 > S 1 > T1 (s l) > T 1 (s l) > —
S i > S2 > r \ (S l, S2) > T2 (Sl , S2) > ...
(6.3)
(6.4)
(6.5)
M o re o v e r , fo r any к = 1, 2, . . . , the se ts de fined by
= { s 1 e (т, 0) I (Sj) > - « } С (т, 0)
= { (S j, s2) e I t | (S j , s 2) > - « } С
(6.6)
a re open , the r e s t r ic t io n m ap s тк| т
The c e lls o f d im e n s io n one of type I I a re
{ Х (2?+ 11 ( 8 г) = Ф+( т 1 (в 1) - * 1 ’ < V \ }
4 V - , i = W 21? - , 1 ( s i ) = ф . ( ^ ( б 1) ■ s ^ ^ ( S i ) ) ! 8 ! 6
3 0 M RICÂ
( 6 . 10 )m u
°-(2? + , к = Ы ^ . к ^ = ' P + ^ S l ) - ^ ( S j ) ; x (2 ). tk . i ( s 1 ) ) | s 1 G q f k }
° ¿ J - ,k = H V - . k ( s i ) = <P-(Tik( S l ) - T k - M s j ) ; G <#\}
fo r к = 2 , 3 , . . .
The c e lls o f d im e n s io n two of type I are
t f i f l . i = b (i2)+, i (s1( s2) = cp+(s2 - S p x ^ K s O ik s i , s 2) 6 <&}
CTi ? - , i = ( S i - . S a ) = 9 . ( s 2 - s 1 ; ( s i - s 2> G < ? f }
; ; ; ; ; ; ; ; ; ; ; ; ; ( б - ю \ 2)>1
°l,+ ,k = ^Xi, + ,k (s l> s2 = tP+(s 2 ’ T1 (sl ) ; xV-.k-l (s l)H ^S1 ‘ S2 S ^ 1)
CTi2)- , k = ix (i%,k <S1« s2) = Ф-(s2 ~ Tí ' 1(s1); x ^ .k - i ( s x))I (S;L> S2) e W*}
fo r к = 2 , 3, . . .
The c e lls o f d im e n s io n two o f type II are
°-2?1.1 = ix l22)+, i ( si- s2) = s2) - s2; x'l,5- j iS p s2))| (S l, S2) 6 < # \ )
4 !- . ! = S2> = Ф - ^ 3 ! ' S2) ' S2 ; * f , + , l (S l ‘ s 2)> I (s i> s2) € 2}
........................... ( 6 . 1 0 ) (2) ,ц
ст22 + ,к = {Xî? + ,k(s l» s2)
= 4>+{ r l { s v s2) - r l - 1 ( s 1 , s 2); x(22)-,k-l <s l- S2^ | ( S1 * s2> e
° 2?-,k = b í ^ - k K * s2) = ф - ^ г ^ р s2))| (S l, s2) e < r k}
fo r к = 2, 3 , , . . .
The c e lls of d im e n s io n th re e (of type I) are
o fp i = {x+ !i(s !, s 2, S3 ) = <p+(s3 - s 2; x(i2)- ,i(s ! , s 2))| (S l, S 2)
e <¡g, Sg G (T 2( S 1; S2) , S j ) }
o-ifi = { x ^ i f s j , s 2, S3) = ф_(s3 ■ s2; Xi2)+ii ( si< s2))| (S l, s 2)
G <3?, S3 G ( t 2( s j , s 2 ) , s2 ) }
(6.io),3)
IA E A -SM R -17/46 31
CT) % = í x ^ k í S i , s 2, Sg) = 9 +( s 3 - s 2) ; x 2 >- , k - i ( s i « s 2>M i s i - s 2>
6 ^ 2 _1 S 3 ^ ( t ^ ( s i , S 2) , S2)
= Î X - , k ( S l , s 2, s 3) = ф . ( в 3 - Я Г г ' ^ 8 ! - S 2) ; X 22l , k - l ( s l* S 2) ) | ( S 1 - S 2 >
£ ^ 2 X< S3 G (T 2 ( S l , S2 ) , t 1 - 1 ( S i , S2 ) ) }
fo r k = 2, 3, . . .
W e de fine the se ts P °, P 1, P 2, P 3, N , of the sy n th e s is as fo llow s :
P ° = a ‘+0) = CT-0) = i ° > . p l= p ° U °\Ц+ U a\i>_ U ( U (a™ k и ст^ k Лк ~ 1 ' * ' /
P 2 = P* u U a £ ) . u u 4 ? +.k U < - , k 0 <6 Л 1 )
p 3 = P 2 и ( Z (a (3), U a (3) )\ N = фVk = l +,k J
The t im e - o p t im a l fe edback c o n tro l v :P 3 -> [-1, +1 ] is d e fined by
v(x) =
+1 if x e U ( 4 ï , k и ff^+ k и &2,+, к и < к ) и . Iк=1
О i f х = О
-1 i f x e U (ст<^ к и ст<^к и ст^_ к и с/3)к ) и о £ 1
(6 . 12 )
The p ro o f o f th e o re m 6.1 fo llow s step by s tep th e way we p roved
th e o re m 5.1 b u t h e re we s h a ll o m it the d e ta ils . W e r e m a r k on ly th a t
th e o re m 6 .2 is an ana logous r e s u lt to le m m a 2.6 c o n ce rn in g the n u m b e r
and the d is t r ib u t io n o f the z e ro s o f the fu n c tio n s f :(*«> , 0 ) -» R de fin ed by
(6 .7 ). A n ana logous r e s u lt to le m m a 5.4 fo llo w s fr o m th e o re m 6.2 .
R e m a rk 6.3
A n o th e r d e s c r ip t io n o f the "sw itc h in g s u r fa c e " of the sy n th e s is fo r
th e s am e p ro b le m w as g ive n in [19] by M o ro z b u t he d id no t p ro ve th a t the
c o n s tr u c t io n y ie ld s a r e g u la r s y n th e s is .
P r o o f o f th e o re m 6 .2
W e r e m a r k f i r s t th a t fr o m the g e n e ra l th eo ry o f h ig h e r- o rd e r l in e a r
d if fe r e n t ia l e qua tio ns w ith c o ns tan t c o e ff ic ie n ts i t fo llo w s th a t a r e a l fu n c tio n
f : R -» R is a s o lu t io n to the e qua tio n de fined by the m a t r ix - A e L (R 3, R 3)
if and o n ly i f th e re e x is t som e r e a l co ns tan ts k j , k 2, k3 £ R such th a t
f(t) = k xe at + e flt (k2cos 7 t+ k 3 s i n 7 1) (6.13)
w here - a G R and ± Í7 (j3 e R , 7 > 0) a re the e ig enva lu e s of the m a t r ix - A .
32 MIRICA
O n the o th e r hand , fr o m the co m p le te c o n tro l la b i li ty o f the p a ir (A ,b )
it fo llow s th a t any such s o lu t io n is o f the fo rm f(t) = f(t;X ) = <(X, e 'Atb)>
w here X € R 3 (see fo r e x am p le le m m a 5.3 ).
It is easy to see th a t any fu n c tio n of the fo rm (6.13) fo r k 1( k2, k 3 £ R
m a y be w r it te n in the fo rm
f(t) = c x e"a t- c 2 e_et s in (-yt - c2), Cj € R , c2 ê 0, c 3 G t0,2îr) (6.14)
F u r th e r on, s in c e the fu n c tio n t -> e"at does no t change the s ig n , it
fo llow s th a t we can re p la c e in the s ta te m e n t of th e o re m 6 .2 the fu nc tio ns
f : (- °o, 0) -» R de fined by (6.14) by the fu nc tio ns g : (- 00, 0) -» R de fined by
g(t) = с - e(“ " 6^ s in (Y t- k ) (6.15)
w here the ran g e o f the p a r a m e te rs с and к is d e fined by
c e R , к е [ 0 , 2тг) (6.16)
In o rd e r fu r th e r to s im p lify the fu n c tio n s (6.13) we c o n s id e r the fo llow in g
change o f v a r ia b le :
s = Yt (6.17)
w h ich does not a ffe c t the s ta te m e n ts o f th e o re m 6 . 2 .
I f we denote
P = ^ - l i (6.18)
th en we m ay r e s t r ic t o u rse lv e s to the study of the change of s ig n of the
fu n c tio n s h ( .;c ,k ) : (- °°,0) -» R de fin ed by
h (s ;c ,k ) = с - eps s in (s - k ) , (c ,k ) 6 R X [0,27t) (6.19)
In o th e r w o rd s , in o rd e r to p rove th e o re m 6.2 i t is s u ff ic ie n t 'to p rove
the s ta te m e n ts in th is th e o re m , w here X G R 3 and f( .;X ) d e fined by (6.7)
a re re p la c e d by (c ,k )G R X Í 0 , 2 n ) and h ( .;c ,k ) d e fined by (6 .19 ), r e s p e c t iv e ly .
F r o m the s ta te m e n t of th e o rem 6.2 i t fo llow s th a t r G [*«>,0) in th is
th e o re m sho u ld have the p ro p e r ty : т = in f {s i < 0 such th a t (c ,k) G RX[0,27r)
e x is ts and e > 0 such th a t the r e s t r ic t io n m ap h ( .;c ,k ) | (sx- e ,0 ) changes the
s ig n on ly at S j) .
T h e re fo re , if we denote by A ^ i s ^ the se t of a l l p a ir s (c ,k) € R X [0,27r)
such th a t h ( .;c ,k ) changes the s ig n only at s, on the in te r v a l [ s ,,0) then
t = in f { s ! < O lAofSj) f </>}.
F i r s t we s h a l l d e te rm in e fo r each S jG ('«>,0) the se t A ( s x) = {(c ,k ) G R
X [0 ,27г) I h ( .;c ,k ) changes s ig n at S j} . S ince
h'Sl(s i;c ,k ) = -eps> «J 1 +pii, s in ( s i + 9 * k )
IAEA-S M R-17/46 33
w here в E ( 0 , ж) is g iv e n by
Л (6 . 20 )
the s e tA fs j ) is g iv e n by:
A (s i) = {(c ,k ) j с = e ^ 1 s in (s j - k ), s in fs j + Q * k) f 0 , к E l0,27r)} (6 . 2 1 )
W e c o n s id e r now n e {0 ,1 ,2 , . . . , } and
S ince h ( . ;c ,k ± 7r) = -h (.;- c ,k ), it fo llow s th a t we m ay re p la c e A (s j) f r o m
(6. 2 1 ) by
A (s i) = { (c ,k ) I с = е<* s in ts j - к ), к £ ( s i+ 0 +пя-, s : +0 + (n + 1 ),7г)}
(6.23)
and th e re fo re ( c ,k )G A ( s 1) i f and on ly if S j+ 0 - k G (-(п+1)тг, -П7г) and
s j 6 ( - Ш Г - в , - (n- 1)тг - в ) . I f fo r any (c ,k ) E A ( s x) we denote
hSi (s;k) = e ps‘ s in (s j - k) - eps s in (s - k) (6.24)
it is easy to see tha t the e x tre m e po in ts o f hSj ( ,;k ) a re :
t m = к - 0 ' ттг, m G Z (6.25)
w here m = n , n- 1 , . . . fo r t m > s x and m = n + l , n + 2 , . . . fo r t m < s 1; the
e x tre m e va lue s o f h Sj (.;k ) a re g ive n by:
it is o bv ious th a t the changes o f s ig n o f the fu n c tio n s h Si(.;k ) depend on the
b e h a v io u r o f the fu n c t io n F ( .;p ) d e fin ed by (6 .27).
The r e s u lts needed in the seq ue l fo r the s tudy of the fu n c tio n s h s¡ (.;k)
a re co n ta in ed in the fo llo w in g le m m a whose p ro o f is t r iv ia l .
L e m m a 6.4
(1) F (s ;0 )*0 fo r any s 6 R \ { 2 m 7r| m E Z } and F (2m îr;0 ) = 0 fo r any
in te g e r , m E Z ;
Sj E [-П7Г - 9 , — (n— (6 . 22 )
I f we denote
F (s ;p ) = eps (p s in s - c o s s) +1 (6.27)
34 MIRICA
(2) И p < O th en th e re ex is t the nega tive r e a l n u m b e rs 0 = po > p ^ > p 2> . . .
su ch th a t pm G ( ' (m + l ) n , - т 7г) fo r m = 1 , 2 , . . . and such th a t F ( .;p ) d e fined
by (6.27) changes s ig n on ly at p1 > p j > . . . . M o re o v e r , F (s ;p ) > 0 fo r
s G m4 0 (P2m+l - p2m ) U (0, + « ) , F ( s ;p ) < 0 fo r s e P2m.i) and
F (s ■ m7r; p) (-1)™ > 0 fo r s e (*7Г, р г +тг], m = 1, 2 . . .
(3) If p > 0 th en th e re ex is t the p o s it iv e r e a l n u m b e rs 0 = q0 < q x< q 2< . . .
s u ch th a t дт £ (тэт (m + l ) 7r ) f o r m = 1, 2, . . . and such th a t F ( . ; p ) changes
s ig n on ly at q j > q2 > . . . . M o re o v e r F (s ;p ) > 0 fo r s € mQ 0(q2m> q2m + l-,(_oo.0)
and F ( s ;p ) < 0 fo r s e 0 ( q ^ ^ , q2m).m = 1
F u r th e r m o re , th e re ex is t the sequences {em} С (0 ,0 ), { e ¿ ,} C (0 + q x),
{бт } С (0 ,7г), {ó¿,} С (9-тг, - ir) such th a t F ( Q = F (e ^ ) = 1 - e'P1™ , F ( 6m )
= F ( 6^|) = 1 + e~prmr fo r m = 1 , 2 , . . . , and e m /•в, 6m^ 0 , e'm в + it, 6 'm ^ 0 - 7r.F r o m the le m m a it fo llow s th a t if pgO then h s (.;k ) does not change
s ig n on the in te rv a l (s- , 0) i f and on ly if
t n_j ê 0 and h (0 ;k) {- l)n S 0 (6.28)
w here tm a re de fined by (6.25); if s x< "2 i t (and p ë 0) t h e n A 0(Sj) = ф and
th e re fo re t ê " 2 п.If we denote
LflfSj) = { (c ,k ) I с = e P s * s in ( s j " к ), к e (sj + 0 + П7Г, s i + 0 + ( n + l ) 7r),
h Sj (0 ;k) (- 1 )" è 0 }
L i ( s j ) = { (c ,k ) I с = ePsi s in fs j ■ к ), к e (0 + (n - l ) i r , s i +0 + (n+ l ) 7r),
h s (0 ;k) (- 1 ) " ё 0}
(6.29)
th en fo r p i 0 we have
L 0(s i) i f s j £ ["tt,0)
L 1(s1) i f s i e(-27r ,-ir)A ) ( s i ) (6.30)
I f p < 0 th en h s¡(.;k ) does no t change the s ig n on ( s ^ 0) if and on ly if
e ith e r (6.28) o r the fo llo w in g c o nd it io n s ho ld :
tn-i < 0 and p 2 < -27Г < s i - t n-i s P i (6.31)
I f we denote :
L 2(s i) = {(c ,k ) I с = epSl s in ( s ! - к ), к € [sj + 0 + (п+1 )я- - p x, 0 + (п-1 )тг)}
(6.32)
L 3(s i) = { (c ,k ) I с = epSl s in (s i - к ), к 6 [ Si + 0 + (п-1)тг - p x, s j + 0 + (n+ l)7r)
IA E A -SM R -17/46 35
t h e n f o r p < 0 , t h e s e t A 0( s j ) i s g i v e n b y : A 0{ s i ) i s L g i s j ) i f S j G [ - 7 r , 0 ) , i s L ^ S i ) i f S j G [ p i , - 7 r ) i s L ^ s j ) U L 2( s 1) i f s x G ( - 2 7 T ,p 1) a n d i s L g f s j ) i f S j 6 ( - » , -27Г ] .
In o rd e r to check w he the r the se ts L ^S ]) and L ^ S j) a re em pty we c o n
s id e r the fu n c tio n H S[ (k) = h s,(0 ;k ) • (-1)"; one can p rove e a s ily th a t
t = ~ 2 n i f p = 0 , т = -qi if t > 0 and т = - °° i f p < 0; m o re o v e r , the sets
LofSj) and L ^ S j ) a re g iven by
L o ts J = {(c ,k ) I с = eps‘ s in (s 1 - k) |k G [k^s-jh Sj + 0 + (n + l ) 7r)}
LqÍS]^) if S jG ( t , 0) and p S 0 and i f S jG (px, 0) and p < 0
{(c ,k ) I с = epSl s in f s j - к ), к € [0 + (n - l)7T, Sj +0 (n + 1)ît)}
if S jG (-27T,p1) and p < 0
L i ( s j ) (6.33)
w here k ^ S j) G (sj +0 + n n , S j +0 + (n + l)jr) a re the ze ro s o f the fu n c tio n H s(.).
I t fo llow s a lso th a t if p s 0 then fo r any k G (s j +0 +mr, S j + 0 + (n+ 1 )7r) such th a t (e i&i s i n ^ - k ), k) G A 0(S]) and fo r any m G {n + l , n + 2, . . . } th e re
e x is ts an sm_n+1(k) G ( tm+1, tm) such th a t h Sl( .;k ) changes s ig n at s m_n+1 (k).
I f p > 0 th en we have the fo llo w in g cases
(i) if k G (sx +0 +П7Г, S j + 0 +(n+ 1 )7Г - 6X] then h s (.;k ) does not change s ig n
on (-°», S j);
( ii) i f k G (sx + 0 + (n + 1 )7T - ógf.-j , s x +0 + (n + 1 )7Г - 62r+1) th en h s (,;k ) changes
the s ig n at s m-n+i(k) fo r m = n + l , n + 2 , . . . n + 2r ;
( ii i) i f к = Sj + ín + lJ ír then h s (.;k ) changes the s ig n a t sm_n+1(k) fo r any
m = n + l , n + 2 , . . . ;
(iv) if k G ( s x + 0 + (n+ 1)тг - e2r+2, S j + 0 + (n + l)îr - e2r), r = 1, 2, . . . , then
h s (.;k ) changes the s ig n at s m_n+1(k) fo r m = n + l , n + 2 , . . . n + 2 r + l ;
(v) i f k G [s1 + 0 + (n + l ) i r - e 2, s 1 + 0 (n + l) ir ) th en h s (.;k ) changes the s ig n
on ly a t s2 G (tn+2, tn+]).
I t fo llo w s a lso th a t the im p lic it- fu n c t io n th e o re m is a p p lic a b le to the
equa tio n
h ^ f s ^ k j s epSl s in f s j- k ) - epssin(js-k) = 0 (6.34)
w h ich de fines the ze ro s sm_n+i(k) < S jo f the fu n c tio n h Si ( .;k ) , m = n + l , n + 2 , . . .
C o m p u tin g the d e r iv a t iv e s o f ¡^ .„ .^ (к ) w ith r e s p e c t to k , and tak ing
in to accoun t the rang e of the p a r a m e te r к we ob ta in :
(a) if p < 0 th en fo r s x G tp i,0 ) , t J í s J , i = 1, 2, . . . , a re su c ce s s iv e roo ts
le s s th an Sj of the equa tio n
(eps's in s i * e ^ s in s j f e ^ ic o s S j- 1) - (é p ic o s s i - ePscos s) ePsis in s x = 0
(6.35)
and fo r s j G ( - °°, p j) , T jfs j) , Í = 1, 2, . . . , a re g iven by
T Í Í S j) = S j - P j + P j ^ (6.36)
36 MIRICÂ
w here р { 6 ( - ( £ +1 )7г, -£тт) a re de fin ed by le m m a 6.4; fo r any s j G (т,0)
= (- °°, 0) and any s2 G ( t ^ S j) , S j), Tgisj, s 2), £ = 1, 2, . . . a re su ccess iv e
ro o ts le s s th an s2 o f the equa tio n
(eps‘ sins-L - epss in s ) (eps‘ cos s x - epSz cos s2)
(eps‘ cos Si - ePscos s) (ef®* s in Sj - eP^ s in s 2) = 0 (6.37)
and Totsj) = tJ(s1) fo r SjG ( r ,0).
(b) if p = 0 then we have t ^ ' ^ S j ) = -2£ir, t ^ S j ) = sx - 2£ ir, r 2í "1(s1, S2 )
= S j - 2 £ it, T ^ ÍS j, s2) = s 2 * 2 £ n, and TgfSj) = t J ( s x) fo r any S jG (t ,0)= (-27Г.0), S2 G (tJÍSjJ.Sj).(c) if p > 0 th en fo r S iG [¿i - 6J,0) we have T j(Sj) = £ = 1, 2, . . . , t 0(s1)
= S j - гг - 6: ; fo r
S i e U 2r+1- 6 5 r+1, ¿ 2, - i - ^ r - i ) , t 0( S i ) = t J í B j ) , r \ ( S l ) .......... r xr+ ( S l )
a re the s u c ce ss iv e ro o ts o f (6.34) and Tjr+2(s1) = Tjr+3 (s1) = . . . = - 00; fo r
Sj G (т,-тг], Tofsj) = T i i s ^ , T ffs j), . . . a re a lso de fin ed to be ro o ts o f E q .(6 .3 4 ) .
It is easy to see now th a t the fu n c tio n s тк, т2 so de fin ed have the d e s ire d
p ro p e r t ie s and so the th e o re m is p roved .
R e m a rk 6.5
As a s id e r e s u lt , th e o re m 6.2 an sw ers the p ro b le m fo rm u la te d and
so lved in R e f . i l ] : to f in d the le ng th L of the m a x im a l in te r v a l on w h ich
any s o lu t io n o f the l in e a r d if fe r e n t ia l s y s te m d x /d t = -A*x has at m o s t two
z e ro s , th a t is , the le ng th L o f the m a x im a l in te r v a l on w h ich the b ilo c a l
p ro b le m has a un ique s o lu tio n . F r o m th e o re m 6.2 it fo llow s th a t
L = s u p { (s 1 - TqÍSj), Sj E (tjO )} and fr o m the above p ro o f we o b ta in the s am e
r e s u lt as in R e f. [l ].
R e m a rk 6.6
The fo rm u la s g iv e n in the p ro o f o f th e o re m 6.2 fo r the fun c tio ns
t Tj, Tk and fo r t G [-°o, 0), w h ich de fine the sy n th e s is m a y be used fo r
a p o s s ib ly n u m e r ic a l p ro ce d u re fo r c o m p u tin g the sy n th e s is .
R E F E R E N C E S
[1 ] ARAMA, O ., RIPIANU, D ., Asupra problemei polilocale, I, Stud. Cercet. Mat. Acad. R. P. R. t Fil Cluj8_1 (1957) 37.
[2 ] BOLTYANSKII, V .G ., Mathematical Methods in Optimal Control, Holt and Rinehart, N. Y . (1971).[3 ] BRUNOVSKY, P ., "The closed-loop tim e-optimal control, I” , Optimality, to appear in SIAM J. Control.[4 ] BRUNOVSKY, P ., "The closed-loop tim e-optimal control, II", Stability, to appear in SIAM J. Control.[5 ] BRUNOVSKY, P ., MIRICA, S ., Classical and Filippov solution for the tim e-optimal feedback control,
to appear in Rev. Roum. Math. Pures Appl.[6 ] FILIPPOV, A. F ., On certain questions in the theory of optimal control, SIAMJ. Control 1_(1963) 76.[7 ] FILIPPOV, A. F ., Right-hand-side discontinuous differential equations, Trans. Am. Math. Soc. 2
42 (1964) 199. ~~
IA E A -SM R -17/46 37
[8 ] HAJEK, О ., Geometric theory o f tim e-optimal control, SIAMJ. Control 9 (1971) 339.[9 ] HERMES, H ., Discontinuous vector fields and feedback control, Symp, Diff. Equations and Dynamical
Systems, Acad. Press (1967) 155.[10] HERMES, H ., LASALLE, J .P ., Functional Analysis and Tim e Optimal Control, Academic Press,
N. Y. (1969).[11 ] ISAACS, R ., Differential Games, Wiley (1965).[12] LEE, E. B ., MARKUS, L ., Foundations o f Optimal Control Theory, Wiley, N .Y . (1967).[13 ] KUTUZOV, V .K ., ’.’Sintez optimal’nogo bystrodeistviya sistem vysshego poiyadka", Izv. A .N .
SSSR, Tehn. Kib. 3_(196S) 96.[ 14] MIRICA, S ., An admissible synthesis for control systems on differentiable manifolds, R. I. R. O .,
R-l(1971) 73.[15 ] MIRICA, S ., An algorithm fo i optimal synthesis in control problems, R .l. R. O ., R-2 (1971) 55.[16] MIRICÂ, S ., Universal surfaces in regular synthesis of certain tim e-optim al control problems, Publ.
Math. Bordeaux 1_3 (1973) 1.[17] MIRICA, S ., The tim e-optimal feedback control and the regular synthesis, I, to appear in Rev. Roum,
Sci. T e c h ., Sér. Electrotech. Energ.[ 18] MIRICÂ, S ., The time-optimal feedback control and the regular synthesis, II, to appear in Rev. Roum.
Sci. T e ch ., Ser. Electrotech. Energ.[19 ] MOROZ, A .I . , Sintez optim al’nogo bystrodeistiya dlya lineinykhsistem tret'ego poryadka, Avtom.
Telemekh, 5, 7, 9 (1968).[20] PONTRIAGIN, L .S ., BOLTYANSKII, V. G. et a l . , Mathematical Theory o f Optimal Process,
Interscience (1962).[21] POPOV, V. M ., Hyperstability o f Control Systems, Academic Press (1973).
IAEA-SM R-17 /47
APPLICATIONS OF FUNCTIONAL ANALYSIS TO OPTIMAL CONTROL PROBLEMS
K. MIZUKAMIDepartment of Electrical Engineering,
University of Hiroshima,
Hiroshima, Japan
Abstract
APPLICATIONS OF FUNCTIONAL ANALYSIS TO OPTIMAL CONTROL PROBLEMS.Some basic concepts in functional analysis, a general norm, the Holder inequality, functionals and the
Hahn-Banach theorem are described; a mathematical formulation o f two optimal control problems is introduced by the method of functional analysis.
The problem o f time-optimal control systems with both norm constraints on control inputs and on state variables at discrete intermediate times is formulated as an L-problem in the theory o f moments. The simplex method is used for solving a non-linear minimizing problem inherent in the functional analysis solution to this problem. Numerical results are presented for a train operation.
The second problem is that o f optimal control of discrete linear systems with quadratic cost functionals.The problem is concerned with the case o f unconstrained control and fixed endpoints. This problem is formulated in terms of norms of functionals on suitable Banach spaces.
1. IN T R O D U C T IO N
In fu n c t io n a l a n a ly s is we c o n s id e r v e c to r spaces re s u lt in g fr o m a m e rg in g
of g e o m e try , l in e a r a lg e b ra and a n a ly s is . It has undergone c o n s id e ra b le
d eve lo pm e n t and has a lo ng h is to ry o f in v e s t ig a t io n .
F u n c t io n a l a n a ly s is p ro v id e s a u n if ie d fr a m e w o rk , in som e sense , fo r
c o n s id e r in g o p t im a l c o n tro l p r o b le m s , how ever, i t m ay no t be such a po w e rfu l
te chn iq ue th a t enab le s us to so lve im p o r ta n t p ro b le m s beyond the re a ch of
s im p le r m a th e m a t ic a l a n a ly s is .
A m o n g the m o s t v a lu a b le p ap e rs of fu n c t io n a l a n a ly s is co nce rne d w ith an
a p p lic a t io n to o p t im a l c o n tro l th e o ry a re th ose by B an ach [l] and K re in [2] .
B an a ch d is c u s s e d the n e ce s s a ry and s u ff ic ie n t c o nd it io n s fo r the ex is tence
o f a l in e a r fu n c t io n a l on a n o rm e d v e c to r space s a t is fy in g c o n s tra in ts and not
exceed ing a g iven bound on its n o rm . K re in goes in to the p ro b le m m o re
deep ly and d is c u s s e s the p r o p e r t ie s of so lu tio n s fo r p a r t ic u la r spaces .
T h is is c a lle d K r e in 's L - p ro b le m in the th e o ry of m o m e n ts .
O ne of the f i r s t a p p lic a t io n s of B a n a c h 1 s and K r e in 's w o rks to o p t im a l
c o n t ro l th eo ry w as d e s c r ib e d by K r a s o v s k i j [3]. He deve loped a g e n e ra l
f o rm of the o p t im a l c o n tro l fu n c tio n s fo r the t im e o p t im a l o f a l in e a r sy s tem
w ith a m p litu d e - c o n s tra in e d c o n tro l in p u ts .
K r a s o v s k i j1 s in v e s t ig a t io n s w ere fu r th e r extended by K u lik o w s k i [4]
who d e m o n s tra te d th a t K r e in 's r e s u lts can be d ir e c t ly a p p lie d to a m o re
g e n e ra l type of c o n s tr a in t on the in p u t.
E s p e c ia l ly in the p a s t decade th e re h as been a g ro w ing in te re s t in
a p p ly in g s om e of the te c h n iq u e s o f fu n c t io n a l a n a ly s is to a v a r ie ty o f c o n tro l
p r o b le m s . S e v e ra l d if fe re n t p r o b le m s in the f ie ld of t im e o p t im a l c o n tro l
s y s te m s w ere fo rm u la te d as a b s t ra c t p ro b le m s in te rm s of fu n c t io n a l
39
40 MIZUKAMI
a n a ly s is by K ra n c and S a ra c h ik [5], S w ige r [6], K a tz a n d K ra n c [7], K r e in d le r [8],
and S ilm b e rg and M iz u k a m i [9].
I t is the pu rp ose of th is p a p e r to in tro d u ce the m e th od o f fu n c tio n a l
a n a ly s is and to in d ic a te its a p p lic a t io n to p ro b le m s of " t im e o p t im a l c o n tro l
s y s te m s w ith n o rm c o n s tra in ts on sta te v a r ia b le s at in te rm e d ia te t im e s " [ l0]
and " d is c r e te l in e a r o p t im a l c o n tro l s y s te m s w ith q u a d ra t ic cost
fu n c t io n a ls " [1 1 ].
I t is in tended to d e s c r ib e som e b as ic concep ts of fu n c t io n a l a n a ly s is
re la te d to the p ro b le m s of se c t io n s 3 and 4 , bu t the m a th e m a t ic a l
b a ck g ro u nd in tro d u c e d in the fo llo w in g sec t io n s is in s u ff ic ie n t . A re a d e r
who is in te re s te d in d e ta ils of these concep ts is r e fe r re d to R e fs [12, 13, 14]
and o the r p a p e rs of th ese P ro c e e d in g s .
2. S O M E F U N D A M E N T A L C O N C E P T S
In th is se c t io n we b r ie f ly d e sc r ib e a few of the im p o r ta n t concep ts of
fu n c t io n a l a n a ly s is used la te r on.
2 .1 . A g e n e ra l n o rm
One m e a s u re of the s ize o f a v e c to r used in o rd in a ry g eo m e try is the
E u c lid e a n le n g th o r squa re ro o t of the s um of the s q u a re s o f its o r th og o na l
co m po nen ts . The concep t o f le ng th is g e n e ra liz e d to the concep t of the
n o r m of a v e c to r in a l in e a r v e c to r space o r fu n c tio n space . The n o rm is
d e fin ed to be a non-negative re a l- v a lu e d fu n c tio n o f the v e c to r u w h ich
s a t is f ie s the fo llo w in g th re e ax iom s :
The n o rm of u = || u || :
u fo r к = r e a l n um be r
The n o rm of the c o n tro l v e c to r is the m e a s u re of the c o n tro l e ffo r t w h ich
is used h e re . The n o rm m u s t be de fined so as to have the p h y s ic a l s ig n if i
cance d e s ire d fo r the d e s ig n , and m u s t s a tis fy the th re e a x io m s above.
The p- no rm of the c o n tro l v e c to r u = ( u j , . . . , u r) d e fined by E q . ( l ) s a t is f ie s
a l l r e q u ir e m e n ts , (a), (b) and (c)
(a) t|u|| - 0
(b) ||ku|H
(c) Il U + W IIspace .
| u « p0 i = i
1/P
J ( 1 )
w here p i 1 and u¡(t) is d e fin ed on the in te r v a l [0,T].
W ith , e .g . p = 2 , th is n o rm has the p h y s ic a l s ig n if ic a n c e of the squa re
r o o t o f to ta l energy o f the c o n tro l fu n c tio n s Uj(i = l , . . . , r ) . F o r p = l and p=°o
we have the so - ca lle d a re a (fue l) and a m p litu d e n o rm s o f the v e c to r u ,
r e s p e c t iv e ly .
IAEA-S M R-17/47 41
P h y s ic a l c o n tro l c o n s tra in ts can be re p re se n te d by c o n s tra in ts on the
n o rm of (Iu||p i n t im e o p t im iz a t io n p ro b le m s .
C o n s id e r a s e p a ra te c o n s tr a in t on each com ponen t o f the c o n tro l v e c to r
in the fo rm
i "p Hi (t) I ‘ dt
i/i>-(2 )
w here p ; i 1 , fo r i = 1 , . . . , r .
N ote th a t p j and C¡ m ay be d if fe re n t fo r each i. T h is m e an s th a t each
co m po nen t o f the c o n tro l v e c to r m ay have a d if fe re n t type of c o n s tr a in ts .
F o r e x am p le , i f the c o n tro l in p u t u¡ is to be c o n s tra in e d in m a g n itu d e , then
Pj = °°; i f it is to be c o n s tra in e d in e nergy , th en p t = 2 , e tc .
I t w i l l be conven ien t to c o n s id e r a m e ans of r e p re s e n tin g the set of
r c o n s tra in ts (2) as a s in g le c o n s tr a in t co nd it io n . T h is is r e a d ily done by
d e fin in g a n o rm of u as
Il u II = m ax [||u j /C j] i Fi
N ote th a t , when the s in g le co nd it io n
(3)
(4)
is s a t is f ie d , then it is a p p a re n t th a t the r co nd it io n s g iven by E q .(2 ) a re a lso
s a t is f ie d . T h is m e an s th a t the r c o n tro l in p u t c o n s tra in ts can be im p o se d by
r e q u ir in g th a t the s in g le c o nd it io n (4) be s a t is f ie d . U n fo r tu n a te ly , the n o rm
de fin ed in E q .3 is in co nven ien t fo r d ire c t a p p lic a tio n ; a s im p lif ic a t io n in
the s o lu tio n is a ch ieved by e m bedd ing the c o n s tr a in t c o nd it io n s in a s t i l l
m o re g e n e ra l fo rm of n o r m on u.
T h is m o re g e n e ra l n o rm can be g iven by
C.'P II u.
1/p(5)
i=l
w here p ê 1, and || u £ ||p. is d e fin ed in E q .(2 ). N ote th a t the n o ta tio n ||u ||p of
E q .(5 ) is d if fe re n t fr o m th a t o f E q . ( l ) .
I t is a pp a re n t th a t in the l im i t o f p ^ ° ° the s o lu t io n w ith ||u||p c o n s tra in e d
a pp ro ache s the s o lu tio n w ith ||u|| c o n s tra in e d so th a t the c o n s tra in ts of E q .(2 )
c an be re p la c e d by the c o n s tra in t
Ilu IIp s 1 . ( 6 )
The s o lu t io n o f an o r ig in a l ly s ta te d p ro b le m w ith E q .(2 ) c o n s tra in e d is
th u s o b ta in ed by f i r s t s o lv in g the m o re g e n e ra l p ro b le m w h ich h as a
c o n s tr a in t g iv e n by E q . ( 6) and th en le tt in g p-*°°.
T h is m e th od of u s in g a g e n e ra l n o rm is a p p lie d w hen so lv in g a t im e
o p t im a l p ro b le m w ith n o rm c o n s tr a in ts on sta te v a r ia b le s in se c t io n 3 .
F o r fu r th e r in fo r m a t io n see R e fs [6, 15].
42 MIZUKAMI
L e t us in tro d u ce an in e q u a lity w h ich w i l l be im p o r ta n t fo r c e r ta in
a p p lic a t io n s to c o n tro l p ro b le m s . I t is in s tr u c t iv e to f i r s t c o n s id e r a
s im p l if ie d v e rs io n of c o n tro l p ro b le m s . A s s u m e a t im e - in v a r ia n t l in e a r
s y s te m w ith an im p u ls e re spo nse w(t) w h ich can be a c tiv a te d by a c o n tro l
in p u t Ujft) and w h ich has a s in g le ou tput Xj(t). The re spo n se (the to ta l
ou tput c o n s is ts of the re sp o n se xa(t) and the re spo nse due to in i t ia l c o nd it io n s
w h ich is a s su m e d to be know n) at t im e t to the in p u t u jft), w h ich is app lie d
a t t = 0, is g ive n by
2.2 . The H older inequality
£i(t) = J w (t-T )Ul(T)drо
(7)
L e t us c o n s id e r the p ro b le m of m a x im iz in g X jfT ), fo r a s p e c if ie d va lue
of T , w ith the c o n s tr a in t th a t the c o n tro l in p u t is l im ite d by a c o n s tr a in t of
the fo rmT
“ lMp J " j Uj(t) I P dt
i/pS C (8 )
w here p ê 1 .
I t is conven ien t at th is p o in t to in tro d u ce an in e q u a lity c a lle d H o lde r
in e q u a lity [15] w h ich can be s ta te d as
D L>
J f j(t ) g j(t) dt| sj I fx(t) gx(t) I dt= H f J J g J (9)
w here tw o fu n c tio n s fj and g j a re de fined in the in te r v a l [a,b], and p i 1 , qS 1 and l / p + l / q = l .
The e q u iv a le n t H ô ld e r in e q u a lity fo r sum s [13] is
n n1Л>
nr -1 iAi
[ I o?] ( 1 0 )
i= l i=l i=l
w here the e q u a lity h o ld s i f and only if
Qi = k | £ i |4’ 1 sgn [j3j], i = 1 , . . . , n ( 1 1 ) -
w here к is an a r b i t r a r y cons tan t.
I t fo llow s f r o m the H o lde r in e q u a lity (9) a p p lie d to E q .(7 ) th a t
LA EA-S M R -17/47 4 3
The in e q u a lit ie s o f E q .(1 2 ) becom e e q u a lit ie s i f and on ly if
I U j ( t ) Ip = к I w ( T - t ) 14
w here к is a p o s it iv e s c a la r q u an tity and
s g n [ u a (T)] = s g n [ w ( T - t )]
fo r a l l t in the in te r v a l [О, Т].
So the o p t im a l c o n tro l in p u t u * (t) can be g iven by
u* (t) = k 14>|w(T-t)|4/i> sgn [w(T-t)]
(13)
(14)
(15)
W e m u s t now f in d k. A s s u m in g the e q ua lity s ig ns to h o ld in E q .(1 2 )
we have
x ,(T ) = I J |w(T-r)| dr
о
l/qJ I u x(r ) |P dT
l/b(16)
A s s u m in g U j ( t ) to be g iv e n by E q .(1 3 ) , E q .(1 6 ) becom es
X j ( T ) = w(T-r) dr
l/q
о
к J |w(T-r)| dT
о
i/t>(17)
S in ce l / p + l / q = 1, we can w r ite E q .(1 7 ) as
T
х х( Т ) = k1Æ Г |w ( T - t )|4 dT (18)
U s in g the r e q u ir e m e n t th a t E q . ( 8 ) b eco m es an e q ua lity fo r o p t im a l
c o n tro l to m a x im iz e X j ( T ) , we can w r ite E q .(1 6 ) in the fo rm
x *(T ) = w ( T - t ) d rl/q
(19)
K now ing C p and Т , x * (T ) is found fr o m E q .(1 9 ). T hen , w ith E q .(1 8 ) , к is
o b ta in ed f r o m
.1Л> -:*(T )
I | w ( T - t ) I dT
(20)
I t is in te re s t in g to note f r o m E q .(1 9 ) th a t, i f T is f ix e d and C p is v a r ia b le ,
the m a x im u m a ch ie v ab le v a lu e o f x,(T) is d ir e c t ly p r o p o r t io n a l to C p.
T he re a re th re e b a s ic fo rm s in w h ich the s am e o p t im iz a t io n p ro b le m can
be s ta ted [16]:
44 MIZUKAMI
(a) F o r a fix e d t im e T and a s p e c if ie d v a lue of C p, f in d the c o n tro l in p u t
u if(t) w h ich m ak e s Xj(T) a m a x im u m (or m in im u m ) . T h is is the fo rm
of the p ro b le m c o n s id e re d above .
(b) F o r a fixed t im e T and a s p e c if ie d va lue of x-jjT), f in d the c o n tro l in p u t
”u * (t) w h ich m in im iz e s ||u|| . T h is is the p ro b le m of f in d in g the s m a lle s t
p o s s ib le v a lue o f к in E q .(1 5 ) such th a t х г(Т ) is the d e s ire d va lue .
(c) F o r a s p e c if ie d f in a l s ta te x d and a s p e c if ie d v a lue of C p, f in d the c o n tro l
in p u t Uj(t) w h ich a llo w s x d to be re ache d in m in im u m t im e T * . T h is
c o u ld be done by f ix in g w hat v a lue of T w ou ld be re q u ir e d to m ake
x * (T ) =xd. The r e s u lts c o u ld be ob ta ined by p lo tt in g the g ra p h o f F i g . l .
L e t us c o n s id e r case (c) o f o p t im iz a t io n p ro b le m s w h ich is a t im e
o p t im a l p ro b le m and d e te rm in e the o p t im a l c o n tro l in p u t u*j(t).
W e can f in d T* as b e fo re , by a g ra p h of the fo rm of F i g . l , to re a c h the
d e s ire d s ta te xd in m in im u m t im e . H av ing found T * , the c o n tro l in p u t is
g iv e n by
a i f m
x i
T0 T*
FIG. 1. Graph of xs,'(T ) .
w(T*-t)|4/P sgn[w (T *- t)] (2 1 )
/ |w(T*-t)|q dtо
F r o m E q .(1 9 ) we f in d the qu an tity
T *
0
to be
( 22)
о
S u b s t itu tin g E q .(2 2 ) in E q .(2 1 ) we o b ta in
X, . IQ/D
u* ( t ) = T— fq |w(T*-t) I sgn[w (T *- t)]
' D(23)
S in ce q /p = q - 1 we have
u*(t) = C p4w (T *-t)
q-lsgn [w(T* -t)] (24)
In the fo re g o in g we a s su m e d xd to be p o s it iv e . In o rd e r to o b ta in a
s o lu t io n v a lid fo r e ith e r p o s it iv e o r negative x d, E q .(2 4 ) sho u ld be m o d if ie d
as fo llow s :
u * (t) = Cpw (T*-t)
q-ls g n [ w (T * - t) /x d] (25)
To o b ta in an ex tens io n o f H o ld e r 's in e q u a lity to an in te g r a l o f the fo rm
b b n
(26)L I
a i=i
J f(t) • g (t )d t =J f . ( t ) g . ( t ) d t
w here f= ( ^ , . . .Д П) and g= (g j , . . . , gn) a re fu n c tio n s de fined in the in te r v a l [a ,b] ,
o b se rve th a t
b n
J ^ f . ( t ) g . ( t ) d t J f . ( t ) g . ( t ) d t
a i=i(27)
and th a t the e q ua lity s ig n in E q .(2 7 ) w i l l h o ld if and on ly if
f . ( t ) g , ( t ) i 0 , o r f ( t )g .( t ) S 0 (28)
fo r a S t s b and a l l i = 1, . . . ,n .
Now app ly in g the r e s u lts of the s im p le H o ld e r in e q u a lity as e xp re ssed in
E q .(9 ) we see th a t
D
J f . ( t ) g . ( t ) d t « llf jlp . I|gj II q. (29)
w here P jS l , 4¡ S 1 and l / p . + l / q ^ l when |f¡(t)| 1 and |g¡(t)| 1 a re in te g ra l,
and w here ||f¡||p. and ||g¡(]q. a re de fin ed by
46 MIZUKAMI
F r o m E q s (13) and (14) i t fo llow s th a t the in e q u a lity in E q .(2 9 ) becom es an
e q u a lity i f and only i f
f j( t) = k ; I g t (t)|Ql 1 sgn[g .(t)] , a s t s b
S u b s t itu t in g E qs (26) and (29) in to E q .(2 7 ) g ives
f(t) g(t) dt
n
I I I fillp; M q , i= l
(31)
(32)
N ote th a t if E q .(3 1 ) h o ld s fo r a l l i = l , . . . , n and i f a l l k¡ have the sam e s ig n ,
th en c o nd it io n s (28) a re s a t is f ie d . T h is m e a n s th a t the e q ua lity in E q .(3 2 ) is
v a l id if and only i f E q .(3 1 ) ho ld s fo r a l l i = l , . . . , n , and a l l k j have the sam e
s ig n .
Now i f
f'. =•II f ill p .
(33)
w here th e C¡ a re p o s it iv e co ns tan ts ; th en , s in ce f! and gV a re po s it iv e
we see th a t
I f; g ; = I f,ij (34)
i=l i=l
and , u s in g the r e s u lts of E q s (10) and (11), we o b ta in
1
I ' l ' iK D ’!1’] [£li= l i= l i= l
j ' lqi/q
(35)
w here p ï 1 , q ë 1 and l / p + l / q = l , and th a t e qua lity ho lds if and on ly i f
fl = к 1 I g! |4 1 sgn [gj] fo r a l l i (36)
w here k 1 is an a r b i t r a r y p o s it iv e co ns tan t. H ow ever, s in ce f'¡ and g'j a re
p o s it iv e , E q .(3 6 ) can be exp re sse d as
fj = k> g l '4' 1’ (37)
w here k' is a p o s it iv e co ns tan t. W hen E q .(3 3 ) is su b s t itu te d in E q .(3 4 ), the
r e s u lt in g equa tio n c o m b in e d w ith E q .(3 2 ) g ive s
IAEA-SM R-17/47 47
In E q .(3 8 ) e q u a lity h o ld s i f and on ly i f the c o nd it io n s fo r the e q ua lity in
E q s (32) and (35) a re s a t is f ie d . T h is r e q u ir e s th a t a l l k¡ have the sam e
s ig n and th a t E q s (31) and (37) ho ld .
C o n s id e r in g c o nd it io n (37) we ob ta in
q < n * . i p . < I K (t)r *1/,pi ( q-l) q - lM ,,q-l
k 'g ! = k 'C , ||g.||q (39)
S u b s t itu tio n of f j( t ) , as g iven by E q .(3 1 ) , in to the in te g ra l of E q .(3 9 ) y ie ld s
c > j i i g j i ^ ^ k . c ^ i i g i i ; ; 1 (40)
S o lv ing (40) fo r |k;| we o b ta in
I k. I = k 'C ? II g , Ц4 4i f o r a l l i = l , . . . , n . . (41)' i ' i " i " q .
T h e re fo re , i f k j in E q .(3 1 ) is a lw ays chosen so th a t E q .(4 1 ) h o ld s , th en
E q .(3 7 ) w i l l a lso be s a t is f ie d . It s ho u ld a lso be noted th a t kj can be p o s it iv e
o r nega tive p ro v id e d th a t a l l k¡ have the sam e s ig n . T h is m e a n s th a t E q .(4 1 )
c an be r e p la c e d by
k. = k C^Hgj I qZ*1 fo r a l l i = l , . . . , n (42)
w here к is an a r b i t r a r y co ns tan t.
S u b s t itu t in g th is r e s u lt in to E q .(3 1 ) g ive s the f in a l fo rm of the n e ce ssa ry
and s u ff ic ie n t c o nd it io n s fo r e q u a lity in E q .(3 8 ) as
f ¡ ( t )= k C ? [I g t I)q.411gt(t) |4i 1 sgra [g i (t)] (43)
fo r a l l a s t s b and a l l i = l , . . . , n .
2 .3 . F u n c t io n a ls
In th is se c t io n we s h a l l in tro d u c e a fu n c t io n a l conven ien t fo r d e s c r ib in g
the s im p le t r a n s fo rm a t io n s e nco un te red in the la t te r se c t io n s .
L e t X and Y be tw o B an a ch spaces (com p le te , n o rm e d l in e a r spaces)
w here Y is the set of r e a l n u m b e rs . A fu n c t io n a l f is ju s t a m ap p in g of
e lem en ts ( x G X ) fr o m the space X in to the space Y of r e a l s c a la r s , denoted
by y = f(x ), y G Y .
I t can be show n [13] th a t the se t o f a l l l in e a r fu n c t io n a ls d e fined on a
B an ach space is a B an a c h space it s e lf . I t is c a lle d the con ju ga te space of X
and is deno ted by X * .
T he n o rm of a bounded l in e a r fu n c t io n a l is the s m a lle s t n u m b e r С
w h ich s a t is f ie s ||f(x)|| <c||x|| fo r a l l x e X . I t is denoted by ||f|| and is
g iv e n by [13]
||f|| = sup l f(xi l = sup |f(x)|
X* 0 II XII llxll =1
48 MIZUKAMI
To i l lu s t r a te the fo rm of l in e a r fu n c tio n a ls on som e B an a ch spaces and
th e ir n o rm s in the con juga te space , c o n s id e r the fo llo w in g two e x am p le s .
In the n - d im e n s io n a l v e c to r space E n the g e n e ra l fo rm o f a l in e a r
fu n c t io n a l can be show n to be
w here x = ( x j , . . . , ^ ) , and X; a re a r b i t r a r y n u m b e rs and the fu n c t io n a l has the
n o rm
T hus it is obv ious th a t E * is the n - d im e n s io n a l v e c to r space E n and th a t each
e le m e n t f o f th is space can be re p re se n te d as an n-vec to r X = (A p .. . ,/^ ) .
In the B an a ch space L p, the g e n e ra l fo rm of a l in e a r fu n c t io n a l is
w ith l / p + l / q = l .
T hus the co n ju ga te space L * is ju s t the B an ach space L q w ith q = p / (p - l) .
E a c h l in e a r fu n c t io n a l on L p can th us be re p re s e n te d by a t im e fu n c tio n h jit )
d e fin ed on the in te r v a l [a,b].
T he g e n e ra l fo rm of a l in e a r fu n c t io n a l on the B an a ch space L p is
n
(44)
(45)
b
(46)
a
I ts n o rm in L * is
(47)
a
b n
(48)
a i= l
The n o rm in L " is
(49)
a i= i
w here l / p + l / q = l .
2 .4 . The H ahn- B anach th e o re m
L e t f be a l in e a r fu n c t io n a l d e fin ed on a subspace M of a v e c to r space X .
A l in e a r fu n c t io n a l F is s a id to be an ex tens io n o f f i f F is d e fined on a
IAEA-SM R-17/47 49
subspace N w h ich p ro p e r ly co n ta in s M , and if , on M , F is id e n t ic a l w ith f.
In th is c a se , we say th a t F is an ex tens ion of f f r o m M to N .
In s im p le te r m s , the H ahn- B anach th eo rem [14] s ta te s th a t a bounded
l in e a r fu n c t io n a l f d e fin ed only on a subspace M of a n o rm e d space can be
extended to a bounded l in e a r fu n c t io n a l F de fined on the e n tire space , w ith a
n o rm equa l to the n o rm of f on M ; i.e .
и ................ I f ( x ) [Il F ü =||f|| = sup —г— r > x e M (50)
x*o ||x II
A m o re g e n e ra l v e rs io n o f the H ahn- B anach th e o re m is s ta ted be low ,
w here the te r m n o rm is re p la c e d by s u b lin e a r fu n c t io n a l.
L e t us f i r s t in tro d u ce a s u b lin e a r fu n c t io n a l. A re a l- v a lu e d fu n c tio n g
d e fined on a r e a l v e c to r space X is s a id to be a s u b lin e a r fu n c t io n a l on X if
( 1 ) g (x j+ x 2) = g (x1) + g (x2), fo r a l l X p X ^ X
(2) g (»x ) = ffg(x), fo r a l l a ï 0 and x € X
O b v io u s ly , any n o rm is a s u b lin e a r fu n c t io n a l.
The th e o re m is s ta ted as fo llow s :
L e t X be a r e a l l in e a r n o rm e d space and g a co n tin uo us s u b lin e a r
fu n c t io n a l on X . L e t f be a l in e a r fu n c t io n a l d e fin ed on a subspace M of X
s a t is fy in g f(x) S g(x) fo r a l l x G M . T hen th e re is an ex tens io n F of f f r o m
M to X such th a t F (x ) S g (x), x e X on X .
3. T IM E O P T IM A L C O N T R O L SYST E M S W IT H N O R M C O N ST R A IN T S ON
S T A T E V A R IA B L E S A T IN T E R M E D IA T E T IM E S
3 .1 . In tro d u c t io n
In re c e n t y e a rs m u c h a tte n tio n has been devoted to the f ie ld of t im e
o p t im a l c o n tro l s y s te m s . One of the le ad in g app ro ache s to these p ro b le m s is
the m e th o d of fu n c t io n a l a n a ly s is [3, 5, 8, 9, 15]. H ow ever, th e re is a non
l in e a r m in im iz in g p ro b le m in h e re n t in the fu n c t io n a l- a n a ly t ic s o lu t io n to the
m in im a l t im e c o n tro l p ro b le m . T h is p ro b le m in m o s t case s p re se n ts
c o m p u ta t io n a l d if f ic u lt ie s .
R e c e n tly , K ra n c and S h ilm a n [17] c o n s id e re d the p ro b le m of t im e
o p t im a l c o n tro l o f l in e a r d is c re te sy s te m s w ith c o n tro l and output c o n s tra in ts
and ob ta in ed it s s o lu t io n in a s tr a ig h tfo rw a rd m a n n e r . T h e ir m e th od is
b ased on the K a tz and K ra n c fo rm u la t io n [7] in te rm s of n o rm s of fu n c tio n a ls
on s u ita b ly c o n s tru c te d B anach spaces fo r the p ro b le m of co n t in uo u s l in e a r
s y s te m s w ith an in te r io r ou tpu t c o n s tr a in t .
In th is p a p e r we show how the K a tz and K ra n c fo rm u la t io n can be
extended to a m o re g e n e ra l p ro b le m of c on tin uous t im e l in e a r sy s te m s w ith
s e v e ra l n o rm c o n s tra in ts on the s ta te v a r ia b le s at d is c re te in te rm e d ia te
t im e s ; the c o m p u ta t io n a l p ro b le m of a m u lt i- d im e n s io n a l m in im iz a t io n of
a n o n - lin e a r fu n c tio n c an be so lv ed by m e an s of the s im p le x m e th od [18]
w ith the a id of a d ig it a l c o m p u te r .
50 MIZUKAMI
C o n s id e r the p ro b le m of t im e o p t im a l c o n tro l of a con tin uous t im e
l in e a r s y s te m gove rned by the se t o f d if fe r e n t ia l equa tio ns
3.2. D e sc r ip t io n o f the sy stem
x (t) = n - d im e n s io n a l s ta te v e c to r
A (t) = n X n - d im e n s io n a l s y s te m m a t r ix
B (t) = n X m - d im e n s io n a l in p u t o r c o n tro l m a t r ix
u (t) = m - d im e n s io n a l in p u t v e c to r
w ith n o rm c o n s tra in ts im p o s e d on the u k co m po nen ts of the in p u t v e c to r
w here C k and C'k a re p o s it iv e co ns tan ts and x k(tj) is the к- th co m ponen t of
som e d e s ire d t r a je c to r y xd at t im e t^; the sy s te m is thus le d f r o m an in i t i a l
s ta te
(T b e ing the f in a l t im e ) in m in im a l t im e in te r v a l T - t0.
The c o n s tra in e d n o rm s of E q s (2) and (3) c a n be de fin ed to have a
c e r ta in p h y s ic a l s ig n if ic a n c e fo r som e v a lu e s of p k and p 'k . W ith , e .g .
p'k= oo, the n o rm of E q .(3 ) ac ts as a c o n s tr a in t on the m a x im u m dev ia tio n
of the a c tu a l tr a je c to r y fr o m the d e s ire d one . I f xk(t¡) =0 th is beco m es an
a m p litu d e c o n s tr a in t on xk(tj) . F o r p'k=2 , i t is a c o n s tra in t on the squa re
e r r o r of the d e v ia tio n .
T he s o lu t io n o f E q . ( l ) is
x (t) = A (t)x (t ) + B (t) u(t) (1)
w here
(2 )
and on the x. s ta te v a r ia b le s a t in te rm e d ia te t im e s t.к i
(3)
x ( t0) = x 0
(tQ is the in i t i a l t im e ) to a d e s ire d f in a l s ta te
x (T ) = x f
(4)
(5)
Tx (t) = Ф (T ,t0)x 0 + / a (T , t )B ( t ) u ( t ) d t■/ ( 6 )
w h ich c an a ls o be e xp re sse d as
T
7 (T ,t0) = J H (T ,t) u (t) dt
l0
(7)
IAEA-SM R-17/47 51
w here
® (T ,t0) = t r a n s it io n m a t r ix fo r the sy s tem
® (T ,t) = s (T , t 0)<e-1 ( t ,t0)
7 (T ,t0) = x f - i ( T , t 0) x 0H (T ,t) = ® (T ,t )B (t ) = im p u ls e re sp o n se o f the sy s te m
if u(t) is the in p u t w h ich le ad s the sy s te m to x (T ) =xf at the t im e T .
In the fo llo w in g , w ithou t lo s s of g ene ra lity , i t is s u ff ic ie n t to c o n s id e r a
doub le-ou tpu t, s in g le- in pu t sy s te m w ith a c o n s tr a in t of the fo rm
im p o se d on the second com ponen t o f the s ta te v e c to r w here p 's 1 , С ' is a
p o s it iv e co ns tan t and w is a f in ite n um be r o f in te rm e d ia te t im e in s ta n ts t¡ at
w h ich the s ta te v a r ia b le s a re s u b je c t to c o n s tr a in t . The c o n tro l c o n s tra in t
is take n as
w here p 2 1, and С is a p o s it iv e co ns tan t.
T h is la t te r p ro b le m can be r e fo rm u la te d to postpone the d e te rm in a t io n
o f the s m a lle s t T - t0 by te n ta tiv e ly f ix in g T and a sk in g fo r the c o n tro l fu n c tio n
u w h ich m ak e s [7]
I f the m in im u m in E q .( lO ) is u n ity , then the c o r re s p o n d in g c o n tro l is a
s o lu t io n to the o r ig in a l p ro b le m w ith the c o n s tr a in ts o f E q s (7) - (9). I f th is
v a lu e is g re a te r th an u n ity , th en the o r ig in a l p ro b le m has no so lu tio n .
3 .3 . The L - p ro b le m and p ro du c t spaces
1/P 's C (8 )
(9)
( 10 )
w h ile s a t is fy in g E q .(7 ), i .e .
T
( 1 1 )T
T he p ro b le m of m in im iz in g E q .(1 0 ) sub je c t to E q . ( l l ) c an be r e fo rm u la te d
as an L - p ro b le m in the th eo ry o f m o m e n ts [2 ] by m e a n s of the p ro duc t
space [7] of s u ita b le B an a ch sp a ce s . In the tr e a tm e n t of th is p ro b le m the
52 MIZUKAMI
c o n tro l u in Lp is a s so c ia te d w ith a s u ita b le bounded l in á ^ r fu n c t io n a l in the
d u a l space L q. A c c o rd in g ly , we in tro d u ce the in d ic e s con juga te to p ,p ',
r e s p e c t iv e ly , by
- + - = 1 , - , + - , = 1p q p' q
(W )N ow we c o n s id e r the C a r te s ia n p ro du c t space В =Lt]X i q, c o n s is t in g of
c o m p o s ite v e c to rs cp by p a ir in g a fu n c tio n f ( t ) in the B an a ch space L q[t0,T]
w ith s c a la r sequences g (i) in the B an ach spaces i q” :
<p= { { f , g } : f 6 L q , geij,!0} ( 1 2 )
(13)
(14)
I t is now easy to show th a t these co m p o s ite v e c to rs cp fo rm a B an ach space
w ith the n o rm [13]
H I =C||f||q + C ’ ||g||q. (*5)
T he fo llo w in g can now be show n [7, 13, 17].
(a) A ny bounded l in e a r fu n c t io n a l F on c o m p o s ite v e c to rs cp in В m ay ¡be
c o n s tru c te d in the fo rm
T w
F(q>) = J ' f ( t ) u ( t ) d t +
to i=l
w ith u in L p and v ( i) G .(b) The n o rm of the fu n c t io n a l F o f E q .(1 6 ) d e fined as
I
w here the n o rm s of f and g a re de fin ed by
T
||f||q = ( J |f ( t)|4 dt)1/4<°°, l s q s ° °
to
/ ■ \1 /q ’,,q, ) |g(i)|4 J <00» i = q'
i=l
w here || u||p is .defined in E q .(9 ) and
H ip . = ( D v(i)‘l P' ) 1/P (19)i=l
To app ly th is to the p ro b le m of E q s (10) and (11), we c o n s tru c t co m p o s ite
v e c to rs of the fo rm (12) f r i j in th ç e le m e n ts o f H (T ,t):
фг = {Н г(Т ,t) , 0}
ф2 = { H 2(T , t ) ,0 i (2-0)
flj = { H 2(t¡ ,t), t0 S t fi T
IAEA-SMR-1Í7/47 5 3
w here
, 1 , t. =t.
ei < V = { 1 10 , t .¥ = t . , i = l , . . . ,w , 1=1 , . . . ,w
’ J ' l ’
and H ¡(T ,t), (i = 1,2 ), is the i- th row of the im p u ls e re spo n se m a t r ix H (T ,t)
and is a s su m e d to be bounded and p ie ce w ise co n tin uo us .
B y r e q u ir in g th a t a fu n c t io n a l F o f the fo rm (16) s a t is f ie s
F (* ) = -y (T .t )
(2 1 ) F ( V =T2 (T ,t0)
i t r e q u ir e s s im p ly th a t the u- com ponen t o f F s a t is f ie s E q . ( l l ) . I f we fu r th e r
r e q u ir e th a t F s a t is f ie s
F ( 0j ) = X jC tj)- (0 ( t j , t o)x o)2, i = l , . . . ,w (2 2 )
and th a t
v (t .) = ( i ( t . , t 0)x 0)2 + J"H 2( t . , t )u ( t ) d t - x 2(t. )
x„(t.) - xd(t.) , i = l , . . . ,w (23)
the n o rm o f F , as g iv e n in E q .(1 8 ) , then take s the fo rm
Il F II = m ax j- i ||u||p , || x 2(tj) - x^(tj) ||pt| (24)
w h ich is the m in im u m in E q .(1 0 ) .
54 MIZUKAMI
I t is now c le a r th a t the a b s tra c t p ro b le m of f in d in g the bounded l in e a r
fu n c tio n o f m in im u m n o rm on В
w h ich m ap s the g iven e lem en t ф1( ф2 , , i = l , . . . ,w , of th is space in to the
g iv e n fixed s c a la r s s p e c if ie d by E qs (21) and (22), is e qu iv a len t to so lv ing
the v a r ia t io n a l p ro b le m of E qs (10) and (11); the m in im u m - t im e so lu t io n is
o b ta in ed w hen th is fu n c t io n a l has its n o rm e qua l to un ity . T h is a b s tra c t
p ro b le m is s o m e t im e s c a lle d the L - p ro b le m in the th eo ry o f m o m e n ts [2, 17].
3 .4 . E x is te n ce and fo rm of the so lu tio n
I f the s y s te m unde r c o n s id e ra t io n is to be c o m p le te ly o u tp u t- co n tro lla b le ,
ф1 , ф2 and 0j ( i = l , . . . ,w ) a re l in e a r ly independen t and span a w+2 - d im e n s io n a l
l in e a r space B jG B . T hen the fu n c t io n a l F j , b e ing l in e a r and s a t is fy in g
E q s (21) and (22), can be de fined on th is space , fo r a r b it r a r y s c a la r s a 1 , a 2 , 3 j( i = 1 , . . . , w ), by
(26)
i= l
w here
(27)
Il Fi II = SUP
W
k C 1 + « 2 C 2 + 2 е лi= l
(28)sup
otL, ct2, e ¡
not all=0
o r , s ince
i = l
does no t g ive the la rg e s t v a lu e o f th is r a t io ,
IAEA-SM R-17 /47 55
ll*i 11= ------------------------------ ------------------ ------------- (2 9 )
i n f | | ^ 1 + ^ 2 + 2 з ; а||
Q jC i + a 2C 2 + 2 3 ¡ d j - 1 l= l
U sing the e x p lic it f o rm of the n o rm of cp in the p ro d u c t space B , g iv e n by
E q s (13)- (15), we ob ta in
11 Fi 11 = ----------- r - *— ------------ ;----------- — — . v ,in f j c [ |'|a1H1(T ,t )+ a 2H2( T , t ) + 2 f dt j C ’ [ ¿ J P j " ] }
° l C l + a 2 C 2 + ? , 0 i d i = 1
(30)
A c c o rd in g to the H ahn- B anach th e o re m [13], the fu n c t io n a l Fj , d e fin ed by
E q .(2 6 ) , c an be extended o ve r the w hole p ro d u c t space В of v e c to rs (12)
w ithou t in c re a s e in n o rm . S ince any ex tens ion of F j o ve r В m u s t have its
n o rm equa l to o r g re a te r th an th a t o f F j , a fu n c t io n a l o f m in im u m n o rm w i l l
c e r ta in ly e x is t s a t is fy in g E q s (21) and (22) w h ich has its n o rm ju s t e qua l to
th a t g iv e n by E q .(3 0 ) .
A s s ta te d p re v io u s ly , the m in im u m t im e T* is a tta in ed w hen || F-J =1.
W e now tu rn to a d e r iv a t io n o f the a c tu a l fo rm of an o p t im a l c o n tro l in p u t.
L e t a , = a * , a„ = a * , B. = B * , (i = l , . . . ,w ) be s u ch v a lu e s th a t the in f im u m inl l 2 2 ' le x p re s s io n (30) is a tta in e d at T = T* w ith
K C i + a 2 C 2 +
W
d i l = :i= l
F o r fu n c t io n a ls F of the fo rm (16) s a t is fy in g E qs (21) and (22) and o p e ra tin g
on c o m p o s ite v e c to rs cp o f th e fo rm (12), H o ld e r 's in e q u a lity g ives
IF (ф) I i H F И ||ф|| (31)
B u t
w w
1 = I F ( о * фг + a * ф2 + )T|3*e.)| S J F ü ||ar* Ф1 + a * Ф2 +^Г|3* в. || = 1 (32 )
1=1 i= l
56 MIZUKAMI
w here the la s t e q ua lity is o b ta ined f r o m E q .(2 9 ) u s ing the fac ts th a t the
in f im u m is a tta in e d at the va lue s a 1 = a f , a 2 = c t \ , (3¡ =Э* > (i = w), T =T*
and th a t || F .JI = || F || . T he re fo re
W VY
F ^ a * ) ^ + а * Ф 2 + ^ / 3 * 0 .^ = ü F ü ¡ctr*^ + а * ф 2 + 3 * 0 ¡ || (33)
i=l i=l
U s ing E q s (9), (15)- (19) and
a *H (T * t)+ a *H 2(T *,t) + l ? t H2 (t ,,t) = K *(t)
i=l
x 2( t . )- x d2(t.) = g (t .) , i = l , . . . ,w
we can w r ite E q .(3 3 ) in the fo rm
(34)
T * w
J K* (t) u (t) dt + T 6*g (t. ) =max || u ||p , ^ || g ||p,|
tn i = l
=max ] p II u ||D , — Il g ||D. f {C||K*||q + C ||6*||q,}
(35)
S ince I a + b| â| a| + | b|
T *
J K* (t)u(t)dt + 0*g(tj) s J K* (t)u{t)dt + ^0*g(t.to i=l to i=l
(36)
F r o m H o ld e r 's in e q u a lity we ob ta in
K * (t)u(t)d t S K* u = C K* ^ uIl llq II Ир II II q С 11 "P (37)
W
^ 3 f g ( t . ) s||p*||q,||g||p ,= c | | 0 * | | q,
i=l
A d d in g the above two in e q u a lit ie s y ie ld s
g iip.
K * (t)u(t)d t
F r o m
a b + c d s [max {b ,d}][a + c], a ,b ,c ,d > 0
(38)
w
g(ti)U С IIK* Hq A I u||p+C II 0* |„- ¿T II slip. <39)i=l
(40)
IA E A -S M R -n /4 7 57
c||K*||q-i|Mip+c'||Hq. ¿HI g lip. smax{è IIu l ’é IIg I 1
X {С II К* ||q + С 1 II 6* j!q. } (41)
C o m p a r in g E q .(3 5 ) w ith E qs (36)- (41) we see th a t a l l in e q u a lit ie s in
E q s (36)- (41) m u s t be s a t is f ie d w hen the e q ua lity s ig n h o ld s . A n e ce ssa ry
co nd it io n fo r e qua lity in E q .(3 7 ) is
u(t) = K | K *(t)| sgn [K*(t)] (42)
w here the co ns tan t К w i l l be d e te rm in e d be low . A n e ce ssa ry c o nd it io n fo r
e q u a lity in (40) is b = d; so fo r equa lity in (41)
è i i - i P = é r i u i i P. (43)
E q u a t io n (43) is tru e p ro v id e d || /3* ||q. > 0; th is c o nd it io n ho ld s w henever
the ou tpu t c o n s tra in ts a re a c tiv e , i .e . the c o n s tra in e d output s o lu tio n d if fe rs
f r o m the u n co n s tra in e d one . T h e re fo re ,
¿ N I P = m a x { è I N I ? - и I I e l i p , } = I I г II = 1 ' ( 4 4 )
U sing the r e la t io n (1 /p ) + ( l / q ) = 1 we o b ta in f r o m E qs (42) and (44)
\i/p(| K * (t) |4 T d
to
it fo l lo w s that
I u||p = k ( y ,(|K*(t)|4‘ V d t ) ’/P=K|K*(t)||J' 1 = С I F II (45)
К =С N К * (t) II1-4 (46)и Iiq
u (t) = C ¡K * (t)| | J '4 |K*(t)|4' :L sgn[K *(t)] , tQs t s T * (47)
as the e x p lic it f o rm of the o p t im a l c o n tro l in p u t, w here
w
K *(t) = « * Н а(Т *Д ) + a * H 2(T * ,t) Э* H 2( t .,t)
i= l
In the c a se w hen the c o n tro l in p u t has a c o n s tr a in t on it s a re a ( p = l ,q = °°),
E q .(4 7 ) m u s t be g ive n a p ro p e r in te rp re ta t io n [17]. H e re we re v e r t d ire c t ly
58 MIZUKAMI
back to the H o ld e r equa lity c o nd it io n s . In e q u a lity (37) s t i l l m u s t be s a t is f ie d
by e qua lity ; th e re fo re i t is u n d e rs to od th a t the c o n tro l is e x p re ssed by a
D ir a c d e lta fu n c tio n 6 (t). I f K * (t) a tta in s it s m a x im u m m ag n itu d e fo r
r G [ t . ,T * ] , i .e .
I К* (t ) I = m ax IK * (t) I , t G [tQ,T* ]
th en the c o n tro l in p u t u is g iven by
M t -т)u (t) =
(48)
(49)K *(t)
S ince E q .(4 4 ) r e m a in s tr u e , the only r e q u ir e m e n t on the in p u t is th a t
T *
I I u (t) I dt = С (50)
T h e re fo re any c o n tro l in p u t s a t is fy in g E qs (49) and (50) is an o p t im a l in p u t.
3 .5 . N u m e r ic a l r e s u lts o f a t r a in o p e ra tio n
C o n s id e r the m o tio n of a t r a in [19] w h ich is a p p ro x im a te ly d e s c r ib e d by
X ^ t ) 0 1 X j ( t )+
0
¿ 2 (t ) 0 - 1 x 2( t ) 1u(t) (51)
w here X j denotes the n o r m a liz e d t r a in p o s it io n and x2 the n o rm a liz e d
v e lo c ity of the t r a in and u is a n o r m a liz e d c o n tro l v a r ia b le ; u> 0 m e ans
the e x e r tio n o f the t r a c t io n fo rce due to the a c tio n of a p r im e m o v e r and
u < 0 m e a n s the a p p lic a t io n of a b rake ; u is a s su m e d to take a v a lue in the
range
(52)
w h ich m e an s th a t p = 00 and С = 1 in E q .(9 ).
T he end p o in t (a next s ta tio n ) c o nd it io n is such th a t
X jfT *) 1
x 2(T* ) 0(53)
w ith an u n s p e c if ie d m in im u m - t im e T* and the in i t i a l s ta te o f t r a in m o tio n is
p la c e d on
[54]х г(0 ) 0
x 2(0 )
tl о 1 1
IAE A-SM R-17/47 59
The p ro b le m is to f in d the c o n tro l u s a t is fy in g the c o n s tr a in t
(52) w h ich d r iv e s the t r a in f r o m the in i t i a l s ta te (one s ta tio n ) to the next
s ta t io n o f E q .(5 3 ) in m in im u m - t im e T* and a t s p e c if ie d in s ta n ts
t¡ , 0 < t ¡ < T * , i = 1 , . . . , w, the t r a in is r e q u ir e d to have a speed l im i t С '
l im í I x 2(ti ) |P J (55)
P 1=1
or
m ax {x2(tj)} â C\ i = l , . . . , w J 0 < t ; < T * (55 ')4
w h ich m e an s p' = °° and х^СЦ) = 0 in E q .( 8 ).
Хг
( a ) S t a t e t r a j e c t o r y .
-1
1 . 0
= 2 .1 7 0 2 0JL¿ . 0
( b ) C o n t r o l
F IG .2. O ptim al state trajectory and control without output constraint, (a ) State tra jectory ; (b ) Control.
60 MIZUKAMI
( a ) S t a t e t r a j e c t o r y
1 . 0
T * = 2 . 18152 __I__2 .0
( b ) C o n t r o l
FIG.3. Optimal state trajecoty and control with C ' = 0 .7 at t = 1 . 5 . (a) State trajectory ; (b) Control.
F i r s t o f a l l le t us i l lu s t r a te (F ig .2) the o p t im a l s ta te tr a je c to r y and
c o n tro l u w hen the r e q u ir e m e n t o f E q .(5 5 ) is a b sen t. N ote th a t the sw itc h in g
of u o c cu rs once at t = 1.583. In th is r e s u lt the h ig h e s t speed of the t r a in
o c cu rs n e a r t = 1.5. T hen we w i l l try to r e s t r ic t the speed l im i t С 1 = 0.7 at an
in s ta n t t = 1.5 such th a t
x 2 ( 1 . 5 ) s 0 . 7 ( 5 6 )
F o r th is case the o p t im a l s ta te tr a je c to r y and c o n tro l a re show n in F ig .3 ;
the c o n tro l sw itches th re e t im e s at the fo llo w in g t im e in s tan ts :
t = 1.462, 1 .501, 1.636
Now c o n s id e r the case th a t a r e s t r ic t io n of speed l im i t is g iven at two
s p e c if ie d in s ta n ts t j= 1 .4 , t 2 =1 .5 as
X g ( 1 . 4 ) S 0 . 7 , X 2 ( 1 . 5 ) S 0 . 7 (57)
IAEA-SM R-17/47 61
(a } S t a t e t r a j e c t o r y
u
T * = 2 . 18750
(b ) C o n t r o l
FIG.4 . Optimal state trajectory and control with C ' = 0 .7 at t= 1 .4 and 1 .5 .(a) State trajectory; (b) Control.
N u m e r ic a l r e s u lts fo r th is case a re show n in F ig .4 in w h ich the o p t im a l
c o n tro l u fias fiv e sw itc h in g t im e s :
t = 1.379, 1 .400, 1 .480, 1 .501, 1.641
N ote th a t the s ta te t r a je c to r y at t = 1.5 r e s u lts in a v a lue of x2 = 0.6737 w h ich
is s m a l le r th an the r e q u ir e d l im i t of 0 .7 , bu t x2 =0 .7070 at t = 1 .4 . I t is
expected t h â t t h e x 2 v a lu e s at b o th t = 1.4 and 1.5 s h o u ld b e 0.7. T h is d iffe ren ce
m a y co m e about because , when c o n s id e r in g s e v e ra l n u m e r ic a l r e s u lts , the
sw itc h in g t im e s o b ta in ed m ig h t not be the tr u e ones because of d ig ita l
c o m p u ta t io n a l e r r o r s in seek ing the in f im u m of the d e n o m in a to r in E q . (30).
In th is d ig ita l c o m p u ta tio n the s im p le x m e thod deve loped by N e ld e r and
M e ad [18] h as been u se d in the m u lt i- d im e n s io n a l s e a rc h fo r o p t im a l
v a lue s a * , a % , , i = l , . . . ,w , and the S im p so n r u le fo r the in te g r a l of the
d e n o m in a to r in E q .(3 0 ).
1
- 1 -
62 MIZUKAMI
A t im e o p t im a l p ro b le m of a con tin uous t im e l in e a r s y s te m w ith a
c o n tro l c o n s tr a in t and output c o n s tra in ts at d is c re te in te rm e d ia te t im e s was
fo rm u la te d as the so- ca lle d L - p ro b le m in the th eo ry of m o m e n ts and its
s o lu t io n was p re se n te d in te r m s of a m u lt i- d im e n s io n a l m in im iz a t io n of
a n o n - lin e a r fu n c tio n .
The m a in d if f ic u lty in a pp ly ing th is p ro ce d u re is the c o m p u ta tio n a l
p r o b le m in d ic a te d in E q .(3 0 ). It is show n th a t the s im p le x m e th od can be
u se d e ffe c t iv e ly and g ive s re a so n ab ly good r e s u lts in the s e a rc h fo r the
m in im u m d e n o m in a to r in E q .(3 0 ) by u s in g a d ig ita l c o m p u te r .
A s m e n tio n e d e a r l ie r , the p ro b le m p re se n te d in th is p ap e r w as suggested
in the p ap e r by K a tz and K ra n c who d is c u s s e d the s im p le case of an in te r io r
ou tpu t c o n s tr a in t . T h e re fo re , i t m ay be no ted th a t the deve lopm en t d e sc r ib e d
in th is p ap e r is a g e n e ra liz a t io n o f the p ro b le m fo rm u la te d by K a tz and
K ra n c and th a t the n u m e r ic a l r e s u lts fo r o u r t r a in o p e ra tio n p ro b le m show
how the s im p le x m e thod m ay be used e ff ic ie n t ly in the fu n c t io n a l- a n a ly t ic a l
s o lu t io n to the t im e o p t im a l p ro b le m w ith c o n tro l and output c o n s tra in ts .
3.6. C on c lu s ion s
4. D IS C R E T E L IN E A R O P T IM A L C O N T R O L SYST E M S W IT H
Q U A D R A T IC C O ST F U N C T IO N A L S
4 .1 . In tro d u c t io n
T he o p t im a l c o n tro l p ro b le m s of d is c re te s y s te m s h as been s tu d ie d by
m an y au th o rs [20-29] fo r v a r io u s types of p ro b le m s .
R e c e n tly , K a tz and K ra n c [7], c o n s id e r in g the le a s t- t im e c o n tro l p ro b le m
of c o n tin uo us t im e sy s te m s w ith an in te r io r ou tput c o n s tr a in t , fo rm u la te d
the p ro b le m in te rm s of n o rm s of fu n c tio n a ls on s u ita b ly c o ns tru c te d B an ach
s p a ce s . K ra n c and S h ilm a n [17, 30], b ase d on K a tz and K r a n c 's fo rm u la t io n ,
o b ta in ed the fu n c t io n a l- a n a ly t ic a l s o lu t io n to bo th the t im e o p t im a l p ro b le m
and the t e r m in a l c o n tro l p ro b le m fo r d is c re te sy s te m s w ith c o n tro l and
output c o n s tr a in ts .
T h is p ap e r p re se n ts a fu n c t io n a l- a n a ly t ic a l s o lu tio n to the d is c re te
o p t im a l c o n tro l p ro b le m w ith a q u a d ra t ic co st fu n c t io n a l. T h is m e th od of
s o lu t io n is b ased on a d eve lo pm e n t o f K a tz and K r a n c 's a p p ro a ch app lie d
to the le a s t- t im e c o n tro l p ro b le m . H ow ever , th e re is a n o n - lin e a r m u l t i
d im e n s io n a l m in im iz a t io n p ro b le m in h e re n t in the func tiona l- a n a ly t ic a l
s o lu t io n to the d is c re te o p t im a l c o n tro l p ro b le m w ith a q u a d ra t ic cost
fu n c t io n a l as w e ll. I t is show n th a t the s im p le x m e th od deve loped by N e ld e r
and M ead [18] can be e ffe c t iv e ly used to so lve th is n o n - lin e a r m in im iz a t io n
p ro b le m .
4 .2 . S y s te m d e s c r ip t io n
C o n s id e r a l in e a r s y s te m w hose b e h av io u r is d e s c r ib e d by the d is c re te
o p e ra to r equa tio n [17]
N
(1)i=0
IAEA-SM R-17/47 63
w here
y(N ) = n - d im e n s io n a l s ta te v e c to r a t the N -th step
u ( i) = m - d im e n s io n a l in p u t v e c to r a t the i- th step
H (N ,i) = nX m - d im e n s io n a l im p u ls e re sp o n se m a t r ix fo r the sy s te m
y0 (N) = n - d im e n s io n a l v e c to r at t im e N due to non- ze ro in i t i a l c o nd it io n s
N = n u m b e r of s tages
T he p ro b le m of a c o n tro l- s y s te m des ign is co nce rne d w ith d e te rm in in g
the c o n tro l in p u t v e c to r u ( i) , i = 0 , . . . ,N , w h ich tr a n s fe r s the sy s tem of E q . ( l )
f r o m a g ive n in i t i a l c o nd it io n to a s p e c if ie d t e r m in a l s ta te
y (N )= y d (2)
w here yd is som e d e s ire d t e r m in a l v a lu e , and w h ich a lso m in im iz e s the
p e r fo rm a n c e fu n c t io n a l
N N
J (N ) = ^ u ( i ) ' P u ( i ) + ^ y ( i ) 'Q y ( i ) (3)
i=o i =o
w here P and Q a re a r b i t r a r y p o s it iv e d e fin ite m a t r ic e s , and the p r im e
denotes the tra n sp o se d .
B y s u ita b ly s c a lin g and r e d e f in in g the v a r ia b le s , the q u a d ra t ic cost
fu n c t io n a l J (N ) o f E q .(3 ) can be w r it te n in the fo rm
iN I n
J (N ) = ^ u ( i) 'u ( i )+ ^ T y ( i) 'y ( i )
i =0 i =0
Il II2 II II2 , A \= U + I y II (4 )
W ith o u t lo s s of g e n e ra lity , the p e r fo rm a n c e index is a s su m e d to take the
fo rm as g ive n in E q .(4 ).
The p ro b le m w i l l read :
F in d the c o n tro l u ( i) , i = 0 , . . . ,N w h ich , fo r J (N )> 0, m ake s
( Il u II2 + [| у II2 )1/2 = m in (5)
w h ile m a in ta in in g
N
ydj " y0j ^ = ^ jH .(N , i ) u ( i ) , . (6 )
i = 0
w here ydj and y0j a re the j- th com ponen ts of the v e c to rs yd and yo(N),
re s p e c t iv e ly , and H j(N ,i) is the j- th row of the m a t r ix H (N ,i) .
4 .3 . The L - p ro b le m and p ro du c t spaces
The above v a r ia t io n a l p ro b le m w ith E q s (5) and (6 ) c an be r e fo rm u la te d
in te rm s of n o rm s of fu n c t io n a ls on s u ita b le B a n a c h spaces [7, 30].
64 MIZUKAMI
L e t us in tro d u ce (N + 1 )m - d im e n s io n a l r e a l n u m b e r sequences b(i):
{b (i) , i = 1 ,. . . ,(N + l ) m } = {b( 1 ) , . . . , b (m ) j , . . . , j b (N m + 1 ) , . . . , b (N m + m )}
= { a (0 ) ;a ( l ) j — j a (N )} (7 )
w ith a n o rm :
(N+l)m
I b (i).2 \
i=l
N m 1/2
=( Z У iaj (i)i2) <o° (8)i=0 j=l
w here
a ( i ) = an m - d im e n s io n a l row ve c to r c o n s is t in g o f the com ponen ts
a j( i) , j = l , . , . , m , i .e .
= ( a ^ i ) , a 2( i ) , . . . , a m ( i ) } , i = 0 , . . . ,N
I t can be show n th a t the space of the b (i) sequence is a B anach space denoted
by i<2N + 1>m .
S im i la r ly , le t d (i) in i 2N+1 n be
{d (i) , i = l , . . . , (N + l) n } = {c (0 ) j c ( l ) j — i с (N )} (9)
w ith
Il d i =( Y I d (i) 12 )i —1
N n n1/2
| c ( i )|2N n 1/2
=( Z Z ic(i)0 <o° (io)w here
с (i) = {C j(i) , C2( i ) , . . . , Cn( i ) } , i = 0 , . . . ,N
D e fin e c o m p o s ite v e c to rs e fo rm e d by c o m b in in g b (i) and d(i)
(N +l)m (N +l)ne = { b ( i ) , b ( i ) e i 2 ; d (i) , d (i) e &2 } ( 1 1 )
and the n o rm o f v e c to rs e as
Il e ü = (||b||2 + ||df )1/2 ( 1 2 )
A g a in i t is easy to show th a t the p ro d u c t space В
(N + l)m (N+1) nв = f 2 X i2
c o n s is t in g o f v e c to rs e is a B an a ch space [13].
IAE A-SM R-17/47 65
O n the B - space o f c o m p o s ite v e c to rs e any bounded l in e a r fu n c t io n a l f
c an be co n s tru c te d as [13]
( N +l)m (N +l)n
f (e ) = У b ( i ) r ( i ) + ^ d (i) s (i)
1=1 i=l
N N
= a ( i) u ( i) + У c (i) v ( i) (13)
i=0 i=0
w ith r ( i ) e f [ N tl,m and s ( i) e j?2N+1'n c o n s is t in g of m - d im e n s io n a l v e c to rs
u ( i) , i = 0 , . . . ,N and n - d im e n s io n a l v e c to rs v ( i) , i = 0,-...,N , re s p e c t iv e ly , i .e .
{ r ( i) , i = 1 ,. . . ,(N + l )m } = {u(0) u ( l ) . . . . u (N )}
(14)
(s ( i ) , i = 1 ,. . . ,(N + l ) n } = {v(0) v ( l ) . . . . v (N )}
F u r th e r m o r e , the n o rm of the fu n c t io n a l f of E q .(1 3 ), d e fin ed as
II f I = sup ^ — (15)
e * 0 ü e I
w ith re s p e c t to the n o rm (1 2 ), is g iven by
l l f l l = ( N I 2 + Il s II 2 ) 1 / 2 ( i 6 )
w here
(N +l)mм | 2 \ 1/2
(i)
(17)
;i —1 i=0 j=l
<N ^ n \ l /2 х Д A 0 / 2
( l i“(i)i ) - С Е Х м О i=l i=0 j=l
and w here U j(i) and v¡ (i) a re the j- th com ponen ts o f the c o lu m n v e c to rs u ( i)
and v ( i) , r e s p e c t iv e ly .
In o rd e r to app ly these d eve lopm en ts to the p ro b le m sp e c if ie d by
E q s (5) and (6 ), the com ponen ts of H (N ,i) a re a s su m e d to be f in ite fo r a l l
N and i. Then the fo llo w in g c o m p o s ite v e c to rs (fcj and <//jk can be c o ns tru c te d
as o f the fo rm ( 1 1 )
<J>. = {X .; 0 } j j
(18)
6 6 MIZUKAMI
w here
X . = {H j (N ,0 ) !H j ( N , l ) i . . . . jH . ( N ,N ) } , j = l , . . . , n
0 is an (N + l)n - d im e n s io n a l row v e c to r w ith a l l ze ro co m po nen ts , and
’/' jk “ W ( 1 9 )
w here
Y.. = { H .(k ,0 ) :H ( k , l ) ! . . . . !H . ( k ,N ) } , j = l , . . . , n , k = 0 ,. . . ,NJ K J 1 J 1 1 J
is an (N + l)n - d im e n s io n a l row ve c to r w ith a l l z e ro co m po nen ts , w ith
the on ly excep tion of the (kn + j)- th com ponen t b e ing un ity .
S u b s t itu t in g <jij fo r e in E q .(1 3 ) , we have
N
f f y ) ^ H j f N . i l u t i ) , j = l , . . . , n (2 0 )
i= 0
and by the co nd it io n
) =ydj - y0j (N), j = l , . . . , n , (2 1 )
we r e q u ir e s im p ly th a t the u ( i) v e c to r of f m u s t s a t is fy E q . ( 6).
S im i la r ly , s u b s t itu t in g E q s (19) in to (13), we have
N
fW'jk) H j ( k ,i) u ( i) " Vj (k), j = к = (22)
i=o
and by r e q u ir in g th a t f m u s t s a tis fy
fO/'ik) = -y0 j(k ). j = l , . . . , n , k = 0 , . . . ,N (23)
we r e q u ir e th a t the j- th co m ponen t of the v e c to r v (k) be g iv e n by
N
v. (k) =yo j(k) H . (k , i) u ( i )
i=0
= (k), j = l ....... n , к = 0 , . . . ,N (24)
The n o r m of f, as g iv e n in E q .(1 6 ) , th en beco m es
Ilf II = (II u l|2 + | y f ) 1/2 (2 5 ^
w h ich is the le ft-hand s id e of e x p re s s io n (5).
A s a consequence of the above tre a tm e n ts the a b s tra c t p ro b le m of
f in d in g the bounded l in e a r fu n c t io n a l f o f m in im u m n o rm on В
II f I = m in (26)
IAEA-SM R-17/47 67
w h ich m ap s the g iven e le m e n ts ф. , , j = l , . . . , n , к = 0 , . . . ,N of E q s (18) and (19)
of th is space to the g iven fixed s c a la r s sp e c if ie d by E q s (21) and (23), is
e qu iv a le n t to s o lv in g the v a r ia t io n a l p ro b le m of E q s (5) and (6). T h is
a b s tra c t p ro b le m is s o m e t im e s c a lle d the L - p ro b le m in the theo ry of
m o m e n ts [2 ].
4 .4 . O p t im a l s o lu tio n
A s s ta te d in the p re v io u s sec t io n , the o r ig in a l p ro b le m of E qs (5) and (6)
can be red uce d to the L - p ro b le m . A s the s o lu t io n to the L r p ro b le m is w e ll
know n [2 ], we s h a l l b r ie f ly d e sc r ib e the m e th od of so lu tio n .
I f the c o n tro lle d s y s te m u nd e r c o n s id e ra t io n is c o m p le te ly o u tp u t-
c o n tro lla b le , the v e c to rs ф■ and ф-к , j = 1 , . . . ,N , k = l , . . . ,N , a re l in e a r ly
independen t [17] and th en they span an n(N + 2 )-d im e n s io n a l space B 1( B jG B,
and a l in e a r fu n c t io n a l f j s a t is fy in g E q s (21) and (23) can be de fined on th is
space , fo r a r b i t r a r y s c a la r s or- and 7jk , by
n N
j=l k=o+II Vik) =Z «,• I kfi<v
j= l j=l k=0 j=l
n n N
= î ° j gi +î Х 7Лj=l k=0
w here
gj =ydj - yoj (N), j =1......N
h jk = -yoj (k), j = l , . . . , n , k = 0 , . . . ,N
The n o rm of f2 on th is n+ n (N +1)-d im e n s io n a l space , a c c o rd in g to the
d e f in it io n (15) and by v ir tu e o f (27 ),is g iven by
i > j g j + 2 2 7 j k h jk j=l j=l k=0
(27)
(28)
sup
not all = 02 a j + 2 2 V jk
(29)
j= i j= l k=o
o r , s in ce
11 U IN
I ^ h i k = ° j= l i= l k=0
does not g ive the la rg e s t v a lu e o f the ratio .,
1f,
in f
^Jk
n n N
2 “ ¡* j3=1
+ 2j= i
2 V * k=0
(30)
68 MIZUKAMI
A p p ly in g now the e x p lic it fo rm o f the n o rm in В g iven by E qs (8 ), (10) and (12)
to the v e c to rs ф ., ф.к of E q s (18) and (19) g ives
a
n N
i n f j 2 2 |k (u| + 2 2 hjkri'- T-, L J= l l=0 j= l k=0J ' rjk
2 S Tjk h j k = ij= l j= l k=0
w here
11 11 IN
K (i) = ^ o . H j (N ,i) + T y ^ H j i k . i ) , i = 0.......N (32)
j= l j =1 k=0
and K j( i) is the j- th co m po nen t of K ( i) .
A c c o rd in g to the H ahn- B anach th e o re m [13] it fo llo w s th a t the fu n c t io n a l fj
d e fin ed by E q .(2 7 ) can be extended to a l in e a r fu n c t io n a l f o ver the whole
space В o f v e c to rs (11) w ithou t an in c re a s e of n o rm . S ince any ex tens ion of
f j o ve r В m u s t have its n o rm equa l to o r g re a te r th an th a t o f fj, th e re is
c e r ta in ly a fu n c t io n a l of m in im u m n o rm on B , s a t is fy in g E qs (21) and (23),
w h ich has it s n o rm ju s t e qua l to th a t g iv e n in E q .(3 1 ). The n o rm eva lu a ted in
(31) is thus the d e s ire d m in im u m n o rm sough t in (26) w ith the co nd it io n s
(21) and (23), bu t the d e s ire d m in im u m is g iven in te rm s of a m u lt i- d im e n s io n a l
s e a rc h w h ich m ay need an e ff ic ie n t c o m p u ta t io n a l a lg o r ith m . A s show n in
the next s e c t io n the s im p le x m e th od deve loped by N e ld e r and M ead [18] can
be used in the s e a rc h fo r th is m in im u m n o rm .
N ow le t us tu rn to a d e r iv a t io n o f the a c tu a l fo rm of an o p t im a l c o n tro l
sequence . .
Suppose th a t the in f im u m in the e x p re s s io n (31) d e fin in g the m in im u m
n o rm IjfjH is a tta in e d a t the m u lt ip l ie r v a lue s c j = a * and 7 jk = 7^ w ith
j= l j= l k = 0= 1 (33)
A ls o le t f be an a r b i t r a r y fu n c t io n a l of m in im u m n o rm s a t is fy in g
E q s (21) and (23). F r o m d e f in it io n (15), fo r a fu n c t io n a l f o p e ra tin g on
c o m p o s ite v e c to rs e, we have
I f(e) I ^ Ilf II ü ell (34)
H ow ever,
n N n N
^ а * ф . + У £ т ^ к) | ф | | X f k ^ j kj= l j= l k=0 j=l j= l k=0
= 1 (35)
IAEA-SM R-17/47 69
w here the la s t e qua lity is o b ta ined fr o m E q .(3 0 ) and the fa c t th a t the in f im u m
is ach ieve d a t the v a lu e s a- = a * , 7^ = 7 jk>. j = l , . . . , n , k = 0 , . . . ,N , and th a tI f j j| = ¡I f ¡I • T he re fo re
n n N
= fj= l j= l k=0
F r o m the e q ua lity o f (36) we have
а * ф . +
j = l j=l k=0
у * ф jk jk
N
J ^ K * ( i ) u ( i ) + ^ 7* ( i ) y ( i ) = ( II u (12 + I y ||2 )1/î2 { I К* ||2 + ||y* ||2 )1/2 i=o i=owhere
7 * (i) = { 7 * ( i ) , . . . ,7 * ( i ) , . . . ,7 * ( i )}, 7 Ÿ( i ) = 7 * , j = i = 0 ,. . . ,N
Since | a + b| ë | a| + |b|
_N N N N
^ K * ( i ) u ( i ) + ^ 7 * ( i) y ( i ) ^ ^ K * ( i ) u (i) + ^ 7* ( i) y ( i )
i=0N
K* (i) u (i)i=oN
i =0
^ * U ) y Ui=0 i )
A d d in g the above two in e q u a lit ie s y ie ld s
N N
S ince ab + cd s (a2 + c 2)1^2 (b2 + d 2)1//2, a ,b ,c ,d > 0
I K * Il I I u I I + Il y * Il II у I I s ( I I u i f + Il у l|2 )1 / 2 ( I I K * f + II - v * ||2 )1 / 2
(36)
(37)
(38)
(39)
(40)
^ K * ( i ) u ( i ) + ^ 7 * ( i ) y ( i ) s II K* II u II + 1 T* II 1 У 1 (41)i=0 1=0
(42)
(43)
C o m p a r in g E q s (37) w ith (38)- (43), we see th a t a l l in e q u a lit ie s in (38)- (43)
w i l l be s a t is f ie d by th e equa lity s ig n . A n e ce ssa ry c o nd it io n fo r e qua lity
in (39) is
U j ( i ) = K |K *(i)| sgn[K Ÿ (i)] , j = l , . . . ,m , i= 0 , . . . ,N (44)
w here a co ns tan t К w i l l be d e te rm in e d in the fo llo w in g . A s a c o nd it io n ad = cb
in (42) g ive s e q ua lity in (42), so fo r e q ua lity in (43)
I u II II r * I = II K* II II у II (45)
70 MIZUKAMI
F u r th e r m o r e , s in ce
N 1 = ( 1 М 1 2 + Ы 1 2 ) 1 / 2 ( 4 6 )we o b ta in f r o m (45) and (46)
II 112 II K * Г II f I PU H--------- ,2 „2 ( 4 7 >II K * 1 1 + II У* II
F r o m (46) we have
I u f = (j K* ||2 K2 (48)
th en , u s in g (47) and (48) we a r r iv e at
K ' 1|К*|Ы|т*Г <4S)
T he re fo re
U j(i) = —--- — -- -- r |K *(i)| sgn[K *(i)] , j = l , . . , n , i = 0 , . . . ,N (50)
K * “ + 7* I 1 1
is o b ta ined as the e x p lic it f o rm of the o p r im a l c o n tro l sequence . H e re K ? ( i) is
the j- th com ponen t o f K * ( i) g iv e n by Eq .32 w hen а = а * and 7jk = 7*jk .
4 .5 . N u m e r ic a l exam p le
To i l lu s t r a te the above p ro ce d u re , c o n s id e r the s in g le- ou tpu t, s in g le
in p u t s y s te m d e s c r ib e d by the fo llo w in g d iffe re n ce equation :
x (k+ 1 ) =x(k) +u(k)
y (k ) =x(k) (51)
w ith the in i t i a l cond it io n :
x(0) = 0 (52)
and a d e s ire d endpo in t at t im e N
y(N ) = l (53)
T he d is c re te o p e ra to r e qua tio n fo r th is s y s te m co rre s p o n d in g to E q . ( l )
b ecom es
N
y(N ) = ^ H ( N , i ) u ( i ) ’ (54)
i =0
IAEA-SM R-17/47 71
T A B L E I. O P T IM A L INPUTS AND O UTPUTS
N = 10
OOII2
N = 6 N = 4 N = 2
u(0) 0.47540 X 1 0 -3 0.89621 X l t r 3 0 .7 1 8 6 2 x 10“ 2 0.47739 X 10-1 0.33313
u ( l ) 0 .4 3 6 2 6 X 1 0 -3 0.21413 X 1 0 -2 0 .1 3 7 6 3 X 1 0 - 1 0 .9 5 3 7 8 X 1 0"1 0.66629
u(2) 0.1017 2 X 1 0 - 2 0 . 4 5 9 6 7 X 1 0 -2 0 .3 4 5 6 8 X 10-1 0.23788 0
u(3) 0.51563 X 1 0 -2 0.12908 x 10“2 0.90158 x 10’ 1 0.61874
u(4) 0.13049 X 10_1 0.34588 X 10“ 2 0.23583 0
u(5) 0.34416 X 10"1 0 .9 0 2 8 5 X 10-1 0.61809
u(6) 0. 90344 X 1 0"1 0.23608 0
u(7) 0.23621 0.61816
u(8) 0.61788 0
u(9) 0
У(0) 0 0 0 0 0
yd) 0.47540 X 1 0 ' s 0.89621 X 10-3 0.71862 x 10- 2 0.47739 X К Г 1 0.33313
У(2) 0.91166 X 10‘ 3 0.30375 X 10“ 2 0.20949 X 1 0"1 0.14312 0.99942
У(3) 0.19289 X 1 0 - 2 0.76342 x 1 0 -2 0 .5 5 5 1 8 X 1 0 - 1 0.38100
У(4) 0.42556 X 1 0 " 2 0.20542 X 1 0 ' 1 0.14568 0.99974
У(5) 0.94120 X 10- 2 0.55130 X 1 0"1 0.38151
У(6) 0 .2 2 4 6 1 X 1 0"1 0.14541 0.99960
yCl) 0.56877 X 1 0"1 0.38150
У(8) 0.14722 0.99966
У(9) 0.38343
У(Ю) 1.0013
j(N) 1.6180 1.6180 1.6182 1.6190 1.6667
w here
lo, N Si
The q u a d ra t ic co st fu n c t io n a l J (N ) is a s su m e d to be
N
J(N) = ||u||2 + ||y||2 = ^ | u ( i ) 2 + y(i)2| (5 5 )
i=0
The n u m e r ic a l r e s u lts a re l is te d in T ab le I fo r s e v e ra l f ix e d v a lu e s of N. The s im p le x m e th od , w ith the a id of a c o m p u te r , is em p lo ye d in the s e a rc h
fo r a m in im iz a t io n o f the d e n o m in a to r in E q .(3 1 ) . O th e r n u m e r ic a l r e s u lts
72 M IZUKAMI
1 . 0
0.8
0 . 6
0 . 4
0.2
0 -L. JL N0 1 2 3 4
(a ) Input when N=3 (b ) Output when N=3
( c ) I np ut when N=5 (d) Output when N=5
F IG .5. Optim al inputs and outputs for N = 3 and 5. (a ) Input with N = 3 ; (b) Output with N = 3; (c ) Input with N = 5 ; (d) Output w ith N = 5.
w h ich a re no t c o m p ile d in T ab le I a re i l lu s t r a te d g ra p h ic a lly in F i g . 5 fo r
N = 3 and 5. N ote th a t the 11-d im e n s io n a l s e a rc h fo r the in f im u m in E q .(3 1 )
w as p e r fo rm e d by the s im p le x m e th od fo r N = 10 in th is s im p le sy s te m .
4 .6 . C o n c lu s io n s
A m e th od of s tudy ing d is c re te l in e a r o p t im a l s y s te m s w ith q u a d ra t ic
c o s t fu n c t io n a ls w as p re se n te d . I t was show n th a t the o r ig in a l p ro b le m w ith
E q s (1)- (3) can be red uce d to the so - ca lle d L - p ro b le m in the th eo ry of
m o m e n ts and th a t its s o lu tio n is o b ta in ed by the use of the H ahn- B anach
th e o re m and the e q ua lity c o nd it io n s of H o ld e r 's in e q u a lity .
T he re is , how eve r, a c o m p u ta t io n a l d if f ic u lty in the fu n c t io n a l- a n a ly t ic a l
s o lu t io n in the m u lt i- d im e n s io n a l s e a rc h fo r a m in im iz a t io n of the non
l in e a r fu n c tio n o f E q .(3 1 ) . T he s im p le x m e th o d , w ith the a id of a c o m p u te r ,
can be used e ffe c t iv e ly to so lve m in im iz a t io n p ro b le m s as in d ic a te d in (31).
R e s e a r c h co nce rne d w ith an e x tens io n o f the m e th od o u tlin e d in th is
p ap e r to so lu tio n s fo r the case o f bounded c o n tro lle r s is in p ro g re s s .
IAEA-SM R-17/47 73
R E F E R E N C E S
[1 ] BANACH, S ., T h eorie des opérations linéaires, 2nd é d . ,Chelsea, New York (1955 ).[2 ] AKHIEZER, N .I . , KREIN, M ., Som e questions in the theory o f moments, Trans.M ath. M ono. £ 4
A m . M ath .S oc. Publ. (1962) 175.[3] KRASOVSKII, N .N ., ’ On the theory o f optim al control, A u t.R em .C ontrol 18 (1957) 1005.[4] KULIKOWSKI, R ., On optim um control with constraints, B u ll .A c a d .P o lo n .S c i. , S e r .S c i .T e c h . 7
(1959) 285.[5 ] KRANC, G .E ., SARACHIK, P .E ., An application o f functional analysis to the optim al control problem ,
Trans. ASME 8 5 (1 9 6 3 ) 144.[6] SWIGER, J .M ., A p plication o f the theory o f m inim um -norm ed operators to optim u m -control-system
problem . Advances in Control Systems, V o l .3 , A cad em ic Press (1966 ).[7 ] K A T Z . S . , KRANC, G .M ., On the least tim e control problem with interior output constraints, IEEE
T ran s.A u t.C on trol 14 (1969) 255.[8] KREINDLER, E ., Contributions to th e theory o f t im e-op tim a l control, J.Franklin Inst. 275 (1963) 314.[9] SILMBERG, J ., MIZUKAMI, K ., M in im al-tim e control o f linear systems with energy constraints on the
input com ponents, J. Franklin Inst. 292 (1971) 245.[10] MIZUKAMI, K ., NEYAMA, J ., McCAUSLAND, I ., T im e -op tim a l control systems with norm constraints
on state variables at interm ediate tim es, J.Franklin Inst. 297 (1974) 17.[11] MIZUKAMI, K ., I^IEYAMA, J ., McCAUSLAND, I., "D iscrete linear optim al control systems with
quadratic cost functionals'', Proc. 1973, IEEE, C on f.on D ecision and Control, San D iego, C al.(1973) 614.
[12] TAYLOR, A .E ., Introduction to Functional Analysis, W iley, New York (1958 ).[13] LIUSTERNIK, L .A . , SOBOLEV, V .J . , Elements o f Functional Analysis, Ungar, New York (1961).[14] LUENBERGER, D .G ., O ptim ization by V ector Space Methods, W ile y (1 9 6 9 ).[15 ] SARACHIK, P .E ., KRANC, G.E., "On optim al control o f systems with m ulti-norm constraints", Proc.
IFAС Congress, Basel, Butterworth, London (1963) 306.[16] McCAUSLAND, I . , Functional Analysis, Lecture note. School o f Graduate Studies, Univ. o f Toronto
(1966-1967).[17] KRANC, G .M ., SHILMAN, M .B ., An application o f functional analysis to t im e -op tim a l control o f linear
discrete systems with output constraints, J.Franklin Inst. 290 (1970) 137.[18] KOWALIK, J ., Methods for Unconstrained O ptim ization Problems, Elsevier, New York (1968 ).[19] ICHIKAWA, K ., A p plication o f optim ization theory for bounded state variable problem s to the operation
o f train. Bull. JSME ЗЛ (1968) 857.[20] CHYUNG, D .H ., D iscrete linear optim al control system with essentially quadratic cost functionals,
Trans. IEEE. A u t. Control 11 ( 1966) 404 .[21] DELEY, G .W ., FRANKLIN, G .F ., O ptim al bounded control o f linear sam pled-data systems with
quadratic lo ss .. J.Basic Eng., T ran s.ASME 87 (1965) 135.[22] LEE, E .B ., Recurrence equations and the control o f their evolution , J. Math. Anal. A ppl. 7 (1963) 118.[23] EATON, J .H ., An online solution to sam pled-data tim e optim al control, J. E lectron.C ontrol 15 (1963) 333.[24] KOEPCKE, R .W ., A solution to the sam pled m inim um tim e problem , J .A .C .C o n f . (1963) 94.[25] POKOWSKI, J ., An analysis schem e for suboptimal m in im u m -tim e sam pled-data systems, J .A .C .C o n f .
(1968) 270.[26] ГГОН, U ., O ptim al control o f the discrete linear system with the bounded controller and the quadratic
cost function ( in Japanese), J.IEE Japan 91 (1971) 521.[27] HALKIN, H -, Optim al control for systems described by difference equations, Advances in Control
Systems l^LEONDES, C .T . ,E d . ) , A ca d em ic Press, New York ( 1964).[28] BUTKOVSKU, A .G ., The necessary and sufficient conditions for optimality of discrete control systems,
A u t.R em .C on trol 24 (1963) 963.[293 JORDAN, B .W ., POLAK, E ., Theory o f a class o f discrete optim al systems, J .E lectron .C on trol 17
(1964) 697.[30] SHUMAN, M .B ., KRANC, G .M ., On a term inal control problem . Int.J. Control 15 (1972) 561.
NON-LINEAR FUNCTIONAL ANALYSIS AND APPLICATIONS TO OPTIMAL CONTROL THEORY
C. VARSAN
Institute of Mathematics,
Bucharest, Romania
Abstract
NON-LINEAR FUNCTIONAL ANALYSIS AND APPLICATIONS T O OPTIMAL CONTROL THEORY.In five sections the paper deals with differentiability (Fréchet), Taylor's formula, the theorem of implicit
functions, optimization problems, sufficient conditions, convexity, separation theorems, higher-order necessary conditions, applications to cpntrol theory, Pontrjagin's maximum principle, point-wise higher-order necessary conditions and controllability problems o f non-linear control systems. Many explicit proofs are given and examples are discussed.
IAE A-SM R-17/30
1. D IF F E R E N T IA B IL IT Y , T A Y L O R 'S F O R M U L A , T H E O R E M S O F
IM P L IC IT F U N C T IO N S
F i r s t le t us r e c a l l som e p ro p e r t ie s o f F r é c h e t and G ateaux d if fe re n t ia ls
fo r n o n - lin e a r o p e ra to rs w h ich a c t on l in e a r n o rm e d sp ace s .
T he e qua tio n of the type T(x) = 0 o r , in a p e r tu rb a t io n fo rm , T (x ,y ) = 0,
w i l l be d is c u s s e d in d e ta il. A n im p o r ta n t p la ce w i l l take the b ifu rc a t io n
e qua tio ns .
The g e n e ra l r e s u lts o b ta in ed w i l l be used la te r in c o n tro l th eo ry to
o b ta in n e ce ssa ry and s u ff ic ie n t c o nd it io n s fo r n o n - lin e a r c o n tro l la b i li ty .
T h ro ugh ou t th is c hap te r X ,Y ,Z w i l l be l in e a r n o rm e d spaces and " || -Ц"
denotes the n o rm .
1.1 . D if fe r e n t ia b il ity
L e t T :U->Z be a co n tin uo us n o n - lin e a r m a p , w here U Ç X is an open set.
D e f in it io n 1. W e say th a t T h as a F r é c h e t d if fe r e n t ia l a t x06 U i f th e re
e x is ts a l in e a r co n tin uo us o p e ra to r L :X - »Z s a t is fy in g the fo llow ing :
fo r every e > 0 th e re e x is ts 6 (e )> 0 such th a t
II T (xQ+x) - T (x 0) -Lx I s e ü x D fo r ||x|| S 6 (e)
D e f in it io n 1 is e q u iv a le n t to the fo llow ing :
T he re e x is ts a l in e a r c o n t in uo u s o p e ra to r L s a t is fy in g
T (x Q + x) - T (xQ) = L x + ||x||u(x0,x)
w here l im w(xn;x) = 0. x-* 0 0
75
76 VÂRSAN
A s a consequence of d e f in it io n 1 we see th a t, i f T has a F r é c h e t d if fe re n t ia l
at x Q, th en the o p e ra to r L is un ique and is c a lle d the F r é c h e t d if fe r e n t ia l o f
Suppose th a t th e re e x is t tw o l in e a r o p e ra to rs L j , L 2, (L j L 2) such th a t
d e f in it io n 1 is s a t is f ie d fo r each o f th em .
T hen , by d e f in it io n , fo r every e > 0 it fo llow s || (L 1 -L2 )(x)|| se fo r a l l
x G X , (I x (j =1 , c o n tra d ic t in g L j - L 2 ф 0.
The o p e ra to r L is a lso c a lle d the " f i r s t v a r ia t io n " o f T at x0 and we
denote dT (x 0;x) = L x . In o rd e r to c a lc u la te a F r é c h e t d if fe re n t ia l , ano the r
type o f d if fe r e n t ia l is u s e fu l, n am e ly the so- ca lle d G ateaux d if fe re n t ia l .
D e f in it io n 2. W e say th a t T has a G ateaux d if fe re n t ia l a t x0 if
e x is ts fo r every h , w here t is a r e a l p a r a m e te r .
W e denote by D T (x0;h ) the v a lue o f the G ateaux d if fe r e n t ia l at x0 fo r a
f ix e d h.
F r o m the d e f in it io n we see th a t D T (x0;h ), as a fu n c tio n o f h , is hom ogeneous
geneous. In g e n e ra l, D T (x Q; . ) : X ^ Z is not a l in e a r m ap .
I f D T (x 0;.) is a l in e a r con tin uous m a p th en we say th a t T has a G ateaux
d e r iv a t iv e at x .
P ro p o s it io n 1
I f T has a F r é c h e t d if fe r e n t ia l dT (xQ;.) th en T has a G ateaux d e r iv a t iv e
D T (x0; .) and dT(x0;.) = D T (x 0; .) .
The p ro o f o f th is p ro p o s it io n fo llo w s d ir e c t ly f r o m the d e f in it io n s .
T h e o re m 1. L e t Z be a B an a ch space . I f th e re e x is ts a G ateaux
d if fe r e n t ia l DT(x;7)~fo r every x G X , | | x -x n || < r , and D T (x ;h ) is con tinuous as
fu n c tio n on S(xn,r ) u n ifo rm ly w ith re sp e c t to ||h[| a 1 , th en th e re e x is ts a
F r é c h e t d if fe r e n t ia l dT (x ;.) fo r every x G S(x0,r ) and dT (x ;.) = D T (x ;.) .
L e t x G S(x0,r ) be fix e d and le t h G X be such th a t < r (x ) , w here
S (x ,r (x ) )C S (x 0, r ) . F o r every tG [0,1] th e re e x is ts a G ateaux d if fe re n t ia l
D T (x t ;h ), w here x t =xQ + th . By d e f in it io n
T a t x0.
P ro o f
D T (x t;h) = l imT (x t + rh )- T (x t )
T
= l im r - 0
T (x 0 + t h +rh ) - T (xQ + th ) = | T(Xo+th)
W e s h a l l p rove th a t D T (x ;h ) is a d d itiv e in h.
IAEA-SMR-П /З О 77
Indeed ,
D T (x ;h 1 + h 2)= l im Т (Х+?(Ь 1+ М ~ T (x )
t-»o t= l im T tx + t t h i+ M - T tx + th , ) | U m T(x + th i) - T(x)
t-*o * t-+0 t
. 11„ T b - K lb . - M - T b - m . , ! + DT(X ,
t O t 1
T he fu n c tio n f(r) = T (x+ th j+ T th2 ), тб[0 ,1], take s its v a lue s in the B anach
space Z and is of c la s s C 1. T hen , th e re e x is ts a R ie m a n n in te g ra l
о
and
f ( j t îi7) dT
I
f ( 1 ) - f (0) =f ( j ^ îl '7 '))dTb
W e have
f ( 1 ) - f(0) =t J 1 DT (x +th j +Tth2;hz) d r ^ tD T fx ; !^ ) +tu(t)о
w here
1
u (t) = J[D T (x + t(h j + T h2); h2) - D T (x ; h 2)] dr
and l im u (t) = 0 .
T h e re fo re
l im T (x +t(h-[ + h2)) - T(x + tht) = DT
t-»0 t 2
and the a d d it iv ity p ro p e r ty is p ro ved .
N ow , in o rd e r to f in is h the p ro o f we have to show th a t
T (x + h) - T (x) = D T (x ;h ) + Il h II u (x ;h )
w here l im u (x ;h ) = 0. W e have h-”0
T (x + h) -T (x ) = y Q ^ T ( x + t h j ) d t DT (x + th; h) dt
о о
= D T (x ;h ) + ||h||u(x;h)
78 VÂRSAN
w here
w(h) DT x+ th :h
Il h|■ DT x:
N Idt
S ince
l im D T (x + th ,- ^ — )- D T (x ;
h^O \ |lh |r l l^ l= 0
u n ifo rm ly w ith re s p e c t to t e [ 0 , l] , i t fo llo w s l im u (x ,h ) = 0 and the p ro o f is
c o m p le te . h_>0
1.2 . T a y lo r 's fo rm u la
A n n - lin e a r s y m m e tr ic fo rm is a m a p a: X X . . . X X -*Z(n times)
v e r ify in g the fo llo w in g p ro p e r t ie s
(1) ahj.- .hjj is l in e a r and co n tin uo us w ith re spe c t to every h ¡ 6 X ,
(2 ) ahQ( . . . = a h j . . . hj, fo r every p e rm u ta t io n a o f ( l , . . . , n ) .
The m a p a h . . . h = ahn is c a lle d a hom ogeneous fo rm of n th o rd e r if a
is an n - lin e a r s y m m e tr ic fo rm .
A p o ly n o m ia l o f deg ree n is a s um of hom ogeneous fo rm s
П
Ik = 0
akhk
W e say th a t T has a F r é c h e t d if fe r e n t ia l of o rd e r n at x0 i f th e re ex is ts
p o ly n o m ia l o f degree
n
n , Pn (h ) = ^ a k h k
k=l
s u ch th a t
(3) T (x Q + h) - T (x0) =Pn (h) + I) h f wn (x0;h)
w here l im u (x„;h) = 0 . h - o n 0
T he n - o rd e r F r é c h e t d if fe r e n t ia l is d e fin ed as n ! a n. D eno tin g an
n th - o rd e r F r é c h e t d if fe r e n t ia l by cÍT ÍXgjh), we have
d nT (x0;h ) = n ! anhn .
Now we g ive ano the r d e f in it io n fo r the F r é c h e t d if fe r e n t ia l o f o rd e r n.
Suppose th a t th e re e x is ts a f i r s t- o r d e r F r é c h e t d if fe r e n t ia l dT(x;-) in a
n e ig hb ou rho o d G of x 0. D enote dT (x ;h) = T '(x )h . Suppose th a t the m a p
T '(x ):G - > L (X ,Z ) on its tu r n has a F r é c h e t d if fe r e n t ia l d T '(x ,h ) . D enote
d T ' ( x i h j ) = T " ( x ) h 1( a n d i t f o l l o w s d i d T f x ; ! ! ) ; ^ ) = T " ( x ) h j h . I n t h e s a m e w a y w e c a n d e f i n e t h e ( n - l ) - o r d e r d i f f e r e n t i a l
d ( d . . ( d T ( x ; h 1 ) ; h 2 ) . . . h n . 1 ) = T i n - 1 ) ( x ) h n . 1 . . . h 1
I f T n ' 1 ( x ) h a s a l s o a F r é c h e t d i f f e r e n t i a l , t h e n w e d e f i n e
T (n ) ( x ) h n = d ( T <n_1) ( x ) ; h n )
a n d i t f o l l o w s
T (n ) ( x ) h n . . . h x = d ( d . . . d T U i h j ) ; h 2 ) . . . ; h n )
F r o m t h e d e f i n i t i o n i t f o l l o w s t h a t
T ( “) ( x)hù ...h1 = gî - g ^ - g i- T ( x + t1h i + t2h 2 + ...+ tn h n) ^
B e c a u s e t h e r i g h t s i d e r e m a i n s u n c h a n g e d w h e n w e t a k e a p e r m u t a t i o n i n t h e n t h - o r d e r d e r i v a t i v e , i t f o l l o w s t h a t T n ( x ) h n . . . h 1 i s a n n - l i n e a r s y m m e t r i c f o r m .
D e f i n i t i o n 3 . W e s a y t h a t T i s c o n t i n u o u s l y F r é c h e t d i f f e r e n t i a b l e o f o r d e r n a t _ x 0 i f t h e r e e x i s t T f n ) ( x ) c o n t i n u o u s i n a n e i g h b o u r h o o d o f x 0 .
P r o p o s i t i o n 2
L e t T b e c o n t i n u o u s l y F r é c h e t d i f f e r e n t i a b l e o f o r d e r n a t x 0 . T h e n T h a s a F r é c h e t d i f f e r e n t i a l o f o r d e r n a t x 0 a n d d nT ( x 0 ; h ) = T ( I ^ ~ ( x o ) h n.
P r o o f
W e h a v e
l
T ( x 0 + h ) - T ( x „ ) T ( x Q + t h ) ) d t
r ° h t 2=J T 1 ( x Q + t h ) h d t = J | T ' ( x 0 ) h + t T " ( x 0 ) h 2 + | y T " ' ( x 0 ) h 3
0 0
ï £ ï 7 T T " ” < V h"
IAEA-SMR-17/30 7 9
+
w h e r e
d t + h R ( x ; h )n и П 0
I \ , ( x 0; h ) t h ) d t ' ' h
II HI
a n d l i m u ( x n ; t h ) = 0 u n i f o r m l y w i t h r e s p e c t t o t € [ 0 , l ] . h^o u
8 0 VÂRSAN
T h e r e f o r e
T ( x 0 + h ) - T ( x 0) = T ' ( x 0) + ^ T " ( x 0 ) h 2 + . . + ^ - T ( n ) ( x ) h n + (j h | | n R n ( x 0 ; h )
w h e r e l i m R n ( x 0 , h ) = 0 ,
n
XïïT T Î k ) ( x o ) h k
i s a p o l y n o m i a l o f d e g r e e n , i t f o l l o w s t h a t T h a s a n n t h - o r d e r F r é c h e t d i f f e r e n t i a l a t x 0 . B y d e f i n i t i o n
a n d t h e p r o o f i s c o m p l e t e .
1 . 3 . E f f e c t i v e c o m p u t a t i o n o f d i f f e r e n t i a l s f o r c e r t a i n o p e r a t o r s
I n c o n t r o l t h e o r y w e a r e u s u a l l y c o n c e r n e d w i t h p r o b l e m s o f t h e
f o l l o w i n g a b s t r a c t f o r m . T h e r e a r e g i v e n t w o l i n e a r n o r m e d s p a c e s - X , Y , a n d t h e r e a r e g i v e n a l s o s o m e f u n c t i o n a l s c p . : X - * R , i G { 0 , l , . . . , m } a n d a n o p e r a t o r T : X ^ Z . U s u a l l y t h e s p a c e X i s d e f i n e d b y a l l p a i r s
w h e r e C f t g . t p R 11) i s t h e s p a c e o f c o n t i n u o u s f u n c t i o n s x : [ t ^ t - J - > R n a n d Ф/ i s t h e
s p a c e o f p i e c e w i s e c o n t i n u o u s f u n c t i o n s u : [ t ^ t - J -► R1 .W e c o n s i d e r t w o t y p e s o f f u n c t i o n a l s :
A s t h e s p a c e Y w e c h o o s e Y = C ( t 0 , t 1 : R n ) a n d t h e o p e r a t o r T : X - > Y i s d e f i n e d b y
I n o r d e r t o c a l c u l a t e F r é c h e t d i f f e r e n t i a l s f o r s u c h m a p s w e n e e d s o m e p r o p e r t i e s f o r g , F a n d f .
S u p p o s e t h a t g ( x , y ) : R X R - » R i s c o n t i n u o u s l y d i f f e r e n t i a b l e u p t o t h e o r d e r m , a n d f ( t , x , u ) : [ t 0 , t ;L] X R n X R r - > R n , F ( t , x , u ) : [ t 0 Jt 1 ] X R n X R 1 - R a r e c o n t i n u o u s l y d i f f e r e n t i a b l e w i t h r e s p e c t t o ( x , u ) u p t o o r d e r m , u n i f o r m l y i n t .
( x ( . ) , u ( . ) ) e C ( t 0 , t l ; R n ) X ^ ( t 0 , t 1 ; R r )
t
IAEA-SMR-17/30 8 1
L e t ( x 0 ( . ) , u 0 ( . ) ) S X b e f i x e d . T h e m a p s cp2 , T a r e c o n t i n u o u s l y F r é c h e t d i f f e r e n t i a b l e o f o r d e r m a n d
w h e r e d x u m e a n s t h a t t h e d i f f e r e n t i a l i s t a k e n o n l y w i t h r e s p e c t t o ( x , u ) G R n X R r .
I n o r d e r t o o b t a i n s u c h f o r m u l a s w e c a l c u l a t e G a t e a u x d i f f e r e n t i a l s o f c o r r e s p o n d i n g o r d e r a n d t h e n v e r i f y t h a t t h e G a t e a u x d i f f e r e n t i a l s a r e c o n t i n u o u s i n a n e i g h b o u r h o o d o f x 0 = ( x 0 ( . ) , u 0 ( . ) ) .
N o w l e t u s c o n s i d e r a n o t h e r t y p e o f m a p . F o r e v e r y u ( . ) 6 ^ ( t g . t p R 1 )
I n t h i s w a y w e o b t a i n a m a p i n g o n R ” X <%/ ( t ^ t ^ R 1 ) t o C f t ^ t ^ R 11) . W e d e n o t e t h i s m a p b y A .
L e t x 0 = x 0 ( . ) , u 0 ( . ) ) G X a n d p 0 € R n b e f i x e d s u c h t h a t T ( x 0 ) = 0 . T h i s m e a n s A ( p 0 , u 0 ( . J ( t ) = x 0 ( t ) , t e [ t g . t j .
w h e r e ^ ( t ) ( n X n ) i s t h e f u n d a m e n t a l m a t r i x f o r t h e h o m o g e n e o u s s y s t e m
S i n c e t h e G a t e a u x d i f f e r e n t i a l i s c o n t i n u o u s i n a n e i g h b o u r h o o d o f ( p 0 , u 0 ( . ) ) i t f o l l o w s f r o m p r o p o s i t i o n 1 t h a t A h a s a F r é c h e t d i f f e r e n t i a l a t ( p 0 , u j . ) ) a n d t h e s a m e i s t r u e f o r t h e G a t e a u x d i f f e r e n t i a l .
d c p a ( x 0 ; x ) = < | | ( x 0 ( t 0 ) , x o t t j ) ) , x ( t 0 ) > + < | y ( x 0 ( t 0 ) , X o i t j ) ) , x ( t j ) >
dm 9l (xQ;x ) = dm g(x0(t0 ), xQ (ti ) ; x(tQ ), x(t1 ))
to4
a n d p G R n t h e r e e x i s t x ( . ) G C i t ^ t ^ R " ) s u c h t h a t T ( x ) = 0 , x = ( x ( . ) , u ( . ) ) a n d x ( t 0 ) = p .
W e s e e t h a t t h e G a t e a u x d i f f e r e n t i a l f o r t h e m a p A i s g i v e n b y
t
D A ( p 0 , u 0 ( . ) ; p , u ( . ) ) ( t ) = c / > ( t ) ( p + J ф'1 ( s ) 1 ^ - ( s , x 0 ( s ) , u 0 ( s ) ) u ( s ) d s )
F i n a l l y , l e t u s c o n s i d e r A 1 : R n X < ^ ' - » R , A 2 : R n Х - й / - R d e f i n e d a s f o l l o w s :
Ajíp.uU) = gip.Afp.ui.Ktj))
8 2 VÂRSAN
4
A 2 ( p , u ( . ) ) =J F ( t , A ( p , u ( . ) ) ( t ) , u ( t ) ) d t
L e t p 0 e R n , ( x 0 ( . ) , U q Í J J s X b e g i v e n s u c h t h a t
A ( p 0 , u 0 ( . ) ) ( t ) = x 0 ( t ) , t e [ t 0 , t j ]
I t i s e a s y t o s e e t h a t G a t e a u x d i f f e r e n t i a l s f o r s u c h f u n c t i o n a l s a r e g i v e n b y
D A i(p0, u 0 (.);p,u (.)) = (pQ i x 0 (tj)), p >
+ <f f ( V x o V * D A ( p 0 , u 0 ( . ) ; p , u ( . ) ) ( t t ) >
D A „
-1
( p 0 , u Q( . ) ; p , u ( . ) ) = j ' < ^ - ( t , x 0 ( t ) , u Q ( t ) ) , D A ( p 0 , u 0 ( . ) ; p , u ( . ) ) ( t ) >
+ < | ^ ( t , x 0 ( t ) , u 0 ( t ) ) , u ( t ) > d t
O t h e r f o r m s o f t h e d i f f e r e n t i a l s D A j a n d D A 2 a r e t h e f o l l o w i n g :
ti
D A ^ (p 0, u „ ( . ) ; p ,u ( . ) ) = < | ^ (pQ .X gftj)), p > + J (i//j ( t ) , | ^ ( t , x 0(t ) , u 0( t )) u ( t )> d t
" i
D A 2(p,,u0(.);p,u(.)) =J2 9 u u 0 '
+ < | ^ ( t , x Q( t ) , u 0 ( t ) ) , u ( t ) > d t
w h e r e Ф1,Ф2 - [ t o . t j H R 11 v e r i f y
IAEA-SMR-17/30 83
W e c a n s e e t h a t D A X a n d D A 2 a r e c o n t i n u o u s i n a n e i g h b o u r h o o d o f
( p 0 , u 0 ( . ) ) a n d f r o m p r o p o s i t i o n 1 i t f o l l o w s t h a t D A j a n d D A 2 a r e i n f a c t F r é c h e t d i f f e r e n t i a l s f o r t h e f u n c t i o n a l s A j a n d A 2 .
1 . 4 . N o n - l i n e a r e q u a t i o n s
W e s h a l l r e c a l l s o m e m e t h o d s f o r s o l v i n g n o n - l i n e a r e q u a t i o n s o f t h e t y p e F ( x ) = 0 w h e r e F i s a m a p p i n g o f s o m e d o m a i n i n a B a n a c h s p a c e o n t o t h e s a m e s p a c e .
I n c o n n e c t i o n w i t h t h i s w e s h a l l s t u d y t h e c a s e o f a n F w h i c h d e p e n d s a l s o o n t h e p a r a m e t e r y a n d , s u p p o s i n g t h a t F h a s a s o l u t i o n f o r a p a r t i c u l a r y 0 , w e a s k w h e t h e r o n c h a n g i n g y s l i g h t l y t h e e q u a t i o n F ( x , y ) = 0 c o n t i n u e s t o h a v e a s o l u t i o n .
P a r t i c u l a r a t t e n t i o n w i l l b e p a i d t o t h o s e t h e o r e m s o f i m p l i c i t f u n c t i o n s i n w h i c h t h e u s u a l c o n d i t i o n o f s u r j e c t i v i t y d o e s n o t h o l d .
S u c h t h e o r e m s w h i c h a r e c o n n e c t e d w i t h b i f u r c a t i o n t h e o r y a r e u s e f u l i n n o n - l i n e a r c o n t r o l l a b i l i t y .
F i r s t o f a l l w e r e c a l l t h e i m p l i c i t - f u n c t i o n t h e o r e m i n t h e ( x , y ) p l a n e f o r a s c a l a r f u n c t i o n f ( x , y ) d e f i n e d i n a n e i g h b o u r h o o d o f t h e o r i g i n w i t h
f ( 0 , 0 ) = 0 . W e w i s h t o s o l v e f ( x , y ) = 0 f o r x i n t e r m s o f y w h e n y i s s m a l l .T h e i m p l i c i t - f u n c t i o n t h e o r e m a s s e r t s t h a t , i f
3 f— ( x , y ) i s c o n t i n u o u s i n a n e i g h b o u r h o o d o f ( 0 , 0 ) a n dO X
3 £ \( 0 , 0 ) ф 0 , t h e n a u n i q u e s o l u t i o n e x i s t s f o r y s u f f i c i e n t l y s m a l l .
T h e u s u a l p r o o f c o n s i s t s i n w r i t i n g { h e e q u a t i o n i n t h e f o r m
0 = f ( x , y ) = x ( 0 , 0 ) + R ( x , y )
j p (0,0) Э х
R ( x , y )
w h i c h , f o r s u f f i c i e n t l y s m a l l y , c a n b e s o l v e d b y i t e r a t i o n s . T h e p r o o f i s b a s e d e s s e n t i a l l y o n t h e f a c t t h a t
3 f— ( 0 , 0 ) x : R - > R i s a s u r j e c t i o n . I f s u c h a h y p o t h e s i s i s n o t v e r i f i e d , t h e
e q u a t i o n m a y h a v e n o s o l u t i o n o r m o r e t h a n o n e s o l u t i o n .F o r e x a m p l e , s u p p o s e t h a t f ( x , y ) i s o f c l a s s C 3 a n d h a s a T a y l o r
e x p a n s i o n a t t h e o r i g i n o f t h e f o r m
f ( x , y ) = a x 2 + 2 b x y + c y 2 + || x , y | | 2 ш ( х , у ) , а Ф 0
b 2 > a c , l i m u ( x , y ) = 0 , J| x , y || =-Jx¿ +y'¿(x ,y ) -0
84 VA RSA N
S i n c e — ( 0 , 0 ) = 0 , t h e a b o v e h y p o t h e s i s i s n o t s a t i s f i e d a n d w e c a n n o t9 f
a p p l y t h e u s u a l i m p l i c i t - f u n c t i o n t h e o r e m . H o w e v e r , t h e e q u a t i o n f ( x , y ) = 0
h a s t w o s o l u t i o n s x 1 ( y ) , x 2 ( y ) f o r y s u f f i c i e n t l y s m a l l .I n d e e d , t h e e q u a t i o n f ( x , y ) = 0 f o r у ф 0 i s e q u i v a l e n t t o t h e f o l l o w i n g :
T h e r e f o r e , F ( o , y ) = 0 i s s a t i s f i e d b y ( o j . O ) , ( a 2 , 0 ) . O n t h e o t h e r h a n d , t h e f u n c t i o n F ( o f , y ) n e a r (a?1 , 0 ) a n d ( a ^ , 0 ) s a t i s f i e s t h e u s u a l c o n d i t i o n s i n v o l v e d i n t h e i m p l i c i t - f u n c t i o n t h e o r e m s t a t e d a b o v e . H e n c e , t h e r e e x i s t
O j ( y ) , a 2 ( y ) f o r у s u f f i c i e n t l y s m a l l , y e ( - 6 , 6 ) , s u c h t h a t o ^ O ) = a 1 , a 2 ( 0 ) =a2 a n d F ( o j ( y ) , y ) = 0 , F ( o r 2 ( y ) , y ) = 0 f o r a l l y e ( - 6 , 6 ) . D e n o t i n g x j ( y ) = ах(у)у, x 2 ( y ) = o 2 ( y ) y w e m a y o b t a i n t h e a s s e r t i o n .
L e t u s n o w c o n s i d e r t h e g e n e r a l c a s e .L e t X a n d Z b e B a n a c h s p a c e s a n d l e t Y b e a n o r m e d s p a c e . W e i n t e n d
t o s o l v e t h e e q u a t i o n F ( x , y ) = 0 n e a r ( 0 , 0 ) f o r x , i n t e r m s o f y , w h e r e F ( 0 , 0 ) = 0 a n d F : X X Y - * Z i s a n o n - l i n e a r m a p w h i c h h a s a F r é c h e t d i f f e r e n t i a l
a t ( 0 , 0 ) w i t h r e s p e c t t o x .
T h e o r e m 1 . S u p p o s e F i s d e f i n e d i n a n e i g h b o u r h o o d o f t h e o r i g i n i n
X X Y a n d h a s a F r é c h e t d i f f e r e n t i a l a t ( 0 , 0 ) w i t h r e s p e c t t o x . S u p p o s e F c a n b e w r i t t e n i n t h e f o r m F ( x , y ) = A x + R ( x , y ) w h e r e
( a ) R ( x , y ) i s c o n t i n u o u s( b ) I) R ( x 2 , y ) - R ( x j , у ) I S e I x 2 - X j I , f o r | | x a | | , | | x 2 | | , | | y | | s 6 ( e ) ,
0 < 6 ( e ) s e .
S u p p o s e A h a s a b o u n d e d i n v e r s e A - 1 d e f i n e d o n a l l Z . T h e n , f o r у s u f f i c i e n t l y s m a l l , F ( x , y ) = 0 h a s a u n i q u e c o n t i n u o u s s o l u t i o n x = x ( y ) n e a r t h e o r i g i n .
W e w r i t e F ( x , y ) = 0 i n t h e f o r m x = - A - 1 R ( x , y ) . L e t e > 0 b e s u f f i c i e n t l y s m a l l s u c h t h a t || A " 1 | e < 1 . I t f o l l o w s t h a t A ’ 1 R ( x , y ) G S ( 0 , 6 ( e ) ) f o r a l l x G S ( 0 , 6 ( e ) ) a n d y , || y | S 6 j ( e ) , w h e r e S ( 0 , 6 (e ) ) = { x : j| x | | S6 ( e ) } , a n d 61(e) É 6 ( e )
i s s u f f i c i e n t l y s m a l l .F o r e v e r y s u c h y , - A " 1 R ( x , y ) : S ( 0 , 6 ( e ) ) - * S ( 0 , 6 ( e ) ) i s a c o n t r a c t i o n i n a
B a n a c h s p a c e .U s i n g t h e c o n t r a c t i o n t h e o r e m w e s e e t h a t f o r e v e r y y G S ( 0 , 6 ( e ) ) t h e r e
e x i s t s a u n i q u e x ( y ) s u c h t h a t F ( x ( y ) , y ) = 0 . T h e s o l u t i o n x ( y ) i s o b t a i n e d a s
a l i m i t w i t h t h e f o l l o w i n g s e q u e n c e
b e c a u s e а ф 0 a n d b 2 > a c .
F o r x = a y i t f o l l o w s
F ( a , y ) = aa2 + 2Ъа + c + ( 1 +a 2 ) ш ( а у , у ) = 0
T h e e q u a t i o n aa2 + 2ba + с =0 h a s t w o r e a l d i s t i n c t s o l u t i o n s 0 ф а1 Ф a2
P r o o f
xn+l(y ) = -A‘1R(xn(y),y), X0(y)=0
IAEA-SMR-П/ЗО 85
From.
x ( y ) = " A " 1 R ( x ( y ) , y )
w e h a v e
£ ( y ) . “ з с ( у 0 ) II S ¡ A ' 1 II || R ( x ( y ) , у ) - R ( x ( y 0 ) , y 0 ) ||
A'-1 II R(x(y), y) - R(x(y0), y)|| + II R(x(y0 ), y) - R(x(y0 ),y0)||
-1 € II x(y ) - x(y0 ) II +)7 ( Il у -y0 II )
w h e r e l i m r¡(t ) = 0 . t->o
M o r e o v e r , i t f o l l o w s t h a t
||x(y ) -x (y 0)||<ri(b - - y ° .ll)1 - Il A ” 1 II € )
a n d t h e p r o o f i s c o m p l e t e .A g e n e r a l i z a t i o n o f t h i s t h e o r e m , w h i c h i s d u e t o G r a v e s ( D u k e J . Г 7
( 1 9 5 0 ) , p p . 1 1 1 - 1 1 4 ) , i s t h e f o l l o w i n g :L e t G b e a m a p t o Z o f a n e i g h b o u r h o o d o f t h e o r i g i n i n X w i t h G ( 0 ) = 0 .
W e w i s h t o s o l v e G ( x ) = z f o r x , i n t e r m s o f z .S u p p o s e t h a t G ( x ) h a s a c o n t i n u o u s F r é c h e t d i f f e r e n t i a l i n a n e i g h b o u r h o o d
o f t h e o r i g i n . T h e s e f a c t s i n v o l v e t h a t G c a n b e w r i t t e n a s G ( x ) = L x + R ( x ) , w h e r e ( a ) R ( x ) i s c o n t i n u o u s , L : X - » Z i s l i n e a r a n d c o n t i n u o u s ;
( 8 ) | | R ( x 2 ) - R ( x 1 ) | | s e | | x 2 - x 1 || f o r | | x j , | | x 2 | | s ó ( e ) , 0 < 6 ( e ) S e
W h e n L i s a b i j e c t i o n , t h e s o l u t i o n x ( z ) f o r G ( x ) = z n e a r t h e o r i g i n i s g i v e n b y t h e o r e m 1 a n d w e k n o w n s u c h a s o l u t i o n t o b e u n i q u e a n d c o n t i n u o u s . I f L i s o n l y a s u r j e c t i o n w i t h o u t b e i n g a n i n j e c t i o n , t h e s o l u t i o n x ( z ) f o r G ( x ) = z w i l l e x i s t , b u t i n t h i s c a s e i t i s n o t n e c e s s a r i l y u n i q u e a n d c o n t i n u o u s .
T h e o r e m 2 . L e t G s a t i s f y (a) a n d ( $ ) . S u p p o s e L X = Z . T h e n , f o r e v e r y z s u f f i c i e n t l y s m a l l , t h e r e e x i s t s a s o l u t i o n x ( z ) o f G ( x ) = z , || x ( z ) || s 6 ( e ) .
P r o o f
D e n o t e N = {xe X : L x = 0 } . D e f i n e X a s t h e q u o t i e n t B a n a c h s p a c e X / N o f a l l c l a s s e s x = { x + x 0 ^ x 0 € N } , | | x | | = i n f { | | y || : y e x } . L e t t h e l i n e a r c o n t i n u o u s o p e r a t o r L : X - * Z b e d e f i n e d b y L x = L x . T h e n L i s a b i j e c t i o n a n d h a s a b o u n d e d i n v e r s e L " 1 . T h e r e f o r e , t h e r e w i l l e x i s t a c o n s t a n t
C j > 0 s u c h t h a t || x || S c j || z || . F o r e v e r y x t h e r e e x i s t x G x s u c h t h a t || x || S c || z | | , w h e r e L x = z , с = 2 c j . S u p p o s e e < l / c , e < l . F o r a n y z , w i t h | | z || S 6 ( e ) ( l - e c ) / c ,
8 6 VA RSA N
w e s h a l l c o n s t r u c t a s o l u t i o n f o r G ( x ) = z w i t h | | x | | s 6 ( e ) . I n t h e f o l l o w i n g a n i t e r a t i v e p r o c e d u r e i s u s e d .
W e c h o o s e x 0 = 0 a n d d e f i n e x 2 e X s u c h t h a t || X j || S с |j z || , L x x = z . F o r n > 1 , x n i s c h o s e n a s a s o l u t i o n o f
L (x n ' x n - l ) = L (x n - l * x n -2) ‘ ( G ( x n_ i ) - G ( x n _ 2)) = z
s a t i s f y i n g j] x n - x n . 1 || — С I) z n _ j | | .W e s h a l l p r o v e b y i n d u c t i o n t h a t | | x n || â Ô ( e ) ( l - ( c e ) ) a n d | | x n - x n - 1 1|
S ( c e ) 11" ^ ( e ) ( l - c e ) .F o r n = 1 t h e s e c o n d i t i o n s a r e s a t i s f i e d b y d e f i n i t i o n .S u p p o s e
| | x n - f x j | S ( c e ) n - 2 6 ( e ) ( l - c e ) , || x n _ 1 || ë ô ( e ) ( 1 - ( c e ) " " 1 )
T h e n
= Il Ь <х п - Г X n-2> - (G <x n- l ) ■ G (x n-2» Il = e II x n - l ■ X n -2 IIZ_
B y d e f i n i t i o n , || x n _ x n _ 1 1| â c || z n _ 1 1 | , a n d w e h a v e
II x n ' x n - l II á c e II x n - l " X n—2 11 ~ ( « I " ' 1 6 ( e ) ( 1 - c e )
x n á Ô ( e ) ( 1 - c e )
, 4n - l -1 n - 1 + i ^ ( c e ) _
1 - c e= 6(e)(l-ce) )
S i n c e { x n } i s a G a u c h y s e q u e n c e i n a B a n a c h s p a c e , t h e r e w i l l e x i s t x £ X , (I x (I s 6 ( e ) s u c h t h a t l i m x n = x . B y d e f i n i t i o n l i m z n = 0 .
O w i n g t o t h e c o n s t r u c t i o n G ( x n ) = z - z n a n d w e o b t a i n G ( x ) = z . T h e p r o o fi s c o m p l e t e .
N o w w e c o n s i d e r t h e s i t u a t i o n d e s c r i b e d i n t h e o r e m 1 w i t h o u t a d m i t t i n gt h a t A h a s a b o u n d e d i n v e r s e o r i s a s u r j e c t i o n .
I n t h i s c a s e t h e e q u a t i o n F ( x , y ) = 0 h a s n o t n e c e s s a r i l y a s o l u t i o n o r t h e r e m a y b e m o r e t h a n o n e s o l u t i o n . W e a d m i t t h a t A i s a l i n e a r c o n t i n u o u s o p e r a t o r w i t h c l o s e d r a n g e R ( A ) i n Z . M o r e o v e r , w e s u p p o s e t h a t N ( A ) , i t s n u l l s p a c e , a n d R ( A ) s p l i t t h e b a s i c s p a c e s X a n d Z i n t o a d i r e c t s u m X = N ( A ) © X j , Z = R ( A ) © Z j . T h e e q u a t i o n A x + R ( x , y ) = 0 t a k e s t h e f o r m
A x x + p r R ( A ) R f X g + X j . y ) = 0 a n d p r z R ( x Q + x 1 , y ) = 0
T o t h e e q u a t i o n A x j + p r R ( A R ^ + x ^ y ) = 0 w e c a n a p p l y t h e o r e m 1 b e c a u s e A ^ j - ’ R i A ) i s a b i s e c t i o n .
I n t h i s w a y w e o b t a i n a u n i q u e s o l u t i o n x j f x ^ y ) n e a r ( 0 , 0 ) i n N ( A ) X Y .I n o r d e r t o s o l v e t h e p r o b l e m w e h a v e t o p r o v e t h a t p r ^ R Í X g + x - ^ X g . y J . y )
h a s a s o l u t i o n x Q( y ) n e a r t h e o r i g i n . T h e e q u a t i o n p r ^ R f x o + X j f x g . y ) , y ) = 0 i s c a l l e d b i f u r c a t i o n e q u a t i o n .
W h e n w r i t t e n i n s u c h a g e n e r a l f o r m w e c a n n o t s a y a n y t h i n g a b o u t t h e
s o l u t i o n s o f t h e b i f u r c a t i o n e q u a t i o n .
IAEA-SMR-17/30 8 7
L e t u s s u p p o s e
d im Z x = 1, R ( x ,y ) = b x m + ^ c ky k + d (x ,y ) , Cj =/=(), m S 2
m
k=l
w h e r e b x m , c k y k a r e h o m o g e n e o u s f o r m s a n d
d ( t x , t y )
t - * o t
u n i f o r m l y w i t h r e s p e c t t o ( x , y ) i n a b o u n d e d s e t .S u p p o s e t h a t t h e r e e x i s t Xj, x 2 G N ( A ) s u c h t h a t p r Z l b 3 c ™ > 0 , p r Z l b x ™ < 0 .
W e a r e l o o k i n g f o r s o l u t i o n s o f t h e f o l l o w i n g t y p e :
x ( e , 6 , y ) = e í ó X j + ( 1 - 6 ) x 2 ) + e m X j ( e , 6 , y ) f o r F ( x , e m + 1 y ) = 0
I n R ( A ) w e h a v e t h e e q u a t i o n
e m ( A X j + p r R ( A ) Ь ( б х х + ( 1 - 6 ) x 2 ) m + u ( e , 6 , y , x )) = 0
W h e r e
II u ( e , 6 , y , x " ) - u ( e , 6 , y , x 1 ) Il g 1 7 ( e ) | | x " - x 1 |
a n d
l i m r¡( e ) = 0
U s i n g t h e o r e m 1 w e s e e t h a t t h e r e e x i s t s a u n i q u e s o l u t i o n Xj(e, 6 , y ) o f t h e a b o v e e q u a t i o n f o r e , 6 , y s u f f i c i e n t l y s m a l l .
O n o r d e r t o h a v e F ( x ( e , ó , y ) , e m + 1 y ) = 0 w e m u s t v e r i f y p r Z l R ( x ( e , 6 , y ) , e m + 1 y ) = 0 o r , i n e q u i v a l e n t f o r m ,
e m p r Z i ( b ( 6 x 1 + ( 1 - 6 ) х 2 ) ш + ф ( e , 6 , y )) = 0
w h e r e
l i m ш ( е , б , у ) = 0 e-*o
u n i f o r m l y w i t h r e s p e c t t o ( б , у ) s u f f i c i e n t l y s m a l l .L e t e 0 > 0 a n d r 0 > 0 b e s u c h t h a t
+ ф ( е > J P r Z ! b S r
p r Z i ( b x ™ + ф ( е , 0 , y ) ) P r Z l b x “
f o r a l l e € ( 0 , e 0 ] , | | y | | g r 0 .
8 8 VARSAN
Ф(е,&, y ) = p i ’z | b ( ê x 1 + ( l - 6 ) x 2 ) m + cp(e , ê , y ))
T h e c o n t i n u o u s f u n c t i o n ф ( е , 6 , y ) s a t i s f i e s ф ( е , 1 , у ) > 0 , ф ( е , 0 , у ) < 0 f o r a l l е е ( 0 , е 0 ] , | | у | | 5 г 0 .
T h e n , f o r e v e r y у w i t h || y || S r Q t h e r e e x i s t s a 6 ( y ) e ( 0 , 1 ) s u c h t h a t Ф(е0, ô ( у ) , у ) = 0 . I n t h i s w a y w e o b t a i n a n o n - z e r o s o l u t i o n
x ( y ) = e 0 ( ê ( y ) x 1 + ( 1 - 6 ( y ) ) x 2 ) + e ” X j i e g , 6 ( y ) y )
f o r s u f f i c i e n t l y s m a l l y .
Denote
2 . O P T I M I Z A T I O N
2 . 1 . A p p l i c a t i o n s t o o p t i m i z a t i o n p r o b l e m s . S u f f i c i e n t c o n d i t i o n s
I n o r d e r t o d e s c r i b e t h e p r o p e r t i e s o f a n o p t i m u m e l e m e n t w e u s e n e c e s s a r y o r s u f f i c i e n t c o n d i t i o n s f o r o p t i m a l i t y . S u c h c o n d i t i o n s i n v o l v e u s u a l l y t h e e x i s t e n c e o f F r é c h e t d i f f e r e n t i a l s a n d a n i m p l i c i t - f u n c t i o n t h e o r e m .
L e t x 0 e X b e s u c h t h a t f ( x 0 ) = m i n f ( x ) , w h e r e x i s a l i n e a r n o r m e dx eX
s p a c e a n d f i s a c o n t i n u o u s r e a l f u n c t i o n a l o n X . S u p p o s e f h a s a G a t e a u x d e r i v a t i v e a t x 0 . T h e n
D f ( x 0 ; h ) = 0 f o r a l l h e X
T h i s f a c t i s a c o n s e q u e n c e o f t h e r e l a t i o n
0 S f ( x 0 + t h ) - f ( x 0 ) = t D f ( x 0 ; h ) + 0 x ( t )
w h e r e
t - 0 t
T h e c o n d i t i o n D f ( x 0 ; h ) = 0 f o r a l l h e x i s c a l l e d a f i r s t - o r d e r n e c e s s a r y c o n d i t i o n f o r t h e u n c o n s t r a i n e d m i n i m u m p r o b l e m m i n f ( x ) .
xe xI n g e n e r a l , i f f h a s a F r é c h e t d i f f e r e n t i a l o f o r d e r m , t h e n a n e c e s s a r y
c o n d i t i o n o f o r d e r m f o r o p t i m a l i t y i s d m f ( x o ; h ) ê 0 f o r a l l h e x w h i c h s a t i s f y d i f ( x 0 ; h ) = 0 , j = l , . . . , m ~ l .
S u c h a c o n d i t i o n i s o b t a i n e d f r o m t h e r e l a t i o n
4> 2 л 4* m ,. ^0 ^ f ( x 0 + t h ) - f ( x Q) = t d f ( x 0 ; h ) + — d f ( x 0; h ) + . . . + — d f ( x 0 ; h ) + 0 m ( t )
IAEA-SMR-17/30 8 9
w h e r e
l i m M ) = о t — о t m
I f m i s o d d t h e n d m f ( x 0 ; h ) = 0 f o r a l l h e X w h i c h s a t i s f y d^ f ( x 0 , h ) = 0 , j = 1 , . . . , m - l .
I n o r d e r t o o b t a i n s u f f i c i e n t c o n d i t i o n s f o r o p t i m a l i t y w e n e e d a s t r o n g e rf o r m o f t h e n e c e s s a r y c o n d i t i o n s . L e t x Q b e f i x e d s u c h t h a t d f ( x Q; h ) = 0 a n dd 2 f ( x o ; h ) S 6 0 | | h | | 2 f o r a l l h G X , w h e r e 6 0 > 0 .
T h e n X o i s a l o c a l l y o p t i m a l e l e m e n t . T h i s m e a n s , t h e r e e x i s t s as p h e r e S c e n t e r e d a t x Q s u c h t h a t f ( x ) > f ( x Q ) f o r a l l x e S , x = ^ x 0 .
L e t m b e e v e n . A s u f f i c i e n t c o n d i t i o n o f o r d e r m f o r o p t i m a l i t y i sd j f ( x 0 ; h ) ê 0 , j = 1 , . . . , m - 1 , d m f ( x Q; h ) ê ¿ 0 || h | | m f o r a l l h £ X , w h e r e 6 0 > 0 .L e t u s n o w c o n s i d e r a c o n s t r a i n e d o p t i m u m p r o b l e m m i n f ( x ) w h e r e t h e s e t
x e LL Q X i s d e f i n e d b y L = { x G X ; T ( x ) = 0 } , a n d T : X - Y i s a n o n - l i n e a r m a p . S u p p o s e X , Y a r e B a n a c h s p a c e s a n d f , T h a s a F r é c h e t d i f f e r e n t i a l o f f i r s t o r d e r . L e t x £ L b e s u c h t h a t f ( x . ) = m i n f ( x ) ; x 0 i s c a l l e d o p t i m a l e l e m e n t .
0 x e L
T h e o r e m 1 . L e t x Q G L b e a n o p t i m a l e l e m e n t . S u p p o s e T h a s a c o n t i n u o u s F r é c h e t d i f f e r e n t i a l i n a n e i g h b o u r h o o d o f x n , d T ( x 0 ; . ) : X - ’- Y i s a s u r j e c t i o n a n d X = N ( d T ( x n ; . )) Ф X v T h e n t h e r e e x i s t s а ц б У * ( t h e c o n j u g a t e
s p a c e o f Y ) s u c h t h a t d f ( x 0 ; h ) + й ( d T ( x 0 ; h ) ) = 0 f o r a l l h £ X .
P r o o f
F i r s t w e s h o w t h a t f o r e v e r y h € X , d T ( x 0 ; h ) = 0 , i t f o l l o w s t h a t d f ( x Q ; h ) = 0 . S u p p o s e t h e e x i s t e n c e o f a n E G N ( d T ( x 0 ; . ) ) s u c h t h a t d f ( x 0 ; h ) < 0 .
B y h y p o t h e s i s t h e c o n d i t i o n s o f t h e o r e m 1 ( s e e s e c t i o n 1 ) a r e s a t i s f i e d a n d w e s e e t h a t f o r e v e r y e t h e r e e x i s t Xj(e) € X j s u c h t h a t T Í X Q + e K + X j j e ) ) = 0 a n d
F o r e s u f f i c i e n t l y s m a l l i t f o l l o w s t h a t
f ( x 0 + e h + X j f e ) ) - f ( x 0 ) = e d f ( x o ; h ) + 6> ( e ) < 0 i n c o n t r a d i c t i o n t o t h e p r o p e r t y o f x 0 .
D e f i n e M G Y * a s f o l l o w s :
- ц (у )= d f ( x Q; h ) w h e r e у = d T ( x Q; h )
T h e f u n c t i o n a l i s l i n e a r a n d w e l l d e f i n e d b e c a u s e f o r
h j ^ h g , d T ( x 0 ; h j ) = d T ( x 0 ; h 2 ) = y i t f o l l o w s d f ( x 0 ; h : ) = d f ( x 0 ; h 2 ) . D e n o t e A = d T ( x 0 ; . ) | X 1 a n d w e s e e t h a t A ^ j - ' Y i s a b i j e c t i o n . T h e r e f o r e a b o u n d e d i n v e r s e A " 1 : Y - » X a w i l l e x i s t a n d f o r a s e q u e n c e y n - » 0 i t f o l l o w s t h a t t h e r e e x i s t { x l n } £ X x s u c h t h a t y n = d T ( x Q; x l n ) a n d l i m x l n = 0 . H e n c e ц i s c o n t i n u o u s a n d t h e p r o o f i s c o m p l e t e .
9 0 VA RSA N
S e c o n d - o r d e r s u f f i c i e n t c o n d i t i o n s f o r a c o n s t r a i n e d o p t i m u m p r o b l e m i n v o l v e t w o e s s e n t i a l f a c t s .
F i r s t o f a l l i t i s n e c e s s a r y t o s o l v e t h e e q u a t i o n T ( x ) = 0 i n a n e i g h b o u r h o o d o f a f i x e d e l e m e n t a n d t h e n w e h a v e t o p r o v e t h a t a H e s s i a n i s p o s i t i v e - d e f i n i t e . W e p o i n t o u t t h a t f o r a s a t i s f a c t o r y t h e o r e m i t i s n e c e s s a r y t o u s e t w o d i f f e r e n t n o r m s o n t h e s p a c e s X a n d Y . T h e e s s e n t i a l c o n d i t i o n s c o n c e r n i n g t h e H e s s i a n a r e s t a t e d w i t h r e s p e c t t o a s e c o n d n o r m f o r w h i c h X a n dY a r e n o t n e c e s s a r i l y B a n a c h s p a c e s .
T h e u s e o f t h i s n o r m m a k e s i t e a s i e r t o v e r i f y t h e p o s i t i v i t y p r o p e r t y ; t h i s c a n b e s e e m i n t h e a p p l i c a t i o n s t o t h e c o n t r o l t h e o r y .
B y II . I w e d e n o t e t h e n o r m i n X a n d Y g i v e n b y d e f i n i t i o n . S u p p o s e Y i s a B a n a c h s p a c e . S u p p o s e w e c a n c h o o s e o n X a n d Y o t h e r n o r m s | | | . |||
s u c h t h a t ¡I x (К S С || x If ( ||| Y ||| S С || у || ) . I n t h e s a m e w a y w e d e n o t e t h e n o r m o n X a n d Y b u t t h i s d o e s n o t m e a n t h a t t h e y a r e i d e n t i c a l . L e t x Q 6 X b e f i x e d s u c h t h a t T ( x 0 ) = 0 .
H y p o t h e s i s i n
f a n d T a r e c o n t i n u o u s l y F r é c h e t - d i f f e r e n t i a b l e , o f s e c o n d o r d e r i n a n e i g h b o u r h o o d ^ V o f x n , V = { x G X : || x - x n || < e } a n d
( a ) | d 2 f ( x 0 ; h , k ) | s c j k | | | HI h H I , ||| d 2 T ( x Q; h , k ) ||| s C l | | | h | | | | | | k | | |
f o r e v e r y h , к £ X w h e r e c 0 , C j > 0 a r e c o n s t a n t s ;
( b ) I d 2 f ( x ; h , k ) - d 2 f ( x Q; h , k ) | É r i 0 ( | | x - x 0 || ) | | | h | | | | | | k | | |
HI d 2 T ( x ; h , k ) - d 2 T ( x 0 ; h , k ) | | | s ( || x - x j ) | | | h | | | | | | k | | |
f o r a n y x € V , h , k G X , w h e r e l i m r) ( t ) = l i m r j ( t ) = 0t->0 t-»o
H y p o t h e s i s Í t
( a ) X m a y b e d e c o m p o s e d i n t o a d i r e c t s u m X = N ( d T ( x n ; . ) ) ® X , , w h e r e X , i s a B a n a c h s u b s p a c e a n d d T ( x „ ; . ) : X ^ -» Y i s b i j e c t i v e ;
^ ( b ) ( d T ( x Q; . ) ) _ 1 : Y - * ' X 1 i s c o n t i n u o u s i n t h e t o p o l o g y i n d u c e d b y t h e n o r m
W h e n HI . I = II . II , t h e n c o n d i t i o n s ( a ) , ( b ) i n h y p o t h e s i s i Q a n d c o n d i t i o n ( b ) i n h y p o t h e s i s i j f o l l o w f r o m t h e o t h e r c o n d i t i o n s s t a t e d a b o v e .
T h e o r e m 2 . L e t x Q , T ( x 0 ) = 0 , b e s u c h t h a t t h e h y p o t h e s e s i 0 a n d i -, a r e s a t i s f i e d . L e t 6 0 > 0 a n d l e t ц G Y * b e c o n t i n u o u s w i t h r e s p e c t t o t h e n o r m | | | . HI a n d s u c h t h a t t h e f o l l o w i n g c o n d i t i o n s a r e s a t i s f i e d :
( c j ) d f f x j j ; h ) + M ( d T f x g ; h ) = 0 f o r e v e r y h € X ,
( c 2 ) ( f f ( x Q ; h ) + M ( d 2 T ( x 0 ; h ) ) s 6 Q HI h | | | 2 f o r e v e r y h G N ( d T ( x 0 ; . ) )
T h e n x 0 i s l o c a l l y o p t i m a l ; i . e . t h e r e e x i s t s a n e i g h b o u r h o o dV 0 = { x £ X : I x - x 0 || < e 0 } s u c h t h a t f ( x ) > f ( x 0 ) f o r e v e r y x Ф x 0 , x G V 0 , T ( x ) = 0 .
2.2 . Suffic ient cond it ions o f se co n d o r d e r f o r optim ality
IAEA-SMR-n/30 91
L e t F : N ( d T ( x 0 ; . ) © Y b e d e f i n e d b y F ( h , x x ) = T ( x 0 + h + X j ) . W e w i s h t o s o l v e F f h . X j ) = 0 f o r X j i n t e r m s o f h .
U s i n g h y p o t h e s i s i 2 w e s e e t h a t t h e c o n d i t i o n s i n t h e o r e m 1 ( s e e s e c t i o n 1 ) a r e s a t i s f i e d . T h e r e f o r e , t h e r e e x i s t s a u n i q u e c o n t i n u o u s f u n c t i o n X j ( h ) d e f i n e d f o r h s u f f i c i e n t l y s m a l l s u c h t h a t
P r o o f '
T ( x Q + h + x x ( h ) ) = 0
W e s h a l l p r o v e t h a t
(1)
l i m l l l x l ( h ) l l l = g I M h o (J h m
F r o m ( 1 ) w e o b t a i n
0 = d T ( x Q; h + x 2 ( h ) ) + / t d T ( x Q + T t z ( h ) ; z ( h ) d T | d t
о о
(2 )
w h e r e z ( h ) = h + X j ( h ) .S i n c e d T ( x 0 ; h ) = 0 , w e o b t a i n f r o m ( 2 )
l 1
d T ( x 0 ; X l ( h ) ) = - t d T ( x Q + Ttz(h) ; z ( h ) ) dT dt ( 3 )
a n d , u s i n g h y p o t h e s i s i 1 ( w e o b t a i n f r o m ( 3 )
l l
0 ■ 0
I d T ( x Q + T t z ( h ) ; z ( h ) ) )|| d T d t ( 4 )
F u r t h e r , u s i n g h y p o t h e s i s i „ , w e o b t a i n f r o m ( 4 )
III ( h ) ¡II s k 1 ||| z ( h ) H I 2 = k j ( (I) h | | | + | | | X j ( h ) | | j ) 2 ( 5 )
F r o m ( 5 ) i t f o l l o w s t h a t
I J K M I Ikl ( I I I h III + 2 III M III + III x i ( h ) III — Г „ 1 ' (6 )
a n d t h e r e f o r e w e o b t a i n
x , ( h )
( 1 - k i l l l x i ( h ) III J ^ V l l N I I + 2 I I I х ! M U ) (V
92 VÂRSAN
U s i n g I x (у s C ü x [j w e f i n d
l i m | | | h | | | = l i m Kl x . ( h ) ¡У = 0
Il h | | -0 l lh l h o
a n d f r o m ( 7 ) w e o b t a i n
III X 1 ( h ) IIIl i m — -------- ---- = 0 ( 8 )
llh||-0 III h | | |
C o n d i t i o n ( 8 ) w i l l t a k e a n i m p o r t a n t p l a c e i n t h e f o l l o w i n g . L e t c p : X - » R b e d e f i n e d a s f o l l o w s :
c p ( x ) = f ( x ) + n ( T ( x ) ) ( 9 )
w h e r e i s g i v e n i n t h e s t a t e m e n t o f t h e t h e o r e m .B y h y p o t h e s i s i 0 cp i s c o n t i n u o u s l y F r é c h e t - d i f f e r e n t i a b l e , o f s e c o n d
o r d e r a t x 0 , a n d w e o b t a i n
1 2ф ( х ) - ф ( х 0 ) = - d Ф ( x 0 + t x ( x - x 0 ) ; x - x 0 )
1 2 1 2 2 = - d ф ( х 0 ; x - x 0 ) + - [ d ф ( x 0 + t x ( x - x 0 ) ; x - x „ ) - d ф ( x Q; x - x Q) ] ( 1 0 )
S i n c e M i s a c o n t i n u o u s f u n c t i o n a l i n t h e t o p o l o g y d e t e r m i n e d b y | | | . ||| , u s i n g ( i 0 ) w e o b t a i n
I d 2 9 ( x 0 + t x ( x - x 0 ) ; x - x 0 ) - d 2 ф ( x 0 ; x - x 0 )) | ë n ( || x - x 0 || ) | | | x - x 0 | | | 2 ( 1 1 )
w h e r e
l i m r)(t ) = 0 t-*o
T h e r e f o r e , ( 1 0 ) b e c o m e s
' Ф ( x ) - ф ( х 0 ) ё i d % ( x 0 ; x - x 0 ) - n ( ] | x - x j ) ||| x - x Q | | | 2 ( 1 2 )
F r o m ( 1 ) a n d ( 9 ) f o l l o w s
f ( x ) - f ( x 0 ) = ф ( х ) - ф ( х 0 ) ( 1 3 )
f o r e v e r y x = x 0 + h + x 1 ( h ) , w h e r e h G N ( d T ( x 0; . ) ) i s s u f f i c i e n t l y s m a l l .
IAEA-SMR-17/30 93
F r o m (12) and (13) we obtain
( 1 4 )
B y h y p o t h e s i s
d 2 9 ( x 0 ; h ) i 5 0 HI h I|2
( 1 5 )
d 2 c p ( x 0 ; h , k ) § C | | | h | | | HI к ( 1 6 )
a n d u s i n g ( 8 ) w e s e e t h a t
2
2
i s b o u n d e d a s a f u n c t i o n o f h . U s i n g ( 1 5 ) a n d ( 1 6 ) , w e o b t a i n f r o m ( 1 4 )
f o r e v e r y x = x 0 + h + x 1 ( h ) w i t h | | h | | s u f f i c i e n t l y s m a l l . N o w , s u p p o s e x 0 i s n o t a l o c a l l y o p t i m a l e l e m e n t . T h e n t h e r e e x i s t s a n { x n ) € X х п ф х 0 , x n - + x 0
s u c h t h a t T Í X j , ) = 0 a n d f ( x n ) - f ( x 0 ) s 0 .A n i n d e x N 0 s u f f i c i e n t l y l a r g e w i l l e x i s t s u c h t h a t x n = x 0 + h n + x 1 ( h n) f o r
n S N o c o n t r a d i c t i n g ( 1 7 ) . T h e p r o o f i s c o m p l e t e .
2 . 3 . A n a p p l i c a t i o n
I n t h i s s e c t i o n , u s i n g t h e o r e m 2 , w e o b t a i n s u f f i c i e n t c o n d i t i o n s o f s e c o n d o r d e r f o r a n o n - l i n e a r c o n t r o l p r o b l e m .
L e t t h e f u n c t i o n a l a n d t h e c o n t r o l s y s t e m b e d e f i n e d a s f o l l o w s :
f ( x ) - f ( x 0 ) ê - ^ - HI h2
( 1 7 )
to
w h e r e x Qe R n t ^ i j G R a r e f i x e d a n d F ( t , x , u ) , f ( t , x , u ) , b e i n g f u n c t i o n s i n ( x , u ) £ R n X R r a r e c o n t i n u o u s l y d i f f e r e n t i a b l e o f s e c o n d o r d e r .
94 VÂRSAN
T h e c l a s s o f a d m i s s i b l e c o n t r o l s u ( . ) i s t h e s p a c e d o f a l l p i e c e w i s e c o n t i n u o u s u : [ t g . t j ] - * R r . F o r e v e r y u l . j e ^ w e o b t a i n a u n i q u e s o l u t i o n x ( . ) G С ( t Q, t j ; R n ) f r o m t h e c o n t r o l s y s t e m .
L e t ( x 0 ( . ) , u 0( . ) ) b e f i x e d , s a t i s f y i n g t h e c o n t r o l s y s t e m .D e n o t e b y || . || t h e u s u a l n o r m i n t h e s p a c e o f c o n t i n u o u s f u n c t i o n s .
T h e o r e m 3 . L e t ( x n ( . ) , u n ( . ) ) b e f i x e d s a t i s f y i n g t h e c o n t r o l s y s t e m . S u p p o s e 6 0 > 0 t o e x i s t s u c h t h a t
Э Н( C j ) t — ( t , x 0 ( t ) , u 0 ( t ) ) = 0 , t G [ t 0 , t j ]
9 u
ti
(Co) ( t , x 0 ( t ) , u 0 ( t ) ) x ( t ) , x ( t ) > + 2 < ^ - S - ( t , x 0 ( t ) , u 0 ( t ) ) x ( t ) , u ( t )
4
( t , x 0 ( t ) , u 0 ( t ) ) ü ( t ) , ü ( t ) > | d t g 60J | u ( t ) | 2 d t
f o r e v e r y
( x ( . ) , u ( . ) ) G C ( t Q , t ^ R ^ X s a t i s f y i n g x ( 0 ) = 0
w h e r e
H ( t , x , u ) + <C0 ( t ) , f ( t , x , u ) ^ + F ( t , x , u )
a n d ф(.) s a t i s f i e s
= 0 , = ( t , x 0 ( t ) , u 0 ( t ) ) + g ( t , x 0 ( t ) , u 0 ( t ) )
T h e n ( x 0 ( . ) , u Q( . ) ) i s l o c a l l y o p t i m a l ; i . e . t h e r e e x i s t s a n e n > 0 s u c h t h a t f o r e v e r y ( x ( . ) , u ( . ) ) w h i c h s a t i s f i e s t h e c o n t r o l s y s t e m a n d
0 < i f x ( . ) - x 0 ( . ) | | + | | u ( . ) - u 0 ( . ) | | < e 0
t: 'i
^ F ( t , x ( t ) , u ( t ) ) d t > J F ( t , x 0 ( t ) , u 0 ( t ) ) d t
t0 to
P r o o f
L e t X a n d Y b e n o r m e d s p a c e s d e f i n e d a s f o l l o w s :
X ^ ( t y t p R 11) X ^ , Y = C(t0, tj ; r ")
\/
IAEA-S MR-17/30
D e n o t e b y ¡|| . j|| t h e n o r m i n d u c e d b y t h e s p a c e
95
L 2 ( t 0 , t 1 ; R ) o n C ( t f l , t ^ R ) a n d & , r e s p e c t i v e l y .
L e t f : X - » R , T : X - ” Y b e d e f i n e d a s f o l l o w s :
f ( x ( . ) , u ( . ) ) = J V ( t , x ( t ) , u ( t ) ) d t
*0t
T ( x ( . ) , u ( . ) ) ( t ) = x ( t ) - X0 - J f ( s , x ( s ) , u ( s ) ) d s , t e [ t 0 , t j ]
( 1 8 )
( 1 9 )
B y h y p o t h e s i s f a n d T a r e c o n t i n u o u s l y F r é c h e t - d i f f e r e n t i a b l e , o f s e c o n d o r d e r a t ( x 0 ( . ) , u 0( . ) ) , a n d a s w e k n o w ( s e e s e c t i o n 1 . 3 )
-I
d f ( x Q( . ) , u Q( . ) ; x ( . ) , u ( . ) ) =J ^ Э х ( t ’ X ° ( t ) ’ u 0 ( t ) ) , x ( t ) >
+ < | ^ ( t , x Q( t ) , u 0 ( t ) ) , u ( t ) > d t (20 )
d T ( x 0 ( . ) , u 0 ( . ) ; x ( . ) , u ( . ) ) ( t ) = x ( t ) - J j | - ( s , x 0 ( s ) , u Q( s ) ) x ( s )
+ § £ ( s , x 0 ( s ) , u 0 ( s ) ) u ( s ) d s (2 1 )
B y d e f i n i t i o n N ( d T ( x Q( . ) , u Q( . ) ; . ) ) i s t h e s e t o f a l l ( x ( . ) , u ( . ) ) w h i c h s a t i s f y
u 0 ( t ) ) x + - ( t , x 0 ( t ) , u 0 ( t ) ) u ( t ) , te [t0, t x ] ( 2 2 )
W e u s e t h e n o t a t i o n ( X j C X ) X i = C ( t 0 , t j ; R n ) X { 0 } .T h e l i n e a r o p e r a t o r d T ( x 0 ( . ) , u Q( . ) ; . ) : X j - * Y i s o f V o l t e r a t y p e a n d i t
f o l l o w s t h a t d T ( x 0 ( . ) , u ( . ) ; . ) , a c t i n g o n X j t o Y , i s a b i s e c t i o n a n d i t s i n v e r s e i s c o n t i n u o u s i n t h e t o p o l o g y i n d u c e d o n C C t g . t j j R 11) b y t h e n o r m || . ||| . A s a c o n s e q u e n c e t h e b a s i c s p a c e X i s s p l i t i n t o a d i r e c t s u m X = N ( d T ( x 0 ( . ) ,Uq ( . ) ; , ) © X j a n d s o h y p o t h e s i s i j i s v e r i f i e d .
9 6 VÂRSAN
O n t h e o t h e r h a n d , h y p o t h e s i s i g i s a d i r e c t c o n s e q u e n c e o f t h e d i f f e r e n t i a b i l i t y p r o p e r t i e s a s s u m e d f o r t h e f u n c t i o n s w h i c h d e t e r m i n e t h e c o n t r o l p r o b l e m .
L e t 6 Y * b e d e f i n e d a s f o l l o w s :
4
i ( C ( . ) ) = J c ( t ) > d t ( 2 3 )
w h e r e ф(.) i s g i v e n i h t h e s t a t e m e n t o f t h e t h e o r e m . B y d e f i n i t i o n a n d
f r o m ( C j ) i t f o l l o w s
d f ( x 0 ( . ) , u 0 ( . ) ; x ( . ) , u ( . ) ) + n ( d T ( x 0 ( . ) , u 0( . ) ; x ( . ) , u ( . ) ) )
ti
^ “ Э х ' b ' A o ' L ' ’ u 0( t ,x0( t ) ,u0(t)) + ^ , x ( t ) > + < ^ | ( t , x 0( t ) ,u 0( t» ,u( t )> d t
*■1
= ( t, x 0(t), u0(t)), u(t)> dt = 0 V ( x ( . ) , u ( . ) ) 6 X ( 2 4 )
O n t h e o t h e r h a n d , u s i n g ( C 2 ) , w e o b t a i n
d 2 f ( x 0 ( . ) , u 0 ( . ) ; x ( . ) , u ( . ) ) + M ( d 2 T ( x 0 ( . ) , u 0 ( . ) ; x ( . ) , u ( . ) ) )
2
( W t ) , U o ( t ) ) x ( t ) , x . ( t ) >
J! n+ 2 <9x3u(t, X0(t), ^ t ) ) x ( t ) , u(t)>
,э2нd t 2 6 0 u ( . ) ( 2 5 )
f o r a l l ( x ( . ) , u ( . ) ) s a t i s f y i n g ( 2 2 ) .U s i n g G r o n w a l l ' s i n e q u a l i t y , w e o b t a i n f r o m ( 2 2 )
| | | u ( . ) | | | S C HI S ( - ) HI ( 2 6 )
a n d t h e r e f o r e ( 2 5 ) b e c o m e s
d2f(x0( . ) , u 0( . ) ; 5 ( . ) , û ( . ) + M(d2T ( x 0(.),Uo(.);x(.),ïï(.)))ê6'(|||x(.)|||2 + |||ïï(.)|||2)
( 2 7 )
a n d t h e h y p o t h e s e s o f t h e o r e m 2 a r e v e r i f i e d .H e n c e ( x Q( . ) , u Q( . ) ) i s l o c a l l y o p t i m a l a n d t h e p r o o f i s c o m p l e t e .
IAEA-SMR-17/30 9 7
3 . C O N V E X I T Y . S E P A R A T I O N T H E O R E M S , H I G H E R - O R D E R
N E C E S S A R Y C O N D I T I O N S
3 . 1 . C o n v e x i t y a n d s e p a r a t i o n t h e o r e m s
L e t X b e a l i n e a r r e a l s p a c e a n d l e t Q Ç X b e a c o n v e x s e t . A n a l g e b r a i c h o m o g e n e o u s h y p e r p l a n e i s a s u b s p a c e X j C X s u c h t h a t t h e r e e x i s t s t h e s u b s p a c e X 2 C X o f d i m e n s i o n 1 w i t h t h e p r o p e r t y t h a t e v e r y x £ X c a n b e
w r i t t e n a s x = x x + x 2 w h e r e х г £ X - , , x 2 £ X 2 .A n a l g e b r a i c h y p e r p l a n e i s a s e t o f t h e f o r m x 0 + X j w h e r e x 0 £ X i s f i x e d
a n d X j C X i s a n a l g e b r a i c h o m o g e n e o u s h y p e r p l a n e .
P r o p o s i t i o n 1
N e c e s s a r y a n d s u f f i c i e n t f o r X ' j £ X t o b e a n a l g e b r a i c h y p e r p l a n e i s t h a t t h e r e e x i s t a r e a l n u m b e r a a n d a l i n e a r n o n - v a n i s h i n g f u n c t i o n a l o n X s u c h t h a t X ' j = { x € X : f ( x ) = a } .
P r o o f
B y h y p o t h e s i s X ' j = x 0 + X 1 , w h e r e X j i s a n a l g e b r a i c h o m o g e n e o u s h y p e r p l a n e . W r i t e X 2 £ X a s X 2 = { x £ X : x = X x 2 , A s R } w h e r e x 2 £ X i s f i x e d X g g X r D e f i n e a n o n - v a n i s h i n g l i n e a r f u n c t i o n a l b y î(x^) = 0 , f o r a l l x 1 £ X 1 , a n d f ( x 2 ) = X f o r a l l x 2 £ X 2 .
B y d e f i n i t i o n X = X 1 + X 2 , a n d t h e r e f o r e f : X - * R . L e t f ( x Q ) = a . I t f o l l o w s t h a t f ( x ) = a f o r a l l x £ X J a n d f ( x ) = £ a f o r a l l x í X ¡ ; t h u s n e c e s s i t y i s p r o v e d . N o w , l e t f b e a n o n - v a n i s h i n g l i n e a r f u n c t i o n a l o n X .
I t f o l l o w s t h a t f ( X ) = R a n d , f o r a n a r b i t r a r y a £ R , t h e r e e x i s t x Q £ X s u c h t h a t f ( x 0 ) = a .
T h e s e t X j = { x G X : f ( x ) = 0 } i s a n a l g e b r a i c h o m o g e n e o u s h y p e r p l a n e b e c a u s e X = X 1 + X 2 , w h e r e X 2 = { x £ X : x = X x , X e R } a n d f ( x ) = 1 . T h e r e f o r e t h e s e t { x £ X : f ( x ) = a } i s t h e s a m e w i t h X q + X j a n d t h e p r o o f i s c o m p l e t e .
L e t X b e a l i n e a r n o r m e d s p a c e a n d H a c l o s e d h y p e r p l a n e i n X o f
e q u a t i o n f ( x ) = a , w h e r e f i s a l i n e a r c o n t i n u o u s f u n c t i o n a l o n X .T h e s e t s X j = { x £ X : f ( x ) á a } , X 2 = { x £ X , f ( x ) > a } a r e a p a r t i t i o n o f
X , X = X j U X 2 , X j n x 2 = ф.T h e s e t s o f t y p e X j a r e c a l l e d c l o s e d h a l f - s p a c e s a n d t h e s e t s o f t y p e X 2
a r e c a l l e d o p e n h a l f - s p a c e s . I f t h e s e t s А , В £ X h a v e t h e p r o p e r t y A C Х г , B Ç X 2 f o r a g i v e n c l o s e d h y p e r p l a n e H , t h e n w e s a y t h a t A a n d В a r e s e p a r a t e d b y H .
W e s t a t e w i t h o u t p r o o f t w o p r o p e r t i e s f o r a c o n v e x s e t Q i n a l i n e a r n o r m e d s p a c e .
P r o p e r t y a
I f t h e s e t Q i s c o n v e x t h e n t h e c l o s u r e Q i s a l s o c o n v e x .
P r o p e r t y b
L e t Q b e a c o n v e x s e t s u c h t h a t i n t Q i s n o n - e m p t y .
98 VÂRSAN
I f x € Q a n d y € Q t h e n t x + ( l - t ) y e i n t Q f o r t e [ 0 , 1 ] .D e n o t e b y c o n v M t h e s m a l l e s t c o n v e x s e t w h i c h c o n t a i n s M .
P r o p o s i t i o n 2
L e t Q j , Q 2 b e t w o d i s j o i n t c o n v e x s e t s i n a l i n e a r s p a c e . T h e n f o r e v e r y x e X a t l e a s t o n e o f t h e s e t s
d e f d e fA j = Q 2 П c o n v ( { x } U Q j ) a n d A 2 = Q x O c o n v ( { x } U Q 2 )
i s e m p t y .
P r o o f
S u p p o s e t h a t t h e r e e x i s t y x € A j a n d y 2 6 A 2 . B y d e f i n i t i o n y j e Q 2 , Y 2 € Q 1 (
a n d t h e r e e x i s t Z j G Q j a n d z 2 € Q 2 s u c h t h a t y j G [ x , z j ] , y 2 e [ x , z 2 ] , w h e r e
iyx - У 2 1 = { t y i + ( 1 - t ) y 2 : te [ 0 , 1 ] }
H e n c e f o l l o w s [ у г , z 2 ] Ç Q 2 , [ y 2 , z j С Q j . W e p r o v e t h a t [ у х , z 2 ] a n d [ y 2 , Z j ] h a v e a n o n - e m p t y i n t e r s e c t i o n .
W e c o n s t r u c t t h e t r i a n g l e { x , z 1 ; z 2 } a n d r e m a r k t h a t t h e s e g m e n t s [ y j , z 2 ]
a n d [ y 2 , z - J m u s t i n t e r s e c t a t a p o i n t q w h i c h b e l o n g s t o Q j П Q 2 c o n t r a c t i n g Q j Г1 Q 2 = ф.
P r o p o s i t i o n 3
L e t Q j , Q 2 b e t w o d i s j o i n t c o n v e x s e t s i n a l i n e a r s p a c e X . T h e n t h e r e e x i s t t w o c o n v e x s e t s X j , X 2 £ X w i t h p r o p e r t i e s Q j Q X j , Q 2 £ X 2 a n d
X = X j U X 2 , X ^ X j = </>.
P r o o f
L e t J ^ b e t h e f a m i l y o f a l l p a i r s o f c o n v e x s e t s ( X j , X 2 ) w i t h t h e p r o p e r t y
X ¡ 2 Q . , i = 1 , 2 , Х а П Х 2 = ф .
Ж i s n o n - e m p t y b e c a u s eI n ^ w e d e f i n e a n o r d e r b y ( X “ , x“)< ( X ® , X ® ) i f x “ С X ® a n d x “ c x 2 .L e t Щ £ ¡Pi Щ = { ( X j , X 2 ) } a e j , b e a t o t a l l y o r d e r e d s e t .T h e p a i r ( U X , U X . ) w i l l b e a m a j o r a n t f o r & , и X ? , i = 1 , 2 , a r e
a e j 1 a e j ¿ 1 a e j
c o n v e x s e t s a n d ( U X “ ) П ( и X ? ) = ф .a e j a e j
T h e r e f o r e ( U X ? , U X ? ) G & a n d , u s i n g Z o r n ' s t h e o r e m , w e s e e t h a t a e j 1 a<=j ¿
t h e r e e x i s t s a m a x i m a l e l e m e n t i n Ж T h e m a x i m a l e l e m e n t ( X j , X 2 ) a l s o p o s s e s s e s t h e p r o p e r t y X j U X 2 = X . I n d e e d , s u p p o s e x e X \ ( X 1 U X 2 ) . T h e n u s i n g t h e p r o p o s i t i o n 2 i t f o l l o w s t h a t o n e o f t h e s e t s X j П c o n v ( { x } U X 2 ) , X 2 П c o n v ( { x } U X x ) i s e m p t y . L e t X 2 П c o n v ( { x } U X j ) = ф. D e n o t e X j = c o n v ( { x } и X j ) a n d t h e p a i r ( X j , X 2 ) w i l l b e i n ^ s u c h t h a t ( X 1 , X 2 ) < ( X ' j , X 2 ) a n d X j C X ' j c o n t r a d i c t i n g t h e p r o p e r t y o f ( X 1 , X 2 ) . T h e
p r o o f i s c o m p l e t e .
IAEA-SMR-17/30 9 9
T h e o r e m 1 . L e t Q , , b e t w o d i s j o i n t c o n v e x s e t s i n a l i n e a r n o r m e d s p a c e X s u c h t h a t i n t Q i =fc<¡>. T h e n t h e r e e x i s t s a c l o s e d h y p e r p l a n e w h i c h s e p a r a t e s i n t Q j a n d Q 2 ; m o r e p r e c i s e l y , t h e r e e x i s t s a n o n - v a n i s h i n g l i n e a r c o n t i n u o u s f u n c t i o n a l f a n d a r e a l c o n s t a n t a s u c h t h a t f f x j ) ë a ë f ( x 2 )
f o r a l l x , e Q j , x 2 £ Q 2 .
P r o o f
L e t X v X 2 b e t w o d i s j o i n t c o n v e x s e t s g i v e n b y p r o p o s i t i o n 3 . B y d e f i n i t i o n Q j Ç X j , Q 2 C X 2 . D e n o t e H = Xj П X 2. H i s a c l o s e d c o n v e x s e t b e c a u s e X j a n d X 2 a r e c l o s e d a n d c o n v e x a n d H i s n o n - e m p t y b e c a u s e i t i s i m p o s s i b l e t o h a v e X = A U B А П В = ф w i t h A a n d В c l o s e d s e t s .
W e h a v e H О i n t X j = ф a n d t h e r e f o r e H = £ X .W e p r o v e t h a t H = x 0 + F , F b e i n g a s u b s p a c e . I t i s s u f f i c i e n t t o s h o w
t h a t f o r e v e r y x , y € H t h e l i n e d e t e r m i n e d b y x , y b e l o n g s t o H . S i n c e H i s a c o n v e x s e t i t f o l l o w s t h a t [ x , y ] С H .
L e t z b e a p o i n t s i t u a t e d o n t h e r i g h t o f y a n d o n t h e l i n e d e t e r m i n e d b y x , y . I f z d o e s n o t b e l o n g t o H , t h e n i t b e l o n g s a t l e a s t t o o n e o f t h e s e t s i n t X j , i n t X ^ : s a y , z £ i n t X * . U s i n g p r o p e r t y b w e s e e t h a t ( x , z ] с i n t X j , a n d у £ i n t Х г c o n t r a d i c t i n g y £ H ( y £ i n t X j , у 6 H m e a n s t h a t t h e r e e x i s t s f e x , n X 2 ) . L e t x j £ H b e f i x e d . D e f i n e d F = H - x 1 a n d i t f o l l o w s t h a t F i s a l i n e a r c l o s e d s u b s p a c e ( b e c a u s e 0 £ F , i t i s s u f f i c i e n t t o n o t e t h a t t Z j + ( 1 - t ) z 2 £ F f o r a l l t G E i f z 1 , z 2 € F ) .
L e t x 0 e i n t X 2 ( i n t X 2 i s n o n - e m p t y s i n c e a l i n e d e t e r m i n e d b y X j . y ,X j £ i n t X j , y £ H , h a s p o i n t s i n X 2 w h i c h d o n o t b e l o n g t o X j ) .
L e t L = s p ( { X o } U H ) b e t h e l i n e a r s p a c e g e n e r a t e d b y x 0 a n d H . S u p p o s e L ^ = X . T h e n L Ф i n t X j ( a l i n e a r s u b s p a c e w h i c h h a s a n o n - e m p t y i n t e r i o r i s t h e e n t i r e s p a c e ) a n d t h e r e e x i s t s a n X j € i n t X j s u c h t h a t x t £ L .
T h e r e f o r e [ x 0 , X j ] h a s n o p o i n t i n c o m m o n w i t h H c o n t r a d i c t i n g X q € i n t X 2 , X j £ i n t X i ( [ x 0 , x j ] c a n n o t b e e n c l o s e d i n X ^ o r i n X 2 a n d m u s t h a v e p o i n t s i n Х г П X 2 ) . B y d e f i n i t i o n L = s p ( { x 0 } U F ) a n d t h e r e f o r e H i s a h y p e r p l a n e . D e f i n e a l i n e a r c o n t i n u o u s f u n c t i o n a l f : X - > R a s f o l l o w s : f ( x ) = 0 f o r a l l x G F a n d f ( A x o ) = X f o r a l l X £ R . S i n c e e v e r y x £ X h a s t h e f o r m x = A x 0 + x F i t f o l l o w s t h a t f i s a l i n e a r c o n t i n u o u s f u n c t i o n a l d e f i n e d o n X .
S i n c e H = x 1 + F , l e t a = f ( x 1 ) . I t f o l l o w s t h a t f ( x ) = a f o r e v e r y x £ H a n d
f ( x j ) S a , f ( x 2 ) 6 a V x j £ X 1(x 2 £ X 2
a n d t h e p r o o f i s c o m p l e t e .
3 . 2 . H i g h e r - o r d e r n e c e s s a r y c o n d i t i o n s
L e t X , Y a n d Z b e n o r m e d s p a c e s . A s s u m e t h a t i n Y a n d Z a r e g i v e n t h e c o n v e x c o n e s К С Y , С Z s u c h t h a t Г2 i s a n o p e n s e t .
I n t h e f o l l o w i n g w e c o n s i d e r t h e o p t i m u m p r o b l e m m i n f ( x ) w i t h t h e c o n s t r a i n t s ф ( x ) £ Г 2 , T ( x ) £ K , w h e r e f : X - > R , ф-.Х-’ Z, T : X - Y . L e t x 0 £ X b e l o c a l l y o p t i m a l . T h i s m e a n s t h a t x 0 i s o p t i m a l w i t h r e s p e c t t o a n e i g h b o u r h o o d S o f X o - T h e p r o p e r t y o f x 0 i s e q u i v a l e n t t o А П В =ф w h e r e t h e s e t s A C R X Z X Y a n d В С R X Z X Y a r e d e f i n e d b y В = R _ X f i x K , ( R _ = { t £ R : t < 0 } )
A = i a = (У0 » z > y ) : У0 = f ( x ) " f ( x 0 ) , z = ф ( х ) , у = T ( x ) , x 6 S }
100 VÂRSAN
I f f , ф a n d T a r e l i n e a r t h e n A a n d В a r e c o n v e x s e t s a n d t h e g e o m e t r i c p r o p e r t y А Л В =<j> c a n b e e x p r e s s e d i n a n a n a l i t i c a l f o r m u s i n g a s e p a r a t i o n t h e o r e m f o r c o n v e x s e t s . I f f , ф a n d T a r e n o t l i n e a r , t h e n t h e s e t A m a y b e n o t c o n v e x a n d w e c a n n o t d i r e c t l y a p p l y a s e p a r a t i o n t h e o r e m f o r A a n d B .
I n t h i s c a s e w e h a v e t o a p p r o x i m a t e A b y a c o n v e x s e t С s u c h t h a t С Л В i s e m p t y b e c a u s e А Л В =ф. I n f a c t , n e c e s s a r y c o n d i t i o n s f o r o p t i m a l i t y , o f a r b i t r a r y o r d e r , i n v o l v e a n i m p l i c i t - f u n c t i o n t h e o r e m , i n o r d e r t o p r o v e t h a t С Л В i s e m p t y , a n d a s e p a r a t i o n t h e o r e m f o r t w o c o n v e x s e t s i n l i n e a r n o r m e d s p a c e s .
S u p p o s e f , ф a n d T a r e c o n t i n u o u s l y F r é c h e t - d i f f e r e n t i a b l e o f o r d e r
p ( p § 1 ) . U s i n g T a y l o r ' s f o r m u l a w e s e e t h a t ( s e e p r o p o s i t i o n 2 i n s e c t i o n 1 ) .
A ( x q + h ) = A ( x 0 ) + d A ( x Q; h ) ^ y d 2 A ( x 0 ; h ) + . . . + ^ ¡ - d p A ( x 0 ; h ) + || h ||P w ( x 0 ; h )
(1)
w h e r e l i m u ( x ; h ) = 0 , A = ( f , ф, T ) .h-0 0
L e t H p ’ 1 £ X b e d e f i n e d f o r p i 2 a s f o l l o w s :
H p _ 1 = { h 6 X : d ^ f i x ^ h ) S 0 , d ^ ( x Q; h ) G ? 2 , d 1 T ( x Q; h ) G К V i = l , 2 , . . . , p - l }
(2)
a n d H ° = { 0 } ( f o r p = 1 ) .F o r e v e r y x G H p _ 1 l e t C X G R X Z X Y b e t h e c o n v e x s e t d e f i n e d b y
С = C x = j c = ( y 0 , z , y ) ; y 0 = d f ( x 0 ; x ) + ^ y d p f ( x 0; x )
z = ф(х0) + йф(х0; х ) + ^ y d рф ( x 0 ; x )
y = T ( x Q) + d T ( x 0 ; x ) + ^ j - d PT ( x 0 ; x ) , x G x j - ( 3 )
L e m m a 1 . L e t X , Y b e B a n a c h s p a c e s a n d l e t A : X ^ Y b e c o n t i n u o u s l yF r é c h e t - d i f f e r e n t i a b l e a t x n £ X . T h e n f o r e v e r y s : ( O . l j - ^ X s u c h t h a tl i m s ( e ) = 0 a n d e v e r y U £ X b o u n d e d e-^o ------------- ----------
A ( x Q + s ( e ) + e px ) = A ( x 0 + s ( e ) ) + e p d A ( x 0 ; x ) + e p u ( e , x )
w h e r e
II u ( e , x " ) - u ( e , x ' ) Il § a ( e ) || x " - x ' II , a (e) -*■ 0 , f o r x ' , x " G Uii e _ , 0
P r o o f
F o r e v e r y e G ( 0 , 1 ) , x ' , x " G U d e f i n e F e x , x „ ( t ) = A ( x 0 + s ( e ) + e p ( x ' + t ( x " - x ' ) ) .B y d e f i n i t i o n , F € , x „ ( t ) i s c o n t i n u o u s l y d i f f e r e n t i a b l e i n t a n d t h e r e f o r e
IAEA-SMR-17/30 1 0 1
1
Fe, x ' , x "
(4)о
w h e r e
( t ) = e p d A ( x Q + s ( e ) + e p ( x ' + t ( x " + x ' ) ) ; x " - x ' )
I t f o l l o w s
(5)
w h e r e
l
ш ( е , x " ) - ы ( е , x ' ) = / [ d A ( x Q + s ( e ) + e p ( x ' + t ( x l ! - x ' ) ; x " - x ' )
о
- d A ( x 0 ; x " - x ' ) ] d t
W e o b t a i n
|| u ( e , x " ) “ u ( e , x 1 ) Il S a ( e ) || x " - x ' || , w h e r e l i m a ( e ) = 0
a n d t h e p r o o f i s c o m p l e t e .
L e m m a 2 . L e t X g b e l o c a l l y o p t i m a l . S u p p o s e f , ф a n d T c o n t i n u o u s l y F r é c h e t - d i f f e r e n t i a b l e o f o r d e r p a t x n . L e t d T ( x n , . ) : X - > Y b e a s u r j e c t i o n . T h e n f o r e v e r y x £ H p ‘ i , С ^ П В = ф.
P r o o f
_ P“1 ЗГS u p p o s e t h a t t h e r e e x i s t x E H s u c h t h a t С П В i s n o n - e m p t y . B y
d e f i n i t i o n X j G X , y 0 < 0 , U j S f i a n d k j £ К w i l l e x i s t s u c h t h a t
S i n c e f , ф a n d T a r e c o n t i n u o u s l y F r é c h e t - d i f f e r e n t i a b l e , w e o b t a i n f r o m l e m m a 1 .
A ( x 0) + d A ( x , ) ; X j J + ^ ¡ - d p A ( x 0 ; x ) = ( y 0 , u v Ц ) = a 1 (6 )
w h e r e
A ( x ) = ( f ( x ) - f ( x 0) ) , ф ( x ) , T ( x ) )
A ( x 0 + e x + e p x ) = A ( x 0 + e x ) + e p d A ( x 0 ; x ) + e p u ( e , x ) (V)
w here
u ( e , x " ) - u ( e , x ' ) II S » ( e ) II x " - x ' || , l i m a(e) = 0
102 VA RSA N
Again using the property o f d i f fe re n t ia b i l i ty we obtain
p - l
A ( x 0 + e x ) = A ( x q ) + ^ e 1 à . + e p Q y d pA ( x Q ; x ) + R ( x Q ; e ^ ) ( 8 )
i=l
w h e r e
âi = (ÿ0i ' Ч > kj), ÿ0i s o, Ü¡e ñ, k¡e кU s i n g ( 6 ) a n d ( 7 ) w e o b t a i n
A ( x 0 + e x + e p x ) = A ( x Q ) + a ( e ) + c p ( a a - A ( x Q))
+ e p ( d A ( x 0 ; x - x 1 ) + Z(e,x)) ( 9 )
w h e r e
p- l
5 ( e ) = ^ e1 а . е ( - « о ] X S ) X K , а 1 е ( - » 0 ) X Г 2 Х К
i=l
D e n o t i n g L = d T ( x Q; . ) i t f o l l o w s t h a t f o r e v e r y e > 0 t h e h y p o t h e s e s o f t h e o r e m 3 ( s e e s e c t i o n 1 ) a r e s a t i s f i e d f o r t h e m a p G ( x ) = L x + й 2 ( е , х г + x ) ,
w h e r e и = ш 0 , u ^ , ш 2 ) .T h e r e f o r e , f o r e v e r y e s u f f i c i e n t l y s m a l l , t h e r e e x i s t s a n x e s u c h t h a t
| | x e || á 6 ( e ) a n d
d T f x g ; x e ) + Í J 2 ( e , X j + x e ) = 0 , l ú n 5 ( e ) = 0
I n t h i s w a y w e o b t a i n
T ( x Q + e x + e p ( x 1 + x £)) = T ( x Q ) ( l - e p ) + k ( e ) e K ( 1 0 )
f o r e > 0 a n d s u f f i c i e n t l y s m a l l .S i n c e y 0 < 0 , U j S Q w e o b t a i n f r o m ( 9 )
ф ( x 0 + e x + e f f x j + x ^ ) ) = ( 1 - е р ) ф ( х 0 ) + ü ( e ) + e p ( w ! +r)1(e))eQ ( 1 1 )
f ( x 0 + e x + e p ( x a + x € ) ) - f ( x 0 ) = ÿ 0 ( e ) + e p ( y 0 + r j 0 ( e ) ) < 0 ( 1 2 )
f o r e s u f f i c i e n t l y s m a l l , c o n t r a d i c t i n g t h e p r o p e r t y o f x 0 . T h e p r o o f i s c o m p l e t e .
F o r a n a r b i t r a r y c o n v e x c o n e L С Z l e t L d e n o t e t h e s e t o f a l l l i n e a r c o n t i n u o u s f u n c t i o n a l s f w h i c h s a t i s f y f ( í ) S 0 , J 6 L .
IAEA-SMR-17/30 103
T h e o r e m 1 . L e t x 0 b e l o c a l l y o p t i m a l . A s s u m e t h a t f , ф a n d T a r e c o n t i n u o u s l y F r e c h e t - d i f f e r e n t i a b l e o f o r d e r p a t _ x 0 , a n d d T j x o j . h X - Y i s a s u r j e c t i o n . T h e n f o r e v e r y x € H p ~ 1 t h e r e e x i s t ^ 8 0 , v x e Q , ц 5 е к s u c h t h a t
( a ) a x d f ( x Q; x ) + vx ( d < /> (x Q; x ) ) + цх ( d T ( x 0 ; x ) ) = 0 , f o r a l l x £ X ,
( b ) « * + Ц г ^ Ц f 0 , »*(ф(х0)) = 0 , M*(T(x0)) = 0
(c) o-xdpf(x0;x) + г х (с1Рф (xQ; x)) + ц x(dPT ( x Q; x)) ê 0
P r o o f
B y v i r t u e o f o u r h y p o t h e _ s i s t h e c o n d i t i o n s o f l e m m a 2 a r e s a t i s f i e d . T h e r e f o r e t h e c o n v e x s e t s C x a n d В a r e d i s j o i n t f o r e v e r y x G H 5 " 1 . S i n c e H p _ 1 i s a c o n e , C x a n d В a r e d i s j o i n t , w h e r e
C X = | ( y 0 , z , y ) : ( y 0 , z , y ) = A ( x 0) + d A ( x 0 ; x ) + ^ y d PA ( x 0 ; x ) , t > 0 , x s x j
T h e f o l l o w i n g , s t e p i s t o s h o w t h a t w e c a n a p p l y a s e p a r a t i o n t h e o r e m t o t h e c o n v e x s e t s C x a n d B . W e u s e t h e n o t a t i o n
T h e s e t L i s c o n v e x a n d C x О В =ф i s e q u i v a l e n t t o P x n ( R _ X f 2 ) =ф, w h e r e t h e c o n v e x s e t P x C R X Z i s d e f i n e d b y
P x = { ( y 0 , z ) : ( y 0 , z ) = A j f X j , ) + d A 1 ( x Q; x ) + d A ^ x ^ x ) , ( r , x ) e L , j
a n d
A ( x ) = ( A j ( x ) , T ( x ) )
F r o m i n t Í 2 фф i t f o l l o w s i n t ( R _ X Q ) фф a n d , u s i n g t h e s e p a r a t i o n t h e o r e m i n R X Z f o r P * a n d R . X Г2 w e s e e t h a t t h e r e e x i s t a n o n - v a n i s h i n g l i n e a r c o n t i n u o u s f u n c t i o n a l c p e ( R X Z ) * a n d a c o n s t a n t с w i t h t h e p r o p e r t y
ф ( р г ) S c s c p ( p 2 ) v P i e P X> p 2 e ( R _ X f 2 ) ( 1 3 )
S i n c e 0 6 P X П ( R . X Í 2 ) , i t f o l l o w s f r o m ( 1 3 ) t h a t с = 0 . B y d e f i n i t i o n a x £ R , í^ 6 Z * w i l l e x i s t s u c h t h a t ф = ( a x , vx ) a n d t h e r e f o r e ( 1 3 ) b e c o m e s
a x r 1 + P x. ( z 1 ) è 0 V i ^ . z ^ e P * ( 14 )
a x r 2 + v x ( z 2) S 0 V ( r 2 , z 2) S ( R _ X f 2 )
1 0 4 VÂRSAN
« X| + K I I ( 1 5
F o r z 2 = 0 , i t f o l l o w s f r o m ( 1 4 ) t h a t a x 2 _ 0 . F o r x = 0 a n d т = 0 i t f o l l o w s vx(ф ( x 0 ) ) a 0 a n d , c h o o s i n g r 2 = 0 , w e o b t a i n v x ( u ) s 0 f o r a l l a n d t h e r e f o r e
H e n c e
a ê 0 , V х (ф ( x 0 )) = 0 , Л Я ' , a x + || V х || ф 0
L e t F : R X X - * R b e d e f i n e d a s f o l l o w s :
( 1 6 )
D e n o t e b y Q C R X Y t h e c o n v e x s e t d e f i n e d b y
Q = - j ( r , y ) : r = F ( t , x ) + c , У = T ( x 0 ) + d T ( x 0 ; x ) + - y d p T ( x 0 ; x ) L * P •{'e > 0 , t > 0 , x E X = }
U s i n g ( 1 4 ) w e s e e t h a t Q П ( R _ X K ) i s e m p t y .W e s h a l l p r o v e t h a t Q h a s a n o n - e m p t y i n t e r i o r .L e t S ( 0 , 1 ) £ X b e t h e s p h e r e c e n t r e d i n t h e o r i g i n . S i n c e d T ( x 0 ; . ) : X - ,- Y
i s a s u r j e c t i o n , t h e s e t U = d T Í X j , ; S ( 0 , 1 ) ) h a s a n o n - e m p t y i n t e r i o r i n Y .S i n c e F ( 0 , x ) i s a l i n e a r c o n t i n u o u s f u n c t i o n a l t h e r e w i l l e x i s t a
c o n s t a n t k > 0 s u c h t h a t | F ( 0 , x ) | â к f o r a l l x E S ( 0 , l ) .T h e r e f o r e ( k , ° ° ) x u i s c o n t a i n e d i n Q a n d i n t Q i s n o n - e m p t y . U s i n g
a g a i n t h e s e p a r a t i o n t h e o r e m w e s e e t h a t t h e r e e x i s t a c o n s t a n t s a n d a y * s u c h t h a t
o ( F ( t , x ) + e ) + цх ( T ( x Q) + d T ( x Q; x ) + d PT ( x Q; x ) ) g 0 ( 1 7 )
f o r a l l e S 0 , t È 0 , x € X
a r + / ( k ) s 0 V r i O , k G K ( 1 8 )
a I + ||mx || ф 0 ( 1 9 )
I f к = 0 , i t f o l l o w s f r o m ( 1 9 ) t h a t s i 0 .I f a = 0 t h e n , w i t h ( 1 7 ) , i t f o l l o w s t h a t A<x = 0 w h i c h i s i n c o n t r a d i c t i o n
w i t h ( 1 9 ) . T h e r e f o r e a> 0 a n d d i v i d i n g ( 1 7 ) b y i t w e o b t a i n
F (т, x) + ц х (T(x0) + dT(xQ; x) + ^ - dT(x0; x)) ë 0 ( 2 0 )
f o r a l l t ê 0 , x G X .
IAEA-SMR-17/30 105
F r o m ( 1 8 ) i t f o l l o w s t h a t a n d f r o m ( 2 0 ) , f o r т = 0 , x = 0 , w e o b t a i nK 5 ( T ( x 0 l i O .
H e n c e
f / ( T ( x 0 )) = 0 , Й е к ' ( 2 1 )
F r o m ( 2 0 ) , f o r т = 0 , w e o b t a i n t h e c o n c l u s i o n ( a ) s t a t e d i n o u r t h e o r e m a n d f o r x = 0 c o n c l u s i o n ( c ) f o l l o w s .
C o n c l u s i o n ( b ) i s a c o n s e q u e n c e o f ( 1 6 ) a n d ( 2 1 ) .T h e p r o o f i s t h u s c o m p l e t e .C h o o s i n g x = 0 i n t h e o r e m 1 w e o b t a i n f i r s t - o r d e r n e c e s s a r y c o n d i t i o n s
f o r o p t i m a l i t y .C h o o s i n g p = 2 , 3 , . . w e o b t a i n f r o m t h e o r e m 1 n e c e s s a r y c o n d i t i o n s o f
s e c o n d o r d e r , t h i r d o r d e r a n d s o o n .
T h e c o n c l u s i o n s i n t h e o r e m 1 h a v e t h e f o l l o w i n g g e o m e t r i c a l m e a n i n g .
P r o p o s i t i o n 4
L e t x 0€ X b e f i x e d a n d a s s u m e t h a t f , ф, T a r e c o n t i n u o u s l y F r é c h e t - d i f f e r e n t i a b l e o f o r d e r p a t _ x 0 . S u p p o s e d T ( x Q; . ) : X - * Y i s a s u r j e c t i o n . T h e n t h e c o n d i t i o n s ( a ) , ( b ) a n d ( c ) o f t h e o r e m 2 a r e s a t i s f i e d i f a n d o n l y i f t h e s e t
D 5 = |
i s e m p t y f o r e v e r y x 6 H P_1, w h e r e A ( x ) = ( f ( x ) ) - f ( x 0 ) , ф ( x ) , T ( x ) ) .
P r o o f
S u p p o s e ( a ) , ( b ) a n d ( c ) a r e s a t i s f i e d a n d a s s u m e t h a t x 1 6 D x f o r a f i x e d X € H P _ 1 . F r o m ( a ) a n d ( c ) f o l l o w s
a ! r j + v V j ) + M ^ k j ) ê 0 (22)
w h e r e
( r j , U j , k j ) = A ( x 0 ) + d A ( x Q; X j ) + d PA ( x Q; x ) e R _ X f 2 X K
B u t a* r x s 0 , v * ^ ) s 0 , ju x ( k j ) S 0 a n d h e n c e f o l l o w s a* = 0 , v* ( u . , ) = 0 w h i c h i m p l i e s а * + Ц и ^ Ц = 0 , i n c o n t r a d i c t i o n w i t h c o n d i t i o n ( b ) .
A s t o n e c e s s i t y , n o t e t h a t t h e s e t D * i s e m p t y b e c a u s e A * O B i s a n e m p t y s e t ( s e e t h e p r o o f o f t h e o r e m 1 ) .
D e f i n i t i o n . L e t x n€ X b e f i x e d . W e s a y t h a t x 0 i s _ ( f , ф, T ) e x t r e m a l o f o r d e r p i f t h e s e t D x i s e m p t y f o r e v e r y x e H p ~ I .
T h e d e p e n d e n c e o f х 6 Н р - 1 o n t h e m u l t i p l i e r s ax, ii*,v* i s a s p e c i a l s i t u a t i o n w h i c h o c c u r s i n h i g h e r - o r d e r n e c e s s a r y c o n d i t i o n s . W e c a n g i v e r e s u l t s i n w h i c h t h i s d e p e n d e n c e w i l l d i s a p p e a r .
x ê X : A ( x 0) + d A ( x 0 ; x ) + — d A ( x 0 ; x ) £ R _ X Í 2 X K
T h e o r e m .2 . A s s u m e ф ( x ) = 0 a n d t h e h y p o t h e s e s o f t h e o r e m 1 a r e s a t i s f i e d . T h e n t h e r e e x i s t s а у £ K " , ju ( T ( x Q)) = 0 s u c h t h a t
(a) d f ( x Q; x ) + n ( d T ( x 0 ; x ) ) = 0 V x £ X
( 0 ) d p f ( x 0 ; x ) + / u ( d p T ( x 0 ; S ) ) g 0 V x E H 15' 1
T h e o r e m 2 i s a d i r e c t c o n s e q u e n c e o f t h e o r e m 1 .
R e m a r k . S u p p o s e t h e h y p o t h e s e s o f p r o p o s i t i o n 4 a r e s a t i s f i e d . I f x n i s a n o p t i m a l e l e m e n t t h e n i t i s a n ( f , < f > , T ) e x t r e m a l o f o r d e r p . T h e c o n v e r s e i s n o t t r u e .
T h e o r e m 3 . L e t x n b e ( f , ф, T ) e x t r e m a l o f o r d e r p s u c h t h a t t h e h y p o t h e s e s o f p r o p o s i t i o n 4 a r e s a t i s f i e d . S u p p o s e t h e r e e x i s t
ц0е к , v 0 e n ‘
Д 0 ( Т ( х 0 )) = 0 , ^ 0 ( Ф ( х 0 )) = 0
s u c h t h a t
d f ( x Q; x ) + v 0 ( d < i > ( x 0 ; x ) ) + n 0 ( d T ( x 0 ; x ) ) = 0 V x G X
a n d d T ( x 0 ; . ) : L - Y i s a s u r j e c t i o n , w h e r e
L = { x £ X : d f ( x Q; x ) + / J 0 ( d T ( x Q; x ) ) = 0 , d<£ ( x Q; x ) € Q }
T h e n ,
d p f ( x 0 , x ) + m 0 ( d PT ( x 0 ; x ) ) ê 0 f o r a l l x E H 13' 1
t h a t s a t i s f y
v0 ( d p</> ( x 0 ; x ) ) = 0 , d рф ( x 0 ; x ) e ñ
P r o o f
S u p p o s e t h e r e e x i s t
x £ H P \ vQ ( d P< | ) ( x 0 ; x ) ) = 0 , d Pÿ ( x 0 ; x ) e f 2
s u c h t h a t d p f ( x Q; x ) + f i 0 ( d p T ( x 0 ; x ) ) < 0 .F r o m t h e h y p o t h e s i s i t f o l l o w s
(a) d f ( x 0 ; x ) + d p f ( x 0 ; x ) + / ^ 0 ( d T ( x 0 ; x ) + d pT ( x 0 ; x ) ) < 0
106 VARS AN
IAEA-SMR-И/ЗО 1 0 7
f o r a l l x 6 L , a n d t h e r e e x i s t s a n X j S L s u c h t h a t
( / 3 ) d T ( x 0 ; X l ) + ^ - d p T ( x 0 ; x ) = 0
T h e r e f o r e
d f ( x j j ; X j ) + - ^ - d pf ( x 0 ; x ) < 0 P '
a n d
T ( x 0 ) + d T ( x 0 ; x 1 ) + ^ y c f T Í X ^ x ) = T ( x 0 ) € K
ф(х0) + d ф(х0; X j ) + ^ y d P< i i ( x 0 ; x ) € f 2
I t f o l l o w s t h a t x 0 i s n o t a n ( f , < f > , T ) e x t r e m a l o f o r d e r p a n d , u s i n g p r o p o s i t i o n 4 , w e o b t a i n a c o n t r a d i c t i o n .
T h e p r o o f i s t h u s c o m p l e t e .
4 . A P P L I C A T I O N S T O C O N T R O L T H E O R Y . P O N T R J A G I N ' S M A X I M U M P R I N C I P L E . H I G H E R - O R D E R N E C E S S A R Y C O N D I T I O N S
4 . 1 . G e n e r a l p r o b l e m o f o p t i m a l - c o n t r o l t h e o r y
A u s u a l c o n t r o l p r o b l e m i s t h a t i n w h i c h w e a r e l o o k i n g f o r t h e m i n i m u m
o f a c e r t a i n f u n c t i o n a l С ( t 0 , t p R n ) X ^ ( t Q , t j j R 1 ) - * - R o n t h e s e t o f a l l p a i r s ( x ( . ) , u ( . ) ) w h i c h s a t i s f y t h e c o n t r o l s y s t e m
a n d t h e f i n a l c o n d i t i o n s
c p j M t j ) ) â 0 , i 6 { - í , . . . , - l }
c P j M t j ) ) = 0 , i e { l , . . . , k }
I n t h e f o l l o w i n g w e s u p p o s e . _ ^ -( x ( . ) , u ( . ) ) = c p 0 ( x ( . ) ) . T h e s p a c e t j ; R r ) c o n t a i n s a l l t h e f u n c t i o n s u : [ t 0 , t 1 ] - ’- R r b e i n g p i e c e w i s e c o n t i n u o u s .
B e s i d e s w e m a y h a v e a l s o a n o t h e r r e s t r i c t i o n o f t h e t y p e u ( t ) 6 U ( t ) , t s t t Q . t j ] , w h e r e t h e c o n t r o l s e t U ( t ) Q R r i s a g i v e n s e t f o r e v e r y t e [ t g . t j ] .
E v e r y u ( . ) e ^ ( t Q, R r ) i s c a l l e d c o n t r o l a n d x ( . ) s a t i s f y i n g t h e c o n t r o ls y s t e m i s d e n o t e d a t r a j e c t o r y .
A c o n t r o l s a t i s f y i n g u ( t ) e u ( t ) , t e [ t 0 , t j ] i s c a l l e d a d m i s s i b l e c o n t r o l a n d a t r a j e c t o r y w h i c h c o r r e s p o n d s t o a n a d m i s s i b l e c o n t r o l w i l l b e c a l l e d a d m i s s i b l e t r a j e c t o r y . I n t h i s w a y a n o p t i m a l - c o n t r o l p r o b l e m i n v o l v e s t h e m i n i m i z a t i o n o f a f u n c t i o n a l o n t h e s e t o f a l l a d m i s s i b l e p a i r s ( x ( . ) , u ( . ) ) .
T h e s o - c a l l e d o p t i m a l p a i r s r e p r e s e n t a s o l u t i o n t o t h e o p t i m a l - c o n t r o l p r o b l e m .
1 0 8 VÂRSAN
A n o p t i m a l - c o n t r o l p r o b l e m m a y c o n t a i n a l s o r e s t r i c t i o n s o n t h e t r a j e c t o r y f o r t b e l o n g i n g t o a n i n t e r v a l I C [ t 0 , t j ] ( p h a s e c o n s t r a i n t s ) o r r e s t r i c t i o n s o n b o t h t r a j e c t o r y a n d c o n t r o l ( m i x e d c o n s t r a i n t s ) .
E v e r y v a r i a t i o n a l p r o b l e m f r o m t h e c l a s s i c a l c a l c u l u s o f v a r i a t i o n s m a y b e f o r m u l a t e d a s a c o n t r o l p r o b l e m i n w h i c h U ( t ) = R r f o r a l l t S [ t Q , t j ] .
F o r e x a m p l e a L a g r a n g e p r o b l e m w i t h i s o p e r i m e t r i c c o n d i t i o n s i s a c o n t r o l p r o b l e m o f t h e t y p e d e s c r i b e d a b o v e .
I n d e e d , i n a L a g r a n g e p r o b l e m w e a r e l o o k i n g f o r
F 0 ( t , y ( t ) , y ( t ) ) d t
o n t h e s e t o f a l l p i e c e w i s e s m o o t h y : [ t 0 , t j ] - > R n w h i c h s a t i s f i e s
ti
У(t0) =y0 (t,y(t),y(t))dt = C j , ie{l,....k}to
w h e r e
a n d c ; € R , y 0 G R n a r e g i v e n .D e n o t e y = u a n d d e f i n e Ç j : [ t 0 , t 1 ] - > R , i e { l , . . . , k } b y
( t ) = J f j ( s , y ( s ) , y ( s ) ) d s , t e [ t 0 , t j ]
toD e n o t e
h
X = ( y , i ) , J ^ ( x ( . ) ) , u ( . ) ) =J F0 ( t , y ( t ) , u ( t ) ) d t
to
f ( t , x , u ) = ( u , F ( t , y , u ) ) , w h e r e f = ( f r . . . , ? k ) ,
F ( t , y , u ) = ( F 1 ( t , y , u ) , . . . , F k ( t , y , u ) )
T h e L a g r a n g e p r o b l e m i s t h e f o l l o w i n g : m i n J ( x ( . ) , u ( . ) ) o n t h e s e t o f a l l p a i r ( x ( . ) , u i . J e C f t j , t x : ( t 0 , R n ) w h i c h s a t i s f y t h e c o n t r o l s y s t e m
t
x ( t ) = ( y 0 , 0 ) +J f ( s , x ( s ) , u ( s ) ) d s , t G [ t 0 , t ]
to
a n d t h e f i n a l c o n d i t i o n s
| . ( t a ) = C j , i e { l , . . . , k }
W e h a v e s e e n t h a t n e c e s s a r y o r s u f f i c i e n t c o n d i t i o n s f o r a n o p t i m u m p r o b l e m u s e t h e d i f f e r e n t i a l s o f t h e m a p p i n g s w h i c h d e t e r m i n e t h e p r o b l e m .
IAEA-SMR-17/30 1 0 9
I n t h e v a r i a t i o n a l p r o b l e m s s u c h d i f f e r e n t i a l s c a n b e c a l c u l a t e d a n d u s e d i n o r d e r t o c h a r a c t e r i z e t h e o p t i m a l e l e m e n t b e c a u s e o f t h e s p e c i a l s i t u a t i o n u(t) = R r f o r te[t0,tx].
W h e n U ( t ) i s a n a r b i t r a r y s e t , w e c a n n o t c h a r a c t e r i z e t h e o p t i m a l e l e m e n t b y m e a n s o f d i f f e r e n t i a l s i n t h e o r i g i n a l p r o b l e m . I n t h i s c a s e i t i s n e c e s s a r y t o t r a n s f o r m t h e o r i g i n a l p r o b l e m i n t o a n o t h e r o n e i n w h i c h t h e o p t i m a l i t y p r o p e r t y c a n b e c h a r a c t e r i z e d b y d i f f e r e n t i a l s .
S u p p o s e t h a t U ( t ) i s a n a r b i t r a r y s e t . L e t u j ( . ) , . . . , u ( . ) b e a d m i s s i b l ec o n t r o l s . S u p p o s e t h a t ( x 0 ( . ) , u 0 ( . ) ) i s a n o p t i m a l p a i r .
D e f i n e a n e w c o n t r o l s y s t e m a s f o l l o w s :
f ¡ ( s , y ) = f ( s , y , U j ( s ) ) , i G { 0 , l , . . . , m }
^ ( j e ^ t g . t ^ R ) , j G { l , . . . , m }
T h e c o n t r o l w i l l b e a(.) = ( o ^ a m ( . ) ) .T h e c o n t r o l s e t s a r e U ( t ) = U = { u E R m : u ¡ ë 0 , i = l , . . . , m } f o r a l l t S [ t 0 , t j ] .T h e f i n a l r e s t r i c t i o n s a r e t h e s a m e a s i n t h e o r i g i n a l p r o b l e m :
P j i y i t j ) ) - 0 , i G { - i , . . . , - 1 } , Ф j ( У ( t x)) = 0 , i E { l , . . . , k }
A s t h e f u n c t i o n a l w e t a k e cp0 .I t i s e a s y t o s e e t h a t ( x 0 ( . ) , 0 ) i s a n a d m i s s i b l e p a i r f o r t h i s c o n t r o l
p r o b l e m .I n g e n e r a l w e d o n o t k n o w i f ( x o ( . ) , 0 ) i s o p t i m a l , b u t u n d e r c e r t a i n
c o n d i t i o n s o n e c a n p r o v e t h a t i t i s e x t r e m a l . T h i s n e w p r o b l e m c a n b e s t a t e da s a n a b s t r a c t o n e o f t h e f o r m s t u d i e d b e f o r e . I n d e e d , d e n o t e
m
y ( t ) = x 0 +
w here
X = A C ( t 0 , t j ; Rn) x m t 0, t j ; R m )
Y = R k X A C ( t 0 , t 1 ; R n ) , Z = R t X ^ ( t 0 , t j ; R m )
T h e c o n v e x c o n e s Í 2 C Z , K C Y a r e d e f i n e d a s f o l l o w s :
П = í(t1. . . tü,a1( . ) , . . ,a m(. )) : ti <0 , o j ( . ) > 0 , i = l , . . . , j f , j = l , . . , m } , К = { 0 }
D e f i n e
f ( x ) = Ф 0 ( у ( . ) ) , ф ( х ) = ( ç . 4 ( y ( t 1 ) ) , . . . , c ? . 1 ( y ( t 1 ) ) , o j t . ) ,
T ( x ) = (9 1 (y(t1 ) , . . . , p k (y(t1 )), T j ( y (.), o(.))
110 VARSAN
w h e r e
i m
т х ( y ( . ) , a ( . ) ) ( t ) = y ( t ) - x 0 -J f 0 ( s , y ( s ) ) ( s ) ( f j ( s , y ( s ) ) - f 0 ( s , y ( s ) )
t 0 j = l
ds
T h e p r o p e r t y o f a d m i s s i b i l i t y i n t h e s e n o t a t i o n s m e a n s ф ( х ) е Г 2 , T ( x ) € K .I f i s a c o n v e x s e t , f o r e v e r y t e [ t 0 , t j ] t h e r e e x i s t s a n a d d i t i o n a l
o t h e r t y p e o f t r a n s f o r m a t i o n o f t h e o r i g i n a l p r o b l e m . T h e c o n t r o l s y s t e m
w i l l b et m
y(t) = x 0 + J f ( s , y ( s ) , и0(в) + У ^ (s) (u3(s) — U0(s}) )d s , t e [ t Q, tj]to i = 1
A l l o t h e r c o n d i t i o n s r e m a i n u n c h a n g e d .A n d a g a i n u n d e r c e r t a i n c o n d i t i o n s o n e c a n p r o v e t h a t , i f x 0 ( . ) , u 0 ( . ) ) i s
o p t i m a l f o r t h e o r i g i n a l p r o b l e m , t h e n ( x 0 ( . ) , 0 ) i s a n ( f , < i > , T ) e x t r e m a l o f o r d e r p w h e r e f , ф a n d T a r e d e f i n e d i n t h e s a m e w a y a s b e f o r e .
T h e ( f , ф, T ) - e x t r e m a l i t y f o r ( x 0 ( . ) , 0 ) e n a b l e s u s t o o b t a i n h i g h e r - o r d e r n e c e s s a r y c o n d i t i o n s i n t r a n s f o r m e d p r o b l e m s w h i c h b e c o m e h i g h e r - o r d e r n e c e s s a r y c o n d i t i o n s f o r t h e o r i g i n a l p r o b l e m . T h e s e c o n n e c t i o n s w i l l b e d i s c u s s e d i n d e t a i l w h e n w e c o n s i d e r n e c e s s a r y c o n d i t i o n s f o r c o n t r o l
p r o b l e m s .
4 . 2 . P o n t r j a g i n ' s m a x i m u m p r i n c i p l e
W e g i v e n e c e s s a r y c o n d i t i o n s f o r t h e f o l l o w i n g o p t i m a l - c o n t r o l p r o b l e m :
m i n cp0 ( x ( t j ) ) V ( x ( . ) , u ( . ) ) e A C Í t ^ t ^ R 11) X < * / ( t 0 , t ^ R r )
w h i c h s a t i s f y
t
x ( t ) = x + ^ Ц в , x ( s ) , u ( s ) ) d s , t e [ t Q, t j ] , x Q € R n f i x e d ( 1 )
to
<p¡ ( x ( t 1 ) = 0 , i G { l , . . . , k } ( 2 )
u ( t ) e u , t e [ t Q, t 1 l ( 3 )
w h e r e U С R r i s a g i v e n s e t n o t n e c e s s a r i l y o p e n , a n d [ t g , t x ] i s a f i x e d
i n t e r v a l .T h i s c o n t r o l p r o b l e m i s a p a r t i c u l a r c a s e o f t h e p r o b l e m c o n s i d e r e d
i n s e c t i o n 4 . 1 b e c a u s e h e r e t h e c o n s t r a i n t s 9 1 ( x ( t t )) s 0 a r e o m i t t e d a n d t h e
s e t s U ( t ) d o n o t d e p e n d o n t .T h e f u n c t i o n s cp. : R n - » R , i e { 0 , 1 , . . . , k } a r e s u p p o s e d t o b e o f c l a s s C 1 a n d
f ( t , x , u ) : [ t 0 , t 1 ] X R n X R r - R n i s s u p p o s e d t o b e a c o n t i n u o u s f u n c t i o n w h i c h p o s s e s s e s c o n t i n u o u s f i r s t d e r i v a t i v e s w i t h r e s p e c t t o ( x , u ) 6 R n X R r .
I n o r d e r t o o b t a i n n e c e s s a r y c o n d i t i o n s w e m u s t c o n s i d e r t h e s t r u c t u r e o f t h e s e t U . I f U i s a n o p e n s e t , t h e n f i r s t - o r d e r n e c e s s a r y c o n d i t i o n s c a n
IAEA-SMR-17/30 111
b e g i v e n i n a l o c a l f o r m a n d t h e y a r e k n o w n f r o m t h e c l a s s i c a l c a l c u l u s o f v a r i a t i o n s . I n t h i s c a s e f i r s t - o r d e r n e c e s s a r y c o n d i t i o n s a r e t h e f o l l o w i n g :
L e t ( x 0 ( . ) , u Q ( . ) ) b e a n o p t i m a l p a i r . T h e n t h e r e e x i s t s a n
к
a. G R , ^ | л . | = f 0
i = 0
s u c h t h a t
( t , x ( t ) , u ( t ) ) s 0 ( 4 )9 u 0 0
к
*(V S ’ (xo(ti» (5)i = 0
- f w
w h e r e
H ( t , x , u ) = < i / / ( t ) , f ( t , x , u ) /
C o n d i t i o n s ( 4 ) a n d ( 6 ) a r e e q u i v a l e n t t o t h e " E u l e r e q u a t i o n " a n d ( 5 ) i s t h e s o - c a l l e d t r a n s v e r s a l i t y c o n d i t i o n .
I f = 0 , t h e n f i r s t - o r d e r n e c e s s a r y c o n d i t i o n s d o n o t c o n t a i n t h e f u n c t i o n a l t o b e m i n i m i z e d . T h i s s i t u a t i o n c a n o c c u r w h e n t h e v e c t o r s
^ - ( x ^ ) ) , i e { l , . . . , k }
a r e l i n e a r l y d e p e n d e n t o r , m o r e g e n e r a l , w h e n t h e l i n e a r s u b s p a c e
L = - j f i j , . . . , i k ) : J?i = < ^ - ( х 0 ( ^ ) ) , x ( t a ) > , x ( . ) е м |
s a t i s f i e s L ^ R 1* , w h e r e M С A C ( t 0 , t x ; R n ) i s d e f i n e d b y
M = - j x ( . ) : t h e r e e x i s t u ( . ) € ^ ( t 0, t x ; R r ) s u c h t h a t
c ï f = I x ( W t W i » * + f ^ ( t , x 0 ( t ) , u 0 ( t ) ) u ( t ) , x ( 0 ) = o }
A c t u a l l y , i f ЪфВ.к t h e f i r s t - o r d e r n e c e s s a r y c o n d i t i o n s a r e o b t a i n e d
v e r y e a s i l y a n d t h e y e x p r e s s o n l y t h e p r o p e r t y i n t L =<f> w h i c h c a n b e m e t w i t h a n y o t h e r f u n c t i o n a l cpk .
F o r n e c e s s a r y c o n d i t i o n s t o b e s a t i s f a c t o r y w e n e e d a0 = 1 a n d t h i s c a n b e o b t a i n e d i f w e r e q u i r e L = R k .
112 VÂRSAN
I n d e e d , i f L = R k , t h e n t h e c o n c l u s i o n ( 4 ) w h i c h c a n b e w r i t t e n i n t h e f o r m
_k i
) _ Л ( x 0 ( M ' X ( V > = ° V x ( . ) 6 M ( 7 )
i=0
s h o w s t h a t a 0 c a n n o t b e z e r o b e c a u s e i n t h i s c a s e f r o m L = R k i t f o l l o w s t h a t û j = 0 , i € { l , . . . , k } , c o n t r a d i c t i n g t h e p r o p e r t y
к
Z l“ili =0
I n t h e f o l l o w i n g i t i s p r o v e d t h a t t h e f i r s t - o r d e r n e c e s s a r y c o n d i t i o n s c o n t a i n e d i n ( 4 ) , ( 5 ) a n d ( 6 ) , w i t h a0 = 1 , c a n b e o b t a i n e d f r o m t h e o r e m 2 f o r x = 0 ( o r t h e o r e m 1 ) .
S u p p o s e L = R k . T h e n , t h e r e e x i s t ( и : ( . ) , . . . , u k ( . ) ) £ cê/[tv t p R r ) a n d X j ( . ) , . . . , X | (( . ) e M s u c h t h a t
( x o f l ^ x ^ t j ) , . . . . I J ( x 0 ( t j ) х 1< ( % ) ( 8 )
a r e l i n e a r l y i n d e p e n d e n t w h e r e ф = ( q ^ , . . . , Ф к ) . L e t u k + 1 ( . ) 6 ^ ( t 0 , t j ; R r ) b e a r b i t r a r i l y c h o s e n .
D e n o t e
X = A C ( t 0 , t 1 ; R n ) X ^ ( t 0 , t 1 ; R k + 1 ) , Y = R k X A C ( t 0 , t j j R n )
L e t f : X - > R , T : K - * Y b e d e f i n e d a s f o l l o w s :
f(x) = ф0(х(^) ) , T ( x ) = (ф(x(tj )), T j (x ))
w h e r e
t k + 1
T j (x ) ( t ) = x ( t ) - x 0 - J f ( s , x ( s ) , u Q( s ) + У a. ( s ) u . ( s ) ) d s
to i=l
F o r X t o b e a B a n a c h s p a c e b y ty', R k + 1 ) w e u n d e r s t a n d t h e s p a c e o fa l l a(.) = ( a 1 ( . ) , . . , a k + 1 ( . ) ) s u c h t h a t Q j ( . ) i s p i e c e w i s e c o n t i n u o u s w i t h d i s c o n t i n u i t i e s i n t h e s e t o f a l l d i s c o n t i n u i t y p o i n t s o f t h e f u n c t i o n s U j ( . ) , . . . , u k + 1 ( . ) . O n w e c h o o s e t h e n o r m g i v e n b y u n i f o r m c o n v e r g e n c e .
B y d e f i n i t i o n x 0 = ( x Q( . ) , 0 ) i s l o c a l l y o p t i m a l f o r t h e p r o b l e m m i n f ( x ) w i t h t h e c o n s t r a i n t T ( x ) = 0 .
C o n d i t i o n ( 8 ) i m p l i e s t h a t d T ( x 0 ; . ) : X - * Y i s a s u r j e c t i o n . T h e n , u s i n g t h e o r e m 2 ( s e e s e c t i o n 3 ) f o r x = 0 , w e s e e t h a t t h e r e e x i s t s а ц G Y *
s u c h t h a t
df(xQ; x) + M(dT(xQ;x)) = 0 V x G X (9)
IAEA-SMR-17/30 И З
B y d e f i n i t i o n fi h a s t h e f o r m ¡u = ( 0 ^ , . . . , » к , / u 0 ) w h e r e a . G R , i k }
a n d / u 0 e ( A C ( t o , t ¡ ; R n ) ) * . B y d e f i n i t i o n ( 9 ) b e c o m e s
к
i = l
f o r a l l x ( . ) £ A C ( t 0 , t j j R n ) w h i c h s a t i s f y x ( 0 ) = 0 ,
k + i
Ж = ё ( t ' X o( t ) ’ u 0 ( t ) ) x + Z 5 i ( t ) Ï Ï Ü ( t ’ x o ( t ) u o ( t ) ) U i ( t ) i = l
A t a f i r s t s i g h t t h e m u l t i p l i e r s ax, . . . , a k i n ( 1 0 ) s e e m t o d e p e n d o n t h e c h o i c e o f u k + 1 ( . ) E ^ ( t 0 , t j j R r ) . I t i s n o t s o b e c a u s e , i f w e c h a n g e u k + 1 ( . ) , c o n d i t i o n ( 8 ) p e r m i t s u s t o v e r i f y t h a t O j , . . . , a k i n t h e e q u a t i o n ( 1 0 ) w i l l b e t h e s a m e a s b e f o r e .
T h e r e f o r e , c o n c l u s i o n ( 1 0 ) m a y b e f o r m u l a t e d a s f o l l o w s : t h e r e e x i s t o ' 1 , . . . , Q ' k 6 R s u c h t h a t
к
< ’3 ' “ ( x Q Í t j J b x f t j ) ) < ^ L ( x 0 ( t 1 ) ) , x ( t ] ) ) > = О ( 1 1 )
i = l
f o r a l l
( x ( . ) , ï ï ( . ) ) € A C ( t 0 , t 1 ; R n ) X % t ^ R 1 )
w h i c h s a t i s f y
x ( ° ) = 0 , ^ ( f , x „ ( t ) , u ( t ) ) * + f ^ ( t , x „ ( t ) , u 0 ( t ) ) u ( t )
D e f i n e ф: [ t QJ t t ] - ♦ R n b y
к
= S ' ( x o ( t i ) ) + X ° i ë ' ( x o ( t i , ) " ж = ФМ x 0 { 1: ) , u 0 ( t » i = l
s o w e o b t a i n f r o m ( 1 1 )
— ( t , x 0 ( t ) , u Q( t ) ) = 0 , ( 1 2 )
w h e r e H ( t , x , u) = < ( i / / ( t ) , f ( t , x , u ) X C o n c l u s i o n s ( 4 ) , ( 5 ) a n d ( 6 ) a r e t h u s o b t a i n e d .
I f U i s n o t a n o p e n s e t b u t a c o n v e x o n e , t h e n f i r s t - o r d e r n e c e s s a r y c o n d i t i o n s t a k e t h e f o l l o w i n g f o r m : t h e r e e x i s t « ¡ G R , i £ { 0 , 1 , . . . , k } , ê 0 s u c h t h a t
f) T-T^ 3 u ( t ’ X o ( t ) ’ U 0 W , U ' V * » - 0 f o r a U - u e U a n d t £ [ t 0 , tj ] ( 1 3 )
1 1 4 VARSAN
w h e r e H ( t , x , u ) = <C< // ( t ) , f ( t , x , u ) / , a n d ф s a t i s f i e s
к
^ ( V =b i ^ (x0(tiL ' ^ = ,/,S (t’ xo(t)' uo(t)) (14)i=0
J u s t a s i n t h e c a s e o f o p e n U , i f a0 = 0 , t h e n ( 1 3 ) a n d ( 1 4 ) d o n o t c o n t a i n t h e
f u n c t i o n a l cpQ.T h i s s i t u a t i o n m a y o c c u r w h e n t h e c o n v e x s e t L j C R k g i v e n b y
Ц = ^ ( C 1 , . . , c k ) : c i = < ^ L ( x 0 ( t 1 ) ) , x ( t j ) > , x ( . ) е м |
h a s a n e m p t y i n t e r i o r , w h e r e t h e c o n v e x s e t M j £ A C ( t 0 , R n ) i s d e f i n e d
a s f o l l o w s :
M j = | х ( . ) : t h e r e e x i s t s u ( . ) e < ? / ( t 0 , t - ^ U ) a n d
x ( ° ) = 0 , ^ ( t , x 0 ( t ) , u 0 ( t ) ) x ( t ) + | ^ ( t , x 0 ( t ) , u ^ t ) ) ( u ( t ) - « „ ( t » }
I f w e r e q u i r e i n t L j Э 0 t h e n t h e c o n c l u s i o n s ( 1 3 ) , ( 1 4 ) c a n b e w r i t t e n w i t h
£*0 - 1.I n t h e f o l l o w i n g , s u p p o s i n g 0 £ i n t L j a n d u s i n g t h e o r e m 1 w e o b t a i n t h e
c o n c l u s i o n s ( 1 3 ) a n d ( 1 4 ) .I f f ( t , x , . ) : R r - * R n i s a n a f f i n e m a p p i n g c o n c l u s i o n ( 1 3 ) t a k e s t h e f o r m
H ( t , x Q( t ) , u Q( t ) ) = m i n H ( t , x 0 ( t ) , u ) f o r e v e r y t e [ t 0 , t j ] ( 1 5 )u e u
w h i c h i s t h e w e l l - k n o w n P o n t r j a g i n m a x i m u m p r i n c i p l e . W h e n U i s a n a r b i t r a r y s e t t h e n P o n t r j a g i n ' s m a x i m u m p r i n c i p l e r e a d s a s f o l l o w s :
t h e r e e x i s t
к
o . e R , i € { O , l , . . . , k } , a 0 î O , ^ I ûf j I ф 0i=0
s u c h t h a t
H ( t , x 0 ( t ) , u „ ( t ) ) = m i n H ( t , x 0 ( t ) , u ) f o r e v e r y t e [ t 0 , t a ] ( 1 6 )n e U
w here ф: [tQ, t j l ^ R 11 sa t is f ie s
к
i = 0
(17)
IAEA-SMR-17/30 115
H e r e , a s b e f o r e , w h e n a 0 = 0 , P o n t r j a g i n ' s p r i n c i p l e d o e s n o t c o n t a i n t h e f u n c t i o n a l cpQ. I n o r d e r t o o b t a i n a s a t i s f a c t o r y m a x i m u m p r i n c i p l e w e n e e d a 0 = 1 ( o r a 0 > 0 ) . S u c h a c o n c l u s i o n c a n b e o b t a i n e d b y m e a n s o f t h e
f o l l o w i n g h y p o t h e s i s :
H y p o t h e s i s I t
T h e r e e x i s t U j ( . u k + 1 ( . ) G ¿ ? / ( t 0 , t p U ) s u c h t h a t
< | ^ ( x 0 ( t i ) ) ’ x i ( V > , 1 S { l , . . . , k + 1 }
a r e i n a g e n e r a l p o s i t i o n a n d
0 G i n t c Q { d c p ( x Q ) : x 1 ( t 0 ) ) , . . . , d c p ( x 0 ( t j ) ; x ^ + j i t j ) ) }
w h e r e ф = ( ф 1 , . . . , k k ) , x ¡ ( . ) G M c o r r e s p o n d s t o u ¡ ( . ) .T h e s e t M С A C ( t 0 , t a; R n ) i s d e f i n e d b y a l l x ( . ) f o r w h i c h t h e r e e x i s t
u ( . ) G ^ ( t 0 , t p U ) s u c h t h a t
x ( t 0 ) = ° , | ^ ( t , x 0 ( t ) , u 0 ( t ) ) x + f ( t , x 0 ( t ) , u 0 ( t ) ) - f ( t , x 0 ( t ) , u ( t ) ) ( 1 8 )
W e s e e k h y p o t h e s e s w h i c h p r o v i d e aQ = 1 b e c a u s e t h i s i s t h e f i r s t s t e p f o r t h e m a x i m u m p r i n c i p l e t o b e a s u f f i c i e n t c o n d i t i o n f o r o p t i m a l i t y .
4 . 3 . S o m e r e m a r k s a b o u t s u f f i c i e n c y o f t h e m a x i m u m p r i n c i p l e
E v e n i n t h e c a s e o f u 0 ( . ) b e i n g t h e u n i q u e s o l u t i o n o f t h e m a x i m u m p r i n c i p l e i t m a y h a p p e n t h a t u Q( . ) i s n o t a n o p t i m a l s o l u t i o n .
I t i s n e c e s s a r y t h a t t h e m a x i m u m p r i n c i p l e i s s a t i s f i e d i n a s t r o n g f o r m a s i n t h e f o l l o w i n g h y p o t h e s i s :
H y p o t h e s i s I ?
T h e r e e x i s t o - , , . . . a k € R a n d c 0 > 0 s u c h t h a t
U ti
J [ H ( t , x 0 ( t ) , u ( t ) ) - H ( t , x Q( t ) , u 0 ( t ) ) ] d t ê CgJ" | f ( t , x 0 ( t ) , u ( t ) I
to to
- f ( t , x 0 ( t ) , u 0 ( t ) ) | d t
f o r a l l U )
116 VÂRSAN
w h e r e
К
H ( t , x , u ) = f ( t , x , u ) > , ф(t a ) = ^ - ( X j j i t j ) ) + ^ < > i ( X o f t j ) )
' ft" =ФМ H (t’xo(t)'uo(t,)P r o p o s i t i o n 1 . ( U a r b i t r a r y )
L e t ( x 0 ( . ) , u 0 ( . ) ) b e a n a d m i s s i b l e p a i r . S u p p o s e h y p o t h e s i s I 2 i s s a t i s f i e d . T h e n t h e r e e x i s t s a n Eg > 0 s u c h t h a t ср0 ( х 0 ( ^ ) ) S c p o i x t t j ) ) f o r a l l a d m i s s i b l e p a i r s
( x ( . ) , u ( . ) ) , | u ( t ) - u 0 ( t ) | < e 0 - t e l t Q . t j ] ,
P r o o f
W e s h a l l s k e t c h t h e p r o o f . L e t ( x ( . ) , u ( . )) b e a n a r b i t r a r y a d m i s s i b l e p a i r . T h e t r a j e c t o r y x ( . ) m a y b e d e c o m p o s e d i n a u n i q u e m a n n e r a s f o l l o w s :
x ( t ) = x 0 ( t ) + x ( t ) + x ( t , x ( . ) ) ( 1 9 )
w h e r e x ( . ) a n d x ( . ) s a t i s f y
x ( 0 ) = 0 , ^ = g ( t , x 0 ( t ) , u 0 ( t ) ) x + f ( t , x 0 ( t ) , u ( t ) ) - f ( t , x Q( t ) , u 0 ( t ) ) ( 2 0 )
x ( 0 ) = 0 , | ^ ( t , x 0 ( t ) , u Q( t ) ) x + a ( t , £ ( . ) , x ( . ) ) ( 2 1 )
w h e r e
l i m I ( t j 3 é ( . ) , x ( . ) ) [ _ q u n i f o r m i y i n t G [ t 0 , t j ]
x ( . ) . ï ( . ) ^ o | | x ( . ) | |
II a ( . , x 2 ( . ) , x ( . ) ) - a t ^ X j i . ) , x ( . ) ) | | s e || x 2 ( . ) - x a ( . ) | | ( 2 2 )
f o r
II X j ( . ) У s 6 ( e ) , | | x ( . ) | | éô(e)
D e f i n e a l i n e a r c o n t i n u o u s m a p p i n g A : A C ( t Q, t x ; R n ) -*■ A C ( t 0 , t p R n ) b y
t
A ( x ( . ) ) ( t ) = x ( t ) -J ( s , x 0 ( s ) , u 0 ( s ) ) x ( s ) d s , t G [ t Q, t x ]
to
(23)
IAEA-S MR-17/30 117
I t i s e a s y t o s e e t h a t A i s a b i s e c t i o n a n d . f r o m ( 2 2 ) i t f o l l o w s t h a t t h e h y p o t h e s e s o f t h e t h e o r e m 2 ( s e e s e c t i o n 1 ) a r e s a t i s f i e d f o r E q . ( 2 1 ) .
T h e r e f o r e , t h e r e e x i s t x ( . , x ( . ) ) f o r x ( . ) s u f f i c i e n t l y s m a l l s a t i s f y i n g ( 2 1 ) a n d l i m x ( . , x ( . ) ) = 0
x(0-o
U s i n g ( 2 1 ) w e o b t a i n
Ш п J t e M = 0 ( 2 4 )x(0 -o ||x(.)||-
L e t O j , . . . , a k G R b e t h e c o n s t a n t s g i v e n b y o u r h y p o t h e s i s .F o r e v e r y a d m i s s i b l e p a i r ( x ( . ) , u ( . ) ) w e o b t a i n
К
ç 0(x(t1))-cp0(x0(t1)) = ç Q(x (t1) ) - ф ^ х ^ ) ) + ^ a i(ç 1(x (t1) ) - ф . ( х 0( ^ ) ) )
i = lк
= < ^ - (XqÍÍj )), х (Ч ) > + ^ « ¡ < ^ (x o( t i ) )>x(t i ) > + N - ) I M x ( . ) )i= l
w h e r e l i m r | ( x ( . ) ) = 0 .5c (0 — 0
B y d e f i n i t i o n i t f o l l o w s
i= lti
( 2 5 - )
=J [ H ( t , x 0 ( t ) , u ( t ) ) - H ( t , x Q( t ) , u 0 ( t ) ) ] d t ( 2 6 )
U s i n g h y p o t h e s i s I 2 w e o b t a i n f r o m ( 2 5 )
Ф0 ( a c ( t j )) - Ф 0 ( x o ( t x )) ê c0J | f ( t , x 0 ( t ) , u ( t ) ) - f ( t , x 0 ( t ) , u Q( t ) ) | d t
to
+ | | x ( . ) | | r ) ( x ( . ) ) ( 2 7 )
F r o m ( 2 0 ) f o l l o w s
| x ( . ) | | | f ( t , x 0 ( t ) , u ( t ) ) - f ( t , x 0 ( t ) , u 0 ( t ) ) | d t ( 2 8 )
tQ
118 VÂRSAN
a n d a s a c o n s e q u e n c e o f ( 2 7 ) a n d ( 2 8 ) w e o b t a i n
ti
(29)
f o r a l l x ( . ) s u f f i c i e n t l y s m a l l .S u c h s u f f i c i e n t l y s m a l l x ( . ) a r e o b t a i n e d i f | u ( t ) - u 0 ( t ) | , t e t t g . t - J i s
s u f f i c i e n t l y s m a l l , a n d t h e p r o o f i s c o m p l e t e .
P r o p o s i t i o n 2 ( U c o n v e x )
L e t ( x 0 ( . ) , u 0 . ) ) b e a n a d m i s s i b l e p a i r . S u p p o s e t h e r e e x i s t c o n s t a n t s f f j , . . . , t t k e R , Cq > 0 a n d i / / : [ t 0 , t j l - R " s u c h t h a t
i = l
T h e n t h e r e e x i s t s a n e 0 > 0 s u c h t h a t c p ^ X o ^ ) ) S 9 0 ( x ( t j ) ) f o r a l l a d m i s s i b l e p a i r s ( x ( . ) , u ( . ) ) w h i c h s a t i s f y | u ( t ) - u ^ t ) | < e Q, t E [ t 0 , t j .
T h e p r o o f f o l l o w s i n t h e s a m e w a y a s b e f o r e .
4 . 4 . H i g h e r - o r d e r n e c e s s a r y c o n d i t i o n s
I n g e n e r a l P o n t r j a g i n ' s p r i n c i p l e i s u n s a t i s f a c t o r y w h e n w e w a n t t o d e c i d e w h e t h e r a c e r t a i n c o n t r o l i s o p t i m a l o r n o t .
L e t u s c o n s i d e r t h e f o l l o w i n g e x a m p l e . T h e c o n t r o l s y s t e m i s
d x i ix ( 0 ) = 0 , u , U = { u : u S l ) , u 6 R , t 6 [ 0 , l ]
a n d t h e f u n c t i o n a l t o b e m i n i m i z e d i s
f o r a l l u ( . ) E W (t0; t j j U )
к
1
о
I n t h i s c a s e t h e m a x i m u m p r i n c i p l e r e a d s
ф (t)nAt) = m i n i M t ) u u u l u M i
IAEA-SMR-17/30 119
w h e r e tpQ ( . ) i s d e f i n e d b y
ф0( 1 ) = 0 , ^ _ = 2 x 0 ( t ) , t G [ 0 , 1 ]
W e s e e t h a t , a l t h o u g h t h e c o n t r o l u 0 ( . ) = 0 s a t i s f i e s t h e m a x i m u m p r i n c i p l e i t i s n o t t h e o p t i m a l c o n t r o l .
T h e o p t i m a l c o n t r o l i s u „ ( t ) = 1 o r u 0 ( t ) = - 1 .I n t h i s e x a m p l e w e h a v e a c o u n t a b l e s e t o f a d m i s s i b l e c o n t r o l s
{ u " ( . ) } , u “ ( t ) = s g n c o s — n = 1 , 2 , . . , w h i c h s a t i s f y t h e m a x i m u m
p r i n c i p l e a n d a r e n o t o p t i m a l .F o r n = 1 f o l l o w s
u j ( t ) =
' Г \l , t e _ ’ 3,)
1 > Xg(t) = »- i , t e è > 1_з
t £ | 0 ’ 3
23 -t, te Í 1
< ( t ) =
t2 - ± t e 0,:
1 4t о ,- ? + т - л t, f 1 a n d 0 Q ( t ) u ^ t ) = m i n ^ ( t j u l u l = i
A c o n t r o l u ( . ) w h i c h s a t i s f i e s t h e m a x i m u m p r i n c i p l e a n d
H ( t , x Q( t ) , u ( t ) ) = H ( t , x Q( t ) , u Q( t ) ) , t e [ t 0 , t j ] f o r a n a d m i s s i b l e u ( . ) , u ( . ) = £ и 0 ( . )
i s c a l l e d a s i n g u l a r c o n t r o l .
I t i s e a s y t o s e e t h a t f o r a s i n g u l a r c o n t r o l h y p o t h e s i s I 2 i s n o t s a t i s f i e d ( s e e p r o p o s i t i o n 1 ) .
I n g e n e r a l , f o r s i n g u l a r c o n t r o l s t h e m a x i m u m p r i n c i p l e c a n n o t b e a s u f f i c i e n t c o n d i t i o n f o r o p t i m a l i t y .
S o , i n o r d e r b e t t e r t o c h a r a c t e r i z e t h e o p t i m a l c o n t r o l w e n e e d o t h e r c o n c l u s i o n s w h i c h t e l l u s s o m e t h i n g n e w w h e n t h e m a x i m u m p r i n c i p l e g i v e s n o i n f o r m a t i o n .
S u p p o s e t h a t c p ¡ ( x ) a n d f ( t , x , u ) a r e c o n t i n u o u s l y d i f f e r e n t i a b l e o f o r d e r p ( p — 1 ) w i t h r e s p e c t t o ( x , u ) .
F i r s t w e g i v e s e c o n d - o r d e r n e c e s s a r y c o n d i t i o n s .
T h e o r e m 1 ( U o p e n ) . L e t ( x 0 ( . ) , u 0 ( . ) ) b e a n o p t i m a l p a i r s u c h t h a t L = R k . T h e n t h e r e e x i s t c ^ , . . . , a k 6 R s u c h t h a t
(a)Э НЭ и
(t,x0(t),u0(t)) = o v te[t0,ta]
120 VÂRSAN
( b ) y » . d 2 cp. ( x 0 ( t 1 ) ; x ( t a )) +J d (2x u ) H ( t , x 0 ( t ) , u 0 ( t ) ; x ( t ) , û ( t ) ) d t S 0 ( < * 0 = 1 )
i=0 to
f o r a i l ( x ( . ) , ï ï ( . ) ) w h i c h s a t i s f y
( c ) d cP j ( x o t t j í j x í t j ) ) = 0 , i E { l , . . . , k } , x ( t 0 ) = 0
Ж = à x X ° ^ ’ * + x o W ’ u o № “ ( t )
w h e r e H ( t , x , u ) = f ( t , x , u ) X a n d ф s a t i s f i e s
к
( d ) D . - f = * § § ( t , x 0 ( t ) ) , u 0 ( t »
i=0
P r o o f
T h e s i t u a t i o n i s t h a t o f t h e o r e m 2 ( s e e s e c t i o n 3 ) . H e n c e f o l l o w s t h a t t h e r e e x i s t 6 R a n d д E ( A C ( t 0 , t a ; R n ) ) * ( Y = 1 ^ X A C ( t 0 , t j ; R n ))s u c h t h a t
ЧФ 0 ( Х Д ) ; x ( ^ )) + У dcp. ( х Д ) ; x i t j ) ) + M ( d T j ( x Q( . ) , u Q( . ) ; x ( . ) , u ( . ) ) ) = 0 ( 3 0 )
i=l
f o r a l l ( x ( . ) , u ( . ) ) E A C ( t 0 , t j ; R n ) X < ' ( t 0 , t j ; R r ) a n d
d 2 cp0 ( x 0 ( t j ) ; x ( t j ) ) + У а '1 d 2 cpj ( х 0 ( ^ ) ; x f t j ) ) + м ( d 2 T j ( x 0 ( . ) , u 0 ( . ) ; x ( . ) , u ( . ) ) ) g 0
i = 1 ( 3 1 )
f o r a l l ( x ( . ) , ï ï ( . ) ) s a t i s f y i n g ( c )W i t h ( d ) i t f o l l o w s f r o m ( 3 0 )
*1 t!
< i / / ( t 1 ) , x ( t 1 ) > = y ' < < M t ) , x ( t ) > d t = y ^ p j j ( t , x 0 ( t ) , u Q( t ) ) u ( t ) d t = 0 ( 3 2 )
t0 to
f o r a l l u ( . ) € < ? / ( t 0 , t 1 ; R r ) a n d c o n c l u s i o n ( a ) i s o b t a i n e d .F r o m ( 3 0 ) a n d ( 3 1 ) w e d e r i v e ( b ) a n d t h e p r o o f i s c o m p l e t e .
T h e o r e m 2 ( U c o n v e x ) . L e t ( x n ( . ) , u 0 ( . ) ) b e a n o p t i m a l p a i r s u c h t h a t O E i n t L j . T h e n f o r e v e r y x = ( x \ . ) , u ( . ) ) t h a t s a t i s f i e s ( c ) t h e r e e x i s t
t ? k £ R a n d Tp': [ t 0 , t j - > R n s a t i s f y i n g ( d ) s u c h t h a t
Q tT( a ) < g — (t, X0(t), u0(t)), u-u0(t)>è 0 f o r a l l u e U , t E [t0, t j
_k_ 11
( b ) ^ û j d 2 c p j ( x 0 ( t ]_); x í t j ) ) + J d 2 X iU )H ( t , x 0 ( t ) , u 0 ( t ) ; x ( t ) , ï r ( t ) ) d t ê 0 ( 5 0 = 1 )
i=0 ^
I n t h i s c a s e w e h a v e
( c ) d c p j ( x 0 ( t a ) ; x ( t a )) = 0 , i 6 { 0 , l , . . , k }
x ( t 0 ) = 0 , | | ^ , х 0 и ) , и 0 ( ^ х + 0 ( 1 ) | | и , х 0 ( 1 ) , и 0 Ш ) ( u ( t ) - u 0 ( t ) ) ,
â(t)so ï ï ( . ) 6 ^ ( t 0, t i; U), â ( . ) 6 ^ ( t 0, t i ; R)
к _
( d ) Mt1) = ^ â l =i - 0
I n a d d i t i o n , i f t h e r e e x i s t ( x ¿( - ) , ( - ) ) , i G { 1 , . . . , к + 1 } s a t i s f y i n g
( 7 ) X j ( t 0 ) = 0 , ^ - = I I ( t , x 0 ( t ) j U o ( t ) ) x . + | i ( t , x 0 ( t ) , u 0 ( t ) ) ( u . ( t ) - u 0 ( t ) )
< | ^ ( t . x 0 ( t ) , U 0 ( t ) ) , U j ( t ) - U 0 ( t ) > = 0 t e [ t 0 , t j ]
a n d
( 6 ) O e i n t c o { d ф ( x 0 ( t 1 ) ; x ^ ) , . . . , d c p t x ^ ) ; х к + 1 ( ^ ) ) }
t h e n c o n c l u s i o n ( b ) w i l l h o l d w i t h t h e s a m e f o r a l l ( x ( . ) , u ( . ) )s a t i s f y i n g ( c ) .
P r o o f
B y h y p o t h e s i s t h e r e e x i s t ( x ; ( . ) , U j ( . ) ) , i e { l , . . . , k + l } s a t i s f y i n g ( 7 ) s u c h t h a t
O S i n t с о { d c p ( x 0 ( t 1 ) ; x 1 ( t 1 ) , . . . , d ( p ( x 0 ( t 1 ) ; x k + 1 ( t 1))} ( 3 3 )
L e t u U e ^ i t o . t j ; U ) b e a r b i t r a r i l y f i x e d a n d l e t ( x ( . ) , u ( . ) ) s a t i s f y ( c ) . D e n o t et h e B a n a c h s p a c e o f a l l p a i r s ( x ( . ) , a ( . ) ) b y X
x( , )eAC( t0, tj ¡Rn), a(.) = a1(.)i. . . ,ak+3(.))eW(t0,t1;Rk+3)
w h e r e a j ( . ) h a s d i s c o n t i n u i t i e s a m o n g t h o s e o f t h e f u n c t i o n s
Uj(. ) , . . . , u k + 1 , u ( . ) , u ( . ) .
IAEA-S MR-17/30 121
122 VÂRSAN
D e n o t e : Y = R k X A C ( t 0 , t I ; R n ) .L e t f : X - > R b e d e f i n e d b y f ( x ) =c p 0 ( x ( t 1 ) ) .
D e n o t e u k + 2 ( . ) = u ( . ) , u k + 3 ( . ) = u ( . ) a n d d e f i n e T ^ X ^ A C ( t „ , t ^ R " ) a s f o l l o w s :
к +3
T j ( x ) ( t ) = x ( t ) - x Q- J f ( s , x ( s ) , u 0 ( s ) + ^ a . ( s ) u . ( s ) ) d s , t e [ t Q, t x ]
t 0 i = l
L e t ф ( x ) = a(.) , a n d d e n o t e T ( x ) = ( ф ( х ( ^ ) ) , T j i x ) ) .
B y d e f i n i t i o n ( x 0 ( . ) , 0 ) i s l o c a l l y o p t i m a l f o r t h e p r o b l e m m i n f ( x ) w i t h t h e c o n t r a i n t s T ( x ) = 0 , ф ( х ) 6 Г 2 , w h e r e
SI = ^ a U e ‘M t 0 , t 1 ; R k + 3 ) : a j ( t ) s 0
C o n d i t i o n ( 3 3 ) p r o v i d e s t h a t d T ( x 0 ( . ) , 0 ; . ) : X - * Y i s a s u r j e c t i o n . T h e n , w i t h t h e o r e m 1 f o r p = 2 a n d x = ( x ( . ) , u ( . ) ) , a(.) = ( 0 , . . . , 0 , 1 ) , x ( . ) s a t i s f y i n g ( d ) ,i t f o l l o w s t h a t t h e r e e x i s t V G Г2 , Д" G Y * a n d a0 i 0 w i t h t h e p r o p e r t i e s
ó 0 d f ( x 0 ; x ) + t r ( d ^ ( x 0 ; x ) ) + / I ( d T ( x 0 ; x ) ) = 0 ( 3 4 )
f o r a l l x G X
«о + II11 0 (35)0 0 d 2 f ( x 0 ; x ) + v ( d 2 < i ) ( x 0 ; x ) ) + ^ ( d 2 T ( x 0 ; x ) ) g 0 ( 3 6 )
B y d e f i n i t i o n d 2 < f i ( x 0 ; . ) = 0 . S u p p o s e aQ = 0 . F r o m ( 3 4 ) w e o b t a i n 7(а{.)) = 0 f o r a l l a(.) G Q a n d t h i s i n v o l v e s 'v = 0 w h i c h i s i n c o n t r a d i c t i o n t o ( 3 5 ) . T h e r e f o r e с ^ > 0 ; d i v i d i n g b y i t w e o b t a i n f r o m ( 3 4 ) a n d ( 3 6 )
d f f x g ; x ) + Т’ (0ф(х0; x ) ) + j T ( d T ( x 0 ; x ) ) = 0 V x G X , ( 3 7 )
d 2 f ( x 0 ; x ) + x ) ) ê 0 ( 3 8 )
B y d e f i n i t i o n o f ¡ I t h e r e e x i s t 5 ^ , . . â ] , G R a n d G ( A C ( t 0 , t j ; R n ) ) * s u c h t h a t W = ( ô j , . . . , ô ^ . i T j ) a n d w e o b t a i n f r o m ( 3 4 ) a n d ( 3 6 )
к
dcp0 ( x 0 ( t 1 ) ; x ( t 1 ) ) + 1 7 ( o f ( . ) ) d c p . f x Q Í t ^ x í t ^ J + M j í d T j í X Q Í x ) ) = 0 ( 3 9 )
i=l
for all x = (x(.), a ( . ) ) G X.
IAEA-SMR-17/30 1 2 3
d ç 0 ( x 0 ( t 1 ) ; x ( t 1 ) ) + d 2 9 0 ( x 0 ( t 1 ) ; x ( t 1 ) (dcp. ( х Д ) ; х ( ^ ) )
i=l
+ d 2 ç . ( x 0 ( t 1 ) ; x ( t 1 ) ) ) + i r i ( d T ( x 0 ; x ) ) + d 2 T 1 ( x 0 ; x » ê О V x G X ( 4 0 )
C h o o s i n g a(.) a n d x ( . ) s u c h t h a t a - ( t ) ê 0 , j e { 1,. . . , к + 3 }
d T j ^ ( x Q ; x ) = 0 , w e o b t a i n f r o m ( 3 9 )
3 H( t , x Q( t ) , u Q ( t ) ) , u ( t ) - u Q( t ) ) > a 0 f o r e v e r y t £ [ t 0 , t j ] ( 4 1 )
w h e r e u ( . ) G ^ / ( t 0 , t j ; U ) w a s a r b i t r a r i l y f i x e d .N o w w e s h a l l s h o w t h a t ( 4 1 ) i s s a t i s f i e d f o r a l l u 6 U f o r a f i x e d t G [ t 0 , t j L C o n c l u s i o n ( 4 1 ) i s a d i r e c t c o n s e q u e n c e o f t h e c o n d i t i o n
к '
ô . d c p . f x ^ ) ; x ( t 1 ) ) ^ 0 ( 4 2 )
i =0
f o r a l l ( x ( . ) , q ( . ) ) G X s a t i s f y i n g o^. ( t ) ë 0 , d T j f X j , ; x ) = 0 , ( q 0 = 1 ) .
D e n o t e
к
Л и = - | ( a 0 , O j , . . . , a k ) G R k + 1 : a0 ê 0 , = 1 , (aQ, .. . . ak) s a t i s f i e s ( 4 2 ^
i=o
f o r e v e r y u(.)e<%/(t0, t j j U ) .E a c h A u i s a c l o s e d s e t i n t h e u n i t s p h e r e o f t h e R k + 1 . T h e f a m i l y
h a s t h e p r o p e r t y o f f i n i t e i n t e r s e c t i o n a n d h e n c e f o l l o w s t h a t ПЛиф=ф.T h e r e f o r e , w i l l e x i s t (a0, av . . . , o k ) G R k + 1 s a t i s f y i n g ( 4 2 ) f o r a l l
u(.) T h u s ( 4 1 ) i s v e r i f i e d f o r a l l u G U w h e n t e [ t 0 , t j ] i s f i x e da n d s t a t e m e n t ( a ) i s p r o v e d .
S t a t e m e n t ( b ) i s a d i r e c t c o n s e q u e n c e o f ( 4 0 ) w h e n w e c h o o s e x = ( x ( . ) , 0 ) s u c h t h a t d T j ( x 0 ; x ) + d ^ j f x f , ; x ) = 0 . I n o r d e r t o o b t a i n t h e f i n a l p a r t o f t h e t h e o r e m w e r e m a r k t h a t o u r a d d i t i o n a l h y p o t h e s i s i s t h e s a m e w h e n d T ( X Q ; . ) : L — Y i s a s u r j e c t i o n i n t h e o r e m 3 ( s e e s e c t i o n 3 ) . T h e h y p o t h e s e s
o f t h e o r e m 3 b e i n g v e r i f i e d , i t f o l l o w s t h a t t h e s e t s A u j , u ( . ) , ï ï ( . ) £ t j j U ) h a v e t h e p r o p e r t y o f f i n i t e i n t e r s e c t i o n w h e r e
к
Л и ,и = | ( a o , a r " * , Q í i ^ е к к + 1 ; % - ° » ^ l a i l = 1 > (a0,...,O!k) s a t i s f i e s ( 4 0 ) , ( 4 2 ) j -i = 0
124 VÂRSAN
A s b e f o r e w e c o n c l u d e t h a t { A ^ h a s a n o n - e m p t y i n t e r s e c t i o n , i . e . c o n c l u s i o n ( b ) h o l d s f o r a l l ( x ( . ) , u ( . ) ) s a t i s f y i n g ( c ) w i t h t h e s a m e c o n s t a n t s a 1 , . . . , o k . W h e n U i s a n a r b i t r a r y s e t , w e c a n u s e a s i m i l a r p r o c e d u r e a s i n t h e p r e c e d i n g t h e o r e m s s o t o o b t a i n
T h e o r e m 3 . ( U a r b i t r a r y ) . L e t ( x 0 ( . ) , u 0 ( . ) ) b e a n o p t i m a l p a i r s u c h t h a t t h e h y p o t h e s i s I i s s a t i s f i e d . T h e n f o r e v e r y x = ( x . ) , u ( . ) ) s a t i s f y i n g( c ) a n d ( d ) t h e r e e x i s t G R a n d Щ.) s a t i s f y i n g ( e ) w i t h t h e f o l l o w i n gp r o p e r t i e s :
( a ) H ( t , x Q( t ) , u Q( t ) ) = m i n H ( t , x 0 ( t ) , u ) f o r e v e r y t G [ t Q, t ^ ]
k n 'i
( b ) d 2 <pi ( x 0 ( t 1 ) ; x ( t 1 ) ) + J [ c ^ H ( t , x 0 ( t ) , u 0 ( t ) ; x ( t ) )
i=0 'о
+ 2 ( d x H ( t , x 0 ( t ) , u ( t ) ; x ( t ) ) - d x H ( t , x 0 ( t ) , u Q( t ) ; x ( t ) ) ) ] d t ê 0 , . { ô 0 = 1 )
I n t h i s c a s e
( c ) d c p . t x ^ t j ) ; x ( t j ) ) = 0 , i G { 0 , 1 , . . . , k }
( d ) x ( t 0 ) = 0 , ^ = | ^ ( t , x 0 ( t ) , u 0 ( t ) ) x + f ( t , x 0 ( t ) , u ( t ) ) - f ( t , x 0 ( t ) , u Q( t ) )
a n d
к __
( e ) ф(t j ) ^ ( x ^ t j ) ) , - ^ = 5 T | | ( t , x 0 ( t ) , u 0 ( t ) )
i=0
M o r e o v e r , i f t h e r e e x i s t ( x . ( . ) , u ¡ ( . ) ) , i e { 1 , . . . , к + 1 } s a t i s f y i n g ( d )
H ( t , x Q( t ) , U j ( t ) ) - H ( t , x Q( t ) , U g ( t )) = 0
a n d
( f ) 0 G i n t c o { d 9 ( x 0 ( t 1 ) ; x 1 ( t 1 ) ) , . . . , d c p ( x Q( t 1 ) ; x k + 1 ( t 1))}
t h e n c o n c l u s i o n ( b ) w i l l h o l d f o r a l l ( x ( . ) , u ( . ) ) s a t i s f y i n g ( c ) a n d ( d ) .
R e m a r k 1
I f t h e c o n t r o l p r o b l e m d o e s n o t c o n t a i n t h e c o n s t r a i n t s cp. ( x ( t j )) = 0 ,i G { 1 , . . . , k } t h e n t h e o r e m 2 ( t h e o r e m 3 ) t a k e s t h e f o l l o w i n g f o r m :
L e t ( x 0 ( . ) u 0 ( . )) b e a n o p t i m a l p a i r . T h e n c o n c l u s i o n s ( a ) a n d ( b ) a r e
s a t i s f i e d f o r a l l ( x ( . ) ) , u ( . ) ) s a t i s f y i n g ( c ) , ( ( c ) a n d ( d ) ) .O f c o u r s e i n t h i s c a s e t h e f i n a l p a r t o f t h e o r e m 2 ( 3 ) i s u n n e c e s s a r y .
C o n d i t i o n s ( c ) a n d ( d ) i n t h e o r e m 3 a r e t h e s a m e a s t h o s e w h e r e a l l t h e
IAEA-SMR-17/30 125
R e m a r k 2
c o n d i t i o n s o f ( c ) a n d ( d ) a r e m a i n t a i n e d , e x c e p t f o r d c p 0 ( х 0 ( 1^ ) ; x f t j )) = 0
w h i c h i s r e p l a c e d b y
A c c o r d i n g t o r e m a r k 2 s e c o n d - o r d e r n e c e s s a r y c o n d i t i o n s a r e s t a t e d f o r a p a i r ( x ( . ) , u ( J f o r w h i c h P o n t r j a g i n ' s p r i n c i p l e g i v e s n o i n f o r m a t i o n o n u 0 ( . ) .
4 . 5 . P o i n t w i s e h i g h e r - o r d e r n e c e s s a r y c o n d i t i o n s
O n e s e e s t h a t i n t h e t h e o r e m s s t a t e d a b o v e f i r s t - o r d e r n e c e s s a r y c o n d i t i o n s ( s e e ( a ) ) h a v e a p o i n t w i s e f o r m , w h i l e s e c o n d - o r d e r n e c e s s a r y c o n d i t i o n s ( s e e ( b ) ) a r e s t i l l i n a n i n t e g r a l f o r m .
W h e n w e w a n t t o t e s t t h e p r o p e r t y o f n o n - o p t i m a l i t y f o r a c e r t a i n a d m i s s i b l e c o n t r o l i t i s u s e f u l t o h a v e a p o i n t w i s e f o r m a l s o f o r ( b ) .
F i r s t , s u p p o s e t h a t t h e c o n t r o l p r o b l e m d o e s n o t c o n t a i n t h e c o n s t r a i n t s
H ( t , x 0 ( t ) , u ( t ) ) - H ( t , x Q( t ) , u Q( t ) ) = 0
Ф; (х) = 0 , i e { l , . . . , k } .
L e t Ф : [ Л 0 , t j J - L Í R 11, R n ) b e d e f i n e d b y
w h e r e t h e p a r t i a l d e r i v a t i v e s a r e t a k e n a t ( t , x 0 ( t ) , u 0 ( t ) ) . T h e n c o n c l u s i o n ( b ) i n t h e o r e m s 1 , 2 a n d 3 b e c o m e s
f o r a l l u € R r s a t i s f y i n g
Э2 Н< " Э й 2 " x o ( s ) ’ U 0 * S ^ u , u > = 0 , s e [ t , t + e )
f o r a l l u e U s a t i s f y i n g
s e [t, t +e )
126 VÂRSAN
(b3) ( f ( t , X0(t), u) - f ( t , X0(t), u0(t))T Ф ( Ю ( f ( t , x Q( t ) , u) - f ( t , x 0 ( t ) , u 0(t)) )
f o r a i l u 6 U s a t i s f y i n g H ( s , x „ ( s ) , u ) - H(s, x 0(s), u0(s)) = 0 , s E [t, t + e ) .T h e p a r t i a l d e r i v a t i v e s a r e t a k e n a t ( t , x 0 ( t ) , u 0 ( t ) ) .
C o n c l u s i o n s ( b j ) , ( b 2 ) a n d ( b 3 ) a r e e a s i l y o b t a i n e d f r o m t h e c o r r e s p o n d i n g t h e o r e m s s i n c e i n t h e a b s c e n c e o f t h e c o n s t r a i n t s cpj = 0 w e n e e d n o t s a t i s f y t h e c o n s t r a i n t s d c p ¡ ( x 0 ( t i ) ; х О ^ ) ) = 0 , i G { l , . . . , k } . W e s e e t h a t t h e m a t r i x M ^ t ) i s o b t a i n e d a s a s o l u t i o n t o a d i f f e r e n t i a l e q u a t i o n a n d , i n o r d e r t o s a t i s f y ( b j ) , ( b 2 ) o r ( b 3 ) , w e m u s t k n o w t h e f u n c t i o n H ( t , x , u ) a n d i n t e g r a t e a m a t r i x d i f f e r e n t i a l e q u a t i o n .
I n t h e f o l l o w i n g w e g i v e a p o i n t w i s e f o r m f o r t h e c o n c l u s i o n i n t h e o r e m 2 w i t h o u t r e q u i r i n g t h e k n o w l e d g e o f t h e m a t r i x ' ï ' ( t ) .
L e t . 5 ^ ( t ) b e a m a t r i x d e f i n e d b y
D e f i n i t i o n
A p a i r ( t , u ) G ( t 0 , % ) X U p o s s e s s e s t h e p r o p e r t y ( S ) i f f o r e v e r y e , t t - t S e > 0 , t h e r e e x i s t s a u * ( . ) : [ t , t + e ) - > R r p i e c e w i s e c o n t i n u o u s s u c h t h a t
= E , - I I ( t , x 0 ( t ) , u 0 ( t ) )
L e t G b e a m a t r i x d e f i n e d b y
u £t ( t ) = u , u * ( s ) = a*{s ) ( u ( s ) - u Q( s ) ) , u ( s ) G <%¿a*{s ) ê 0
a n d
I GL£T(s ) ^ ( s , x Q( s ) , u 0 ( s ) ) u €t ( s ) d s = 0 ( e ) , l i m = 0
t
IAEA-SMR-n/30 1 2 7
s G [ t , t + e )
q 2 tt( i 4 ) ^ 3 ^ 2 ( s ' x 0 ( s ) , u 0 ( s ) ) u j ( s ) , u * ( s ) > = 0 f o r a i l s G [ t , t + e )
T h e m a t r i x G -1 i s c o m p o s e d b y ( n - k ) l i n e a r l y i n d e p e n d e n t v e c t o r s w h i c h a r e
o r t h o g o n a l t o t h e v e c t o r s i n G .L e t U ( t ) C U b e d e f i n e d b y
U ( t ) = { u G U : ( t , u) p o s s e s s e s t h e p r o p e r t y ( S ) }
T h e o r e m 2 ' . L e t t h e h y p o t h e s e s o f t h e o r e m 2 b e s a t i s f i e d ( i n d u c t i n g 7 ) . T h e n f o r e v e r y t G ( t n , t , ) s u c h t h a t U ( t ) фф
f o r a l l u e U ( t ) .
P r o o f
L e t u G U ( t ) b e f i x e d , a n d l e t e > 0 , U j f . ) b e g i v e n b y t h e p r o p e r t y S . L e t x £ ( . ) : [ t Q, t . j ] - » R n b e d e f i n e d a s f o l l o w s
f o r a l l u G U ( t ) .I n a d d i t i o n , i f
^ y i ( s , x 0 ( s ) , u Q( s ) ) = 0 s G [ t , t + e ) , d 2 <pi ( x 0 ( t 1 ) ; . ) = 0
i e { 0 , l , . . . , k } t h e n
<(
0 s e [ t 0 , t ]
x £ ( s ) S G [ t , t j ]
w h e r e x 6 ( . ) s a t i s f i e s t h e f o l l o w i n g s y s t e m :
B y h y p o t h e s i s i t f o l l o w s t h a t G & ( t) x*(t) = 0 f o r a l l т s t + e , e > 0 . I n d e e d , b y
d e f i n i t i o n
t + €
G ^ ( t + e ) x e ( t + e ) = J G & { s ) Ц ( s , x Q( s ) , u Q( s ) ) u ^ ( s ) d s = 0
t
a n d
-7 - G&~(t) x* ( т ) = 0 f o r t ê t + e d r c
T h e r e f o r e 0 = G ¿ ^ ( t a ) x * ( t j ) = G x * ( t j ) a n d t h e c o n s t r a i n t s
d c p j f X j j i t j ) ; X g f t j ) ) = 0 , i G { l , . . . , k } , a r e s a t i s f i e d .
I n o r d e r t o v e r i f y d ф 0 ( х 0 ( t j ) ; x * ( t 1 )) = 0 i t i s s u f f i c i e n t t o s h o w t h a t
128 VÂRSAN
( s , Xq(s), u 0 ( s ) ) u^(s) = 0 f o r s G [ t , t + e )
t h i s f o l l o w s b y h y p o t h e s i s ( s e e ( i 3 )) .O n t h e o t h e r h a n d , G ^ " ( t ) x * ( t ) , f o r T S t + e , s a t i s f i e s
G ^ ( t + e ) x * ( t + e ) = J G ^ s ) — ( s , x Q( s ) , u Q( s ) ) u 6t ( s ) d s = 0 ( e )
t
a n d
■ ^ G ±¿ í I Í ( t ) x . * ( t ) = 0 f o r Tit+eH e n c e G'l ( t ) x *(t ) = 6 ( e ) f o r a l l - r S t + e .
B e c a u s e t h e m a t r i c e s ( G , G ) a n d 3^(t) a r e i n v e r t i b l e w e o b t a i nx*(t) = 0 ( e ) f o r a l l т й t + e , w h e r e
l i m M = 0e -0 e
N o w , u s i n g t h e o r e m 2 w e o b t a i n
+ f d ( x . < s ' x o(s b u Q( s ) ; x * ( s ) , U j ( s ) ) d s
i =0 *0 t + € 2 _
= 2 J ( s , x 0( s ) , u 0( s ) ) x * ( s ) , u €t ( s ) > d s + 0 ( e 2 ) s о
IAEA-S MR-17/30 129
a n d d i v i s i o n b y e 2 y i e l d s ( w i t h e -» 0 )
( t , x 0 ( t ) , u 0 ( t )) | j j ( t , x 0 ( t ) , u 0 ( t ) ) ( u - u 0 ( t ) ) g 0
I f
( s , x f l( s ) , u Q( s )) = 0 f o r s G [ t , t + e )
t h e a b o v e i n t e g r a l b e c o m e s
t + £ j
t
a n d d i v i s i o n b y e 3 ( w i t h e - » 0 ) y i e l d s t h e l a s t p a r t o f t h e t h e o r e m . S o t h e p r o o f i s c o m p l e t e .
R e m a r k 1 .
I f t h e c o n t r o l p r o b l e m h a s n o c o n s t r a i n t o f t h e t y p e ф ; ( х ( ^ ) ) = 0 t h e n t h e p r o p e r t y ( S ) d o e s n o t c o n t a i n t h e c o n d i t i o n
t + e
w h i c h i s n o t e a s y t o v e r i f y ( s e e t h e f o l l o w i n g L e m m a ) .
H y p o t h e s i s J 2
L e t t G ( t 0 , t j b e f i x e d . T h e r e e x i s t U i , . . , u k + 1 G U , i " > 0 , s u c h t h a t
L e m m a . L e t t G ( t Q, t ^ b e f i x e d s u c h t h a t t h e h y p o t h e s i s J 2 i s s a t i s f i e d . T h e n f o r e v e r y
k + 1
u = ^ a ¡ u i< > 0
1=1
t
( a ) G x^ ( s ) g ^ ( s , x 0 ( s ) , u 0 ( s ) ) ( u j - u 0 ( s ) ) = 0 , s G [ t , t + e )
( b ) v j ( t ) = G & l t ) | ^ ( t , x 0 ( t ) , u 0 ( t ) ) ( u ¿ - u 0 ( t ) ) , i G { l , . . „ k + l }
i n R k a r e i n a g e n e r a l p o s i t i o n a n d 0 G i n t c o { v - ^ t ) , . . . , v k + 1 ( t ) } .
1 3 0 vArsan
t h e r e e x i s t e n > 0 a n d a s m o o t h c o n t r o l u j r ( s ) : [ t , t + e 0 ) - > R r , 0 < e S e 0 , w i t h t h e
p r o p e r t i e s uf(s) = a € ( s ) ( u t ( s ) - u 0 ( s ) ) , u t ( s ) G U , u t ( t ) = u , a n d t h e c o n d i t i o n s ( i j ) , ( i 2 ) a n d ( i 3 ) o f p r o p e r t y ( S ) a r e s a t i s f i e d .
P r o o f
L e t
k+i
u = У o - j U j b e f i x e d .
i = l
B y h y p o t h e s i s t h e r e e x i s t v G R k a n d e 0 > 0 s u c h t h a t
± v G i n t c o i v ^ s ) , . . . , V j ^ f s ) } f o r a l l s € [ t , t + e Q)
A s V j f t ) . . . v k + 1 ( t ) a r e i n a g e n e r a l p o s i t i o n w e o b t a i n t h e a ¡ ( s ) È 0 , i 6 { 1 , . . , к + 1 } s u c h t h a t
k + 1
o-, ( t ) = ai У o- j ( s ) v ¡ ( s ) = v , s G [ t , t + e 0 )
t h e O j i . ) b e i n g s m o o t h f u n c t i o n s .D e n o t e
k + l k + l
ut (s) =Z gi (s)Ui J |31(s) = ~kf i i(S) ’ X ^ i (s) = :1=1 ¿^¡(s) 1
H e n c e f o l l o w s u t ( s ) £ U , s G [ t , t + e 0 ) .D e f i n e
r 0 s G [ t g , t ) и [ t + C p . t j ]
a e ( s ) = 1 21 ' — ( s - t ) s G [ t , t + e Q)
a n d
k + l
. t + e nu * ( s ) = a e ( s ) ^ t t f ( s ï ( u t ( s ) - u 0 ( B ) ) , s G [ t , 1
S i n c e
t+e
J a e ( s ) d s = 0 f o r a l l e > 0 w e o b t a i n
t
t+ e t+e
G&(s) ( s , x Q( s ) , u Q( s ) ) u « ( s ) d s = v J a £{s ) d s = 0
IAEA-SMR-17/30 131
On the other hand,t+ e t+ e
J G ^ s ) ( s , x 0 ( s ) , u Q( s ) ) u £t ( s ) d s =J ae(s ) . 0 d s = 0
t t
B y d e f i n i t i o n
9 f f 1± ^ ¡ j ( s , x 0 ( s ) , U0(s)) ( u t ( s ) - U0(s)) e c o n j — ( s , x 0 ( s ) , u 0 ( s ) ) ( u - U 0 ( s ) ) : u E U ^
a n d t h e s a m e p r o p e r t y r e l a t i o n s a r e s a t i s f i e d w h e n w e r e p l a c e u t ( s ) b y u ^ f s ) .
A n d t h e p r o o f i s c o m p l e t e .
R e m a r k 2
I f U i s a n o p e n s e t t h e n c o n d i t i o n ( i 3 ) o f t h e p r o p e r t y ( S ) d o e s n o t a p p e a r b e c a u s e i n t h i s c a s e w e h a v e
c o n x „ ( s ) , u 0 ( s ) ) ( u - u 0 ( s ) ) : u G u j - = | ^ ( s , x 0 ( s ) , u Q( s ) ) ( R r )
R e m a r k 3
I f t h e c o n t r o l p r o b l e m d o e s n o t c o n t a i n c o n s t r a i n t s o f t h e t y p e 9 i x ( t 1 )) = 0 a n d U = R r t h e n a n y v a r i a t i o n u f ( . ) w h i c h s a t i s f i e s
t+e
J ' U j ( s ) d s = 0 w i l l s a t i s f y t h e c o n d i t i o n s ( i 2 ) , ( i 2 ) a n d ( i 3 ) o f t h e p r o p e r t y ( S ) .
t
A c c o r d i n g t o r e m a r k s 1 a n d 2 w e h a v e t o v e r i f y o n l y ( i j ) . I n t h i s c a s e , i n t e g r a t i n g b y p a r t s y i e l d s
t+e t+e
J ^ s) f ^ ( s . x 0 ( s ) , u 0 ( s ) ) u te ( s ) d s = - J (S,x0(s),u0(sj)t t
s
X J uet(t ) d T d s = 0 ( e )
4 . 6 . S o m e e x a m p l e s
H i g h e r - o r d e r n e c e s s a r y c o n d i t i o n s a r e u s e f u l i n t h e s t u d y o f s i n g u l a r c o n t r o l .
I n t h e f o l l o w i n g w e s h a l l c o n s i d e r t h r e e e x a m p l e s .
( 1 ) ^ • = u ' = x i ( ° ) = х г ( ° ) = 0 - U = { u : | u | á 1 } , t e [ 0 , 1 ] , ф 0 ( х ) = x 2
132 VÂRSAN
W e h a v e n o c o n s t r a i n t o f t h e t y p e c p j i x f t j ) ) = 0 .I t f o l l o w s t h a t H ( t , x , u ) = 0 1 ( t ) u - 0 2 X j U , w h e r e
u 0 ( t ) , = 0 , фг ( 1 ) = 0,ф2 (1) = 1 , a n d h e n c e ф2 ( t ) = 1 .
E a c h u : [ 0 , 1 ] - [ - 1 , 1 ] s u c h t h a t
l
J u ( t ) d t = 0
о
i s a n e x t r e m a l ( s a t i s f y i n g P o n t r j a g i n ' s p r i n c i p l e ) .I n d e e d , l e t u 0 ( . ) b e s u c h t h a t
l
J' u 0 ( t ) d t = 0 ( f o r e x a m p l e u 0 ( . ) = 0 ) . T h e n i / / j ( 1 ) = x 1 0 ( l ) = 0 a n d
о
^ ( ' ¿ i - x 1 0 ) ( t ) = 0
H e n c e f o l l o w s
^ ( t ) = x 1 0 ( t ) , t G [ 0 , 1 ] , H ( t , x 0 ( t ) , u ) = ( i / ^ t ) - x 1 0 ( t ) ) u = 0
a n d P o n t r j a g i n ' s p r i n c i p l e i s u s e l e s s a s i t c a n n o t s a y a n y t h i n g a b o u t t h e o p t i m a l i t y o f s u c h a u 0 ( . ) .
B u t f r o m s e c o n d - o r d e r n e c e s s a r y c o n d i t i o n s ( s e e t h e o r e m 2 ) i t f o l l o w s t h a t
1
J " c i ï ( t ) - u 0 ( t ) ) X l ( t ) d t S 0 f o r a l l
о
u ( . ) G < ! / ( 0 , 1 ; U ) a n d X j i O ) = 0 , ^ 7 - = ( u ( t ) - u 0 ( t ) ) , t G [ 0 , l ]
T h e r e f o r e
1
ï J É ® a n < ^ s o f ° r & H w h i c h i s a0
c o n t r a d i c t i o n b e c a u s e f o r u ( t ) = l , t £ [ 0 , l ] , w e o b t a i n ( x 1 ( l ) f > 0 .T h e r e f o r e , a s e c o n d - o r d e r n e c e s s a r y c o n d i t i o n p r o v e d t h a t a n y c o n t r o l
u 0 ( . ) s a t i s f y i n g
1
J ' u 0 ( t ) d t = 1 c a n n o t b e a n o p t i m a l c o n t r o l .
0
T h e o r e m 2 ' c a n a l s o b e u s e d . I t f o l l o w s t h a t ( u - u 0 ( 0 ) ) 2 ë 0 f o r a l l u G [ - 1 , 1 ] s a t i s f y i n g u 0 ( 0 ) - u G U - u 0 ( 0 ) . C h o o s i n g u = - u 0 ( 0 ) w i t h u Q( 0 ) ф 0 a n d u G U , u = / = u 0 ( 0 ) = 0 w e a r r i v e a t a c o n t r a d i c t i o n .
(2) ^ = v 1 T = U’ ^ r = _xi2’ x1(o )=x2(o)=x3(o) = o, t e [o.i]
U = { u : j u I s i } , cp0 ( x ) = x g ( l )
H ( t , x , u ) =ф1х2 + ф2и-ф3х 2
^ - = 2x1, = 03 ( t) = 1, ф1(1)=ф2(1) = 0T h e c o n t r o l u 0 ( . ) = 0 i s a s i n g u l a r o n e b e c a u s e x 1 0 ( t ) = 0 , x 2 0 ( t ) = 0,ф2 ( t ) = 0
a n d H ( t , x 0 ( t ) , u ) = i//2 ( t ) u = 0 . S e c o n d - o r d e r n e c e s s a r y c o n d i t i o n s a r e ( s e e t h e o r e m 2 )
l
J ( X j f t ) ) d t S 0 V ï ï ( . ) e < ? / ( 0 , 1 ; U ) a n d x ( . )
о
s a t i s f y i n g
Xj(0) = x 2 (0) = 0, % i = x 2, ^ - = ü
w h i c h i s a c o n t r a d i c t i o n b e c a u s e f o r u ( t ) = 1 w e o b t a i n
l
X x ( t ) = у a n d J ( x 1 ( t ) ) 2 d t > 0
о
( 3 ) ^ = X l + X 2 _ X 3 ' ^ = x 2 + u r f r = x 3 + u 2 ’ x ¡ ( 0 ) = 0 , t e l o . u
U = { ( u 1 , U2): I u . I S I }
Ф 0 ( х ) = х 1 , Ф ; ( х ) = х 2 , ф 2 ( х ) = х 3 , H ( t , x , u ) = Ф1(х1 + х\-х1)
+ Ф2 ( х 2 + U j ) +ф3 ( х 3 + U2)
T h e c o n t r o l u Q( . ) = ( 0 , 0 ) i s a s i n g u l a r o n e . I t f o l l o w s t h a t
X i 0 ( t ) s x 2 0 ( t ) = x 3 o ( t ) = 0 , H ( t , x 0 ( t ) , u ) = i / / 2 ( t ) u 1 + ^ 3 ( t ) u 2
IAEA-SMR-17/30 1 3 3
134 VÂRSAN
w h e r e
ápi _ áp2 áP3 _d t * 1 ’ d t 2 ’ d t ^ 3
^ ( 1 ) = 1 , 0 2 ( 1 ) = О, ф3{1)=0
W e o b t a i n Ф2(1) = Ф3№) = 0 a n d t h e r e f o r e H ( t , x Q( t ) , u ) = 0 . S i n c e
9 2 H , . „ Э2 Н , , „- r — j t , x , u ) = 0 , т - r - t , x , u = 0 ,
9 u 2 9 x 9 u
w e o b t a i n t h e f o l l o w i n g s e c o n d - o r d e r n e c e s s a r y c o n d i t i o n s ( s e e t h e o r e m 2 ) :
l
J фг {t ) [ ( x 2 ( t ) f - ( x 3 ( t ) ) 2 ] d t ^ 0 V x 2 ( . ) , x 3 ( . )
0
s a t i s f y i n g
x 2 ( 0 ) = x 3 ( 0 ) = 0 , x 2 ( l ) = 0 , x 3 ( l ) = 0 , ^ • = x 2 + ü 1 , • ^ ■ = x 3 + u 2
F o r e v e r y t G [ 0 , 1 ) t h e c o n d i t i o n s o f o u r l e m m a a r e s a t i s f i e d w h e n w e c h o o s e u 1 = ( 1 , 1 ) , u 2 = ( - 1 , 1 ) , u 3 = ( - 1 , 0 ) .
F r o m t h i s l e m m a w e s e e t h a t t h e v e c t o r s
u = ) a . u 1 , O j > 0 , ) a. = 1
d i s p l a y t h e p r o p e r t y ( S ) f o r e v e r y t G [ 0 , 1 ) . F r o m t h e o r e m 2 ' , f o r t = 0 , w e o b t a i n ( / / j ( 0 ) [ ( ï ï 1 ( 0 ) ) 2 - ( u 2 ( 0 ))2 ] S 0 f o r a l l
3
1 1r "
U j ( 0 ) = Ц - a 2- a 3), й2(0) = ^ a . 4 = ^ + a 2 , a l > 0 , ^ a ¡
S i n c e фг ( 0 ) > 0 ( i / ^ t t ) = e 1 _ t ) , w e o b t a i n , w h e n c h o o s i n g
1 1 1 “ 1 = З ’ ^ ' З , " 3 " 3
(ü1(0))2 - (Ü2 ( 0 ))2 = ( а х ~ а 2 - a 3 f - (o-j + а 2)2 = " 4 < 0
c o n t r a d i c t i n g t h e s e c o n d - o r d e r n e c e s s a r y c o n d i t i o n s . T h e r e f o r e u Q( . ) c a n n o t b e a n o p t i m a l c o n t r o l .
5 . C O N T R O L L A B I L I T Y F O R N O N - L I N E A R C O N T R O L S Y S T E M S
5 . 1 . B a s i c d e f i n i t i o n s a n d f o r m u l a t i o n o f t h e p r o b l e m
IAEA-SMR-17/30 1 3 5
T h r o u g h o u t t h i s s e c t i o n w e s h a l l c o n s i d e r t h e f o l l o w i n g c o n t r o l s y s t e m :
d x— = f ( t , x , u ) , x ( 0 ) = p Q, t ê O , u £ U ( 1 )
w h e r e f : [ 0 , ° ° ) Х Е ? X R r - + R n i s o f c l a s s С .L e t £ L ^ I O , 00) ; ! ! ) b e t h e s e t o f p i e c e w i s e C ” f u n c t i o n s a n d l e t
C ' î / b e d e f i n e d b y p i e c e w i s e c o n s t a n t f u n c t i o n s . L e t R ( t ) ( R 0 ( t ) ) Ç R n b e d e f i n e d a s f o l l o w s :
R ( t ) ( R 0 ( t ) ) = { y E R n ; y = x ( t , u ( . ) ) ,
w h e r e x ( . , u ( . ) ) i s t h e s o l u t i o n o f ( 1 ) c o r r e s p o n d i n g t o u ( . ) } .B y d e f i n i t i o n R ( t ) ( R 0 ( t ) ) i s t h e s e t o f a l l a c c e s s i b l e p o i n t s a t t h e m o m e n t t ,
u s i n g t r a j e c t o r i e s o f s y s t e m ( 1 ) w h i c h c o r r e s p o n d t o a d m i s s i b l e c o n t r o l s u ( . ) € T h e p r o b l e m w e a r e i n t e r e s t e d i n i s t o g i v e c o n d i t i o n s f o ra n a d m i s s i b l e t r a j e c t o r y x 0 ( . ) o f ( 1 ) s u c h t h a t x 0 ( t ) 6 i n t R ( t ) ( i n t R 0 ( t ) ) i s s a t i s f i e d f o r a l l t > 0 .
T h e c o n c l u s i o n x 0 ( t ) e i n t R ( t ) i n v o l v e s t w o f a c t s .F i r s t , R ( t ) h a s t o h a v e a n o n - e m p t y i n t e r i o r i n R n a n d s e c o n d , a c e r t a i n
t r a j e c t o r y m u s t b e i n i n t R ( t ) f o r a l l t > 0 .W h e n s y s t e m ( 1 ) i s l i n e a r i n b o t h v a r i a b l e s ( x , u ) , a n d U = R r , s u c h a
p r o b l e m c o n t a i n s t h e g l o b a l c o n t r o l l a b i l i t y o f t h e l i n e a r s y s t e m
d xx ( 0 ) = 0 , — = A ( t ) x + B ( t ) u , t e [ 0 , 1 ] ( 3 )
f o r e v e r y T > 0 .
C h o o s i n g u 0 ( t ) = 0 , w e o b t a i n x 0 ( t ) = 0 , a n d t h e c o n d i t i o n x 0 ( t ) e i n t R ( t ) m e a n s t h a t a t r a j e c t o r y o f s y s t e m ( 3 ) m a y c o n n e c t t h e o r i g i n w i t h a n y у 6 R n w i t h i n a f i n i t e t i m e .
T h e c o m p l e t e c o n t r o l l a b i l i t y o f s y s t e m ( 3 ) c a n b e e x p r e s s e d b y r e q u i r i n g t h a t t h e m a t r i c e s A ( t ) , B ( t ) s a t i s f y a c e r t a i n c o n d i t i o n .
F o r e x a m p l e , i f A ( t ) = A , B ( t ) = B , t h e n t h e c o m p l e t e c o n t r o l l a b i l i t y f o r ( 3 ) i s e q u i v a l e n t t o t h e f o l l o w i n g c o n d i t i o n :
r a n k ( B , A B , . . . , A n ‘ 1 B ) = n ( 4 )
D e f i n i t i o n 1 . W e s a y s y s t e m ( 1 ) i s l o c a l l y c o n t r o l l a b l e f r o m p Q i f t h e r e e x i s t s a n e i g h b o u r h o o d V o f p „ w i t h t h e p r o p e r t y t h a t e v e r y y £ V c a n b e
r e a c h e d w i t h i n a f i n i t e t i m e b y a n a d m i s s i b l e t r a j e c t o r y w i t h x ( 0 ) = p 0 .A c o n d i t i o n s i m i l a r t o ( 4 ) c a n b e u s e d t o o b t a i n s u f f i c i e n t c o n d i t i o n s f o r
l o c a l c o n t r o l l a b i l i t y .S u p p o s e f ( t , x , u ) = f ( x , u ) a n d
f ( P o > U q ) - 0 , U = R r
136 VÁRSAN
If the m a tr ic e s
(4)
s a t i s f y ( 4 ) t h e n s y s t e m ( 1 ) i s l o c a l l y c o n t r o l l a b l e .A s a c o n s e q u e n c e o f ( 5 ) i t f o l l o w s a l s o t h a t l o c a l c o n t r o l l a b i l i t y o f ( 1 )
i s e q u i v a l e n t t o p QG i n t R ( t ) f o r a l l t > 0 .
R e m a r k 1
I f i n t R ( T ) фф f o r a f i x e d T > 0 t h e n i n t R ( t ) фф f o r a l l t ê T .I n d e e d , l e t x 0( . ) b e a n a d m i s s i b l e t r a j e c t o r y s u c h t h a t x 0 ( T ) G i n t R ( T ) .
F o r e v e r y t S T t h e s y s t e m
d x—i p = f ( t , x , u Q( t ) ) , x ( T ) G R ( T ) , d e f i n e s a n e i g h b o u r h o o d V t s u c h t h a t
x 0 ( t ) G i n t V t . B u t e v e r y c o n t r o l
w h e r e u ( . ) G w i l l a l s o s a t i s f y u j ( . ) G a n d t h e r e f o r e V t Ç R ( t ) f o r a l l t â T . H e n c e x 0 ( t ) G i n t R ( t ) f o r a l l t ê T .
L o c a l c o n t r o l l a b i l i t y o f ( 1 ) i s a c h i e v e d i f t h e r e e x i s t s a T > 0 s u c h t h a t p 0 G i n t R ( T ) .
O u r p r o b l e m x 0 ( t ) G i n t R ( t ) i s m o r e r e s t r i c t i v e .
D e f i n i t i o n 2 . T h e s y s t e m ( 1 ) i s l o c a l l y c o n t r o l l a b l e a l o n g a n a d m i s s i b l e t r a j e c t o r y x 0 ( . ) i f x 0 ( t ) G i n t R ( t ) ( i n t R o ( t ) ) .
O n e s e e s t h a t c o n d i t i o n s ( 4 ) a n d ( 5 ) i m p o s e d o n ( 1 ) i n v o l v e l o c a l c o n t r o l l a b i l i t y a l o n g x 0 ( . ) = p 0 .
I n g e n e r a l w e c a n s a y t h a t , i f w e s o l v e t h e p r o b l e m o f l o c a l c o n t r o l l a b i l i t y a l o n g a f i x e d t r a j e c t o r y , w e c a n d e c i d e a b o u t l o c a l c o n t r o l l a b i l i t y .
5 . 2 . F i r s t - o r d e r l o c a l c o n t r o l l a b i l i t y
I n f a c t c o n d i t i o n ( 4 ) i n v o l v e s t h e s u r j e c t i v i t y o f t h e d i f f e r e n t i a l i n
Ч ) - * ( p o > U ) o f a c e r t a i n o p e r a t o r T t t a k i n g i t s v a l u e s i n R ( t ) ( R 0 ( t ) ) . T h i s f a c t i m p l i e s t h a t t h e c o n d i t i o n s o f t h e i m p l i c i t - f u n c t i o n t h e o r e m a r e s a t i s f i e d a n d a s a c o n s e q u e n c e w e o b t a i n x 0 ( t ) G i n t R ( t ) ( i n t R 0 ( t ) ) . T h i s i s t h e r e a s o n o f t h e l a b e l " F i r s t - o r d e r " . I n o u r c a s e t h e o p e r a t o r T t i s d e f i n e d a s f o l l o w s :
T t ( u ( . ) ) = x ( t , u ( . ) ) , w h e r e x ( t , u ( . ) ) s a t i s f i e s ( 1 ) .
B y d e f i n i t i o n t h e d i f f e r e n t i a l d T t i s
(1 )
t
d T £ ( u 0 ; u ( . )) = J ^ ( b) B u (s ) d s
to
IAEA-S MR-1.7/30 137
w h e r e t h e m a t r i x SC ( t ) s a t i s f i e s
SC ( 0 ) = E , - — = 8C ( t ) A (6 )
a n d
S u p p o s e t h a t ( 4 ) a n d ( 5 ) a r e s a t i s f i e d , a n d d T t i s n o t a s u r j e c t i o n . T h e n a k 0 G R n , k 0 = 1 , w i l l e x i s t s u c h t h a t
B u t t h i s m e a n s k 0 ^ f ( s ) B = 0 o n [ 0 , t ] a n d , t a k i n g t h e d e r i v a t i v e s u p t o t h e o r d e r ( n - 1 ) a t s = 0 , w e o b t a i n
1 ^ = 0 , k g A B = 0 , . . . , k 0 A ( n ' 1 ) B = 0
F r o m ( 4 ) f o l l o w s k 0 = 0 w h i c h c o n t r a d i c t s | k 0 | = 1 .W h e n c o n d i t i o n ( 5 ) i s n o t s a t i s f i e d w e h a v e t o r e p l a c e ( 4 ) b y a s i m i l a r
h y p o t h e s i s .
H y p o t h e s i s J
T h e r e e x i s t u 0 ( . ) , . . . , u m ( . ) 6 ^ ( ^ ) a n d t h e n a t u r a l n u m b e r s n ^ . . , n m s u c h t h a t ( 1 - в) u0(.) + 9 u]-(.)e% '® J) f o r e v e r y 0 6 [ - 1 , 1 ] a n d
R e m a r k 2
I f u n ( . ) , U j ( . ) , . . . , u m ( . )&{Щ> a n d f ( t , x , u ) = f ( x , u ) t h e n t h e r a n k c o n d i t i o n i n h y p o t h e s i s J b e c o m e s
J k 0 № ) B u ( s ) d s = 0 V u (. )е$СЩ)
w h e r e
V j ( t ) = SC(\) b j ( t ) , b j ( t ) = — ( t , x 0 ( t ) , u 0 ( t ) ) ( u j ( t ) - u 0 ( t ) )
111
r a n k ( Y j ( 0 ) , [ a d X , Y j ] ( 0 ) , . . . , [ a d X , Y j ] ( 0 ) , . . . , Y m ( 0 ) , . . [ a d X , Y m ] ( 0 ) ) = n
138 VÂRSAN
w h e r e
9 fX ( x ) = f ( x , u 0 ( 0 ) ) , Y . ( x ) = — ( x , u 0 ( 0 ) ) [ u j ( 0 ) - u 0 ( 0 ) ]
[ a d X , Y ] = ^ X ( x ) - Y ( x ) ( L i e b r a c k e t ) d x d x
R e m a r k 3
I f O G i n t U t h e n h y p o t h e s i s J i s a s u f f i c i e n t c o n d i t i o n f o r l o c a l c o n t r o l l
a b i l i t y o f t h e l i n e a r s y s t e m
fix- f = A ( t ) x + B ( t ) u , x ( 0 ) = 0 , t i O , u G U
T h e o r e m 1 . S u p p o s e ( 1 ) s a t i s f i e s h y p o t h e s i s J . L e t x n ( . ) b e t h e c o r r e s p o n d i n g s o l u t i o n t o u 0 ( . ) i n ( 1 ) . T h e n
x Q( t ) G i n t R ( t ) ( i n t R Q ( t ) ) f o r a l l t > 0
P r o o f
L e t u Q( . ) , . . . , u m ( . ) £ ¿ § / ( ^ ) b e g i v e n b y h y p o t h e s i s J .D e n o t e
j=i
L e t T > 0 b e f i x e d a n d d e f i n e R ( T ) Ç R ( T ) a s t h e s e t o f a l l v e c t o r s x ( T , u Q( . ) ) w h i c h c o r r e s p o n d t o
d t
w h e r e
A ( t ) = — ( t , x Q ( t ) , u 0 ( t ) ) , B ( t ) = - ( t , x 0 ( t ) , u 0 ( t J )
m
m
0 : [ 0 , Т ] — П [ - 1 , 1 ] , 0 ( . )l
a p i e c e w i s e C°° f u n c t i o n , a n d x ( . 0 ( . ) ) s a t i s f y i n g ( 1 ) . D e f i n e T : & ( 0 , T ; © ) - > R n b y
T (0 ) = x ( T , u 0(.)) (7)
IAEA-SMR-17/30 139
w h e r e
Ш.
& = Д [ “ 1 , 1 ] . a n d ( 0 , T ; &) i s t h e s p a c e o f a l l p i e c e w i s e C ° ° f u n c t i o n s1
0 : [ O , T ] ^ © .B y d e f i n i t i o n T ( 0 ) = x 0 ( T ) a n d
m T
d T ( 0 ; 6 ) = ^ - a ( T ) J в. ( t )3T(t) I I ( t , x Q( t ) , u 0 ( t ) ) ( u . ( t ) - u 0 ( t ) ) d t ( 8 )
j=l 0(S(l, ivll _ -, Ш
W e s h a l l s h o w t h a t t h e r e e x i s t 0 ( . ) , . . . , в G < ? / ( o , T ; R ) s u c h t h a t t h e v e c t o r s d T Í O j é ! 1) , i E { l , . . . . , n } , a r e l i n e a r l y i n d e p e n d e n t ; i n o t h e r w o r d s , d T ( 0 ; . ) : 4 / ( 0 , T ; R m ) - > R n i s a s u r j e c t i o n .
D e n o t e
V j ( t ) = &T(t) I I ( t , x 0 ( t ) , u 0 ( t ) ) ( u j ( t ) - u 0 ( t ) )
a n d s u p p o s e d T ( 0 ; ^ ( 0 , T ; R m )) = ^ R n . T h e n a k 0 6 R n , | k 0 | = 1 , w i l l e x i s t s a t i s f y i n g
T
< k 0 , y ' e j ( t ) ^ ( t ) d t > = 0 v j = i ...........r n , v a ¡ ( . ) e ^ ( o , t ; r ) ( 9 )
0
F r o m ( 9 ) f o l l o w s
< k Q, Vj. ( t ) > = 0 V t € [ 0 , T ] , j = l , . . . , m ( 1 0 )
N o w , t a k i n g t h e d e r i v a t i v e s u p t o t h e o r d e r n j a t t = 0 w e g e t f r o m ( 1 0 )
< k Q, V j ( 0 ) > = 0 , . . . , < k 0 , £ - J i ( 0 ) > = 0 , . . . , < k 0 , d . ■ j k ( 0 ) > = 0 ( 1 1 )d t d t
U s i n g t h e r a n k c o n d i t i o n o f h y p o t h e s i s J w e o b t a i n f r o m ( 1 1 ) k 0 = 0 c o n t r a d i c t i n g
I k 0 1 = 1 a n d t h e a s s e r t i o n i s p r o v e d .
L e t V £ R n b e a s u f f i c i e n t l y s m a l l s p h e r e c e n t r e d a t t h e o r i g i n s u c h t h a t
n
J ^ e . a \ t ) e [ - 1 , 1 ] f o r a l l 0 6 V , j e { i ...........m } , t e [ o , T ]
i= l
D e f i n e a n e w o p e r a t o r T:V~>Rn b y T ( 0 ) = x ( T , u 0 ( . ) ) , w h e r e
n m
S e = U o ( - ) + I 0 i & Í ( - ) ( u i ( - ) _ U o ( - » i = i j = i
140 VÁRSAN
B y d e f i n i t i o n T ( 0 ) = x 0 ( T ) a n d d X ( 0 ; . ) : R n ^ R n i s a s u r j e c t i o n . U s i n g t h e i m p l i c i t - f u n c t i o n t h e o r e m w e s e e t h a t t h e r e e x i s t s a s p h e r e S c e n t r e d a t x 0 ( T ) a n d 6 : S ^ V s u c h t h a t T ( 0 ( y ) ) = y f o r a l l y G S . B u t f o r e v e r y y e S w e h a v e
^ 0 ( у ) ( - ) е ^ (Щ,) a n d t h e r e f o r e S Ç R ( T ) ( R 0 ( T | .S o , x 0 ( T ) e i n t R ( T ) ( i n t R 0 ( T ) ) a n d t h e p r o o f i s c o m p l e t e .
5 . 3 . S o m e r e m a r k s a b o u t f i r s t - o r d e r l o c a l c o n t r o l l a b i l i t y a n d f i r s t - o r d e r n e c e s s a r y c o n d i t i o n s
I n t h e c o n d i t i o n s o f t h e o r e m 1 w e c a n a c h i e v e t h a t f o r e v e r y T > 0 a r b i t r a r i l y f i x e d , x 0 ( T ) d o e s n o t s a t i s f y P o n t r j a g i n ' s p r i n c i p l e ( f i r s t - o r d e r
n e c e s s a r y c o n d i t i o n s ) w i t h r e s p e c t t o a n y f u n c t i o n a l < c , x ( T ) ) > , c e R n , | c | ф 0 , w h e n t h e c o n t r o l s y s t e m i s g i v e n b y ( 1 ) f o r t e [ 0 , T ] , S u p p o s e t h a t t h e r e
e x i s t c e R , I c j ф 0 s u c h t h a t x 0 ( . ) s a t i s f i e s P o n t r j a g i n ' s p r i n c i p l e .T h i s m e a n s t h a t Ф: [ 0 , T ] - > - R n w i l l e x i s t s u c h t h a t
,/' ( T ) = C ’ ‘ dt Ü ( t - x 0 ( t ) , u o(-t)) (12)
< 0 ( t ) , f ( t , x ( t ) , u ( t ) ) > = m i n < 0 ( t ) , f ( t , x ( t ) , u ) > , t e [ 0 , T ]u u u E D u
( 1 3 )
W i t h h y p o t h e s i s J , w e o b t a i n f r o m ( 1 3 )
< 0 ( t ) , ( t , x 0 ( t ) , u 0 ( t ) ) ( U j ( t ) - u Q( t ) > = 0 V t € [ 0 , T ] , j e { l , . . . , m } ( 1 4 )
T a k i n g t h e d e r i v a t i v e s u p t o o r d e r n j a t t = 0 w e o b t a i n f r o m ( 1 4 )
<Ф(0 ) , V ( 0 ) > = 0 ...........< 0 ( 0 ) , d - УЧ® > = 0 ..............< 0 ( 0 ) , 1 — Y ü Î 2 1 _ > = o ( 1 5 )1 d t d t m
a n d f r o m t h e r a n k c o n d i t i o n o f h y p o t h e s i s J i t f o l l o w s t h a t 0 ( 0 ) = 0 . T h e r e f o r e , ф(T ) = 0 , t h u s c o n t r a d i c t i n g ф(T ) = c ф 0 . C o n c l u s i o n s ( 1 2 ) a n d ( 1 3 ) a r e t h e s a m e a s
< c , x ( T ) > g 0 ( 1 6 )
f o r a l l x ( . ) s a t i s f y i n g
- f (t, xQ (t ), uQ (t )) ] (17)
IAEA-SMR-17/30 141
where
are arb itrarily chosen.In fact,we have achieved that in the conditions of theorem 1 (16) does not
hold fo r every с € Rn, | с | Ф 0. But if (16) is not satisfied fo r every с e Rn, с фо, it follow s that the convex cone
L (T ) = {y € Rn: у = x (T ), x (.) sa tisfies (17)}
contains the origin in its in terior; hence L (T ) =Rn. T h ere fore , hypothesis J is a sufficient condition fo r L (T ) = Rn, fo r every T >0. The conclusion of theorem 1 rem ains valid if we rep lace hypothesis J by L (T ) =Rn fo r every T> 0.
T heorem 2. Let (x„(.), un(.)) be an adm issible pa irs such that L (T ) =Rn fo r every T > 0. Then x 0(t) 6 intR (t) (intR 0 (t)) fo r all t> 0.
a . ( . ) e < g f ( 0 , T ; [ 0 , « > ) ) and Í ê 1
5.4. H igher-order lo ca l controllability
In general, the hypotheses of theorem s 1 and 2 are not n ecessary for lo ca l controllability along x0(.).
In the follow ing we shall show this by means o f an exam ple. We consider the follow ing con trol system :
■x1 + u, ^ ^ - = - x 2 -x J , Xl(0) =x 2(0) = 0, U = {u G R : |u| S 1} (18)
choosing u0(.) = 0 we obtain x0 (t) = (x1 0 (t), x 20(t)) = 0 .Let c 0 = (0,1) and let T> 0 be fixed. We obtain
<c0,x (T )> = 0 V x ( .) (19)
Satisfying
Xj(0) =x2 (0) = 0, -X j+ ïï, ^ - = -x 2, t e [ 0 ,T ] (20)
where
u:[0,T]->R, ï ï ( . ) e ^ ( 0 ,T ;R )
Conditions (19) and (20) show that the hypotheses o f theorem s 1 and 2 are not satisfied .
How ever, we can show that 0 £ in tR (T ) fo r all T >0 .
142 VA RSA N
F rom (18) it follow s that
T
x 1(T)=e~r J ' e 'uftjdt о
T t
x g(T) =-е~т e2t( ^ e su (s )d s )d t о о
In ord er to prove O e in tR (T ) it is sufficient to show
0 € in tR (T ) where R (T) =eTR(T)
Define ïîjt.J^ïïgt.JitO.Tl^R as follow s:
Uj(t) =-
e_t, t e 0, — ) and U 2( t ) = - U X ( . )
- e _t, t e TP T
By definition
T
J ¿ ïï.(t) dt = 0 , i = 1 , 2
о
On the other hand we have
t t, t
esu 1 (s) ds = -о T -t , t e
t - t , t e > - ? )/ esil2 (s) ds ='
~T — TJ0 t -T . t e T
2 2
( 2 1 )
(2 2 )
By definition
J e J esu1 (s )d sJ dt = 60 > 0
о 0
(23)
and
' e 2t ( j e u2(s) ds J dt = -6Q< 0 (24)
IAEA-SMR-17/30 143
Denote
u ^ t .e .y ) = e2 ^ j - y ^ y j 6 ( - l , l ) , t e [ 0 ,T ]
u(t, e ,6) = «(óü^tJ + U -filu ^ t)), te [o ,T ]
Hence follow s
T
Z e.ô.y ) =J e* (Uj(t, e , yj ) + u(t, e , 6 )) dt = e2yx (25)
о о
for y2e ( - 1 , 1 ).Using (23) and (24) we obtain from (26) that, fo r e0 > 0 sufficiently sm all,
Z 2 (e0, в ,у 1 ,у 2) changes its sign as a function o f 6 € [0,1] for every уг ,y2 € (-1 ,1 ). In this way to every ya, y2 G (-1 ,1 ) there exists a 6 (yj , y2) e (0,1) such that
is included in R (T ) and this finishes the proof o f our assertion .In the follow ing we are concerned with the non-linear controllability
prob lem when the hypotheses o f theorem s 1 and 2 are not fu lfilled.Definitions and notations are the same as before .A s we saw such hypotheses are used in fir s t -o rd e r non-linear con tro ll
ability ; they guarantee that the firs t differentia l of a certain operator is su rjective.
When such a hypothesis is not satisfied we have to use differentials of higher ord er and this is the reason of the label "h ig h er-o rd er" lo ca l controllability . B esides, in order to solve the problem of controllability we have to use a b ifurcation theorem instead of the usual im plicit-function theorem .
We must begin with a condition im plying that the hypotheses of theorem s 1 and 2 cannot be fu lfilled .
Suppose that there exist R , |Xj| = 1, and €o> 0, such that
Z2(e0,ô(y1,y2),yi,y2)=O.T h ere fore , we have achieved that the open set
<X1 ^ '( t ) ( f ( t ,x 0(t), u(t)) - f ( t ,x 0(t), u 0(t))> =0 V t e [0 , e0] (27a)
and u (.)€ < 2 / of c la ss C* where the m atrix Sf{ t) is defined by (6 ).
144 VÂRSAN
Let (x0(.), uQ(.)) be an adm issible pair such that (27a) holds.
Then the hypotheses of theorem s 1 and 2 are not satisfied .
P roo f
We write (27a) in an equivalent form
<X^.^(e0), x(eQ)> = 0, fo r all x: [O,e0 ]-*Rn (27b)
Satisfying
Ix ( 0 ) = 0 , ^ (t.X pM ugttM x+J^ftK fit.X pttbU jit))
j = l
- f ( t ,x Q(t),u 0(t)) ) (28)
where a j(t )s 0, St ê 1 are arbitrarily chosen.But (27) and (28) prove that the hypothesis in theorem 2 is not satisfied
fo r t e [0 , eQ].Suppose that hypothesis J in theorem 1 is satisfied . Then
<\3T(t) (t, x0 (t), uQ(t)) (u. (t) - uQ(t))> = 0 V tG lO .C j]
and j e { l , . . . ,m } , where u^(.) given by hypothesis J is C°° on [0 ,ex] for all j S { 1 >..., m }.
D ifferentiating up to the ord er n at t = 0, fo r every j, we obtain
.nm v<x1v1(0)> = 0,.. <xa, — ¡г-ш-(° )> =0
dt m
and, using the rank condition, we have Xj = 0 and this contradicts (27a). The p roo f is com plete.
Let X j,..., XnG Rn, I Xj| =1 be such that Хг,. . . , Xn is an orthogonal base in Rn, where Xj is given in (27a).
Let Hg(t, x , u) = <Xg, t ) f(t, x , u))>, fo r 0 = 1 , . . . , n.L et U j(.)e <2/, i e {1 , . . . , n} be fixed. F or every p iecew ise C“ or¡(t)S O,
i £ { l , . . . , n } , let x: [0,00) ->Rn be the solution of the system
x (0 ) = 0 ^ =|^-(t,x0 (t),u 0 (t))x
P r o p o s it io n
n
+ ^ 0 . ( t ) ( f ( t , X Q( t ) , U. ( t )) - f ( t , X 0( t ) , Uq ( t )) )
i = l
(29)
i = l
dt ' (30)
F o r every m e N, m S 2 denote
T
y ® ( T , ( x ( . ),<*(.))= J d™H0( t ,x o(t),u o(t); x(t))0
n
+ ( m ! ) ^ a . í t H d ^ H g t t . x ^ t J . U j í t b x í t ) ) i = l
-d “ ^Hgtt, x Q(t), u0(t); x(t)) )
where djH denotes the d ifferential with respect to x.
Hypothesis H
F o r every t > 0 there exist u ^ .), un(.)G a// such that M(t) = Rn"1, where
M(t) = { y £ R n: there exists x (.) satisfying (29), y =x(t)}
Definition 3. We say that the control system (1) has conjugate points of order m , т й 2, along (xn(.), % (.)) if hypothesis H is satisfied and there exist T > 0, x^ t), x 2 (t), t G [0 ,T], solutions of (29) such that
(ij) a\.))> a2 ( . ) )< 0 , x 1 (T) = x 2 (T) = 0
(i2) d^H git^ ft), uQ(t); x 6(t)) = 0, dj 1 He(t, x 0(t), Uj(t); xs(t))
= d^H git.Xgit), U0(t)); x 6(t))
fo r all i = 2 , . . . m - 1 , 0 = 1 , . . . , n, j = l , . . . ,n , t € [0 ,T ], where x s(.) = 6 x1(.)+ ( l - 6 )x 2(.) 6 € [0,1]; T w ill be ca lled a conjugate point of order m .
Theorem 3. Let (x0(.), u0(.)) be an adm issible pair such that (a) is satisfied .
Suppose there exist m g 2, m natural, such that every T > 0 is a conjugate point o f o rd er m.
Then x n(t)€ intR(t)(intR„(t)) for all t> 0.
IAEA-S MR-17/30 145
P roo f
Suppose f( t ,x , U) is a convex set fo r every ( t ,x ) 6 [0,°°)XRn.Denote A= (Xv . . . , ÀJ. In order to show that x 0(t) e intR (t) for all t> 0,
it is sufficient to prove that y0 (t) e in tR (t), w herey (t) = Æ â ^ t )x .(t),R (t) = AT ( t )R ( t ) .
146 VÂRSAN
T o each adm issible pair (x (.),u (.)) we have a corresponding (y (.),u (.J y (t )E R (t) , t§ 0 , satisfying the follow ing con trol system :
y (0) = ЛТр0, ^ = ЛТ<Г(t) f f t ^ t J A y , u(t»dt
- ÁF¿%~(t) Ц ( t ,x 0(t ) ,u 0(t))^ ‘ 1 (t)Ay (31)
Denote by F (t ,y ,u ) the right-hand side of the con trol system (31); then we can write
(t,y 0 (t ),u 0(t ))= O (32)
Let T> 0 be arb itrarily chosen. Without lo ss of generality we choose T S £ 0, where eQ is given by (27a).
By hypothesis U ji.),..., un( . ) e w ill exist such that all the solutionsof the system
n
у (0 ) = 0 , ^L= (t )A ^ (t ) [f (t ,x 0(t),u ¡(t) - f ( t ,x 0(t), u^t))],i=l
t e [0,T],(a.(t)ê 0)' (33)
satisfy
Уха ) = 0, t e [0,T] (34)
R " ' 1 = {(y2 (T ), . . . , y n(T)) : y (.) sa tisfies (33)} (35)
Since T is a conjugate point, or1 (.) = (c^(.), .. . ,a j( .) ) , cP-{.) = (û^(.), ...,a^(.)) and x l( .) , x 2(.) solutions of (29) w ill exist such that
J ^ Í T .x Y ) , a 1 ( . ) )> 0 ,J ? 1 (T ,x2(.), a 2 (.))<0 , x X(T) =x 2(T) = 0 (36)
У((Т, x ó(.), a6(.)) = 0, fo r a ll SL = 2 , . . . ,m - l , в = 1, . . . , n (37)
where
x6(.) = 6 xx(.) + ( l - 6 )x 2(.), a6(.) = 6a\.) + (l- ô )a 2(.)
Let
y !(t) = A ^ lt jx ^ t ) , y2 (t) = A<T(t)x2 (t)
By definition,у 1 (.). i = 1.2, satisfy the system (33).
Denote y6(.) = 6 y l(.) + (1-6 ) y 2(.). F or every p = (p j,..., pn) € Rn, p. г 0, define
« j (t, e , 6 ,p) = etc. (t, 6 ) + em p.
Let y (t, e , 6 , p), t S [0 ,T ], be the solution o f the system
n
y (0 )= V Í t t = uo^® ô ' p) - и£йй ■ F (t ,y , n0(t)) ] (38)i=l
where y0 = Лтр0, and F is defined in (31).F o r e = 0 we have y(t, 0, 6, p) =yQ (t), t G [0 ,Т].Using the sm ooth dependence o f the solution with respect to the
param eter e , we obtain from (38)
y ¡(t ,e , 6 ,p ) =y0i i (t )+ ey 6_.(t)
+ e2 y2 (t ,x0 (.) , ae(-))+ ... + e™'1^ (t,x 6( . ) ,o 6(.))
+ e my¿ ( t ,6 ,p ) + ст. (em , t , 6 ,p ) (39)
i e { l , . . . , n}
where
lim —*- -*■- ’ ^ = 0, uniform ly with resp ect to t e [0,T] and (6 ,p),€ “*0 ç
in a fixed bounded set.Using (26) and (37) we obtain from (39)
у. (T, e , 6 , p) =y0 i(T) + em y. (T ,6 , p )+ 0.(em,T ,6 ,p ) (40)
such that ÿj has the property
y1 ( T , 6 ,p ) = I1m (T ,x 0( . ) ,a6(.)) (41)
(y2 (T ,« ,p ) ...... yn( T , 6 ,pJ).
ДЦ T= ^ J p. A ^ (t) [f(t,xo(t),u.(t)) - f ( t ,x 0(t), u0(t))dt
i = l 0
IAEA-SMR-17/30 147
+Д п(1’ , х 0(.),о?0(.)) (42)
where ...... S * ) .
148 VÂRSAN
Let us re ca ll the ideas used in the bifurcation theorem of section 2 . F rom the im plicit-function theorem we obtain
p(e, 6 , z) = (pa(e, 6 , z), . . . , pn(e, 6, z)), p .(e , 6, z ) iO
such that
(ya(T, 6, p), . . . ,y n ( T , 6 ,p)) + CT(em/.rJ - .6’-P) = z, в =(е2,..,вп) (43)
fo r all e € [0,e“] and z € S0 where S0 Q R 11"1 is a sphere centred at the origin . Denote
у . ( Т ,е ,б ,р ( е , 6 ,г ))= у ; (T ,e , 6 , z)
from (40), (41) we see that y2 (T , e, 6, z) - y0 j(T ) - e m+1Z j changes its sign as a function o f ó € [0,1] fo r every z £ S0, Z j€ (“ 1,1) if e =e* Se is sufficiently sm all.
T h ere fore , fo r every (z j, z ) e (-1 ,1 )X S 0 there exists a 6 (z1( z )€ [0,1] such that
y 1 (T ,e * ,6 (z1, z), z) = (e* ) m + 1 z 1 +y0<1 (T) (44)
(y2(T,e*,S(z1, z), z ) , . . . ,y n(T ,e* , 6 (zp z), z)) = (e*)m z + (y0 2 (T ),.. .,y 0 n (T))
(45)
In conclusion y0 (T)E in tR (T ).The proof is com plete when {f (t ,x ,u ) : u € U} is a convex set for every
(t ,x ). If the convexity condition is not satisfied then, using G am krelidze 'slem m a, we can approxim ate the function
n n
^ a . ( t ) f ( t ,x , (t) = 1 by f(t,x ,u (t)), u ( . ) E ^i=l i
such thatt" n
/ f ( t , x , u ( t )) - V a ( t ) f ( t ) , x , Uj (t) ) d tdV
L__ii = l
is sufficiently sm all fo r all t ' , t" S [0,T],This allows us to treat the non-convex case as a convex one.In this case the proof is m ore com plicated and we shall om it it here.
Sufficient conditions of existence of conjugate points
In theorem 3 an im portant ro le was played by the hypothesis concerning the existence of conjugate points. It is desirable to give such sufficient
IAEA -S M R-17/30 149
conditions which involve the basic param eters of the con trol problem , namely f, U and p0.
Suppose that there exist ^ . . . ^ „ e U such that
Suppose that (27a) is satisfied and let the X ^ ..., Xne Rn be mutually orthogonal, j X. j = i ( where X¡ is given by (27a).
F or every m i 2 define the tensors A®.(t), B®(t) as follow s:
fo r 9 = j = 2 , . . . ,m - l , where H0 (t, x, u) = < Xq,£HT(t) f(t, x , u )X and dj isthe differential with resp ect to the variable x.
Denote R" = {a e Rn: a > 0} .
Hypothesis I m
(ij) There exist X, 6 Rn and u , u n 6 U such that (27a) and .(27c) are satisfied ;
(12) A 17> 0 w ill exist such that A®(t) =B®(t) = 0 fo r te [0,tj] j e {2 , . . . , m - 1 }, в = 1 , . . . , n,
(13) А к a 0, к natural, w ill exist such that either
f(o, p0, uj ) ■ f(o, p0, u0(°)), j e { i , . . . , n }
are in a general position and
f (0 , p0, uQ(0 )) € intrel co { f ( 0 , p0, u 1 ) , . . . , f ( 0 , p0, un)} (27c)
F o r every u¡ thus defined let x ¡(.) be the solution of
x ( 0 ) = f(0 , po, Uj) - f ( 0 , p0, u0(0 )), ^ H (t, xQ(t), u0 (t)) x (2 7 d )
- d j 1 HQ(t, x Q(t), uQ(t); x. ( t ) ,. . . ,x . (t))2 j J lj. i2, .
and
(m times)
150 VÂRSAN
change th e ir s ign s , o r
f 4 0 ) = 0 , i € { 0 , l , . . . , k - 2 } , - J f i 11 (0 ) = 0 , i e { 0 , l , . . . ,k }d^B1
and
change their signs.
T heorem 4. Let (x0(.), Uq(.)) be an adm issible pair such that hypothesis Im is satisfied . Then every T > 0 is a conjugate point o f order m .
P roo f
Let î i ê r J be arb itrarily fixed and let x ( t .e .a ) be defined as follow s:
n
(46)о i=l
where X j ( . ) , i € { l , . . . , n } are defined in (27d), and
By hypothesis
d jH gtt.x^thu^t); x (t,e ,a j) = 0 (47)
n
(48)
Now suppose condition (a) in hypothesis Im to be satisfied . Let S i,S 2 6 R" be given by hypothesis. It follow s that
IAEA-SMR-17/30 151
Define the functions a| (t)s 0, i £ { l , . . . , n } , j G {1 ,2 }, such that
n
A ^ (t )[ f (t , x0 (t), u.) - f(t, x Q(t), u 0(t))]i=l
= ^ A [ f ( 0 ,po, u. ) - f ( 0 , pQf uQ(0 ))] i=l
for all t e [0 , 6], where 6 > 0 is sufficiently sm all. This is possib le because o f (27a) and (27c). We define x J (t, e), j 6 {1,2} as follow s:
Xj ( t , e ) = ( ^ J <p(s, e) d s ^ ^ a j x ^ t ) = t ^ l - 0 ^ a ij x i ( t )
0 i = l i= l
It fo llow s that xJ sa tisfies (46) fo r 0 = 3 ' and we obtain
T m
j £ ( T , x J( . ,e ) ,« J(.))= / { t m( l - 0 <A1m(t)Si ...... S j >0
/ \ / 4m-l
+ <BÍ, (t )71(t)Sí . . . . , S 1>
<Bm (t)S J, . . . ,S J>(m times)
dt
where 7 J(t) = a (t) - and a (.), j 6 {1,2} , are defined by (50). By d irect calculation from (52) we get
j ^ ( T , x j (.,e ),a j (.))= I tn,-3( l - y ( l - Ç - < (B M t)S J...... S J>
+ +tm^ l - ^ <A1m(t)5 i , . . „ S J>j-dt
Expanding B ^ (t) and Aam(t) in T aylor se r ie s , we obtain from (53)
T ,x J(.,e),aJ(.)) =
(50)
(51)
(52)
(53)
K ‘ - 0 ......*■>
+ »»(t))+tm( l 4 T ^ r ( < ^ ^ ( 0 ) S i, . . . , 3 i >+ÔJ(t))}-dt (54)
where lim (t) = lim J* (t) = 0 . t-+ о t-* о
152 VÂRSAN
Choosing T =e, fo r e sufficiently sm all, we obtain
1 'P -1/ / ,k_l
0
- ek+m ^/d^m _ (0 ) o j , . . , 5 J > c 2 (k) + (5 5 )
where c(k)> 0 , lim # (e) = 01 e-> о
f tm( l - i j tk( < ^ (0 ) « j ...... Si > + P (t ))d t
= ek+m 1 (c2 (k )^ —^ g L(0 )g j ,...,a?y + $2(e)), where CgikjX),
lim bo (e) = 0 (56)£-*o
T h ere fore , an e0> 0 w ill exist such that , Xj (,,e0), a^(.)) has thesam e sign as
(■1)к + т ф ( о ) г . . . , г О
Since xi(ep » *%) = 0 (see (51)), we choose x (t, Cq) =0 fo r all tê e0 so to achieve that x ^ t ,^ sa tisfies system (29) corresponding to 5{(t) defined as follow s:
Ï (t) ’аЦt) t e [O ,e 0]
t> e0
the ûfj(t) are defined in (50).Hence follow s
^ ( t . x J( . ,e 0). e J( J = ^ ( е 0 , х Ч ,е 0)<*]'(.)) (57)
for all t s e0, j € { 1 ,2 } .Now, using (56) and (57) we obtain
J ^ (t ,x 1 ( . ,e 0), » 1 ( . ) ) > O ,^ ( t ,x 2 ( .,e 0)Ja (.)) < 0, xj (t, eQ) = 0 (58)
for all t s e0.F inally , (47), (48) and (58) prove that every t ë e 0 sa tisfies conditions
(ij) and (i2) fo r conjugacy. Hypothesis H is thus satisfied .
IAEA-S MR-17/30 153
T herefore every t s e 0 is a conjugate point of order m . But what we did fo r e„ we can do fo r every 0< e §e0 and the p roo f is com plete for case (a).
Now, suppose that the condition (a1) in the hypothesis .#m is satisfied.In the sam e way as in the case (a) we can achieve that , /^ ( T , x-'(., T ), a? (.)) has the sam e sign as
( -D k+m < ^ A m ( ° ) 3' ......& > for a 1 1 T> °-
A s a consequence we obtain
^ ( t . x 1 ( . .c 0) ,S 1( J > 0 , J ^ ( t ,x 2(.,e 0), S2(.))<0
x J(t,e0)=O , j e { l , 2 } (59)
fo r all t è e0.Again using (47), (48) and (59) we can prove that every t i e 0 is a conjugate
point o f ord er m . Repeating fo r every 0< e Se0 what we did for e0 we com plete the p roo f a lso for case (a1 )• The entire p roof is thus com plete.
Some rem arks
Let us con sider the follow ing con trol system :
d x o i i— = h (t ,x )+ u g (t ,x )x (0 )= p 0, tSO, x € R , U = {u eR :| u | s i }
Define by uQ(t) = 0 and xQ(.) the corresponding solution. Suppose that h and g are C~ functions and that there exist 6 > 0, XjS R2, | Xj| = 1, such that
< \ , ^ ( t ) g ( t , x 0(t))>= 0 V t e [0 ,6 ]
where
- ^ ( t ) | | (t ,x 0(t )) ,^ (p )= E
Let us assum e g(O,p0)^ O .F o r such a system the conditions (27a) and (27c) (see the hypothesis
are satisfied when we choose
f( t ,x , Uj) =h(t, x )+ g (t , x ) , and f(t, x, u2) = h(t, x) - g(t, x)
In the follow ing we shall study (i2) and (i3) in the hypothesis In this case the functions (t )» , . . . , o X <A ^ ( t ) a , . . (аг,а2)Е R 2,
take the form cp(t) (o-j - a2)m where cp(t) is a sca lar function. As a consequence, we cannot prove fo r even m that hypothesis holds since cpitHo'j-<*2)m or
154 VÂRSAN
its derivatives with resp ect to t do not change the sign on R| . So, in order to study such system s from the standpoint o f loca l controllability we have totake into account only the ca se o f m being odd.
This situation is strongly connected with the form of the system . In the follow ing we shall apply theorem s 3 and 4 to three exam ples.
Exam ples
dx i dxp o t i(a) ——- = -x .+ u , —г— = -x p - x , , U = (u 6 R : u U l (dt 1 dt 1 1 1
(xL(0 ) , x 2 (0 )) = (0 ,0 )
We choose uQ(.) =0 and it follow s that x 10 (t) = x2 0 (t) = 0, tê 0. Now we shallshow that hypothesis is satisfied (see theorem 4).
Choosing Xj = (0,1), Uj = 1, u2 = -1 we achieve that condition (ij) is satisfied. Condition (i3) is satisfied fo r m = 3, к = 1. Indeed, by definition,
A ^t) = 0, B 2 (t) = 0, A^(t) = 0, B 2 (t) = 0, and <A13(0) a, a, a) = -6(a1-a2)3>0 fo r a = (ûîj, a2) £ R+, Cj <ff2.
Hypothesis being satisfied according to theorem s 3 and 4, OS intR (t) for all t> 0 .
dx-i о о dx 2 dx3 _(b) "dt^ = x 1+ x | - x § , - ^ ^ 2 + Uj,
U= { ( l i j . u ^ R 2: |uj S I } , x i (0 )= x 2 (0 )= x 3(0 ) = 0
We choose uQ(.) = (Ugji^), uQ2(.)) = 0 and it fo llow s that xQ(t) = 0. Define u1 = (1,1), u" = (1, -1 ), u1" = (-1 ,0 ).Condition (i^ of hypothesis is satisfied for
Xj = ( 1 ,0 ,0 ) and Uj = u' U jFu", u3 =u"'
Indeed, we have
e-t 0 0
¿^ (t ) = I 0 e_t 0 I and
<A1 ,áT (t)f(x 0 (t),u(t)) - f ( x 0(t ) ,u 0(t))> = e"1 (^ ¡^ - "
fo r all t ê 0, and u(.) G (<?/0).The vectors f(0,u ' ) - f(0, u 0(0)) = (0, u' ), f (0, u" ) - f(0, uQ(0)) = (0, u" ),
f ( 0 , u '" ) - f ( 0 ,u 0(0 )) = (0 , u '")
IAEA-SMR-17/30 155
are in a general position and
f (0 , u0 (0 )) = (0 , 0 ,0 ) e intrel co { ( 0 ,u' ), (0 ,u" ), (0 , u"' )}
Conditions (i2) and (i3) are satisfied for m = 2, к =0.Indeed, by definition B2 (t )= 0 , and
<A^(0 )a , a) = 2 [(o + a2 ~ » 3 f ~ ( Qj - o 2 f ] fo r a e R^
Choosing 0<S j = a2 = 2g we obtain
< A 2(0 ) S\ 51>>0
On the other hand, choosing o 2 = (S2, S|, S3) such that 3 2 +3| =a\, S2 ,>3|>0, we obtain <A 2 (0) e 2 ,S 2 >< 0 and the verification of hypothesis is finished.
Since x 0 (t) = 0, we obtain from theorem s 3 and 4 that for every t> 0 there exists a sphere St C R 3 centred in the orig in such that every x e St canbe joined with the origin , in the tim e t, by an adm issible tra jectory o f (b).
(c) =u, = 4 + xjXg, xa(0) =x2 (0) = 0, U = {u e R : j u| S 1}
We choose uQ(.) = 0 and obtain x0 1 (t) = 0, xQ2 (t) =4t, ^ ( t ) =E. F or Xj = (0,1), Uj = 1 , u2 = - 1 it follow s that (ix) in hypothesis is satisfied.
Conditions (i ) and (i3) are satisfied for m = 5, к = 2.By definition A j(t)= 0, for j = 2,3 ,4 , and B*(t) = 0 fo r all j g 2. M oreover,
we have A?(t) = 0, Bj (t)= 0 fo r all j g 2.F or j =5 we obtain ^ A ^ ft)» ,..., аУ = (5 ! ) x 0 2(t)(x 1 (t, a))5 where
Xj(0, а) =а1-а2, ^ 1. = 0
T herefore
dA1i ^ (0) a ,..,a У = 4(5 ! )(a1~ a2f and the sign of this expression changes on R2.
The condition of theorem 4 being satisfied , we derive from theorem s3 and 4 that’ fo r every t> 0 there exists a sphere St Ç R2 centred in (0,4t) such that St C R (t).
B I B L I O G R A P H Y
LIUSTERNIK, L .A ., SOBOLEV, V . I . , Elemente der Funktionalanalysis, Akademie-Verlag (1965).
NIRENBERG, L ., Functional Analysis, New York, University.
BERGER, M .S ., Perspectives in Non-Linearity, Benjamin, Inc. (1968).
156 VÂRSAN
VARSAN, С ., "Sufficient conditions for non-linear optimal-control problems", Revue Roum. Math. Pures et Appl. 19 4 (1974).
DUNFORD, N .. SCHWARTZ, I .T . , Linear Operators, Parti.
MARINESCU, G. Tratat de Analiza Functionalâ, 2, Ed. Academiei.
VÂRSAN, C ., Higher-order necessary conditions for extremality, RevueRoum.Math. Pureset Appl., 18, 4 (1 9 7 3 ).
VÂRSAN, C ., General Theory of Extremum Problems with Application to Optimal Control Theory, Ed. Academiei (1974) (in Romanian).
DOBOVITSK1J, A .Y a . , MILIUTIN, A .A . , Extremum problems in the presence of constraints, J. Vîcisl.Mat. i Mat. Fiz., N r.3 (1965).
PONTRJAGIN, L .S ., BOLTYANSKU, V .G ., GAMKRELIDZE, R .V ., MISCHENKO, E .V ., The Mathematical Theory of Optimal Processes, W iley, New York (1962).
LEE, E .B ., MARCUS, L ., Foundations of Optimal Control Theory.
GABASOV, R., KIRILLOVA, F .M ., Singular Optimal Controls, Nauka, Moskow (1973) (in Russian).
VÂRSAN, С ., General Theory of Extremum Problems with Applications to optimal Control Theory, Ed. Academiei (1974) (in Romanian).
VÂRSAN, С ., Pointwise higher-order necessary conditions for optimal control problems, Revue Roum.Mat.Pures et Appl. (in press).
LEE, E .B ., MARKUS, L -, Foundations of Optimal Control Theory.
HAYNES, G. W •, HERMES, H ., Non-linear controllability via Lie theory, SIAMJ. Control 8 (1970), 450.
LOBRY, C ., Contrôlabilité des systèmes non-linéaires, SIAM J. Control £ (1970 ) 573.
SUSSMAN, H .I* . JURDJEVIC, V ., Controllability of non-linear systems, J. Diff. Equations, 12 (1972) 95.
VARSAN, C ., First-order non-linear controllability, Revue. Roum. Math. Pures et A p pl., N o .l0 (1 9 7 4 ).
VÂRSAN, С ., Higher-order non-linear controllability, RevueRoum. Math. Pures et A p pl., N r .1(1975).
GAMKRELIDZE, R .V ., On some extremal problems in the theory of differential equations with applications to the theory of optimal control, SIAM J. Control, 3 (1965).
IA E A -SM R -17/46
SOME TOPICS IN THE M A T H E M A T IC A L TH EORY OF O P T IM A L CO N TRO L
T. ZOLEZZICentre o f Mathematics and Theoretical Physics,Genoa, Italy
AbstractSOME TOPICS IN THE MATHEMATICAL THEORY OF OPTIMAL CONTROL.
The paper deals with mathematical problems of optimal control theory for ordinary differential systems, including a sketch of optimal control for parabolic problems. Using the epsilon method (penalization of the dynamics), the most important results are derived (sometimes in a heuristic manner) for existence, necessary conditions, approximation of optimal controls, synthesis and sensitivity. Some functional-analytic aspects are discussed. A sketch of the time-optimal linear problem, and a detailed discussion of the linear-quadratic problem are given. Some examples and exercises are contained in the text. The present treatment is far from being complete, but many different results as the existence theorem, the maximum principle, and feedback control are derived in a constructive and unified way.
1. FORMULATION OF AN OPTIM AL-CO N TRO L PROBLEM AND SOME EXAM PLES
A sufficiently general problem in optim al-con trol theory, encom passing many im portant applications, is the follow ing: ' m inim ize
b
<p[x(a), x(b)] + J' f ( t ,x , u)dt ( 1 )a
subject to the follow ing constraints: we have a system o f ordinary d ifferential equations
x = g(t, x, u) ( 2 )
a set of initial and final conditions im posed on the state x
(a, x(a), b , x (b ) )£ E (3)
a pointwise constraint on the graph of the state
(t, x(t)) eG (t), a s t s b (4)
and som e pointwise condition on the con trol u
u(t) 6 V(t,x(t)) (5)
157
158 ZOLEZZI
In the above problem , in the follow ing denoted by (P ), a pair (u, x) defined in som e interval [a, b], u = (ux, ur) ', x = (xb .. . , x ^ )', is called adm issib le if the follow ing holds:
u is (L ebesgu e-) m easurable from [a ,b ] to Rr;
x is absolutely continuous from [a ,b ] to Rm;
t -» f(t, x(t), u(t)) is (L ebesgu e-) integrable in [a ,b ];
( 2 ) is satisfied alm ost everyw here in [a ,b );
(3), (4), and a lso (5) hold, fo r alm ost a ll t 6 [a ,b ].
f is (obviously!) a rea l-va lu ed function, g is an Rm-valued vector , both are defined on som e subset of the (t ,x , u) space, that is Rr+m + 1 (and E C R 2m+2). F or a given t, G(t) is a (non-em pty) subset of R m, so that G is a m ultifunction (this m eans a set-valued function). F or a given (t, x), V(t, x) is a given (non-em pty) subset of R r, so that V is a given m ulti-function. The problem (P) is a general form ulation of an op tim al-con trol problem d escribed by a system o f ordinary d ifferentia l equations ( 2 ), that represents the effect of the con trol u on the state x, with the pointwise constraints (3), (4), (5) governing both the states and the controls. The cost (1) we wish to m inim ize is a sum o f a term <p[x(a), x(b)] depending only on the initial and
bfinal states, and an integral term J f(t, x, u)dt. Many im portant exam ples
a
o f op tim a l-con tro l problem s are included in problem (P ). Frequently weare confronted with integral constraints o f the form
bJ h(t, x, u)dt â с
a t They can also be written in pointwise form (adding a com ponent y(t) = / hds
to the state x). Som etim es the dynamic relation connecting the con trol u with the state x, is given in the form o f a boundary-value problem fo r som e partial differentia l equations (this possib ility w ill be brie fly studied at the end of this paper), etc.
L et us show som e im portant sp ecific exam ples of an optim al control prob lem of the form (P).
1.1. L inear tim e optim al problem
Given an initial position x 0 S R m, we seek a con trol u ' such that uv(t) € V is a fixed restra int set, and such that the rendez-vous between a given continuous m oving point z = z(t) and the state x is the solution of
x(t) = A(t) x(t) + B(t) u(t)
x(0) « x , <6>
o ccu rs at optim al tim e; in other w ords, when we denote by x(u) the unique solution o f ( 6 ) fo r a given adm issib le con trol u, we seek an adm issib le u 1', defined in som e interval [ 0 ,t '|C], such that
t* = m in {t г 0 : x(u)(t) = z(t), u adm issib le} (7)
\
IAEA-SMR-17/46 159
The quantity t v in (7) is called the optim al tim e. Let us show that this t im e- optim al problem is a particular instance of problem (P). In fact we have
a = 0 , fixed , b fre e , 6 0 , <p - 0, î = 1,g (t ,x , u) = A (t)x + B(t)u, with A(t) an m X n, B(t) an m X r m atrix,
E = {(0 ,Х о ,Ь ,у ) :b S 0, y e R m}, G(t) = Rmfo r a ll t, V independent o f t, x
(a constant m ulti-function ).
In this problem the dynam ics ( 6 ) is linear, that is , the differential system is linear in x and u. This im plies (under som e integrability condition on A and B) existence and uniqueness o f x, with given u. Som etim es tim e-op tim al problem s involve a lso a term inal constraint x ( b ) €E T 0. Let us mention that in this problem a is fixed but b is fre e : so , in general, in problem (P ), the interval [a ,b ] is not given in advance, but depends on the adm issib le pair (u, x) defined on it.
G eom etrically we can interpret tim e-optim al con tro l as fo llow s:
In other w ords, rem em bering thatt
x(u)(t) = <Mt) [0_1(О)х0 + J '^ (s jB isJ u isM s о
(by the variation o f constants form ula, ф being a fundamental m atrix forx(t) = A(t) x(t)), we can write
tC(t) = {<Mt) [0_:HO) x° + J' ф'Чя) B (s) u(s) ds : u m easurable and u(s) e v )
оThen C(t) is the set of all points of Rm that can be reached at the instant t by using adm issib le con trols (this C(t) is called the set o f attainability at t). Our tim e-optim al problem can be seen as fo llow s: find the firs t instant o f tim e t'1' at which C(t) and z(t) in tersect.
1.2. L inear p rocesses with quadratic cost cr iter ia
A m easurem ent of the quality o f a con tro l system by integrating the system e r r o r squared over a fixed tim e interval provides a cr iter ion of
160 ZOLEZZI
perform ance of great im portance, from th eoretica l as w ell as practical points of view. These optim al control prob lem s, with tim e-optim al control, belong to the best known ones: both can be solved, in a sense, explicitly. The dynam ics is linear, as above:
x(t) = A(t) x(t) + B(t) u(t)
The tim e interval [0, T] is fixed, in contrast to the tim e-optim al problem s. The state x starts from the fixed in itial state xQ,
and is steered by the con tro l u to the term inal restraint set T0, that is ,
Given three rea l square m atrices P, Q, R, continuous and sym m etric on [0, T], Q positive definite, P and R positive sem i-defin ite ; the problem is to m inim ize the cost functional
(the prim e m eans transpose).Obviously, this is an instance o f problem (P ). In fact
(fix) = x 'R x , f(t, x, u) = x 'P (t)x + u'Q(t)u
g (t ,x ,u ) = A(t) x + B(t) u, E = { ( 0 ,x 0, T, y) :y e TQ}
G(t) = R m + 1 fo r all t, V(t, x) = R r fo r all t, x
As a third exam ple we m ention a problem with the m ore com plicated constraints (5) (depending on tim e and the state at that tim e).
1. 3. The follow ing op tim a l-con trol problem a rises in certain econom icm odels o f nuclear rea ctor power system s. M inim ize
b
x (0 ) = x 0
x(T )e T 0
T
0
subject to
with the constraints
0 su(t) s min {x(t) - r(t), h(t)}
IA E A -S M R -n /4 6 161
The above problem is another instance of problem (P ), withm
f(t, x, u) = k(t)u, g(t, x, u) = g 0(t) + b(t) u
E = { (O .x ^ b .y ) : q 1 S x Q § q2, b ê 0, y £ R m}, G(t) = R m + 1 fo r all t
V(t, x) = [0, m in { x - r ( t ) , h(t)} ]
2. THE ABSTRACT PENALIZATION METHOD
Many approaches exist to a m athem atical treatm ent of our problem (P). We shall fo llow a general method that seem s to be particu larly useful as regards the p ractica l and theoretica l purposes; it y ie lds quickly and naturally (at least form ally) som e o f the m ost im portant resu lts.
Let us begin with som e general rem arks. (P ), as form ulated above, is a constrained m inim ization problem : we seek optim al pairs (u'1', x '1') m inim izing the rea l-va lu ed functional (1) under constraints (2), (3), (4), (5). A ssum ing that conditions (2), . . . , (5) are com patible (that is , som e adm issib le pair ex ists), it is appropriate to con sider ( 2 ) as the m ost im portant constraint, and (3), (4), (5) as subsidiary ones. T herefore , problem (P) can be abstractly view ed as fo llow s: we want to m inim ize a rea l-va lu ed functional
subject to the constraints
g(z) = 0, z G U
w here U is a subset o f a given Banach space X ; Y is another Banach space, and g is a mapping from X to Y:
P rec ise ly , in (P) we take X as the cartesian product o f the two Banach spaces L 1 (I) and AC(I), equipped with the standard norm s
b
K z)
g : X - Y
u defined in [a, b]
ьx defined in [a ,b ]
a
I is a large bounded interval containing all domains o f adm issib le pairs: we put u = 0 , x(t) = x(a) when t < a, x(t) = x(b) when t > b , beyond the domain. This ch oice im plies integrability o f adm issib le con tro ls : as we
162 ZOLEZZI
shall see in section 4, this is a natural assumption. M oreover, we take Y = L^I), and
g : 1 Л 1 ) X AC(I) - 1 Л 1 )
defined by
g(u, x) = x - g ( . ,x ,u )
finally
U = {(u, x) e X : (3), (4), (5) valid}
(obviously U is not empty), and
7 : U -» (-», + °°)
Let us now fo r get problem (P ), and con sider the abstract constrained m inim ization problem (M)
m inim ize f (z) subject to
g(z) = 0 and z e U
given the Banach spaces X and Y, as defined above.Am ong the many approaches to problem (M), the penalization method
is one of the m ost interesting, from both num erical and theoretica l standpoints; it is o f specia l im portance fo r us. Replace (M) by the follow ing sequence of " fr e e " m inim ization prob lem s:
M inim ize fk(z) = f(z) + к ||g(z)|| on u (M k)
Intuitively speaking, if zj, is a solution to (Mk) as к ->■ °° zk must be c lo se to som e solution o f the orig inal constrained problem (M) : in fact ||g(zk)|| m ust approach 0 , otherw ise the penalization coe ffic ien t к would prevent7(zk) + k||g(zk)|| to afford a minimum fo r fk on U. In other w ords, points z, fo r which condition g(z) = 0 is seriou sly violated , w ill give r ise to high values o f к ||g(z)|| when к is large , and consequently such z w ill be unsuitable candidates fo r the m inim izing point zk. This suggests the assum ption that, if zk -» z as к then g(z) = 0 and z m inim izes f subjectto this constraint. This conclusion is a lso plausible from a physical point o f view . To quote Courant: "Q uite generally rig id boundary conditions should be regarded as lim iting cases o f natural conditions in which a param eter tends to infinity. This corresponds to the fact that rig id constraints are only idealized lim iting ca ses of very large restorin g fo r c e s " .
We can easily v e r ify the follow ing:
with b
a
Theorem 1. (Mk) has a solution zk, and som e subsequence of {z k} converges weakly to som e solution o f (M ), if
IAE A-SM R-17 /4 6 163
(a) X is a re flex ive Banach space;(b) U is a weakly closed set;(c) f(z) ->+ oo as ||z[| -» +(d) F, ||g ü are weakly sequentially low er sem icontinuous.
Sketch o f proof. F ix к > 0. Since f + к ||g || is weakly sequentially low er sem icontinuous, U is weakly c losed , and X is a re flex ive space, we see that (Mb) has a solution zk. F rom (c) we see that sup ||zk || < + °°. F rom re flex iv ity it follow s that som e subsequence o f {z k}, denoted as the orig inal sequence, w ill converge weakly to z 0. F rom (b), z 0 £U . By (c ) , s e m icontinuity and reflex iv ity , f is bounded from below , say
f(z) s m fo r a ll z
F or a ll к
f(zk) + k ||g(zk) У s f(z) + к ||g(z) ||, z e U (A)
With fixed z^GU, g(z ‘ ) = 0, z'1- being a feasib le point o f the originalproblem (M), we derive from (A)
m + к l|g(Zk) ü § f(zk) + к ||g(zk) I s f{z*)
T herefore
и—í \ II - fU * ) - ml|g<zk> II s -----к------
how ever, since ||g || is weakly sequentially low er sem icontinuous,
||g(z0) ü S lim inf ||g(zk) I = 0
This shows that z 0 is a feasib le point o f (M). Let us show that z 0 is a solution o f this problem . F rom (A) we see that
f(z k) § f (zk) + к ||g(zk) I § f(z) i f z e U and g(z) = 0
and from the weak sequential low er sem icontinuity o f f
f ( z Q) S lim inf f(zk) s f ( z ) , z adm issib le for (M)
E x e rc ise 1. Supply a ll details of the proof o f theorem 1.
E x e rc ise 2. Suppose that U = X , Y are rea l fin ite-d im ensional spaces (Euclidean_spaces). C onsider the follow ing m odified version o f (Mk): m inim ize f(z) + к ||g(z)||2. Show that, under the usual (constraint qualification) assum ptions o f ca lcu lus, the penalization method gives the Lagrange _ m ultip lier ru le. (Hint: if f, g are o f c lass C1, then the gradient_f + k||g||2, evaluated at zk, m ust be zero . C onsider the convergence o f 2k g(zk). )
Let us rem ark that theorem 1, at a fir s t glance, seem s o f no use fo r us, because in the application to problem (P ), we chose X as a n on -reflexive Banach space (L 1, AC are not re flex ive). But this is only a m inor difficu lty,
164 ZOLEZZI
because, as we shall see in section 4, natural assum ptions can be made in ord er to achieve weak convergence in X o f the zx (this was the m ost important use of re flex iv ity assum ption in the p roo f o f theorem 1 ).
Now let us return to problem (P ), form ulated as an abstract constrained m inim ization problem , and try to so lve it using the penalization method.
3. HEURISTIC DERIVATION OF RESULTS ABOUT PROBLEM (P)
We have to consider many im portant problem s about (P). A ssum e consistency o f (P) (that is , som e adm issible pair ex ists : this is often re fe rred to as con trollab ility : som etim es the physical orig in o f the problem m akes it c lear that adm issib le pairs exist, although m athem atically the con trollab ility problem may be quite difficult). We are faced with the follow ing questions:
(a) the existence o f som e optim al control;(b) its uniqueness;(c) the n ecessa ry conditions (that m ust hold fo r every optim al pair);(d) the knowledge of optim al con trols not only as functions o f "t im e" t only,
u* = u*(t) (open loop ), but a lso as instantaneous functions o f t and the state x(t) at tim e t, u"1' = u '^t, x (t)), the feedback optim al con trol (c losed loop ); this is ca lled the synthesis problem . In each optim al control problem our ultimate goal is to synthesize the optim al con tro l by an appropriately designed closed o r feedback loop. The advantage o f such a c lo se d -lo o p con tro l, over an open -loop con trol, is that the p rocess then becom es se lf-ad justing and se lf -co rre ct in g . A feedback control can often co r r e c t fo r unpredictable variations in the environm ent o f the plant or fo r repeated perturbations o r irregu larities in the p rocess .
(e) E ffective num erical algorithm s fo r the determ ination o f optim al con trols ;
(f) the dependence o f optim al states and controls from the data (sensitivity p rob lem ).
In this section we shall use the penalization m ethod, as sketched above, to find resu lts to problem (P ), especia lly as to the questions (a), (b), (c ) , (d). We shall derive part o f such resu lts in an in form al and heuristic m anner, free ly choosing the lim its and assum ing convergences as needed. Then we shall show that, in the particular im portant case o f the linear-quadratic problem , such calculations are m athem atically co rre c t .
The penalization method applied to an optim al con trol problem as described in section 2, was introduced by A .V . Balakrishnan, and is known as the "ep silon m ethod" (because the penalization term к ||g(z)|| is often
written as ^ ||g(z)||, with e -* 0 +).
We denote by
U = {(u, x) : u is m easurable, x is absolutely continuous on som e interval [a ,b ]; (3), (4), (5) hold in [a ,b ]} , and con sider the sequence o f " fr e e " optim ization problem s (Pk), к > 0: fo r a given к m inim ize
b b(8)
a a
IAEA-SMR-17/46 165
on the set o f (u ,x )e U , such that
t ^ g ( t , x(t), u it i le L ^ a .b ) (9)
Som etim es it w ill be useful to con sid er the m odified statement o f problem (Pk) as fo llow s: m inim ize
on the set o f a ll {(u, x), u m easurable, x absolutely continuous}, satisfying (3), (4), (5), and
x - g ( t ,x ,u )€ L 2
This alternative statem ent w ill be denoted by (P 2): note that no essentia l d ifferen ce exists between (Pk) and (P 2) as far as existence and approxim ation problem s fo r optim al con trol are considered . Let us observe that, with (P 2), the term |g(t, x , u) - x | 2 is differentiable in x o r u if g is , and this cannot be true fo r the correspond ing term in (Pk) (this is the main reason fo r introducing (P2)).
This approach to optim al con tro l problem s p ossesses two im portant advantages: it provides useful com putational a lgorithm s, and, above a ll, it m akes it unnecessary to so lve d ifferential equations (2 ) (as required by any other method o f solution).
E x e rc ise 1. V erify that, follow ing section 2, (Pk) is an instance o f penalized problem s (M ^ with constraint (9) added.
4. EXISTENCE OF OPTIM AL CONTROLS
C onsider the existence problem (a). We can use theorem 1 to get a sequence o f pairs (uk, x k) approxim ating an optim al pair o f (P ), under suitable assum ptions. F rom theorem 1 we learned that (uk, x k) w orks w ell if it is a solution to (Pk ). It may be difficu lt to show that (Pk) has solutions: anyway, it is often unnecessary, because, p ractica lly speaking, we effect only approxim ate m inim ization : so we can try to find (uk, x k) defined on som e [ak, bjJ, such that, as к -» oo,
a so ca lled quasi-m in im izing sequence for {P jJ .Such quasi-m in im izing sequences w ill always exist (by definition o f
inf 1^) in contrast to exact solutions o f (P^), if , as we constantly assum e, only (P) is consistent; now we can show (using som e functional analysis and variational argum ents) that (uk, x k) is c lo se (in a weak sense at least) to an optim al solution o f (P) under natural assum ptions. E xp licitly we have the follow ing:
b
a
166 ZOLEZZI
T heorem 2. Given any quasi-m inim izing sequence (uk, xk) fo r (Pk), then there exists an optim al solution (u0, Xq) fo r problem (P ), such that, fo r a subsequence, as к — » ,
xk- x 0 u n i f o r m l y , x k- x 0 i n L 1
bk bo
J f ( t , xk, u k)d t -* J f ( t , x Q, U0)d t = m i n P ak a0
bk
in f Pk - m in P , к J I Xk - g(t, Xk, Uk) I dt “*• 0
ak
i f we assum e that
(i) G(t), E are closed sets ; the pro jection o f E on the firs t (2m + l) co -ord in ates is com pact; V has a closed graph and closed values;
(ii) with Q(t, x) = { (z , g (t ,x , u)) : z È f(t, x, u), u e V (t ,x ) }Q (t,x , e) = _U{Q(t,y) : |y-x|§ e}
then Q(t, x) = П {со Q(t, x, e) : e > 0}
(Hi) fo r any bounded set o f x, we can find C, D such that|g(t, x , u) I S C(t) + D |u|, C e L 1; g m easurable in t and continuous in (x, u);
(iv) <p is continuous, f is B orel m easurable and low er sem icontinuous in (x, u)
f(t, x , u) § ф (t, |u |), with •* + “ as z -> + «
0 (t, z) ê ф0(t) with Ф0€ Ь
Some rem arks on theorem 2 are in order. Under assum ptions (i), . . . ,(iv ), given a quasi-m in im izing sequence (uk, xk) fo r the " fr e e " penalizedproblem s (Pk) (that can be obtained in many w ays: Ritz m ethod, e tc .), asubsequence o f states converges uniform ly to an optim al state of (P ), alongwith weak convergence o f derivatives in L 1; m oreover the term s
bkf(t, x k, uk)dt and the inf Pk both converge to the value o f P , m in P; finally
the penalization term goes to zero as к - « : a ll this applies to a subsequence. Note that theorem 2 gives no inform ation about the convergence o f uk to u0. This is due to the great generality o f (P ); but it can be shown that the orig inal sequence u k- u 0 strongly in L 1 if , fo r exam ple, g is linear in u and f is s tr ictly convex in u.
Let us d igress b rie fly to the m eaning o f the assum ptions in theorem 2. Assum ption (iv) is a substitute o f the reflex iv ity o f X in theorem 1, in the sense that we can ascerta in that
bk bk
J </>(t, |uk |)dt s J f(t, xk, u^dt s L < + 00 ( 1 0 )ak ak
and this essentia lly guarantees weak sequential com pactness of {uk} in L 1.
IAE A-SM R-17/46 167
J |¿k|dt sF F F
s ^ (const + inf Pk) + С dt + D J ju k ¡dtF F
F rom ( 1 0 ) it can be seen that I |uk|dt can be made sm all, uniform ly
in k, if m eas F is sm all; the sam e is obviously true fo r the term / С dt;F
and inf Pk is bounded, because if (u, x) is a fixed adm issible pair fo r (P ), then the penalization term evaluated at (u, x) is zero so that
b
inf Pk s J' f(t, x , u)dt < + °° a
so from (11) we obtain / |x, Idt as sm all as requ ired when m eas F is sm all:F k
taking F = [t1, t"] we get from (11)
t” t”
|xk(t ') - x k(t") |=| J' xk(t)dt I s I J |xk|dt I ( 1 2 )t’ t'
the last term is sm all if |t' - 1" | is su fficiently sm all, uniform ly in k. F rom com pactness o f the pro jection of E (assum ption (1)) we see that the x k are uniform ly bounded; as (12) shows their equicontinuity, we can use A s c o li - A rze là theorem (on com pact sets o f continuous functions), and uniform convergence is obtained fo r a subsequence o f xk. With m ore technical calculations the p ro o f o f theorem 2 can be com pleted. Let us rem ark that, using the penalization m ethod, we obtain from theorem 2 som e general answ er to the existence problem of (P ), which, in a sense, is a constructive one. C om pletely analogous resu lts hold fo r (P k).
Let us now exam ine the meaning o f assum ption (ii). C onsider the very particu lar case o f problem (P) when <p= 0, g(t, x , u) = u, V (t,x ) = Rm, then (P) becom es the follow ing: m inim ize
bJ f(t, x , x)dt a
on all curves x = x(t) (in "n on -p aram etric" form ) subject to constraints(3), (4) (for exam ple, curves connecting two given points). This is the sim plest fre e problem in the c la ss ica l calculus o f variations, w ell known in ph ysics, c la ss ica l analytic m echan ics, etc. In this problem the set
Q (t ,x) = { (z ,u ) : z ë f ( t ,x , u), u e R m}
is sim ply the epigraph o f f as a function o f u only. Assum ption (ii) im plies that, fo r any (t, x ), Q(t, x) is a closed convex set (being an in tersection of such sets ), and this m eans that f is a convex function with resp ect to u;
P r o m (iii) we find, g iven a set F ,
g(t»xk»ukHdt + J V И ( U )
168 ZOLEZZI
z
m oreov er , its epigraph depends on x (for fixed t) in som e "reg u la r" way.In the general op tim a l-con tro l problem (P ), assum ption (ii) is a natural generalization o f such a condition for c la ss ica l variational ca lcu lus, taking into account the "d ifferen tia l constraint" x = g(t, x, u) in the definition of Q(t, x). Such a condition was firs t introduced by C esari, and it is known as "p rop erty Q ".
5. NECESSARY CONDITIONS FOR OPTIM ALITY
Now we con sider the n ecessary conditions (problem (c )), leaving aside the uniqueness problem (b) (which is answered in the affirm ative if, for exam ple, we con sid er the linear tim e-optim al problem in som e particular ca se s , and also in the linear-quadratic problem , as we shall see in section 7).
Suppose (uk,x k) is a solution o f (P2). Then
Let u be a m easurable function on [ak,b k] such that a lm ost everyw here
u(t) G V(t, xk(t))
Then, settingь
Ik(u, x ) = j [f(t, x, u ) + ^ |x - g(t, x, u) j2]dtIa
we can w rite
+ | I Xk - g(t, X k, u) |2] dt
b k
К ,с к|g(t,xk, u) - g (t ,x k, u^ |2dt - J H (t,x k,u , P k) d t (13)к
+ 7 72 / I
IAEA-SMR-17/46 169
where
H(t, x , u, p) = - f ( t ,x ,u ) + g '(tsx ,u )p
is ca lled the Hamiltonian o f the problem , g '(t , x_, u) is the transpose o f g(t, x ,u ), and the new variable p e R m; finally we see that
Pk(t) = k[xk - g(t, xk, i^)] (14)
(We are tacitly using the standard convention o f indicating a ll v ectors as colum ns, and m ultiplying m atrices row by colum n. )Since uk m inim izes
u - Ik(u ,x k)
from (13) it does not follow that uk m axim izes
u - J H(t, xk, u, pk)dt
but all w orks, as if it w ere so, that is , fo r a ll u
bk bkH(t, x k, u, pk)dt s J H(t, xk, uk, pk)dt (15)
ak akF o r s im plicity we assum e that, fo r the m om ent, V is a constant set so that the constraint on the con tro l is
u (t )e V a.e.
A ssum ing that (uk, xk) - (u0, Xq), an optim al solution o f P , and pk - p0, a function o f t, free ly choosing the lim its in (15), we find
b bH(t, x Q, u, p0)dt s J H(t, x Q, u0, p0)dt fo r a ll u (16)
a
A ssum e now fixed v £ V , e > 0, t 0 e (a ,b ) such that a 5 t0 ± e s b. Taking
u(t) = v if |t - t Q| s e, u(t) = u0(t)
we have from (16)
h t0 + Í to + «
H dt = / H (t,x 0, v , p 0)dt + / H dt s / H(t, x 0, u0, p0)dt‘° ' е I to - tl > e to-€
+ J H dt11 —10 1 > e
170 ZOLEZZI
that is ,to + e t0 + e
h J H(t' V v 'P 0)dt s ¿ / H t t ^ u ^ d tto " € to - e
A s € “*■ 0, we see that alm ost everyw here
H (t,x 0(t ) ,v , p0(t)) S H(t, x 0( t ) ,u 0( t ) ,p 0(t)) (17)
that is , there exists p0, ca lled the adjoint state, such that the optim al con tro l u0 m axim izes (a .e .) on V the Hamiltonian as a function o f u, evaluated at t , x 0(t), p0(t).
To obtain m ore inform ation on p0, we assum e differentiability of f and g as needed, and take fo r the mom ent
G(t) = Rm + 1
Let yecj (a^bk), that is , y is a sm ooth function with y(ak) = 0 = y(bk). F or any rea l s, x k + sy sa tisfies the constraint (3); but x k m inim izes
x - Ik(uk,x )
th erefore
^ 7 I k 4 . x k + sy) = 0s = 0
(18)
Differentiating under the integral sign, we obtain from (18)
bk
X k + sÿ - g(t, X k + sy , Ukjftdts = 0
bk
= 0 = J jy'fxi^x iikJ+fclxk-gi^x^UjçilUÿ-gxi^Xiyiikiy] hdt (19) ak
In (19) the follow ing notations are used:
f = gradient v ector o f f with resp ect to Xj, . . . , x ^
g = Jacobian m atrix o f g with resp ect to
’ l X j ’ S l X rVp . . . Xm =
: ш К ' m î t ,
IAEA-SMR-17/46 171
When we take (14) into account, (19) can be written as fo llow s:
I {■y' fx(t* xk> V + Pk' fy - xk> uk)y J / dt = 0
bk
}, Xk, Uk) - p¿gx(t, X k , U k) - p ¡J y d t (20 )
as у is arb itrary ; free ly taking lim its as к -» oo in (2 0 ), we obtain
(21 )
i .e . po satisfies the linear adjoint system (21). The relations (17), (21) are (an essentia l part of) the m axim um princip le of Fontryagin, the m ost im portant (and widely known) single result in op tim a l-con tro l theory: it is an im portant generalization o f the c la ss ica l f ir s t -o rd e r n ecessary optim ality conditions in the calculus o f variations.
Let us now con sider a connected group o f e x e rc ise s about tim e-optim al linear p rob lem s. We begin with the follow ing problem : find the fir s t t ! 0 such that
x(u) (t) = z(t)
where z is a given continuous function on t i 0 , x(u) is the solution of X (0) = x Q and
A , В are integrable m atrices (o f appropriate d im ensions), and the m easurable con tro l u is constrained by
O bserve that (23) can be written as u(t) e V , where V is the cube
{ |uJI S 1 : j = i , . . . , r }
This is a sim ple instance o f a linear tim e-optim al con tro l problem (as in section 1 , (a)).
A s we know, it can be written in the form (P ), with f(t, x , u) = 1. T h ere fo r e we see that the Ham iltonian is
H(t, x , u, p) = -1 + p ' [ A (t)x + B(t)u]
In this problem m uch in form ation can be derived from the m aximum p rincip le , as we can see from the follow ing e x e rc ise s .
x(t) = A(t)x(t) + B(t)u(t) (2 2)
¡u j(t) I s 1 , j = 1 , . . . , r (a .e.) (23)
E x e rc ise 1. Show that tim e-op tim a l con trols exist.
172 ZOLËZZI
E x e rc ise 2. Show that, in the above tim e-optim al prob lem , any adjoint state m ust satisfy
p(t) = а 'ф '1^)
fo r som e a 6 Rn, where Ф is a fundamental m atrix o f y(t) = A(t)y(t).
E x e rc ise 3. Show that, i f u* is a tim e-optim al con trol with optim al tim e t* > 0 , then there exists an a e R m such that a.e. in (0 ,t* )
u?(t) = sg n [a '$ ’ 1 (t)B (t)]]., j = 1, r
where sgn x = i f x ф 0 , sgn 0 undefined.
With such an event, we w rite brie fly
u*(t) = sg n [ a l^"1 (t)B(t)] (24)
It can be shown that а ф 0.
D efinitions, (i) An adm issib le con tro l fo r the above problem is called bang-bang if |u.¡(t) | = 1 fo r a ll j = 1, . . . , r and a .e .t. (ii) The linear control system (2 2 ) is ca lled norm al if fo r a ll j = 1 , . . . , r and fo r a ll а ф 0
m eas {t § 0 : a f <ÿ- 1 (t)b 3(t) = 0 } = 0
where b¡ is the j-th colum n o f the m atrix B.
E x e rc ise 4. Show that, i f A , В are constant m a tr ices , then (22) is norm al iff b j, A b j, . . . , Am _ 1 bj are linearly independent fo r a ll j.
Hint: ф(t) = e At, a 'e ‘ Atbj is an analytic function, so norm ality holds if f a 'e "Atbj = 0 im plies a = 0. If our v ectors are linearly independent, differentiating m tim es and setting t = 0 in
a ' e"At b. = 0 j
we get a = 0. C onversely , take a polynom ial q with q (-A ) = 0 (Cayley - Hamilton theorem ), set f(t) = a 'e "At bj (а ф 0), and compute
q( j l ) f(t) = °* f(k> (0) = °-
E x e rc ise 5. V erify that, if (22) is a norm al system , then there exists a unique tim e-op tim a l bang-bang control.
E x erc ise 6 . F or which rea l к the harm onic o sc illa to r with the external con trolling fo rce
x(t) + k 2 x(t) = u(t)
is a norm al system ?A nsw er the sam e question for the m ore general system
x + kx = u
IAEA-SMR- П /4 6 173
Definition. System (22) is com pletely controllable betw een 1 1 and t2 if anypairs o f points in Rm can be connected by som e tra je cto ry of (2 2 ) (withoutconstraint (23) on u) starting at t and ending at
*2
E x erc ise 7. Let S(u) = <f>(t0) / ф'1 Bu ds, u m easurable and bounded on:tt i, t2]. V erify that (22) is com pletely controllable betw een t x and 1 2 iff S is
a su rjective mapping between the m easurable bounded functions and R m.
E x e r c ise 8 . Show that, i f (2 2 ) is a norm al system , then it is com pletely con trollable between two given instants. 4
Hint: S o f e x erc ise 7 is onto if f |a | = 1 im plies J |a’ ф'1 В ¡ds > О,
that is , sup / a 1 </>_1Ви ds = + °°, |a| = 1, o r sup {a 1 у : y e S (boundedu ^ u
m easurable functions)} = + a>, that is , S m aps the bounded m easurable functions onto R m.
6 . A FORM AL APPROACH TO THE SYNTHESIS PROBLEM
The integrand o f the " f r e e " penalized problem s (P k) is
f ( t ,x , u )+ k | x - g(t, x , u)| (25)
F ix t, x , and m inim ize (25) with resp ect to u on V(t, x (t)), so to obtain a m inim um point
uk= u k(t ,x (t) ,x (t ))
Now integrate with u = uk and m inim ize in x (absolutely continuous), subject to (3), (4)
b
x J' [ f ( t , X , ï ï k ( t , x, x)) + к I x - g(t, x, U k ( t , x ,x ) |] d t (26)a
Let x km inim ize (26), and set
u*(t) = uk(t, x k(t), x k(t)) (27)
In a sense, (26) resem bles a c la ss ica l free problem : from n ecessaryconditions satisfied by xk we can elim inate xk, obtaining from (27) a function
шit o f t, x
such that
и кШ = uk(t» x k(t)) (28)
174 20LEZZI
иГ 1 1
0 ! Г " 1 *ri t i
FIG. 1. Optimal trajectory for uï!i(t) = 1.
Letting (as usual in a form a l way) к - °° in (28), we get (uk,x k) - som e optim al (u0, x0) fo r (P ), (uk,x k) being optim al (by construction) fo r (Pk), and th ere fore we get the optim al con tro l fo r (P) as a closed loop
u|(t) = u(t, x 0(t)) (29)
rea liz in g the synthesis.We shall con sider (in a m athem atically valid way) the synthesis fo r the
linear-quadratic problem in section 7. Let us examine in detail an exam ple o f synthesis fo r a tim e-op tim a l problem . In the follow ing exam ple we use resu lts from ex e rc ise s 1 - 6 in section 5.
E xam ple. C onsider the sca lar tim e-op tim al problem
y(t) + y(t) = u(t), t § 0
|u(t)I s 1. Starting from a given point x 0 = (y0, y0) in the phase plane, we wish to reach the orig in ( 0 , 0 ) in the least tim e, the so -ca lle d specia l problem . Introducing, as usual, the phase co-ord inates
Xj = y , xg = y
we can w rite the con tro l system as
x(t) = A(t)x(t) + B(t)u(t) (30)
, m = 2, r = 1. Since this system is norm al
and ^"1 (t)B = J then with (24), any tim e-optim al con trol is given by
u*(t) = sgn sin (t + a), |a|S7r (31)
F rom (31) we see that u* is p iecew ise constant, and has jumps exactly after 7Г units o f tim e. The tra je cto r ie s with u(t) = 1 (u(t) = -1) are c ir c le s with cen tres o f ( 1 , 0 ) ( ( - 1 , 0 )), and with тг units of tim e a tra jectory is a half c ir c le ; the phase point runs along these c ir c le s counterclockw ise (clockw ise). To achieve the synthesis, we use the fact that (30) is autonomous and integrate (30) backward. Let 0 < a § v. in the last interval o f tim e t on which u*, the optim al con tro l rela tive to x 0= (y0, ÿoh is constant, the optim al tra jectory com es from u*(t) = 1 say, (see F ig .l ) , and the phase point m oves along the
with A = 0 1
- 1 0В =
IA E A -SM R -17/46 175
■x
FIG. 2 . Motion of phase point.
c ir c le , which passes through the orig in and has its centre at (1, 0). The optim al con tro l u*(t) = 1 in the last interval o f constancy, o f length < гг, th ere fore in such an interval the phase point reaches the orig in after having travelled along the arc AO (see F ig .2 ), i.e . less than one half o f the circu m feren ce . The instant corresponding to A is a switch tim e of u*, th erefore the phase point reaches A by m oving, during a tim e interval of duration 7Г, under the influence o f the con trol us,'(t) = - 1 , so that in such a tim e interval the phase tra jectory consists o f the se m ic ir c le BA with centre ( - 1 , 0 ), and so on, until we reach (m oving backwards) the in itial point x0.If in the final tim e interval we have u*(t) = -1 , the phase tra jectory is obtained from that in F ig.2 by re flection through the origin . We see that the infinite number o f se m ic ir c le s o f radius 1 and cen tres ( ± 1 , 0 ), (± 3 ,0 ) and so on, are the switching lo c i у fo r our con trol system : that is , an optim al con trol has a discontinuity at t iff the corresponding tra jectory m eets у at t (see F ig .3).
The synthesis is th erefore obtained as fo llow s: set
(see F ig .3). Then we can find the optim al tra jectory x which steers a given x 0 to the orig in in minimum tim e solving
j ¿ l = x 2> X j(O )= y 0jx2 = -хг + ш (x1;x2), x2(0) = y0 31aand the optim al con trol is u*(t) = u[x(t)j|.
Let us rem ark that (31a) has a discontinuous right-hand side (as a function o f x).
- 1 , otherw ise
1 , i f x is not above у
176 ZOLEZZI
7. THE LINEAR-QUADRATIC PROBLEM
We m inim ize (see exam ple (b), section 1)
T
J' (x 1 Px + u' Qu)dt (32)
T fixed , with dynam ics
x = A (t)x + B(t)u]
x (0 ) = x (33)
the standard linear-quadratic op tim al-con tro l problem with free -en d condition.In this section we assum e P, Q, A , В continuous, P sym m etric positive -
sem i-d e fin ite , Q sym m etric positive-defin ite , that is , fo r all X and t€ [ 0, T]
X 'P (t )X ë O , X'Q(t) X ê a |X|2, a > 0
|*| denotes E uclidean norm , that ism
i—1T
An adm issib le con trol u is such that / u' Qudt < + °° and (by continuity2 °and positivity o f Q) this happens iff u e L (0, T). Note that the continuity
assum ptions on P, Q, A , В can be relaxed.:>2 .The correspond ing P£ problem is : m inim ize
T T
4 / (x 'P x + u'Qu)dt + ^ / |x - A x - Bu 12 dt = Ik(u, x) (3 4 )
on all uGL2(0, T), all xeAC(0 , T) with x(0) = x.
IAEA-SMR- П /4 6 177
Given x£ A C (0 , T) with x(0) = x, we could m inim ize (with t fixed) the integrand
u - j (x 1 Px + u ' Qu) + ^ |x - A x - Bu |2 (35)
on R r and then proceed as in section 6 . But the sam e resu lts can be reached in the follow ing (m ore inform ative) way. Due to the quadratic nature o f Ik,it is easy to show (using standard variational arguments) that fo r any к am inim izing pair
(uk, xk)
exists fo r Ik. In fa ct, fo r a fixed k, let {(u,,, xn)} be a m inim izing sequence fo r Ik. Then
T T T
I / (x'nP x n+ 4 Qun) d t + | f Ix h -A X h -B u J 2 dt ê ! J J u j2 dt0 0 0
th ere fore {u,,} is a bounded sequence in L2 (0, T) and so
un- u0 in L 2(0, T)
fo r som e subsequence. Since
I x n " -A-xn| — 2 I x n — A xn - Bun j + 2 I Bun I
we see that xn are uniform ly bounded. M oreover, rem em bering thatT |2
Ik(un, x,,) and / |un| dt are uniform ly bounded, we obtain, fo r som e large
constant C,
T T T
x n |2 dt s Ik(un, Xn) + С J | x j2 dt + С J |un |2 dt + С + 2 (Ik(un, xn) )2C0 0 0
T 2so that / x dt are also uniform ly bounded.
оT h ere fore , fo r som e subsequence
x nJ y in L 2(0, T)
SinceT
2|xn| dt C , fo r any t ! , t
t"
n( t ' ) - x „ (t " )| ii / i i j dt I â C t ' - t "
178 ZOLEZZI
this shows that the xn are equieontinuous: by the A rz e là -A sco li theorem x n -» x 0 uniform ly. But then
fo r a subsequence. But then
x n - Axn - Bun J x 0 - Axq - Bu0 in L 2(0, T)
and, by the weak sequential low er sem icontinuity o f the norm in a Banach space, we obtain from (36)
Let us now show that (uk, xk) converges to an optim al solution o f (P).In a sense, we shall obtain an explicit solution o f the problem , synthesizing the optim al con tro l fo r (P). R em em ber that, with given k, (u^, xy) m inim izes Ik on the set o f a ll u GL2 (0, T), and x e A C (0 , T) with x(0) = x. T h ere fo re , denoting by s a rea l variab le , given y e C j ( 0 , T) (that is , y is a sm ooth function with y ( 0 ) = y(T ) = 0 ), we get
о 0
th ere fore x 0 = y ; we have shown that
x n -*• xQ uniform ly, xn - xQ in L2 (0, T) (36)
that is , differentiating under the integral sign, and setting
(37)
we a rrive at
T T
(38)0 0
T TIntegrating by parts y ields ! z jÿd t = - / il ydt, and from (38) we obtain
о о
оA s y is arb itrary in cj(0, T),
Pxz k = - A 'z k + — a.e. in (0, T) (39)
Taking y 6 C 1 (0, T ), y(0) = 0 only, and repeating the above calcu lations, the only d ifference being
T T
J Zjiy d t = _z¡¡(T)y(T) -fi[ ydt о 0
by arb itrariness o f y (T ), we see that
z k(T) = 0 (40)л
T herefore
zk(t) = ф~1(t) J ф P xkdsT
and by (37)
(41)
t
x k = A xk + Buk + ^ Ф'1 J' 0 P xkds, a.e. in (0 , T) (42)T
Given v e L 2 (0, T ), fo r any к
th erefore
¿ Ik(uk + s v ,x ¿ | = 0
<s = o.
that is (differentiating under the integral sign)
T
J (ul^Q - к z'k B) v d t = 0 (43)о
Since v is arb itrary in L 2(0 ,T ) , we see from (43) that
Quk—:— = B 1 z k a .e . in (0, T) (44)к K
Com bining (44) and (37) we obtain
tuk = Q_ 1 В ' Ф'1J ф Pxkds (45)
which, incidentally, shows a form o f approxim ate synthesis, but in integral form ; using (42), we a rrive at
180
т i2R em em bering that |xk(t) | C fo r a ll t, we see that sup / |xk| dt < + and by re flex iv ity o f L2 (0, T) k 0
x k^ x Q in L2 (0, T) (47)
fo r a subsequence. By (45)T
J |uk |2 dt < + oosup k 0
so , fo r som e subsequence *
uk u0 in L 2(0, T) (48)
F rom (41) zk -* 0 uniform ly, and, by letting к -*■ °o in (37), using (47), (48), we see that
x k^ A x0+ Bu0 in L 2(0, T)
th ere fore x k - x 0 uniform ly, and
x 0 = A x 0 + Bu0 a. e. in (0, T ), x 0(O) = x
Taking к -* «о in (45), (46), we find t
u 0 = Q’ 1 B l Ф’1 J' ф P x 0dsT t
x 0 = A x 0 + BQ ’ 1 В ' ф'1 J' ф P x 0 ds, x 0(0 ) = x .T
If (u, x) is an adm issib le pair fo r (P ), then
(49)
Ik(uk, xfc) s ik(u, x) = J (x 1 Px + u' Qu)dt0
m oreov er , by (47), (48),
lim inf Ik(uk, x^ È lim inf J (x^ P xk + u'k Quk) dt ê J (x¡, P x 0 + u '0 Q u q ) dt о 0
since the norm is weakly sequentially low er sem icontinuous.
Sum m arizing,
T T
j 1 (x '0 PXq + u '0 Qu0) dt § J'(x 1 pX + u'Qu)dt0 0
that is , u 0 is an optim al con tro l fo r the linear-quadratic problem . Let us rem ark that u0 is the only optim al con trol: in fact, by the assumptions onP , Q ,
IAEA-SMR-17/46 181
x - x 'P x is a convex function on R m
u -*■ u 1 Pu is a str ictly convex function on R r
Owing to the linearity o f the dynam ics (33), i f x(u) denotes the state corresponding to the con trol u,
T
u -* J' (x(u)' Px(u) + u'Qu)dtо
is a str ictly convex functional on L2 (0, T ), th erefore a uniqueness theorem holds. F rom (49), to find the optim al pair (u0, x0), we m ust solve the in tegro -d ifferentia l equation fo r x 0 so to obtain uo. Let us rem ark that (49)y ields a regu larization theorem fo r the optim al con tro l: in fact we see thatu 0 is a continuous function (m uch better than only L 2(0, T ), as at thebeginning of the problem ). F rom the firs t expression (49) we can try to finda differentiable m atrix E that does not depend on x , such that in (0 ,T )
u0 = Q‘ 1 B 'E x 0 (50)
the synthesis form o f the optim al control. By (49) it su fficesth
E x o = 0 -1j ф P x 0ds (51)T
Differentiating (51) and rem em bering (49), we see thattо
Ê x 0 + E x 0 = -А 1 Ф'1 J 0Pxods + P x 0
Tso that
t
-in i л-iEx0 = P x 0 - A 'E x 0 - E A x 0 - EBQ "1 B I ф~1 j ф P xQds
= Px0 - A 'Exq - E A x 0 - EBQ ^ 'E x q
and E(T) = 0.Assum ing th erefore E to be a solution in (0, T) o f the R icca ti equation
J É = P - A 'E - E A - E BQ ‘ 3 B 'E l (52)
[ e (t ) = o Jwe see that the solution of
x = A x + B Q ^B 1 E x, x(0) = x (53)
is just the optim al response x 0, and thus the optim al con tro l is given by (50). Thus the feedback con tro l (50),
u0(t) = Q- 1(t)B ' (t)E (t)x 0(t)
autom atically constructs the optim al response x 0 fo r a ll initial states x.
182 ZOLEZZI
Of cou rse E must be computed from the solution o f a (non-linear) m atrix R icca ti-typ e equation (52), that can be solved by standard num erical m ethods. It can be shown that (52) has a unique sym m etric solution E defined on a ll [ 0 ,T ] .
By using the m atrix E we can com pute explicitly the value o f the problem . In fact, by (53), (52) and (50),
^ x'0 E x 0 = xJ,Ex0 + XqÈxq+ xJjEx 0
= (A x 0+ BQ"1 B 'E x 0) ' E x q + x'0 Px0 - x ^ A ' E X p - x'0E A x 0
- xf)EBQ’ 1 B 'E x Q - x'0E (A x 0+ B Q ^B 'E xq)
= x'0 P x 0 + u{,Qu0
and integration y ields the value o f our problem :
m in P = -x ' E (0)x
F inally, from the above calculations we see that the penalization method gives a com plete solution o f the linear-quadratic prob lem , in a m athem atically valid way. We shall examine brie fly a different approach to the problem in Section 9.
E x e rc ise 1. Show that E , as a solution o f the R iccati equation (52), must be a sym m etric m atrix.
E x erc ise 2. G eneralize the above resu lts to the cost functional
T
x (T ) 'G x (T )+ J (x 'P x + u'Qu)dtо
P, Q as above, G sym m etric and positive-sem idefin ite .
E x e rc ise 3. Show that uQ(t) = — sin t - — co s t t is the optim al con trol for the follow ing problem :
x = y , ÿ = -x + u, x (0 ) = x , y ( 0 ) = y
2тг
x(tir) = у(27г) = 0, m inim izing / u2dtо
(regulation o f an e r r o r x(t) to zero in a finite duration with m inim al energy).
E x e rc ise 4. The dynam ic equation fo r a rotor is x = u (x = angular mom entum , n = sca lar controlling torque about the fixed axis o f rotation). The total
energy expenditure is I a u2 dt, cr > 0. Given the in itial value x and the cost
functional x ( l ) 2 + a J u2dt, synthesize the optim al con tro ller as a feedback
con trol, and compute the value (m inim al cost).
IAEA-SMR-17/46 183
8 . A SENSITIVITY RESULT
In section 3, we have seen that, owing to the various physical in te r pretations o f optim al con trol p ro ce sse s , the very im portant problem a rises , whether optim al con tro l, state and value depend in an appropriate continuous manner on the data (such as coe ffic ien ts , boundary values, param eters).
There exists no unified theory about sensitivity, only specia l results have been published. In this section we shall d iscuss b r ie fly som e results on this top ic, without giving m athem atical details. To begin with, consider a sequence of linear-quadratic problem s IJ, o f the follow ing type:
m inim ize ¡Уn - xn(u) (T) 1 2
where x n(u) is the solution o f
x(t) = A n(t)x(t) + Bn(t)u(t), 0 s t s T
. x (0 ) = x n
and the con tro l is further constrained by
|u.(t) I s C , j = 1, . . . , r
We think of P0 as a given problem , and I},, n = 1, 2, . . . , as perturbations o f P0. T herefore we perturb the desired final state y 0 (which we intend to approxim ate as c lo se as possib le ), the coefficien ts Aq, B 0, and the initial state x 0. The sim plest situation is that o f the sca lar ca se , that is , m = r = 1. Suppose that
A n- A 0, Bn - B 0 in L ^ O .T ) (54)
T T(that is , / An <p ds J A 0 <p ds for every bounded m easurable function tp and
о 0sim ilar ly fo r Bn), and that
Bn(t) ф 0 (55)
except fo r at m ost к isolated points and subintervals o f (0 ,T ) , к independent o f n. Then we can find optim al con trols un fo r P such that
un -*■ uQ, x n(u^ -*■ x j u 0) both (p iecew ise) uniform on (0, T)
m oreover m in Pn ■* m in P0 (56)
In other w ords, fo r the sim ple sca lar ca se , if we con sider "regu la r" perturbations im posed on the coe ffic ien ts in the sense o f (55), weak convergence su ffices already to uniform ly approxim ate optim al con trols , states, and the value of the orig inal problem P0.
A sim ilar theorem holds in the general (n on -sca lar) case ; the inequality (55) changes to (55 ') which can be form ulated as fo llow s: if фп is the principal
184 ZOLEZZI
m atrix o f x n = A,jXn, fo r every choice of axes (yi, . . . , ym) and fo r every pair o f ve rtices p, q o f the constraint cube
{u € R r : |u. I & C, j = 1, Bn(p-q)]'y . 0
except as in (55).C onversely , if (for sim plicity) we fix the unconstrained dynam ics, that
is , A n = A 0 fo r a ll n, and perform perturbations Pn o f Po, and know a p r io r i that by such perturbations we can uniform ly approxim ate the optim al elem ents o f P0 by optim al pairs o f Pn, then, n ecessa rily ,
B n -*■ B 0 in L*(0,T )
In a sen se , optim al elem ents o f P0 are com pact functions o f the coefficien ts B 0 (with the L^O, T) topology).
9. A SKETCH OF OPTIM AL CONTROL FOR THE HEAT EQUATION
Given an adm issib le sca la r con trol u = u(x, t), O S x S a , O s t s T , constrained by
- °° s a § u(x, t) s b s + 00 (56a)
we denote by Q the rectangle 0 s x s a, 0 § t s T, and by z(u) the solution o f the follow ing C auchy-D irich let problem fo r the inhomogeneous heat equation:
9zat"
a29 zЭх2
z (0 ,x ) — z(x), 0 S x § a
z(t, 0 ) = z(t, a) = 0 , O s t s T(57)
where z is som e fixed function, and we try to m inim ize with resp ect to u
J [(z(u) - z * ) 2 + u^dx dt (58)0
where z* is a given function o f x and t.This is a sim ple instance of an op tim al-con tro l problem for a parabolic
partial d ifferentia l equation; in a sense, it is a linear-quadratic problem correspond ing to that of section 8 fo r ordinary differentia l system s. One o f the obvious physical meanings of (57), (58) is the follow ing: we con trol
IA E A -SM R -17/46 185
the distribution u of heat sou rces , constrained by (56a), so to obtain a m inim al deviation of the resulting tem perature to the desired one (z* ), taking into account the number o f heat sou rces.
F rom the case of ordinary d ifferential equations we know that optim al con trols can be discontinuous functions, so we cannot interpret (57) in a c la ss ica l sense, because we m ust only assum e that u is a m easurable function such that
u dx dt
con verges, that is , u € L 2 (Q). T herefore we are fo rced to read (57) in a generalized sense: z is a weak solution o f this boundary value problem iff, fo r every sm ooth <p defined in Q and 0 near the boundary o f Q,
(i) / ( - Z ipt + z 0
x<px)dx dt = / u <pdx dt 0
(ii) J z (x ,t) <Mx)dx -* / z ÿ d x a s t 0 + fo r a ll sm ooth ф о о
and finally, as a function o f x only, z(t, •) has a square sum m able d is tr ibutional partial derivative; it takes the value 0 at x = 0 and x = a in a
ageneralized sense, J z2dx dt con verges, and / z2 (t, x)dx is bounded in t.
0 x 0 The relation (i) can be obtained form ally from (57) by means of an
integration by parts, so that, i f u w ere sm ooth, then z(u) would be a c la ss ica l solution, so that it would satisfy (i), (ii). It can be shown that (i) a lso holds if <p is a tr ia l function o f a broader c la ss , that is with square-sum m able firs t derivatives and zero on the boundary o f Q. A s we can see from the cu m b ersom e definition o f the solution of (57), at this point the m athem atical apparatus: needed to explore our optim al con trol problem is large and some - what sophisticated. T herefore we ignore, in this sketch, many problem s about (57): it can be shown that, fo r a given u 6 L 2 (Q), there exists a unique z(u), and if we choose z , z'1' in L 2 (Q) our optim al problem is perfectly sensib le . Many properties of z(u) are known: fo r exam ple, it can be shown that z(u) is a continuous function. F rom the linear-quadratic nature o f (57), (58), a natural functional-analytic setting fo r our problem is the follow ing:
Let S denote the set of a ll pairs (u, z(u)), u adm issib le , so that we can con sider S as a subset o f the cartesian product o f two L2 spaces on Q, this being a H ilbert space H with the norm
(u,z(u))|£ = ||uf + |z(u) II = J [u2 + z(u)2]dx dt 0
Owing to the linearity o f u -*■ z(u) it is easy to verify that S is a closed convex set in our H ilbert space H. So the op tim al-con tro l problem (57), (58) has an optim al solution (uq, z(u0)) iff this pair m inim izes the (squared) distance between the given point (0, z*) and S. Thus the optim al pair w ill be the orthogonal p ro jection o f (0, z*) on S.At this point we can use the follow ing c la ss ica l abstract theorem :
186 ZOLEZZI
Theorem 3. Let H be a rea l H ilbert space (with points x , y, . . . , and sca lar product <( x , у У ) . Let S be a c losed convex subset o f H, and xQ a given point o f H. ' Then we see that
(i) there exists a unique y0 6 S such that ||xQ -y 0 || = m in {||x0 - y || : y G S };
(ii) any m inim izing sequence yn, that is , yn£ S and ||х0 -Уп11 m in {||x0 - y II : y e S} converges strongly to y0;
(iii) y 0 is the orthogonal pro jection o f x 0 on S iff fo r every y e S < x 0 - y0, У - У0 > - 0
C learly (i) is the existence and uniqueness theorem , (ii) an approxim ation resu lt (both due to R iesz), and (iii) is a n ecessary (and sufficient) condition o f optim ality fo r y 0.
P roo f. Let {у д} be m inim izing, that is ,
||x o -У п II ■* к = inf ||x0- y II
(the inf taken on S). F rom the parallelogram law we obtain
1|Уп-УрЦ2 = 1|Уп - хо + хо-Ур 1|2 = 2(||Уп - х о1|2 + 11х о-УрЦ2>
- 1|Уп-2хо + Ур1Г = 2(1|уп- хо1Г + 11хо-уРИ2) - 4 II ' хо112
s 2 <||УП - х о 1 |2 + Нх о-УрЦ2) - 4к 0 as n, p - °о
This shows that {y n} is a Cauchy sequence in H, th erefore convergent to som e y 0 e S . M oreover, if y 0, ya are m inim izing, then, again using the parallelogram law, we see that
1|У0-У1 f = 2<1|Уо - xol|2 + llxo -У1 lf> - 4 II " xol|2 s 4k2-4k2 = 0
th erefore y0 = y2 and (i), (ii) are proved. Let us now prove (iii). Assum ing that y0 is the p ro jection of x 0 on S, let yG S , and
f(t) = II t y0 + (1 - 1) у - x 0 f , 0 s t § 1
IAEA-SMR-17/46 187
Then f(t) ï f ( l ) , th ere fore f ( l ) § 0, that is
^ < t y 0 + (l - t ) y - x Q, t y Q+ ( l - t ) y - x 0> jt = i
= 2< У0 - x o Уо 'УУ - °- C onversely if y0 6 S and < x 0 - y 0, y - y 0 > S 0 fo r a ll y £ S, then < x 0 - y 0, x 0 - y > 6 < x 0 - y 0, x 0 - y 0> =-:|x0-y 0 ||2 and by Schw arz's inequality we see that ||x0 -y 0 || S ||x0 - y ||, that is y0 is m inim izing, q.e.d.
Let us rem ark that the linear-quadratic prob lem o f section 7 can be approached by a functional analytic method based on theorem 3, at least when m atrix P is (strictly ) positive.
F rom theorem 3 we get ex istence , uniqueness and an approxim ation resu lt about optim al con trol o f (57), (58). M oreover u0 is an optim al control with optim al response z 0 = z(u0) iff (see (iii) and the definition o f H fo r our problem )
J' [u0(u - u0) + (z0 - z*) (z(u) - z0)] dx dt fe 0 (59)Q
fo r every adm issib le u. By using (59) we can represent the optim al pair as solution o f a set o f boundary value problem s and inequalities, introducingthe adjoint state as in section 5. Define (at least form ally) the adjoint stateas the solution of the backward boundary problem
' -Pt - Pxx = z 0 - z * i n Q
p(T , x) = 0 = p(t, 0) = p(t, a) (60)
It can be shown that (60) has a unique (weak) solution.q q ^ 2
Rem ark that - — - — - is the adjoint operator o f — - — 3 in (57), so thatot Эх2 ot Эх^(in a sense) we have s tr ict analogy between (60) and (21). F rom (i) in thedefinition o f z(u), using p as test function, we get (rem em bering (60))
J' [(-z (u ) + Z 0 ) P t + (z(u)x - z 0x)px] dx dt =0
(u -u ^ p d x d t =J p (z(u)t - zot) + px(z(u)x - z 0x) dxdt =0 0
= J (z 0 - z*) (z(u) - z0) dx dt Q
then it w ill follow from (59)
J ' (p + U q) ( u - u0) dxdt S 0 fo r a ll adm issib le u (61)Q
So we see that (57), with u = u0, z = ь0, (60), and (61) ch aracterize the optim al pair. If no constraint is im posed on con tro ls , then from (61) we get u0 = -p , and in this sim ple case we see that the optim al pair is obtained by
188 ZOLEZZI
solving the follow ing system with the unknowns z 0, p, and setting u0 = -p fo r the optim al con trol:
zQ( 0 ,x ) = z(x); z 0(t, 0 ) = z Q(t, a) = 0
p(T , x) = p(t, 0) = p(t, a) = 0
Further inform ation about u0, z 0 can be obtained (in particular we see that uo is m ore "reg u la r" than a sim ple L 2 -function): fo r exam ple, from ( a set o f "R icca ti equations" can be derived.
PONTRYAGIN, BOLTYANSKII, GAMKRELIDZE, MISHCHENKO, The mathematical Theory of Optimal Processes, Interscience, New York (1962).
ATHANS-FALB, Optimal Control, McGraw-Hill, New York (1966).
LEE, MARKUS, Foundations o f Optimal Control Theory, W iley, New York (1967).
LIONS, Contrôle optimal de systèmes gouvernés par des équations aux dérivées partielles. Dunod-Gauthier-Villars, Paris (1968).
VARAIYA, Notes on Optimization, Van Nostrand, New York (1972).
BELLMAN, Introduction to the Mathematical Theory of Control Processes, Academic Press, New York (1967).
BALAKRISHNAN, "The epsilon technique — A constructive approach to optimal control’’ , Control Theory and the Calculus on Variations, UCLA (1968),
BERKOVITZ, Optimal control theory, Applied Math Sci. 12, Springer (1971).
FLEMING, RISHEL, Deterministric and stochastic optimal control^Applications of Math. 1, Springer (1975).
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B I B L I O G R A P H Y
lA E A -SM R -17/49
AN A P P LIC A TIO N OF PONTRJAGIN'S PRINCIPLE TO THE STUDY OF THE O P T IM A L GROWTH OF POPULATION
Vera de SPIN ADEL Department of Mathematics,Faculty of Exact Sciences,University of Buenos Aires,Buenos Aires, Argentina
Abstract
AN APPLICATION OF PONTRJAGIN'S PRINCIPLE TO THE STUDY OF THE OPTIMAL GROWTH OF POPULATION.
This paper examines the consequences of an optimal control of population growth and allows to derive criteria referring to the economic basis of expenditure on population control and to obtain optimal paths for a model in which such a control is possible.
By means of very simple assumptions one can reduce the problem to a two-state variable control problem and, in consequence, apply Pontrjagin's maximum principle to solve it.
INTRODUCTION
One of the fundamental problem s in the theory of econom ic planning is to study the consequences o f an optim al control o f population growth and to derive cr iter ia that may be of relevance to an investigation of the econom ic basis of expenditure on population control.
Population growth is influenced by expenditures for public health, by fam ily allow ances, by governm ent p o lic ies with regard to fam ily planning and by general cultural and re lig iou s positions tow ards the idea o f population control. This con trol can be achieved by encouraging various form s o f birth con trol or em igration to hold down population growth, or by subsidizing births and im m igration to make a population grow. A ll these m easures are expensive and their costs must be com pared with those of investm ent and consumption before making a satisfactory choice o f a determ ined policy to follow .
The econom ic m odel used in this paper is based on the one developed by John P itchford (1] and its soc ia l objective is to control the system so that from som e initial point it is transferred , after som e unspecified tim e, to a final point at which sustainable consum ption per head is m axim ized.At the same tim e, the accum ulated divergence o f welfare from the maximum sustainable level o f w elfare, is m inim ized.
To dim inish the com plexity o f the d iscussion o f this optim ization problem we are forced to make som e assumptions re ferrin g to the econom ic aspects of the form ulation. In this way, we are able to reduce the general problem to a tw o-state variable con trol prob lem , which can be solved by Pontrjagin 's maximum princip le [2 ].
189
190 SPIN ADEL
Let us use the follow ing notations:
W: soc ia l welfare function (m easuring, for exam ple, the "standard ofliving" ),
<j>: aggregate production function (production per unit tim e, taken as afunction of capital and labour),
c : consumption per capita, a function o f tim e,K(t) capital at tim e t, such that K(t) ï 0,N(t): labour fo rce at time t, such that N(t) ê 0,К: investm ent per unit tim e,Ñ: population growth rate.
We seek to manipulate K and Ñ so as to m ove the econom y from an initial point (N0, K 0) to a given term inal point (N, K), at which sustainable consum ption per head is m axim ized, at som e unspecified time T and over this path to m inim ize
M A T H E M A T I C A L F O R M U L A T I O N O F T H E E C O N O M I C M O D E L
(i) К = W j); K(0) = K 0; K(t) = K,
(ii) Ñ = g (N ,I2 , I 3); N(0) = N 0; N(t) = Ñ,(iii) <¡>(N, К) - cN - Ij - I 2- I 3 ê 0,
(iv) C È С,(v) Ij 8 0 , j = 1 ,2 ,3 ,
where с is som e determ ined minimum level o f consumption per head; Ix, I 2
and I3 are the amounts of available incom e spent on increasing capital, increasing population and reducing population, respective ly , and W is the maximum sustainable level of socia l w elfare.
P itchford developed the case where W = с and we shall consider
W = U (c); U '> 0; U" < 0 (2)
where U is the so -ca lled "utility" function, which represents the utility that consum ption gives to the population.
The main difference between the two assum ptions is that, with (2), increm ents of consum ption are given higher weights at low er levels of utility than at higher levels .
C onsider the problem of m axim izing W = U(c) with respect to N and K, where these variab les can be free ly chosen at constant levels , so that
T
(1 )
0
subject to the conditions
с = ф (N, K )/N (3)
IAEA-SMR-17/49 191
It will be assumed later that, in order to hold a population at a given level, som e expenditure w ill be necessary .
If a maximum ex ists, it must satisfy
Фк = 0 (4)
* N = Ф/N (5)
These two equations have the follow ing meaning: Eq. (4) means that capital is used until further additions cease to be productive and Eq. (5) m eans that the work force is in creased up to the point where the average and m arginal products o f labour are equal.
To specify the production function 0 (N, K), we w ill assum e it to be tw ice differentiable with continuous secon d -ord er partial derivatives, while ф/N is taken to have a unique maximum when (N, K) = (N, K) such that N > О, К > 0. A s a consequence, consider m axim izing <í>(N, K) with resp ect to N and К so that (4) and (5) are valid and
;kk t kn| (6 )kn k ,,' n n ~j
192 SPINADEL
T A B L E I. P O S S I B L E P O L I C I E S
II Ï2 h p q r
A 0 0 0 s U 4 c ) £ U 4 c ) i U '( c )
В > 0 0 0 = U ’ ( c ) < U ’ ( c ) ^ U '(c )
С 0 > 0 0 £ U ' ( c ) = U ' ( c ) ^ U ’(c)
D 0 0 > 0 s U '(c ) i U '( c ) = U4c)
E > 0 > 0 0 = U ' ( c ) = U*(c) s U ’(c)
F > 0 0 > 0 = U ’ ( c ) £ U '(c ) — U ’(c)
G 0 > 0 > 0 ^ U ' ( c ) = U ' ( c ) = U ’(c)
must be negative-defin ite at (Ñ, K). It follow s that
(d K /d N )^ = e /a > (dK /dN )#K = 0 (7)
at (N, K). But ф is not str ictly concave because (6 ) is valid only for (N, K) and not for all (N, K). If we draw a c ro ss -s e c t io n of the production function for a given К = К (Fig. 1), we see that there exists an N' that im plies a nonconcave section for N § N 1. This difficulty is overcom e if we define an output above minimum consum ption (or surplus output) as
F(N, К) = ф (N, K) - cN (8 )
Then, provided it_is assumed that ф/ N = с in tersects ф = ф(N, К) at a value of N > N 1 for all К, the function F (N ,K ) may be taken to be strictly concave for all (N, K) such that ф/N S c.
N otice that
F K
II -e- (9)
F = ó - C ( 1 0 )N N
F NN
n
Z F = è - F = àNK VNKJ KK V KK (П )
so that in the region where y = F /N S 0, (6 ) can be assum ed to be negative- definite. It fo llow s that < 0 , фкк < 0 in this region and we w ill further assum e that фык = <i>KN in this region.
With respect to investm ent and population growth, we w ill assum e that capital is not subject to depreciation , so
K= I, (12)
IAEA-SM R-17/49 193
T A B L E II. P O L IC I E S A N D T H E IR C O N S E Q U E N C E S
Policy A All production consumed:
y > c - с - о ô K - 0 ; N = 6 N ; с = у + с
Policy В All surplus invested:
у > с - с - oÓ К = Ij.; Ñ = ôN; с = с + (F - )/N
Policy С All surplus devoted to population expansion:
y < C “ C ' O Ó К = 0; Ñ = ÔN + I2 /о; с = с + (F - I2 ) /N
Policy D Ail surplus devoted to population contraction:
у < с - с + 00 K - 0; Ñ = ÔN - U / 6; с = c + (F - Ia)/N
Policy Е Ail surplus devoted to investment and population expansion:
у < С “ С "О Ô К = Ii ; Ñ = ÔN + I2 /о ; с = с + (F - h - I2 )/N
Policy F All surplus devoted to investment and population contraction:
у < с - S + 66 К = U; N = ÔN - 1з/ б; с = с + (F - h - I3)/N
Policy G All surplus devoted to population expansion and contraction: inoperative
and the work force growth is
Ñ = 6 N + I2 / c t - I 3/3 (13)
where
6 : fixed rate o f growth of the work fo rce ,a: cost of the acquisition of one new m em ber o f the work fo rce , which is
assum ed to be constant,/3: cost of the reduction o f one m em ber o f the work fo rce , which is also
assum ed to be constant.
With these assum ptions, the socia l ob jective is to find c(t), I i(t), I2 (t), I3 (t), and T so as to m inim ize
T
/ lu (e ) - u ( c ) ] dt0
where U' > 0, U" < 0 subject to
(i) К = I 2, K(0) = K 0, K(T) = K,(ii) Ñ = 6 N + I2 / a - I3 /j3, N(0) = N 0, N(T) = Ñ,
(iii) c È c , I j i 0 for j = 1 , 2 , 3 ,(iv) F - (с - с) N - I j - I2 - I3 ë 0.
194 SPIN ADEL
TABLE III. POSSIBLE POLICIES IN UNDERPOPULATED COUNTRIES
II I2 p q
A 0 > 0 ^ U ’ ( c ) = U ’( c )
В > 0 0 — U ’ ( c ) Sü' ( c )
С > 0 > 0 = U ' ( c ) = U ' ( c )
D 0 0 < U ' ( c ) ^ U ’ ( c )
To maintain the population constant, the cost per capita is ¡36 so that the maximum possib le consumption per capita w ill be given by
с = с + F(Ñ, K )/Ñ - (36
= С + у - fió (14)
То m ove to (Ñ, К), it is n ecessary that the surplus output per head be s tr ictly positive, that is
у = ф/ N - c> 0 (15)
We will solve the problem applying the maximum princip le of Pontrjagin, N ecessary conditions for a solution to the problem are given by Pontrjagin 's theorem , as m odified by A rrow 13]. Let us construct the Hamiltonian
H = U(c) + ф2{6 N + I2/<j - Ig/jS) - U (c) (16)
and the Lagrangian
L - H + X I- + X2 2 ^-3 3 ^-4 (c - c )+ A. 6 [F - ( c - c ) N - Ij - I2 - I3 ] (17)
where фг and ф2 are continuous functions o f t and are the socia l p r ices of investm ent and of population growth, in term s o f consum ption, respective ly .
The m axim ization of the Lagrangian with respect to the control variab les с , 1 1( 1% and I 3 and subject to the conditions (i) ,( ii) ,(i ii ) ,( iv ) gives
| i - = Ф г + X: - X5 = 0 (18)
(19)
IAEA-SMR-17/49 195
TABLE IV. CONSEQUENCES OF POLICIES IN UNDERPOPULATED COUNTRY
Policy A All benefits devoted to increasing population:
12 U(c) - U(c) N U'(c)
II О О О O'
Policy В All benefits devoted to increasing capital:
Ii U(c) - U(c) N LI'(c)
V 0 1! О 0 V 01
Policy С All benefits devoted to increasing capital and population:
U + I2 U(c) - U(c) N U '(с)
1A > 0; I2 > 0; с > с
Policy D Irrelevant, Operative only at the final end point (N,K).
9 Lj - ф9s i ; = ^ + w o (2 0 )
— = U '(c ) + X. - A..N = 0 (21)Э с 4 5
This requ ires that
0 1 + * 1 = + x2
= - Jk + A./3 3
= *5
= [U '(c) + X4]/N (22)
F or optim al con trol, it is also n ecessary that
Max H = 0 for all t e [0, T ]I3
so that
01I 1 + 02( ' 5 N + ^ - - ^ = u ( 5 ) - U(c) (23)
196 SPIN ADEL
<УФК- Ф.-0 /N (fi)
= ф -C(0/N) (а)
FIG.2. (a) oOp, - 0 ^ - c(0 /N ) has a positive slope and varies between 00^ = 0^ - O/^I and 00^ = 0^ - c ;
(b) 0^ = 0 /N has a greater slope than 0K = 0;
(c) о 0pr = 0^ - 0 / N varies between 0^ = 0 / N and 0^ = 0 in the region where 0 , > 0.
It resu lts that
*г=-Ш--Х*¥ к
¿ 2 = - f ^ = - V FN - ( c - c ) - 0 26
(24)
If it were assum ed that U '(c) = oo, the minimum consumption w ill never be optim al so that
с - с > 0
which in turn im plies X4 = 0 so that X5 = U '(c ) /N > 0 and
F = ( c - c J N + I j + I 2 + I 3 (2 5 )
lA E A -SM R -17/49 197
TABLE V. "SWITCHING" POSSIBILITIES
A в С
Aфк < о
°®K > ®N "
® к < °
° ® K > ®N “
В 0
С 00K <O
o®K > 0N - c (ф/N )
The possib le p o lic ies are given in Table I, the last three colum ns giving values of the soc ia l p r ice s which are associated with them and are defined as follow s
p = N; q'= i//2 N/ct; r = -
The inform ation obtained about the various p o lic ie s is sum m arized in Table II.
Application to a particular case
Suppose we have an underpopulated country. Then, it is o f no use to include the variable I3 which we consider to be zero . B esides, we assume that the rate o f growth o f population represented by 6 is also zero ; this m eans that the growth o f population depends only on the investm ent devoted to births an d /or im m igration.
The problem can be solved by trying to m ove to (Ñ, K) so as to m inim izeT
f [U (c) - U(c) ] dt 0
where U ’ > 0; U" < 0 with the conditions
(i) К = I : ; K(0) = K; K(T) = K,(ii) Ñ = I j/a ; N(0) = N; N(T) = Ñ,
(iii) Ij í 0 ; I2 S 0 ,(iv) F - (с - с) N - Ij - I 2 í 0 ,(v) с > C.
198 SPIN ADEL
(N,K)
FIG.3. Optimal paths.
N ecessary conditions for the existence of a solution are
H = U (с ) +ф111 + Ф212/а - U(c) = 0
L = H + W + X2I2 + X3 [F - (c - c) N - Ij - I2 ] = 0
(26)
(27)
Then
Э L Э1-, - ^ j + X - ^ - X g - 0
9L3c U '(c) - X3N = 0
(28)
which in turn, means that
X jlj = 0, X2 I2 = 0 , X3 [ F - ( c - c ) N - I 1 - I 2 ]= 0
With Max h = 0, it resu lts from (26) that c.ii, U
‘M l + = U(c) - U(c)
(29)
(30)
IAEA-SMR-17 /4 9 199
M oreover
* 3 f k
3 'XN
(31)
A s X3 = U '(c ) /N > 0, we must have
F = (с - С) N - Ij - I2 (32)
It is interesting to note that (32) can be rew ritten using (8 ) to show that consumption per head is now a function of output per head, which we shall indicate as c($ /N ).
The possib le p o lic ies are com piled in Table III. The inform ation re ferrin g to the various p o lic ies for this ca se is sum m arized in Table IV.
It now rem ains to determ ine the slope and position of the paths follow ed by p o lic ies A , В and C. F igure 2 illustrates this inform ation.
The fact that the socia l p r ice s Фг, must be continuous, m akes it possib le to specify other p o lic ies to which a given po licy m ay switch and the regions in which the switch m ay and m ay not o ccu r. This is essentia l in the construction of the optim al paths for it enables the specification of the "sw itch ing surface" on which one po licy must change to another. The possib le switches for this case are set out in Table V. A zero indicates that no switch is possib le .
Now it is easy to draw the optim al paths for this case , as is seen in F ig. 3.
CONCLUSIONS
If we vary с (ф/N ) in the equation
°<t>K = ФN - С(Ф/К )
we can determ ine the optim al behaviour as fo llow s:If афк > </iN - c ($ /N ) the benefit could be inverted, while if афк <' ФN - c (Ф/N),
the benefit must be devoted to increase population.B esides, from F ig . 3 we conclude that
(a) In the region where афк < 0N - c(^ /N ) the p o licy A can be switched to p o lic ie s В or C. The p o licy С can only switch to policy B.
(b) In the region афк < <£N - c(<£/N) the policy В cannot switch to any other policy .
R E F E R E N C E S[11 PITCHFORD, J.. Population and Optimal Growth, Econometrics 4 0 , 1 (1972).[2 ] PONTRJAGIN, L.S., BOLTYANSKII, V .G ., GAMKRELIDZE, R.V.] MISCHENKO, E.F., The Mathematical
Theory of Optimal Processes, Wiley, New York (1962).[ 3] ARROW, K.J., "Application of control theory to economic growth", Mathematics of the Decision
Sciences, Part 2, A m . Math. Soc.,Providence (1968).
IAEA -SM R-17 /5 0
F U N C TIO N A L AN ALYSIS IN THE STUDY OF D IF F E R E N T IA L AND IN TEG R AL EQUATIONS
G. R. SELLSchool of Mathematics,University of Minnesota,Minneapolis, Minnesota,United States of America
Abstract
FUNCTIONAL ANALYSIS IN THE STUDY OF DIFFERENTIAL AND INTEGRAL EQUATIONS.This paper illustrates the use of functional analysis in the study of differential equations. Our
particular starting point, the theory of flows or dynamical systems, originated with the work of H. Poincaré, who is the founder of the qualitative theory of ordinary differential equations. In the qualitative theory one tries to describe the behaviour of a solution, or a collection of solutions, without "solving " the differential equation. As a starting point one assumes the existence, and sometimes the uniqueness, of solutions and then one tries to describe the asymptotic behaviour, as time + °°, of these solutions.
We compare the notion of a flow with that of a C„ -group of bounded linear operators on a Banach space. We shall show how the concept C0 -group, or more generally a C0 -semigroup, can be used to study the behaviour of solutions of certain differential and integral equations. Our main objective is to show how the concept of a C0 -group and especially the notion of weak5'-compactness can be used to prove the existence of an invariant measure for a flow on a compact Hausdorff space. Applications to the theory of ordinary differential equations are included.
1. INTRODUCTION
Many of the theories of m odern analysis and topology can trace their orig ins, som etim es over many centuries, to the study of ordinary differential equations. The use of functional analysis, which we shall d escribe , is one such exam ple. Our particular starting point, the theory of flows or dynam ical system s, has a m ore recent h istory going back less than a hundred years to H. Poincare [8 ], who is the founder o f the qualitative theory of ordinary differential equations. In the qualitative theory one tr ies to d escribe the behaviour o f a solution, or a co llection of solutions, without "so lv in g" the differential equation. A s a starting point one assum es the existence, and som etim es the uniqueness, of solutions and then one tr ies to d escribe the asym ptotic behaviour, as tim e t -*■ +00, o f these solutions.
We shall b rie fly review the essence o f the fundamental theory of ordinary d ifferential equations, that is , the question of existence, uniqueness and continuity of solutions, that is needed to build a dynam ical system or flow . We shall then com pare the notion o f a flow with that of a C o-group of bounded linear operators on a Banach space. We shall show how the concept C o-group, or m ore generally a C 0 -sem igroup , can be used to study the behaviour o f solutions of certain d ifferential and integral equations. Our main ob jective though is to show how the concept
201
2 0 2 SELL
of a C o-group and especia lly the notion of w eak*-com pactness can be used to prove the existence o f an invariant m easure for a flow on a com pact H ausdorff space. A s we shall see, the existence of an invariant m easure is a very useful feature in the qualitative theory of ordinary d ifferential equations.
2. REVIEW OF THE FUNDAMENTAL THEORY OFDIFFERENTIAL EQUATIONS
The problem we shall study here is the question of existence and uniqueness of solutions of the in itial-value problem
x ' = f(x ,t), x (0 ) = x ( 1 )
where f : W X R - > R n and W is an open set in R n. A solution o f (1) is defined as an absolutely continuous function cp, defined on an interval containing t = 0 , such that cp(0) = x and cp '( t) = f ( c p ( t ) , t ) fo r alm ost all t. P robably the weakest conditions known, which w ill ensure the existence of a solution fo r every in itial-value problem with x e W, are those due to Carathèodory, v iz.
(i) fo r each t, f(x,t) is continuous in x,(ii) fo r each x, f(x,t) is m easurable in t, and
(iii) fo r each com pact set K £ W there is a function m = m(t)in L ,1 (R ,R) such that
lo c
I f(x,t) I S m (t), (x 6 K, t 6 R)
F o r exam ple, if f(x,t) is jointly continuous in x and t then all three conditions above are satisfied.
Of cou rse , the in itial-value problem (1) may have m ore than one solution, as is w ell known. In order to guarantee the uniqueness of solutions, an additional condition is needed. The L ipschitz condition, which reads
(iv) fo r each com pact set К С W there is a function к = k(t) e L^oc (R,R) such that
I f(x,t) - f(y,t) I § k(t) I x -y I, (x ,y 6 K, t 6 R)
is a typ ical sufficient condition fo r uniqueness. Other exam ples of sufficient conditions fo r uniqueness can be found in the standard textbooks on d ifferentia l equations. Let us sim ply note that if f and 9 f/3x are jointly continuous, then fo r every x e W, the in itial value problem (1) has unique solutions.
One can use the solutions of a differential equation to build a dynam ical system or a flow, as described in [9, 11, 12], provided the appropriate d ifferentia l equations satisfy the existence and uniqueness conditions for solutions of the in itial-value problem . In fact one can even drop the uniqueness requirem ent, see R ef. [12]. H owever, in order to avoid undue techn ical details we shall make a further assumption
IAEA-SMR-17/50 203
a b o u t t h e s o l u t i o n s , v i z . t h e g l o b a l e x i s t e n c e p r o p e r t y , i . e . e v e r y s o l u t i o n c p ( t ) i s d e f i n e d f o r a l l t i n R .
I n p r a c t i c e 3 t h e r e a r e t w o g e n e r a l t h e o r e m s w h i c h g u a r a n t e e t h e g l o b a l e x i s t e n c e p r o p e r t y . T h e s e a r e
1 . I f W = R n a n d
| f ( x , t ) I § A ( t ) | x | + B ( t ) , ( x G R n , t G R )
w h e r e A , В G Ь ^ о с ( R , R ) , t h e n a l l s o l u t i o n s o f ( 1 ) a r e d e f i n e d f o r a l l t G R .
2 . I f W i s b o u n d e d a n d t h e v e c t o r f i e l d f ( x , t ) i s t a n g e n t t o t h e b o u n d a r y 9 W a t e a c h p o i n t x G 9 W , t h e n ( 1 ) h a s t h e g l o b a l e x i s t e n c e p r o p e r t y i n W .F o r e x a m p l e , i f W = { x € R n : | x | < 1 } , a n d i f t h e d o t p r o d u c t x ■ f ( x , t ) i s z e r o f o r a l l x w i t h | x | = 1 , t h e n ( 1 ) h a s t h e g l o b a l e x i s t e n c e p r o p e r t y .
O f c o u r s e , i f W i s u n b o u n d e d a n d W / R n o n e c a n c o m b i n e t h e a b o v e t w o c o n d i t i o n s t o g e t a n a p p r o p r i a t e t h e o r e m g u a r a n t e e i n g t h e g l o b a l e x i s t e n c e p r o p e r t y .
3 . D Y N A M I C A L S Y S T E M S
L e t X d e n o t e a H a u s d o r f f s p a c e . ( I n m o s t a p p l i c a t i o n s w e s h a l l b e i n t e r e s t e d i n , X w i l l b e a m e t r i c s p a c e . ) A m a p p i n g ir : X X R - * X i s s a i d t o b e a f l o w o r d y n a m i c a l s y s t e m o n X i f
( i ) т г ( х , 0 ) = x , f o r a l l x G X( i i ) ir{ir(x,t), s ) = 7r ( x , t + s ) f o r a l l x G X , a n d t , s 6 R
( i i i ) ir i s j o i n t l y c o n t i n u o u s i n x a n d t .
A m a p p i n g я- : X X R + - * X i s s a i d t o b e a s e m i f l o w , w h e r eR + = { t G R : 0 s t < ° ° } , i f c o n d i t i o n s ( i ) , ( i i ) a n d ( i i i ) a r e s a t i s f i e df o r t , s G R + .
T h e p r o t o t y p e f o r a f l o w c o m e s f r o m t h e s t u d y o f a u t o n o m o u s d i f f e r e n t i a l e q u a t i o n s x 1 = f ( x ) o n a n o p e n s e t W Ç R n . I f w e a s s u m e t h a t e v e r y i n i t i a l - v a l u e p r o b l e m
x ' = f ( x ) , x ( 0 ) = x
h a s a u n i q u e s o l u t i o n c p ( x , t ) f o r e v e r y x G W a n d t h a t c p ( x , t ) i s d e f i n e d ( a n d r e m a i n s i n W ) f o r a l l t G R , t h e n
7г( X , t ) = < p ( x , t )
d e f i n e s a f l o w o n W .
F o r n o n - a u t o n o m o u s d i f f e r e n t i a l e q u a t i o n s w e s h a l l t a k e a m o r e g l o b a l p o i n t o f v i e w . R a t h e r t h a n c o n s i d e r i n g a s i n g l e f u n c t i o n f : W X R - * R n a n d t h e a s s o c i a t e d d i f f e r e n t i a l e q u a t i o n x ' = f ( x , t ) , w e s h a l l l o o k a t a c o l l e c t i o n F o f s u c h f u n c t i o n s , w h e r e W i s a n o p e n s e t i n R n . I n o r d e r t o u s e t h e s o l u t i o n s o f t h e d i f f e r e n t i a l e q u a t i o n s x 1 = f ( x , t ) , f G F , t o c o n s t r u c t a f l o w , w e s h a l l a s s u m e t h a t F h a s t h e f o l l o w i n g p r o p e r t i e s :
2 0 4 SELL
1 . F i s t r a n s l a t i o n - i n v a r i a n t ( i . e . i f f £ F , t h e n f , £ F f o r a l l т £ R . w h e r e f T ( x , t ) = f ( x , T + t ) ) ;
2 . F s a t i s f i e s t h e C a r a t h è o d o r y p r o p e r t y ( i . e . f o r e a c h x £ W a n d f £ F , t h e r e i s a u n i q u e s o l u t i o n c p ( x , f , r ) o f t h e i n i t i a l - v a l u e p r o b l e m
x ' = f ( x , t ) , x ( 0 ) = x
a n d c p ( x , f , t ) i s d e f i n e d f o r a l l t £ R ) ;3 . F h a s a t o p o l o g y a n d t h e m a p p i n g s o : F X R - * F a n d c p : W X F X R - * - W
g i v e n b y a : ( f ,T ) -» f T a n d cp : ( x , f , r ) -> < p ( x , f , r ) a r e c o n t i n u o u s .
U n d e r t h e s e t h r e e c o n d i t i o n s , t h e m a p p i n g
т г ( х Д , т ) = ( c p ( x , í , t ), f T)
d e f i n e s a f l o w o n W X F .
T h e t o p o l o g i c a l c o n d i t i o n 3 i s d i s c u s s e d a t s o m e l e n g t h i n R e f . [ 7 ] .L e t u s s i m p l y n o t e h e r e t h a t , i f F c o n s i s t s o f a f a m i l y o f c o n t i n u o u s f u n c t i o n s f r o m W X R t o R n t h a t i s t r a n s l a t i o n - i n v a r i a n t a n d s a t i s f i e s t h e C a r a t h è o d o r y p r o p e r t y , t h e n t h e t o p o l o g i c a l c o n d i t i o n 3 i s s a t i s f i e d
i f F h a s t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e o n c o m p a c t s e t s , s e e R e f . [ 9 ] .A s p e c i a l c a s e o f t h e a b o v e c o n s t r u c t i o n o c c u r s w h e n o n e i s
s t u d y i n g l i n e a r d i f f e r e n t i a l e q u a t i o n s , i . e . f ( x , t ) = A ( t ) x i s l i n e a r i n x .F o r t h i s p u r p o s e l e t j < / d e n o t e a c o l l e c t i o n o f ( n X n ) m a t r i x v a l u e d f u n c t i o n s A ( t ) ( t £ R ) w i t h c o n t i n u o u s c o e f f i c i e n t s , a n d c o n s i d e r t h e l i n e a r d i f f e r e n t i a l e q u a t i o n s
x 1 = A ( t ) x ( A E o O O
w h e r e x £ X a n d X = R n , o r C n . A s s u m e t h a t
1 . i s t r a n s l a t i o n - i n v a r i a n t , a n d2 . .stf h a s t h e t o p o l o g y o f u n i f o r m c o n v e r g e n c e o n c o m p a c t s e t s .
L e t c p ( x , A , t ) d e n o t e t h e s o l u t i o n o f t h e i n i t i a l - v a l u e p r o b l e m x ' = A ( t ) x , x ( 0 ) = x . T h e n
7r(x , A , t ) = (cp ( x , A , t ) , A t )
i s a f l o w i n X X jrf. F u r t h e r m o r e с р ( х , А , т ) i s l i n e a r i n x , s o w e c a n d e f i n e a l i n e a r t r a n s f o r m a t i o n Ф ( Д Д ) o n X b y $ ( A , t ) x = c p ( x , A , t ) . ( N o t e $ ( A , t ) i s r e f e r r e d t o a s t h e f u n d a m e n t a l m a t r i x s o l u t i o n o f x ' = A ( t ) x . )
W e s h a l l b e i n t e r e s t e d i n t h e c a s e w h e r e j a ' i s c o m p a c t i n t h e a b o v e t o p o l o g y . B y t h e A s c o l i - A r z e l á t h e o r e m w e s e e t h a t j s ^ i s c o m p a c t i f a n d o n l y i f
( i ) t h e r e i s a n M < + 0 0 s u c h t h a t | A ( t ) | S M f o r a l l A e j ÿ ' a n d a l l t £ R , a n d
( i i ) f o r e v e r y e > 0 t h e r e i s a 6 > 0 s u c h t h a t | A ( t + T > - A ( t ) | S € ( w h e r e v e r | т | S 6 ) f o r a l l A £ a n d a l l t £ R .
IAEA-SMR-17/50 2 0 5
N o w l e t X d e n o t e a r e a l o r c o m p l e x B a n a c h s p a c e . F o r t £ R l e t T t d e n o t e a b o u n d e d l i n e a r o p e r a t o r f r o m X t o X s a t i s f y i n g
( i ) T 0 = I = i d e n t i t y ,( i i ) T t T s = T t+S , f o r t , s £ R ,
T h e n { T t } i s c a l l e d a ( o n e - p a r a m e t e r ) g r o u p ( o f b o u n d e d l i n e a r o p e r a t o r s ) o n X . I f , i n a d d i t i o n , o n e h a s
( i i i ) { T t } i s s t r o n g l y c o n t i n u o u s , i . e . f o r e a c h t o G R a n d x E X
4 . C 0 - S E M I G R O U P S A N D C 0 - G R O U P S
l i m T t x = T t(| xt — to
t h e n { T t } i s c a l l e d a g r o u p o f c l a s s C o , o r s i m p l y a C p - g r o u p .A C o - s e m i g r o u p i s d e f i n e d i n t h e s a m e w a y w i t h R b e i n g r e p l a c e d
b y R * = ( t : 0 S t < 0 0 } t h r o u g h o u t .L e t { T t } b e a C q - g r o u p o n X a n d d e f i n e i r : X X R -> X b 'y
7г ( X, t ) = T t x
W e t h e n a s k , i s ж a f l o w o n X ? L e t u s c h e c k t h e t h r e e c o n d i t i o n s :
( i ) ?r( x , 0 ) = T o x = I x = x( i i ) 7r ( 7r ( x , t ) , s ) = 7r ( T t x , s ) = T s ( T t x ) = T t+S x = 7r ( x , t + s ) .
T h e f i r s t t w o a r e s a t i s f i e d , b u t w h a t a b o u t t h e j o i n t c o n t i n u i t y o f 7r ? B y t h e s t r o n g c o n t i n u i t y o f { T t } w e s e e t h a t f o r e a c h x , 7r ( x , t ) i s c o n t i n u o u s i n t . F u r t h e r m o r e , s i n c e T { i s a b o u n d e d l i n e a r o p e r a t o r , w e s e e t h a t f o r e a c h t , ж(х,t ) i s c o n t i n u o u s i n x . T h a t i s , 7r ( x , t ) i s c o n t i n u o u s i n e a c h v a r i a b l e s e p a r a t e l y , b u t i t n e e d n o t b e j o i n t l y c o n t i n u o u s , s e e R e f s [ 3 , 1 4 ] .
I f { T t : t ê 0 } i s a C 0 - s e m i g r o u p o n X , t h e n 7r ( x , t ) = T t x d e f i n e s a m a p p i n g ж : X X R + -»■ X , t h a t s a t i s f i e s ж(х,0) = x , ж(ж(х,Х), s ) = 7r ( x , t + s )
f o r a l l t , s ё 0 , a n d ir i s c o n t i n u o u s i n e a c h v a r i a b l e s e p a r a t e l y . I t n e e d n o t b e j o i n t l y c o n t i n u o u s .
L e t u s n o w l o o k a t a v e r y s i m p l e e x a m p l e . L e t A b e a b o u n d e d l i n e a r o p e r a t o r o n a f i n i t e - d i m e n s i o n a l B a n a c h s p a c e X a n d f o r m t h e e x p o n e n t i a l
w h i c h i s d e f i n e d f o r a l l t 6 R . I t i s n o t h a r d t o s h o w t h a t T t = e tA i s a C o ' g r o u p o n X . F u r t h e r m o r e o n e c a n r e c o v e r A f r o m T t b y d i f f e r e n t i a t i n g , i . e .
o n e h a s
CC
n = 0
T h e o p e r a t o r A i s c a l l e d t h e i n f i n i t e s i m a l g e n e r a t o r o f { T t } .
2 0 6 SELL
T h e r e l a t i o n s h i p b e t w e e n a C o ' g r o u p , o r m o r e g e n e r a l l y a C o _ s e m i g r o u p , a n d i t s i n f i n i t e s i m a l g e n e r a t o r i s d e s c r i b e d b y a f u n d a m e n t a l t h e o r e m w h i c h w e s h a l l n o w s t u d y .
5 . I N F I N I T E S I M A L G E N E R A T O R
L e t { T t } b e a C 0 - s e m i g r o u p ( o r a C 0 ~ g r o u p ) o n a B a n a c h s p a c e X . D e f i n e
D ( A ) = i x S X : l i m r - ( T h - I ) x e x i s t s ^ h 0+
F o r x e D ( A ) d e f i n e A x b y
A x = l i m 7 - ( T h - I ) x h n
h -» o +
T h e o p e r a t o r A i s r e f e r r e d t o a s t h e i n f i n i t e s i m a l g e n e r a t o r o f { T t } a n d D ( A ) i s t h e d o m a i n o f A . T h e f o l l o w i n g t h e o r e m g i v e s t h e e s s e n t i a l c o n n e c t i o n b e t w e e n t h e C 0 - g r o u p { T t } a n d i t s i n f i n i t e s i m a l g e n e r a t o r A , s e e R e f s [ 3 , 1 4 ] .
T h e o r e m 1 . L e t { T t } b e a C 0 - s e m i g r o u p o n a B a n a c h s p a c e X a n d l e t A b e t h e i n f i n i t e s i m a l g e n e r a t o r o n t h e d o m a i n D ( A ) . A s s u m e t h e r e a r e c o n s t a n t s M > 0 a n d /3 § 0 s u c h t h a t
Il T t H S M e s t , t ë 0
T h e n t h e f o l l o w i n g h o l d s :
( i ) D ( A ) i s d e n s e i n X a n d A i s a c l o s e d l i n e a r o p e r a t o r ,( i i ) A T t x = T t A x f o r a l l t Ш 0 a n d x € D ( A ) .
( i i i ) {X : R e X > (3 } Ç p ( A ) = r e s o l v e n t s e t o f A .
( i v ) A S M l - ln
f o r m = 1 , 2 , a n d a l l n
s u f f i c i e n t l y l a r g e .
T h e r e i s a c o n v e r s e t o t h e o r e m 1 w h e r e o n e s t a r t s w i t h a l i n e a r o p e r a t o r A a n d p r o c e e d s t o c o n s t r u c t a C o - s e m i g r o u p . I t i s t h e f o l l o w i n g :
T h e o r e m 2 . L e t A b e a l i n e a r o p e r a t o r d e f i n e d o n a d e n s e d o m a i n D ( A ) i n a B a n a c h s p a c e X w i t h r a n g e i n X . A s s u m e t h a t ( I - ( l / n ) A ) h a s a b o u n d e d l i n e a r i n v e r s e , f o r n s u f f i c i e n t l y l a r g e , a n d t h a t t h e r e a r e c o n s t a n t s /3 Ш 0 a n d M > 0 s u c h t h a t
S M I
IAEA-SMR-17/50 207
f o r m = 1 , 2 , . . . a n d a l l n s u f f i c i e n t l y l a r g e . T h e n A i s t h e i n f i n i t e s i m a l g e n e r a t o r o f a C 0 - s e m i g r o u p T t , a n d T t s a t i s f i e s || T t || S M e 8 t f o r a l l t ê 0 . O n c e a g a i n w e r e f e r t h e r e a d e r t o [ 3 , 1 4 ] f o r t h e p r o o f .
T h e s t u d y o f t h e s p e c t r u m o f t h e i n f i n i t e s i m a l g e n e r a t o r A l e a d s t o v a l u a b l e i n f o r m a t i o n c o n c e r n i n g t h e s t r u c t u r e o f t h e s e m i g r o u p T t . A v e r y i n t e r e s t i n g e x a m p l e o f t h i s o c c u r s i n t h e s t u d y o f l i n e a r f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n s
w h e r e L i s a b o u n d e d l i n e a r o p e r a t o r f r o m C ( [ - r , 0 ] , R n ) = С t o R n , C ( [ - r , 0 ] , R n ) i s t h e B a n a c h s p a c e o f c o n t i n u o u s f u n c t i o n s f r o m { 0 : - r S В s 0 } t o R n a n d x t ( 6 ) = x ( t + 0 ) , f o r - r ê f t S 0 , i s t h e p r o f i l e o f t h e s o l u t i o n x ( t ) o v e r t h e p r e v i o u s r - u n i t s . I n t h i s c a s e , t h e f u n c t i o n a l d i f f e r e n t i a l e q u a t i o n ( 3 ) g e n e r a t e s a C 0 - s e m i g r o u p T t o n C , f u r t h e r m o r e
f o r t è r , t h e l i n e a r o p e r a t o r T t i s c o m p l e t e l y c o n t i n u o u s . T h e a s s o c i a t e d i n f i n i t e s i m a l g e n e r a t o r A h a s t h e p r o p e r t y t h a t i t s s p e c t r u m c o n s i s t s o f a p o i n t s p e c t r u m o n l y a n d t h a t X i s a n e i g e n v a l u e o f A i f a n d o n l y i f d e t Д ( Х ) = 0 , w h e r e
I t f o l l o w s t h a t f o r a n y r e a l n u m b e r y, t h e s e t o f e i g e n v a l u e s X s a t i s f y i n g R e X 2 y i s f i n i t e . T h i s l e a d s t o a n i n t e r p r e t a t i o n o f t h e s o l u t i o n s o f ( 3 ) a s h a v i n g a c e r t a i n s a d d l e b e h a v i o u r . W e r e f e r t h e r e a d e r t o R e f . [ 2 ] ( p p . 9 4 - 1 0 4 ) f o r m o r e d e t a i l s .
A n o t h e r e x a m p l e o c c u r s i n t h e s t u d y o f l i n e a r V o l t e r r a i n t e g r o - d i f f e r e n t i a l e q u a t i o n s
w h e r e B G L 1 ( 0 , « > ) , s e e R e f . [ 5 ] . L e t C 0 = { f : ( - 0 0 , 0 ] -<• R n : f i s c o n t i n u o u s a n d h a s c o m p a c t s u p p o r t } .
S i n c e B G L 1 ( 0 , « > ) , o n e c a n s h o w t h a t 9 ( f ) i s b o u n d e d f o r t I 0 . L e t II 9 ( f ) IU = s u p { | c p ( f ) ( t ) | : t ï 0 ) . L e t Y o = { ( f ( 0 ) , c p ( f ) ) : f G C 0 } b e t h e
x ' ( t ) = L ( x t ) ( 3 )
0
-r
a n d w h e r e L , h a s t h e R i e s z r e p r e s e n t a t i o n
0
-r
x ' ( t ) = A x ( t ) + J B ( t - s ) x ( s ) d s + F ( t ) , x ( 0 ) = x 0 ( 4 )
0
L e t
0
( t ê o , f e c 0 )
2 0 8 SELL
l i n e a r s p a c e w i t h n o r m || ( f ( 0 ) , c p ( f ) || = | f ( 0 ) | + | | q > ( f ) | | „ a n d l e t Y b e t h e c o m p l e t i o n o f Y 0 . U s i n g t h e c o n s t r u c t i o n o f a f l o w p r e s e n t e d i n R e f . [ 6 ] , o n e c a n s h o w t h a t
t
it ( t , x 0 , F ) = ( ç ( t , x 0 , F ) , F t ( • ) + J B ( s + - ) c p ( t - s , x 0 , F ) d s )
о
i s a s e m i f l o w o n Y , w h e n c p ( t , x 0 , F ) i s t h e s o l u t i o n o f ( 4 ) . F u r t h e r m o r e
T t ( x „ , F ) = T r ( t , x 0 , F )
d e f i n e s a C o - s e m i g r o u p o n Y . W i t h o u t g o i n g i n t o f u r t h e r d e t a i l , l e t u s s i m p l y n o t e t h a t o n e c a n r e l a t e t h e s t a b i l i t y p r o p e r t i e s o f t h i s s e m i g r o u p T t w i t h t h o s e o f t h e i n f i n i t e s i m a l g e n e r a t o r , s e e R e f . [ 5 ] ,
6 . F L O W S O N C O M P A C T S P A C E S
L e t X n o w d e n o t e a c o m p a c t H a u s d o r f f s p a c e a n d l e t ж : X X R -» X b e a f l o w o n X . D e f i n e wt b y 7rt ( x ) = 7 r ( x , t ) . W e t h e n h a v e t h e f o l l o w i n g e l e m e n t a r y p r o p e r t y :
L e m m a 3 . F o r e a c h t € R , irt i s a h o m e o m o r p h i s m o f X a n d
(jrt j ' 1 = v.t .N o w l e t С = C ( X , R ) d e n o t e t h e B a n a c h s p a c e o f c o n t i n u o u s f u n c t i o n s
f r o m X t o R w i t h t h e s u p n o r m , 11 • | ] . N e x t d e f i n e T t : С - * С b y
( T t c p ) ( x ) = (< p -7 T t ) ( x ) = c p ( 7 r ( x , t ) )
w h e r e cp e C .T h e o r e m 4 . T h e f a m i l y { T t : t S R } i s a C o “ g r o u p o f b o u n d e d l i n e a r
o p e r a t o r s o n C . A l s o t h e m a p p i n g
î f ( < p , t ) = T t q>
d e f i n e s a f l o w o n C , i . e . 5? i s j o i n t l y c o n t i n u o u s i n cp a n d t . F u r t h e r m o r e ,Il T t cp II = || cp II f o r a l l t a n d cp, h e n c e || T t || = 1 f o r a l l t . C o n s e q u e n t l y t h e i n f i n i t e s i m a l g e n e r a t o r A h a s t h e p r o p e r t y t h a t t h e s p e c t r u m
c r ( A ) Ç { X : R e X = 0 }
P r o o f : T 0 cp = Ф • ttq = cp • I = cp, s i n c e 7г0 = I = i d e n t i t y . A l s o T t ( T s cp) = T t (cp . 7TS ) = (cp • 7TS ) • 7Tt = cp • U s • 7Tt ) = cp • 7rt+s = T t+ S cp. H e n c e T t T s = T t + s • L e t u s n o w s h o w d i r e c t l y t h a t 5 г ( ф Д ) i s j o i n t l y c o n t i n u o u s i n ф a n d t . ( T h i s f o l l o w s , a s w e s h a l l s e e , f r o m t h e f a c t t h a t t h e f l o w ж o n X i s j o i n t l y c o n t i n u o u s . ) L e t ф п -» cp i n С a n d t n - * t i n R . I n o r d e r t o s h o w t h a t 7 г (ф п Д п ) ->■ 7 ? (ф Д ) i n С , i t w i l l s u f f i c e t o s h o w t h a t
5г(фп , 8 п) ** 5 г (ф , 0 ) = Ф ( 5 )
IAEA-SMR-17/50 2 0 9
w h e r e cpn -» cp a n d s n = t n - t - 0 . N o w ( 5 ) i s e q u i v a l e n t t o s a y i n g t h a t
s u p I ф п ( т г ( х , s n ) ) - ф ( х ) I -> 0 ( 6 )
H E X
a s n -» ° o , w h e r e s n -» 0 a n d s u p |c p n ( x ) - c p ( x ) | -*■ 0 . I f ( 6 ) w e r e f a l s e ,j e X
t h e n w e c o u l d f i n d a s e q u e n c e { x n } Ç x a n d e > 0 s u c h t h a t
|cp n ( 4 x n , s n ) ) - c p ( x n ) | ï e ( 7 )
W i t h o u t a n y l o s s o f g e n e r a l i t y w e c a n a s s u m e t h a t { x n } i s c o n v e r g e n t , s a y x n -» x o . O n e t h e n h a s 7 r ( x n , s n ) ->■ 7г(х0,О) = x q ( b y t h e j o i n t c o n t i n u i t y o f тт), ф ( х п ) - ф ( х 0 ) ( b y t h e c o n t i n u i t y o f <p) a n d ф п ( я - ( х п , 8 п ) ) - ф ( х 0 ) ( b y t h e u n i f o r m c o n v e r g e n c e o f ф п ) . H e n c e ( 7 ) h o l d s i n t h e l i m i t , w h i c h b e c o m e s | ф ( х 0 ) - ф ( х 0 ) I È e > 0 , a c o n t r a d i c t i o n .
N e x t , s i n c e 7 rt i s a h o m e o m o r p h i s m ( b y l e m m a 3 ) , o n e h a s || T t Ф || = | |ф | | f o r a l l t e R a n d ф Е С . T h e r e f o r e || T t II = 1 f o r a l l t . C o n s e q u e n t l y , t h e r e s o l v e n t s e t p ( A ) , w h e r e A i s t h e i n f i n i t e s i m a l g e n e r a t o r o f T t , c o n t a i n s t h e t w o h a l f - p l a n e s { X : R e X > 0 } a n d { X : R e X < 0 } . H e n c e c x ( A ) С { X: R e X = 0 } . Q . E . D .
E x a m p l e . L e t X = { e i 0 : O s 0 < 2 7 r } d e n o t e t h e u n i t c i r c l e i n t h ec o m p l e x p l a n e a n d c o n s i d e r t h e f l o w o n X g i v e n b y j r ( e ie , t ) = e ¡ ( 9 + t > .T h e B a n a c h s p a c e С = C ( X , R ) c a n n o w b e i d e n t i f i e d a s
C = ^ : R - * R ^ i s c o n t i n u o u s a n d 2 7 r - p e r i o d i c }
T h e n t h e C 0 - g r o u p T t b e c o m e s ( Т 4ф ) ( 0 ) = ф ( 0 - К ) . I t i s t h e n e a s i l y s e e n t h a t t h e i n f i n i t e s i m a l g e n e r a t o r A s a t i s f i e s А ф = ф 1 = d ф / d 0 o n t h e d o m a i n
D ( A ) = { ф Е С : ф ' 6 С }
I n o r d e r t o d e t e r m i n e w h e t h e r X S p ( A ) , t h e r e s o l v e n t s e t , w e h a v e t o d e t e r m i n e w h e t h e r t h e e q u a t i o n А ф - Х ф = ф, o r
ф ' - Х ф = ф (г)
h a s a s o l u t i o n ф € С f o r e v e r y ф G С a n d t h a t t h e r e l a t e d m a p p i n g ф - * ф b e a b o u n d e d l i n e a r o p e r a t o r . T h e s o l u t i o n o f ( 8 ) i s s i m p l y
9
ф ( 0) = e x e (ф о + J ' e~Xs ф( s ) d s ^
о
I n o r d e r f o r ф G С w e n e e d ф t o b e 2 7 r - p e r i o d i c . H e n c e ф m u s t s a t i s f y ф ( 0 ) = ф 0 = ф ( 2 7г) , o r
2тг
( e ‘ 2 ï ïX - 1 ) ф 0 = J e ‘ Xs ^ ( s ) d s
0
( 9 )
210 SELL
N o w E q . ( 9 ) a d m i t s a s o l u t i o n ф о , p r o v i d e d X f i n , w h e r e n i s a n i n t e g e r . T h i s ф 0 i n t u r n g e n e r a t e s a s o l u t i o n ф g C . H e n c e c r ( A )= { i n : n = i n t e g e r } .
N o t e : I f 7Г : X X R + -» X i s a s e m i f l o w , t h e n T t , a s d e f i n e d a b o v e , i s a C o - s e m i g r o u p o n С a n d ^ ф i s j o i n t l y c o n t i n u o u s o n С X R + . I n t h i s c a s e o n e c a n o n l y a s s e r t t h a t || T t || § 1 s i n c e vt n e e d n o t b e a h o m e o - m o r p h i s m o f X .
7 . F L O W S O N C O M P A C T S P A C E S C O N T I N U E D .T H E A D J O I N T S P A C E
W e c o n t i n u e w i t h t h e s e t - u p o f t h e l a s t s e c t i o n . X i s a c o m p a c t H a u s d o r f f s p a c e w i t h a f l o w ж a n d С = C ( X , R ) i s a B a n a c h s p a c e o f c o n t i n u o u s f u n c t i o n s o n X . N e x t l e t С ' d e n o t e t h e a d j o i n t s p a c e t o C , i . e . С ' i s t h e c o l l e c t i o n o f a l l b o u n d e d l i n e a r f u n c t i o n a l s f r o m С t o R .L e t u s r e c a l l t h e R i e s z t h e o r e m w h i c h i s a r e p r e s e n t a t i o n t h e o r e m f o r C \ s e e R e f . [ 1 ] .
T h e o r e m 5 . F o r e v e r y b o u n d e d l i n e a r f u n c t i o n a l i G С ' t h e r e i s a s i g n e d B a i r e m e a s u r e ц o n X s u c h t h a t ^ ( ф ) = / ф ( х ) р Ы х ) f o r a l l ф G С .x .
F u r t h e r m o r e , / j fe 0 i f a n d o n l y i f £ g 0 , i . e . £(q>) ê 0 f o r e v e r y ф I 0 .
A l s o ü ( l ) = 1 i f a n d o n l y i f ц{Х) = 1 . F i n a l l y , i f £ i 0 , t h e n || £ || = i ( l ) .T h e f l o w т г ( ф Д ) = Т { ф o n С c a n b e l i f t e d t o t h e a d j o i n t s p a c e С ' b y
( U t -C ) ( Ф ) = Л Т , ф ) ( 1 0 )
w h e r e I G C 1 a n d ф G С . I f o n e u s e s t h e n o t a t i o n < ^ , j O f o r £(<$>) ( w h e r e ф G С a n d £ G С ' ) t h e n ( 1 0 ) c a n b e r e w r i t t e n a s
< Ф , и ^ > = < T t 9 , i > ( 1 1 )
i . e . U t i s t h e a d j o i n t o p e r a t o r o f T t . B e c a u s e o f t h e c o n t i n u i t y o f ^ ф i n t , w e h a v e t h e f o l l o w i n g f a c t :
L e m m a 6 . F o r e a c h ф G С a n d £ G С 1, t h e m a p p i n g t -*■ ( и ^ ) ( ф ) i s a c o n t i n u o u s m a p p i n g o f R i n t o R .
O n e c a n e a s i l y s h o w t h a t U 0 = I a n d U t U s = U t+S , s o i t s e e m s n a t u r a l t o a s k w h e t h e r t h e m a p p i n g
n{£,t) = U t i
d e f i n e s a f l o w o n С ' . A c t u a l l y t h i s q u e s t i o n i s m e a n i n g f u l i n a m o r e g e n e r a l c o n t e x t , a n d , i f o n e u s e s t h e c o r r e c t t o p o l o g y o n t h e a d j o i n t s p a c e , v i z . t h e w e a k * - t o p o l o g y , o n e c a n c o n c l u d e t h a t f i s a f l o w .
IAEA-SMR-17/50 2 1 1
L e t { T t : t è 0 } b e a C o ' S e m i g r o u p o n a B a n a c h s p a c e X a n d l e t X ' d e n o t e t h e a d j o i n t s p a c e o f X . F o r x G X a n d < 6 X ' l e t ( x,£ У = £(x)= v a l u e o f £ a t x . D e f i n e U t o n X ' b y
< x , U t . f > = < T t x , i > ( 1 2 )
T h e w e a k * - t o p o l o g y o n X ' i s d e f i n e d a s f o l l o w s : A g e n e r a l i z e d s e q u e n c e { £ n } i s s a i d t o c o n v e r g e t o £ i n t h e w e a k * - t o p o l o g y o n X ' i f a n d o n l y i f f o r e v e r y x G X o n e h a s
< x , i n > - < x , i >
W e s h a l l n e e d t w o p r o p e r t i e s o f t h i s t o p o l o g y i n t h e s e q u e l . T h e p r o o f s c a n b e f o u n d i n R e f . [ 1 ] .
L e m m a 7 . L e t {£ n } b e a g e n e r a l i z e d s e q u e n c e i n X ' a n d a s s u m e t h a t £n -» £ i n t h e w e a k * - t o p o l o g y . T h e n s u p n || £n || < + ° ° .
L e m m a 8 . ( W e a k * - c o m p a c t n e s s ) . L e t Q b e a s u b s e t o f X ' w i t h t h e p r o p e r t y t h a t t h e r e i s a c o n s t a n t В < + ° ° s u c h t h a t || i || S B f o r e v e r y £ G Q . T h e n Q i s r e l a t i v e l y w e a k * - c o m p a c t , i . e . i f { £ n } i s a n y g e n e r a l i z e d s e q u e n c e i n Q , t h e n t h e r e i s a g e n e r a l i z e d s u b s e q u e n c e { j 0 n >>} s u c h t h a t { j? n >s} c o n v e r g e s i n t h e w e a k * - t o p o l o g y .
W e c a n n o w p r o v e t h e f o l l o w i n g r e s u l t .
T h e o r e m 9 . L e t { T t : t S 0 } b e a С 0 - s e m i g r o u p o n a B a n a c h s p a c e X a n d l e t X J t b e d e f i n e d o n t h e a d j o i n t s p a c e X 1 b y ( 1 2 ) . T h e n t h e m a p p i n g
“ ( i , t ) = U t i
d e f i n e s a s e m i - f l o w o n X 1, w h e n X ' h a s t h e w e a k * - t o p o l o g y , i . e . 5r i s j o i n t l y c o n t i n u o u s i n £ a n d t .
P r o o f : T h e v e r i f i c a t i o n o f 7 r ( i , 0 ) = £ a n d “ ( ^ ( i , t ) , s ) = “ ( i , t + s ) i s s t r a i g h t f o r w a r d . L e t u s n o w s h o w t h a t 5r i s j o i n t l y c o n t i n u o u s . F i x x G X a n d l e t £n ->■ £ i n t h e w e a k * - t o p o l o g y o n X ' a n d t n -*■ t i n R + . B y t h e s t r o n g c o n t i n u i t y o f T ( o n e h a s T ^ x - * T t x . W e n o w c l a i m t h a t
< T t n x , 0 - < T t x , ^ >
I n d e e d l e t
л п = l < T tn x , £ n > - < T t x , i > |
= | < T x , i n > - < T t x , i n > + < T t x , i n > - < T t x , i > | n
= | < T t n X - T t x , i n > + < T t x , i n - £>l
8 . A B S T R A C T I O N O F T H E L A S T S E C T I O N
S | | T t n X - T t x | | • I U J I + | < T t x , i n - i > |
212 SELL
N o w t h e f i r s t t e r m a b o v e a p p r o a c h e s 0 s i n c e || T t x - T t x | | -» 0 a n d s u p n | | j 0 n || < + 00 b y l e m m a 7 . T h e s e c o n d t e r m a p p r o a c h e s 0 b y t h e w e a k * - c o n v e r g e n c e o f Hn. Q . E . D .
O n e m a y s t i l l a s k w h e t h e r t h e s e m i g r o u p o f b o u n d e d l i n e a r o p e r a t o r s U t i s s t r o n g l y c o n t i n u o u s . T h e a n s w e r t o t h i s c a n b e f o u n d i n t h e o r e m 2 . A s s u m e t h a t T t s a t i s f i e s || T t || S M e ® a n d l e t A b e t h e i n f i n i t e s i m a l g e n e r a t o r o f T t . N e x t l e t A 1 b e t h e a d j o i n t o p e r a t o r d e f i n e d o n d o m a i n D ( A ' ) . I f D ( A ' ) i s d e n s e i n X ' , t h e t h e o r e m 2 w o u l d i m p l y t h a t U t i s s t r o n g l y c o n t i n u o u s , i . e . U t i s a C o - s e m i g r o u p o n X ' . I f D ( A ' ) i s n o t d e n s e i n X ' , t h e n b y r e s t r i c t i n g U t t o t h e B a n a c h s p a c e
X + = Ci D ( A ' )
w h e r e t h e c l o s u r e i s t a k e n i n t h e s t r o n g t o p o l o g y o n X ' , o n e c a n a g a i n s h o w t h a t U t i s a C 0 - s e m i g r o u p o n X + , s e e R e f . [ 1 4 ] .
9 . I N V A R I A N T M E A S U R E S
L e t u s r e t u r n n o w t o a f l o w it o n a c o m p a c t H a u s d o r f f s p a c e X . L , e t ц b e a B a i r e m e a s u r e o n X w i t h t h e p r o p e r t y t h a t ц 2 0 a n d j u ( X ) > 0 . W e s h a l l s a y t h a t /и i s a n i n v a r i a n t m e a s u r e f o r t h e f l o w ж i f f o r e v e r y B a i r e s e t A £ X o n e h a s
ц ( A ) = v { w ( A , t ) ) , ( f o r a l l t € R )
T h e e x i s t e n c e o f i n v a r i a n t m e a s u r e s h a s p r o f o u n d c o n s e q u e n c e s i n t h e g e n e r a l t h e o r y o f t o p o l o g i c a l d y n a m i c s . O n e o f t h e i n t e r e s t i n g a p p l i c a t i o n s a r i s e s i n a p r o b l e m o f t h e c o n t r o l o f a s a t e l l i t e i n t h e t i m e - v a r y i n g g r a v i t a t i o n a l f i e l d o f t h e s o l a r s y s t e m , s e e R e f . 1 4 ] . A m o r e h e u r i s t i c d i s c u s s i o n o f t h i s p r o b l e m c a n b e f o u n d i n R e f s [ 7 , 1 3 ] .
O n e o f t h e b a s i c a p p l i c a t i o n s o f t h e e x i s t e n c e o f a n i n v a r i a n t m e a s u r e i s t h e P o i n c a r é - C a r a t h é o d o r y r e c u r r e n c e t h e o r e m :
T h e o r e m 1 0 . L e t X b e a c o m p a c t H a u s d o r f f s p a c e w i t h a f l o w ж a n d l e t /л b e a n i n v a r i a n t m e a s u r e f o r ж. T h e n a l m o s t e v e r y p o i n t x € X i s a P o i s s o n s t a b l e .
A p o i n t x € X i s P o i s s o n s t a b l e i f f o r e v e r y n e i g h b o u r h o o d V o f x ,
t h e t w o s e t s
{ t i 0 : i ( x , t ) £ V ) { t Í 0 : г ( х , т ) 6 V )
a r e u n b o u n d e d i n R . S i n c e V i s a r b i t r a r y , t h i s m e a n s t h a t x = l i m ir(x,Tn ) f o r s o m e g e n e r a l i z e d s e q u e n c e т п -» + ° ° ( o r т п -» - ° ° ) , i . e . 7 r ( x , t ) i s " v e r y c l o s e t o b e i n g p e r i o d i c i n t " .
B y u s i n g t h e R i e s z r e p r e s e n t a t i o n t h e o r e m ( t h e o r e m 5 ) i t i s p o s s i b l e t o c h a r a c t e r i z e a n i n v a r i a n t m e a s u r e i n t e r m s o f t h e f l o w ж o n t h e
a d j o i n t s p a c e С ' .
L e m m a 1 1 . L e t ж b e a f l o w o n a c o m p a c t H a u s d o r f f s p a c e X a n d l e t 7Г a n d ж d e n o t e t h e i n d u c e d f l o w s o n С a n d С ' , a s d e s c r i b e d i n s e c t i o n s 6 - 8 .
T h e n a B a i r e m e a s u r e ц i s a n i n v a r i a n t m e a s u r e f o r t h e f l o w tt o n X i f a n d
o n l y i f t h e a s s o c i a t e d b o u n d e d l i n e a r f u n c t i o n a l £ G C s a t i s f i e s
( i ) I s 0 , i . e . ü (c p ) è 0 w h e n e v e r ф ê 0 , a n d i ( l ) > 0 ,( i i ) £ i s a f i x e d p o i n t f o r ж, i . e .
U t i = I ( f o r a l l t G R )
o r e q u i v a l e n t l y
i ( T tq>) = < T t c p , j O = < < M > = J2(cp)
f o r a l l ф € С a n d t G R .W e c a n n o w p r o v e t h e e x i s t e n c e o f a n i n v a r i a n t m e a s u r e f o r a f l o w
o n a c o m p a c t H a u s d o r f f s p a c e .
T h e o r e m 1 2 . L e t tt b e a f l o w o n a c o m p a c t H a u s d o r f f s p a c e X .
T h e n t h e r e i s a n i n v a r i a n t m e a s u r e ii o n X w i t h p ( X ) = 1 .
IA E A -S M R -1 7 /5 0 2 1 3
P r o o f . L e t P = ( I e C ' : l H a n d i ( l ) = 1 } . T h e n f o r a l l i G P o n e h a s I I I II = 1 , c o n s e q u e n t l y P i s r e l a t i v e l y c o m p a c t i n t h e w e a k * - t o p o l o g y b y l e m m a 8 . F i x Í € P . T h e n f o r T > 0 d e f i n e j? t : С -► R b y
T T
AT ( ф) = L f (U si ) (cp)ds = y J ¿ ( T scp)ds
о 0
F o r ф f i x e d , t h e i n t e g r a n d a b o v e i s c o n t i n u o u s i n s b y L e m m a 6 , s o t h e i n t e g r a l a b o v e i s t h e s t a n d a r d R i e m a n n i n t e g r a l . I t i s e a s i l y c h e c k e d t h a t f o r e a c h T > 0 , i T i s a b o u n d e d l i n e a r f u n c t i o n a l o n C . M o r e o v e r w e c l a i m t h a t £T G P . I n d e e d , i f ф ë 0 t h e n T s ф s 0 f o r a l l s . T h e r e f o r e , s i n c e £ Ш 0 o n e h a s U s ^ . ê 0 f o r a l l s . C o n s e q u e n t l y f T ï 0 . T h e r e l a t i o n s h i p j? t ( 1 ) = 1 f o l l o w s i m m e d i a t e l y f r o m t h e f a c t t h a t T S 1 = 1 f o r a l l s . H e n c e i T G P .
W e n o w u s e t h e w e a k * - c o m p a c t n e s s o f P a n d c h o o s e a s e q u e n c e { T k } s o t h a t T k -► + 0 0 a n d - * Í i n t h e w e a k * - t o p o l o g y . T h i s m e a n s t h a t f o r e v e r y ф G С a n d e v e r y t G R o n e h a s
£r ( ф ) - 2 ( ф ) к
£ т ( Т 1 ф ) - 1 ( Т 4ф ) к
I t i s e a s i l y v e r i f i e d t h a t £ G P . W e w i l l a p p l y l e m m a 1 1 a n d s h o w t h a t Í i s a f i x e d p o i n t f o r t h e f l o w f o n С ' . S i n c e £ G P , t h i s m e a n s t h a t t h e m e a s u r e ц a s s o c i a t e d w i t h Î i s a n i n v a r i a n t m e a s u r e o n X w i t h ц(Х) = 1 .
I n o r d e r t o s h o w t h a t î ( T t ç ) = 1 ( ф ) f o r a l l ф G С a n d t G R , i t w i l l s u f f i c e t o s h o w t h a t
i x ( T t 9 ) - » . ? ( ф ) , a s T k -* +00
214 SELL
f o r a l l cp £ С a n d t £ R . F i x cp € С a n d t G R a n d c h o o s e T = T k > t . T h e n
T ' T T+t
, ' 3 s
о 0 ti x ( T t q>) = y J i ( T s T t c p ) d s = I- J i ( T s + t c p ) d s = 1. J i ( T s c p ) d s
w h e r e i n t h e l a s t s t e p w e m a d e t h e c h a n g e o f v a r i a b l e s s + t -*■ s . H e n c e
T t T + t
i T ( T t cp) = Y J í ( T s c p ) d s - Y J i ( T s c p ) d s + i - J i ( T s c p ) d s
N o w t a k e t h e l i m i t a s T = T k -»■ + < » , O n e t h e n g e t s
T
~ J j 0 (T sc p ) d s -* i (cp)
о
b y t h e w e a k * - c o n v e r g e n c e . H o w e v e r ,
t
^ f - e ( T s c p ) d s s i l l s u p | i ( T , < p ) | S ^ II i l l • | |ф | | - 0
a s T = T k ^ + ° ° . S i m i l a r l y ,
T + t
Y J i ( T s c p ) d s Щ II ill • Ы1 -0T
H e n c e ü T ( T t cp) - * i ( c p ) a s T = T k - * + ° ° . Q . E . D .
1 0 . A P P L I C A T I O N S T O N O N - A U T O N O M O U S D I F F E R E N T I A L E Q U A T I O N S
1 0 . 1 . L i n e a r c a s e
L e t u s r e t u r n t o t h e f l o w g e n e r a t e d b y t h e f a m i l y o f l i n e a r d i f f e r e n t i a l e q u a t i o n s
x 1 = A ( t ) x ( x 6 X , A S j a O
d e s c r i b e d i n s e c t i o n 3 . W e s h a l l a s s u m e n o w t h a t j ÿ ' i s a c o m p a c t t r a n s l a t i o n - i n v a r i a n t s e t o f m a t r i x - v a l u e d f u n c t i o n s w i t h c o n t i n u o u s c o e f f i c i e n t s , a n d w e s h a l l l e t c t ( A , t ) = A T d e n o t e t h e f l o w o n л/, a n d 7r( x , A , t ) = ( с р ( х , А , т ) , с т ( А , т ) ) t h e f l o w o n X X J Í N o w l e t ц b e a n i n v a r i a n t m e a s u r e f o r c t w i t h = 1 . N e x t l e t v d e n o t e t h e L e b e s g u e m e a s u r eo n X .
IAEA-SMR-17/50 215
L e m m a 1 3 . A s s u m e t h a t f o r a l l A e ^ a n d t 6 R o n e h a s
t r a c e A ( t ) = 0
nw h e r e t r a c e A ( t ) = a ¡ ¡ ( t ) a n d A ( t ) = ( a ¡ j ( t ) ) . T h e n f o r a n y L e b e s g u e
m e a s u r a b l e s e t В С X o n e h a s
i / ( B ) = v ( ф ( В , А , т ) )
f o r a l l t e R a n d A €E J&.
P r o o f : T h e m a p p i n g x 0 -» x = ф ( х 0 , А , т ) d e f i n e s a d i f f e r e n t i a b l e h o m e o m o r p h i s m o f В o n t o ф ( В , А , т ) . T h e r e f o r e , b y t h e s t a n d a r d c h a n g e o f v a r i a b l e s f o r m u l a f o r i n t e g r a t i o n , w e g e t
w h e r e J i s t h e J a c o b i a n o f t h e t r a n s f o r m a t i o n x 0 -» x = ф ( х 0 , А , т ) . S i n c e ф ( х 0 , А , т ) i s l i n e a r i n x 0 , o n e h a s
J = d e t Ф ( A , t ) = W ( A , t )
w h e r e Ф ( А , т ) i s t h e l i n e a r t r a n s f o r m a t i o n o f X o n t o X d e f i n e d b y Ф ( А , т ) х 0 = ф ( х 0 , А , т ) . T h a t i s , J i s s i m p l y t h e W r o n s k i a n W ( A , t ) o f t h e f u n d a m e n t a l m a t r i x s o l u t i o n Ф ( А , т ) . N o w i t i s w e l l k n o w n t h a t W ( A , t ) s a t i s f i e s t h e l i n e a r d i f f e r e n t i a l e q u a t i o n
S i n c e t r a c e A = 0 o n e h a s W ( A , t ) = W ( A , 0 ) f o r a l l t . H o w e v e r , Ф ( А , 0 ) = I , t h e i d e n t i t y . T h u s J = W ( A , t ) = 1 f o r a l l t . H e n c e
l = l
< р ( В ,А ,т ) в
j - W ( A , t ) = ( t r a c e A ( t ) ) • W ( A , t )
v ( ф ( В , А , т ) ) = J d x 0 I 1 d x 0 = v ( B )
в в
Q . E . D .
L e t u s n o w t u r n t o t h e f l o w ж o n X X < .e £
T h e o r e m 1 4 . A s s u m e t h a t f o r a l l A e j a ^ a n d t e R o n e h a s
t r a c e A ( t ) = 0
T h e n t h e p r o d u c t m e a s u r e m = v X /u i s a n i n v a r i a n t m e a s u r e f o r t h e f l o w ж o n X X j ÿ l
216 SELL
P r o o f : I t w i l l s u f f i c e t h e s h o w t h a t f o r e v e r y m e a s u r a b l e s e t i n X X j ^ o f t h e f o r m B X C , w h e r e B Ç X , C £ , a í ¡ o n e h a s
m U T ( B X C ) ) = m ( B X C ) , ( r G R )
w h e r e , o f c o u r s e , m ( B X C ) = y ( B ) / j ( C ) .
X
-7TT(B x C )-ф (В ,А ,т )
N o w , ( b y t h e F u b i n i t h e o r e m )
m ( i r T ( B X C ) ) = J 1 d m
тгг (В x C)
1 d i / d m
o ( C . r ) ¡f(B,A, T)
= J v ( ф ( В , А , т ) d / j = J ¡ 4 B ) d j U
o ( C , T) o ( C , r)
( b y l e m m a 1 3 )
= i / ( B ) • iи ( с г ( С , т ) ) = v(B)v ( C )
s i n c e й i s a n i n v a r i a n t m e a s u r e o n Ж Q . E . D .
1 0 . 2 . N o n - l i n e a r c a s e
T h e a n a l y s i s d e s c r i b e d a b o v e e x t e n d s t o t h e n o n - l i n e a r c a s e ,
x ' = f ( x , t ) ( x G W , f G F )
I n t h i s c a s e , t h e c o n d i t i o n o n t h e t r a c e b e c o m e s
d i v x f ( x , t ) = 0
f o r a l l ( x , t ) G W X R a n d f G F , c f . [ 1 0 ] f o r d e t a i l s . T h i s c a n a l s o b e f o r m u l a t e d f o r d i f f e r e n t i a l e q u a t i o n s o n R i e m a n n i a n m a n i f o l d s . I n t h i s s e t t i n g , t h e g e n e r a l i z a t i o n o f t h e o r e m 1 4 , t o g e t h e r w i t h t h e P o i n c a r ê - C a r a t h é o d o r y r e c u r r e n c e t h e o r e m ( t h e o r e m 1 0 ) , b e c o m e s a f u n d a m e n t a l t o o l i n t h e a n a l y s i s o f t h e c o n t r o l o f a s a t e l l i t e i n a t i m e - v a r y i n g
g r a v i t a t i o n a l f i e l d w h i c h w e r e f e r r e d t o e a r l i e r , s e e R e f s [ 4 , 7 , 1 3 ] .
IA E A -S M R -1 7 /5 0 217
R E F E R E N C E S
[1 ] DUNFORD, N .. SCHW ARTZ, J .T . , "Linear Operators, P a rt i" , Interscience, New York (1957).[ 2 ] HALE, J .K . , "Functional D ifferential Equations", Applied Math. Sciences, £ ( Springer, Berlin
(1971).[3 ] HILLE, E. , PHILLIPS, R .S . , "Functional Analysis and Semigroups ”, A m er. Math. Soc.
Colloquium P u b l,, vo l. 31, Providence, R. I. (1957), (1965).[ 4 ] MARKUS, L . , SELL, G. R . , "Control in Conservative D ynam ical Systems: Recurrence and Capture
in A p eriod ic Fields ", J, D iff. Equations, 16 (1974) 472-505.[ 5 ] MILLER, R. K . , "Linear Volterra lntegro-d ifferentia l Equations as Semigroups ", Funk. E kvacioj,
17 (1974) 3 5 -5 1 .[ 6 ] MILLER, R. K . , SELL, G .R ., "Volterra Integral Equations and T o p o lo g ica l D yn am ics", M em oir
A m er. Math. S o c . , N o. 102, Providence, R .I. (1970).[ 7 ] MILLER, R. K . , SELL, G .R . , ’T o p o lo g ica l D ynam ics and Its Relation to Integral Equations and
Nonautonomous System s", in Proceedings o f International Con ference on D ynam ical Systems,Providence, R . I . , August (1974) (to appear),
[ 8 ] POINCARE, H ., "Sur les courbes défin ies par des équations d ifféren tie lles ’’, J. Math. Pures A ppl. 7 (1881) 375 -422 and 8 (1882) 251 -296 .
[ 9 ] SELL, G .R ., "Nonautonomous D ifferential Equations and T op o log ica l D ynam ics I and II", Trans. Am er. Math. S oc. 127 (1967) 241 -283 .
[1 0 ] SELL, G .R ,, ’’Invariant Measures and Poisson Stability ", in T o p o lo g ica l D ynam ics, p p .435 -454 ,Benjamin, New York (1968). /
[ 1 1 ] SELL, G .R . , Lectures on T o p o lo g ica l D ynam ics and D ifferential Equations, Van Nostrand-Reinhold, London (1971).
[1 2 ] SELL, G .R . , "D ifferential Equations Without Uniqueness and Classical T op o log ica l D ynam ics",J. D iff. Equations 14 (1973) 42 -56 .
[1 3 ] SELL, G .R ., "Generic Theories in the Q ualitative Study o f Ordinary D ifferential Equations ",Bui. Un. M at. Ital. (to appear).
[1 4 ] YOSIDA, K ., Functional Analysis, Springer, Berlin (1965).
IA E A -S M R -1 7 /5 1
SPECTRAL THEORIES FOR LINEAR DIFFERENTIAL EQUATIONS
G.R. SELLSchool of Mathematics,University of Minnesota,Minneapolis, Minnesota,United States of America
Abstract
SPECTRAL THEORIES FOR LINEAR DIFFERENTIAL EQUATIONS.The use o f spectral analysis in the study o f linear d ifferential equations with constant coe ffic ien ts is
not only a fundamental technique but also leads to far-reaching consequences in describing the qualitative behaviour o f the solutions. The spectral analysis, via the Jordan can on ica l form , w ill not only lead to a representation theorem for a basis o f solutions, but w ill also g ive a rather precise statement o f the (exponential) growth rates o f various solutions. Various attempts have been m ade to extend this analysis to linear differential equations with tim e-vary ing coe ffic ien ts . The most com p lete such extensions is the Floquet theory for equations with period ic coe ffic ien ts . For tim e-vary ing linear d ifferential equations with aperiodic coe ffic ien ts several authors have attem pted to "extend" the Foquet theory. The precise meaning o f such an extension is itself a problem , and we present here several attempts in this d irection that are related to the general problem o f extending the spectral analysis o f equations with constant coe ffic ien ts . The main purpose o f this paper is to introduce som e problems o f current research. The primary problem we shall exam ine occurs in the context o f linear differential equations with alm ost periodic coe ffic ien ts . W e ca ll it "the Floquet prob lem ” .
1 . I N T R O D U C T I O N
T h e u s e o f s p e c t r a l a n a l y s i s i n t h e s t u d y o f l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h c o n s t a n t c o e f f i c i e n t s i s n o t o n l y a f u n d a m e n t a l t e c h n i q u e b u t a l s o l e a d s t o f a r - r e a c h i n g c o n s e q u e n c e s i n d e s c r i b i n g t h e q u a l i t a t i v e b e h a v i o u r o f t h e s o l u t i o n s . T h e s p e c t r a l a n a l y s i s , v i a t h e J o r d a n c a n o n i c a l f o r m , w i l l n o t o n l y l e a d t o a r e p r e s e n t a t i o n t h e o r e m f o r a b a s i s o f s o l u t i o n s , b u t w i l l a l s o g i v e a r a t h e r p r e c i s e s t a t e m e n t o f t h e ( e x p o n e n t i a l ) g r o w t h r a t e s o f v a r i o u s s o l u t i o n s .
V a r i o u s a t t e m p t s h a v e b e e n m a d e t o e x t e n d t h i s a n a l y s i s t o l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h t i m e - v a r y i n g c o e f f i c i e n t s . T h e m o s t c o m p l e t e s u c h e x t e n s i o n i s t h e F l o q u e t t h e o r y f o r e q u a t i o n s w i t h p e r i o d i c c o e f f i c i e n t s . F o r t h e s e p e r i o d i c e q u a t i o n s , a s w e s h a l l s e e , o n e i s a b l e t o c o m p l e t e l y r e d u c e t h e p r o b l e m t o t h e c o n s t a n t c o e f f i c i e n t c a s e b y m e a n s o f a p e r i o d i c c h a n g e o f v a r i a b l e s .
F o r t i m e - v a r y i n g l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h a p e r i o d i c c o e f f i c i e n t s s e v e r a l a u t h o r s h a v e a t t e m p t e d t o " e x t e n d " t h e F l o q u e t t h e o r y . T h e p r e c i s e m e a n i n g o f s u c h a n e x t e n s i o n i s i t s e l f a p r o b l e m , a n d w e p r o p o s e h e r e t o d i s c u s s s e v e r a l a t t e m p t s i n t h i s d i r e c t i o n t h a t a r e r e l a t e d t o t h e g e n e r a l p r o b l e m o f e x t e n d i n g t h e s p e c t r a l a n a l y s i s o f e q u a t i o n s w i t h c o n s t a n t c o e f f i c i e n t s . T h e f i r s t s u c h a t t e m p t i s p r o b a b l y t h a t o f L y a p u n o v i n 1 8 9 2 , w h e r e h e i n t r o d u c e d t h e c o n c e p t t h e t y p e n u m b e r ( o r c h a r a c t e r i s t i c e x p o n e n t )
2 1 9
220 SELL
f o r a t i m e - v a r y i n g d i f f e r e n t i a l s y s t e m w i t h b o u n d e d c o e f f i c i e n t s . T h e s u b s e q u e n t t h e o r i e s w e s h a l l s t u d y c a n b e v i e w e d a s m o d i f i c a t i o n s o f t h e L y a p u n o v c o n c e p t .
I n t h i s p a p e r w e i n t r o d u c e s o m e p r o b l e m s o f c u r r e n t r e s e a r c h . T h e p r i m a r y p r o b l e m w e s h a l l p r e s e n t o c c u r s i n t h e c o n t e x t o f l i n e a r d i f f e r e n t i a l e q u a t i o n s w i t h a l m o s t p e r i o d i c c o e f f i c i e n t s . W e c a l l i t " t h e F l o q u e t p r o b l e m " .
2 . C O N S T A N T C O E F F I C I E N T S
C o n s i d e r t h e l i n e a r d i f f e r e n t i a l e q u a t i o n w i t h c o n s t a n t c o m p l e x c o e f f i c i e n t s
x ' = A x ( 1 )
w h e r e x G C n . I f X i s a n e i g e n v a l u e o f A a n d x f 0 i s t h e a s s o c i a t e d e i g e n v e c t o r ( i . e . A x = X x ) , t h e n cp ( x , A , t ) = e w x i s t h e s o l u t i o n o f t h e i n i t i a l v a l u e p r o b l e m
x ’ = A x , x ( 0 ) = x
I f { X j , . . . , X k } d e n o t e d i s t i n c t e i g e n v a l u e s o f A a n d { x j , . . . , x k } t h e c o r r e s p o n d i n g e i g e n v e c t o r s , t h e n t h e c o l l e c t i o n { x 1( . . . , x k } i s l i n e a r l y i n d e p e n d e n t . C o n s e q u e n t l y , i f t h e m a t r i x A h a s n d i s t i n c t e i g e n v a l u e s { X p . . . , X n } , a n d i f { x p . . . , x n } d e n o t e t h e c o r r e s p o n d i n g e i g e n v e c t o r s , t h e n { x p . . . . , x n} a s a b a s i s f o r C n a n d
o p t x j . A , t ) = e x i r x ¡ ( i = 1 , . . . , n )
i s a b a s i s o f s o l u t i o n s o f ( 1 ) . I n t e r m s o f t h i s b a s i s t h e f u n d a m e n t a l m a t r i x s o l u t i o n Ф ( А , t ) c a n b e w r i t t e n a s a d i a g o n a l m a t r i x
Ф ( А , t ) = d i a g f e ^ í 1, . . . , e x n r )
W h a t i s t h e s i t u a t i o n i f t h e r e a r e r e p e a t e d e i g e n v a l u e s , i . e . r e p e a t e d r o o t s o f d e t ( A - X I ) = 0 ? I n t h i s c a s e t h e J o r d a n c a n o n i c a l f o r m f o r A n e e d n o t b e d i a g o n a l , a n d c o n s e q u e n t l y t h e e x p o n e n t i a l s o l u t i o n s e x t x n e e d n o t g e n e r a t e a b a s i s o f s o l u t i o n s o f ( 1 ) . I n t h i s c a s e o n e w o u l d l o o k a t t h e g e n e r a l i z e d e i g e n s p a c e
V i = { x G C n : ( A - X ¡ I ) p x = 0 f o r s o m e p = 1 , 2 , . . . , n }
I f { Х х , . . . , X k } d e n o t e s t h e c o l l e c t i o n o f e i g e n v a l u e s o f A w i t h X ¡ f X j w h e ni f j , t h e n t h e s p a c e s { V j , . . . , V k } a r e n o n - t r i v i a l ( i . e . V ¡ f { 0 } ) ,V j Л V . = { 0 } w h e n i f j a n d
C n = V ! + . . . + V k ( 2 )
F u r t h e r m o r e , i f x G V t t h e n t h e r e i s a p o l y n o m i a l p ( t ) , w h i c h d e p e n d s o n x b u t h a s d e g r e e g n - 1 , s u c h t h a t
IA E A -S M R -1 7 /5 1 2 2 1
I ф ( x , A , t ) I = ] p ( t ) I e Re V 1 x j ( 3 )
I n o t h e r w o r d s , f o r e a c h x 6 V ¡ , w i t h x f 0 , t h e s o l u t i o n ф ( x , A , t ) h a s " e x p o n e n t i a l g r o w t h r a t e " = R e X ¡ .
A s w e s h a l l s e e , i t i s t h e s e t o f r e a l n u m b e r s { R e X j , . . . , R e X k } t h a t g e n e r a l i z e s t o t i m e - v a r y i n g e q u a t i o n s . I t i s t h i s s e t , t h e r e f o r e , t h a t w e s h a l l c a l l t h e s p e c t r u m f o r ( 1 ) .
T h e p r o b l e m o f t r y i n g t o e x t e n d t h i s t h e o r y t o t i m e - v a r y i n g e q u a t i o n s n o w h a s t w o a s p e c t s . F i r s t o n e w a n t s t o d e r i v e a d e c o m p o s i t i o n o f t h e p h a s e s p a c e w h i c h g e n e r a l i z e s ( 2 ) , a n d s e c o n d l y , o n e w a n t s t o a r r i v e a t c o r r e s p o n d i n g g r o w t h e s t i m a t e s a s r e p r e s e n t e d b y ( 3 ) . I n t h e p e r i o d i c c a s e , a s w e s h a l l n o w s e e , t h i s p r o b l e m i s c o m p l e t e l y s e t t l e d .
3 . P E R I O D I C C O E F F I C I E N T S
C o n s i d e r n o w a l i n e a r d i f f e r e n t i a l e q u a t i o n
x ' = A ( t ) x ( x e C n , t G R ) ( 4 )
w h e n A ( t ) i s p e r i o d i c , i . e . A ( t + w ) = A ( t ) f o r a l l t a n d s o m e p e r i o d w > 0 .L e t $ ( t ) b e a f u n d a m e n t a l m a t r i x s o l u t i o n o f ( 4 ) ( i . e . $ ( t ) i s a n ( n X n ) m a t r i x t h a t i s n o n - s i n g u l a r f o r a l l t a n d s a t i s f i e s # ' ( t ) = A ( t ) $ ( t ) ) a n d a s s u m e t h a t Ф i s n o r m a l i z e d s o t h a t Ф ( 0 ) = I . L e t R b e a n ( n X n ) m a t r i x t h a t s a t i s f i e s ® ( w ) = e w R , s e e R e f . [ l ] . S i n c e $ ( t + w ) i s a l s o a f u n d a m e n t a l m a t r i x s o l u t i o n o f ( 4 ) , i t f o l l o w s t h a t t h e r e i s a n o n - s i n g u l a r m a t r i x С s u c h t h a t $ ( t + w ) = $ ( t ) C f o r a l l t . B y s e t t i n g t = 0 w e s e e t h a t С = e wR. I t f o l l o w s t h e n t h a t S !? ( t ) = Ф ^ ) е ~ £К i s p e r i o d i c i n t w i t h p e r i o d w . I n o t h e r w o r d s
« ( t ) = ^ ( t ) e tR
i s t h e F l o q u e t r e p r e s e n t a t i o n o f Ф a s t h e p r o d u c t o f a p e r i o d i c m a t r i x a n d a n e x p o n e n t i a l .
N o w l e t V . d e n o t e t h e g e n e r a l i z e d e i g e n s p a c e o f R a s s o c i a t e d w i t h t h e e i g e n v a l u e X ¡ . T h e s e s p a c e s s a t i s f y ( 2 ) . F u r t h e r m o r e i f x e V i t h e r e e x i s t a p o l y n o m i a l p ( t ) d e p e n d i n g o n x b u t o f d e g r e e s n - 1 , a n d a p e r i o d i c f u n c t i o n q ( t ) s u c h t h a t
I <p(x,A,t) I = I p(t) I I q f t j I e ^ V |x|
i . e . | ф ( х , А Д ) | h a s " e x p o n e n t i a l g r o w t h r a t e " = R e X j . W e s h a l l c a l l t h e s e t o f r e a l n u m b e r s { R e X p . . . , R e X k } , ( w h e r e { X j , . . . , X k } d e n o t e t h e e i g e n v a l u e s o f R ) t h e s p e c t r u m o f ( 4 ) .
T h e r e i s a n o t h e r w a y t o v i e w t h e p e r i o d i c c a s e ( 4 ) . I f o n e m a k e s t h e c h a n g e o f v a r i a b l e s x = ,J , ( t ) y i n ( 4 ) , t h e n i t f o l l o w s t h a t у m u s t s a t i s f y
У-' = НУ. (5)a n e q u a t i o n w i t h c o n s t a n t c o e f f i c i e n t s . T h e n t h e s p e c t r u m o f ( 4 ) a n d t h e s p e c t r u m o f ( 5 ) a r e t h e s a m e .
222 SELL
L e t u s n o t e i n p a s s i n g t h a t , w h i l e R i s n o t u n i q u e l y d e t e r m i n e d b y t h e c o n d i t i o n Ф ^ ) = e w R , i t i s t r u e t h a t t h e s p e c t r u m o f ( 4 ) — o r ( 5 ) — a s w e d e f i n e d i t - d e p e n d s o n l y o n A ( t ) a n d n o t o n t h e p a r t i c u l a r c h o i c e
4 . G E N E R A L T I M E - V A R Y I N G C O E F F I C I E N T S
I n t h e l a s t t w o s e c t i o n s w e i n t r o d u c e d a c o n c e p t o f s p e c t r u m f o r l i n e a r e q u a t i o n s w i t h e i t h e r c o n s t a n t o r p e r i o d i c c o e f f i c i e n t s . W e s h a l l n o w g i v e a d e f i n i t i o n o f s p e c t r u m , i n f a c t w e s h a l l d e f i n e f o u r s p e c t r a , f o r g e n e r a l t i m e - v a r y i n g e q u a t i o n s w i t h b o u n d e d c o e f f i c i e n t s , w h i c h i n c l u d e s t h e a b o v e c o n c e p t s a s s p e c i a l c a s e s . T h i s d e f i n i t i o n i s d u e t o A . M . L y a p a n o v [ 3 ] .
L e t ja ? ' d e n o t e a c o l l e c t i o n o f ( n X n ) m a t r i x v a l u e f u n c t i o n s A = A ( t ) , d e f i n e d f o r t G R , w i t h b o u n d e d c o n t i n u o u s c o e f f i c i e n t s . F o r e a c h A G w e c o n s i d e r t h e l i n e a r d i f f e r e n t i a l e q u a t i o n
w h e r e x G X ( a n d X = R n , o r C n ) . F o r a n y ( x , A ) G X Xdaf, l e t c p ( x , A , t ) d e n o t e t h e s o l u t i o n o f t h e i n i t i a l v a l u e p r o b l e m x ' = A ( t ) x , x ( 0 ) = x . L e t u s n o w a s s u m e t h a t j r f i s t r a n s l a t i o n - i n v a r i a n t , i . e . л ? h a s t h e p r o p e r t y t h a t A T G ja f w h e n e v e r A G ja?’ a n d т G R . ( H e r e A T ( t ) = A ( T + t ) ) . W e s h a l l t h i n k o f a s b e l o n g i n g t o s o m e f u n c t i o n s p a c e a n d t h a t ¿ á h a s a t o p o l o g y
w i t h t h e p r o p e r t y t h a t t h e t w o m a p p i n g s
cr : ( A , t ) - » ' A a n d < p : ( x , A , t ) — cp ( x , A , t )
a r e c o n t i n u o u s . T h e n i t f o l l o w s ( s e e R e f . [ 6 ] ) t h a t
7t ( x , A , t ) = ( с р ( х , А , т ) , с т ( А , т ) )
d e f i n e s a l i n e a r s k e w p r o d u c t f l o w o n X X j a f .
W h i l e m u c h o f t h e s u b s e q u e n t d i s c u s s i o n w i l l b e v a l i d i n t h i s g e n e r a l c o n t e x t d e s c r i b e d a b o v e , i t w i l l b e c o n v e n i e n t f o r s o m e p u r p o s e s t o f u r t h e r r e s t r i c t л / a n d a s k t h a t . j a ^ b e a c o m p a c t m i n i m a l s e t f o r t h e f l o w с т . F o r e x a m p l e m a y b e t h e h u l l o f a n a l m o s t p e r i o d i c f u n c t i o n , s e e R e f . [ 9 ] .
N o w l e t ( x , A ) G X X j a f a n d a s s u m e t h a t x f 0 . W e t h e n d e f i n e f o u r
V s b y
f o r R .
x ’ = A ( t ) x (6)
X ! ( x , A ) = l i m s u p - l o gT^4-«o t
II ф ( х , А , т ) (I
T —* + °o T
X ¡ ( x , A ) = l i m i n f - l o g | | ф ( х , А , т ) | |T-*■-<*> ^
IA E A -S M R -1 7 /5 1 223
Then for each A € we define the A (A )-sets by
{ X * ( x , A ) ■ * t 0 }
{ X j ( x , A ) ’■ x f 0 }
{ X ; ( x , A ) : x f 0 }
{ X ' ; ( x , A ) : X f 0 }
U s i n g t h e f a c t t h a t ( 6 ) c a n h a v e o n l y n l i n e a r l y i n d e p e n d e n t s o l u t i o n s , L y a p u n o v s h o w e d t h a t t h e s e t s A j ( A ) a n d A " ¡ ( A ) c o n t a i n a t m o s t n d i s t i n c t p o i n t s . F u r t h e r m o r e , i n t h e c a s e o f c o n s t a n t c o e f f i c i e n t s o r p e r i o d i c c o e f f i c i e n t s a l l o f t h e A ( A ) - s e t s a r e i d e n t i c a l a n d a g r e e w i t h t h e c o n c e p t o f s p e c t r u m d e f i n e d i n s e c t i o n s 2 a n d 3 . T h e r e f o r e a n y o f t h e s e s e t s c a n b e u s e d , a n d h a v e b e e n u s e d , a s a g e n e r a l i z e d n o t i o n o f t h e s p e c t r u m . U n f o r t u n a t e l y t h i s p a r t i c u l a r g e n e r a l i z a t i o n h a s n o t p r o v e d t o b e f r u i t f u l , e x c e p t i n v e r y s p e c i a l c a s e s . T h e m a i n r e a s o n s e e m s t o b e t h a t t h e c o r r e s p o n d i n g n o t i o n o f t h e d e c o m p o s i t i o n s t a t e m e n t ( 2 ) , w h i c h L y a p u n o v f o r m u l a t e d , l a c k s t h e f e a t u r e o f b e i n g u n i q u e l y d e f i n e d . W e s h a l l o m i t t h e s e d e t a i l s .
O n e o f t h e p r o b l e m s t h a t r e s e a r c h e r s h a v e s t u d i e d i s t h e q u e s t i o n o f t h e s t a b i l i t y , o r c o n t i n u o u s d e p e n d e n c y o f t h e A ( A ) - s e t s w i t h r e s p e c t t oA , w h e r e o n e u s e s t h e t o p o l o g y g e n e r a t e d b y t h e s u p - n o r m o n t h e m a t r i x f u n c t i o n s A . F o r e x a m p l e , J . C . L i l l o [ 2 ] h a s s h o w n t h a t i f X * ( x , A ) = X t ( x , A ) f o r a l l x / 0 a n d i f A + ( A ) c o n t a i n s n d i s t i n c t p o i n t s w h e r e n = d i m X , t h e n A + ( A ) v a r i e s c o n t i n u o u s l y i n a n e i g h b o u r h o o d o f A .
V . M . M i l l i o n s 6 i k o v [ 5 , 6 ] h a s l o o k e d a t s e v e r a l c o n c e p t s o f t h e s p e c t r u m . I n [ 5 ] h e s t u d i e s t h e u n i o n o f A + ( A ) o v e r a l l A G л/, w h e r e л/ i n t h i s c a s e i s a c o m p a c t m i n i m a l s e t . H e t h e n d e r i v e d s o m e g e n e r i c p r o p e r t i e s d e s c r i b i n g t h e d e p e n d e n c e o f A + ( A ) o n A . I n a n o t h e r c a s e h e i n t r o d u c e d a n o t i o n w h i c h h e c a l l e d t h e a u x i l i a r y s p e c t r u m [ 6 ] . W e s h a l l n o t p r e s e n t t h e d e t a i l s o f t h e c o n c e p t h e r e , b u t i n s t e a d w e s h a l l l i m i t o u r s e l v e s t o s o m e g e n e r a l c o m m e n t s c o n c e r n i n g t h e b a c k g r o u n d o f t h i s c o n c e p t .
T h e d e f i n i t i o n o f t h e L y a p u n o v - t y p e n u m b e r s , a s g i v e n a b o v e , d o e s h a v e s o m e d e f e c t s . F o r e x a m p l e , i t i s n o t k n o w n , i n t h e a l m o s t p e r i o d i c
c a s e , w h e t h e r t h e s e t y p e n u m b e r s a r e e v e n m e a s u r a b l e f u n c t i o n s o f ( x , A ) .I n d e f i n i n g t h e a u x i l i a r y s p e c t r u m , M i l l i o n s c i k o v u s e s t h e e i g e n v a l u e s o f B * B w h e r e В = Ф ( А , т ) t o d e f i n e n u m b e r s v £ ( A ) , i = 1 , . . . , n . A s a c o n s e q u e n c e o f h i s d e f i n i t i o n t h e s e f u n c t i o n s o f A a r e b o u n d e d m e a s u r a b l e f u n c t i o n s w h i c h a r e c o n s t a n t a l o n g t r a j e c t o r i e s i n л/. I n t h e a l m o s t p e r i o d i c c a s e o n e c a n t h e n a p p l y t h e B i r k h o f f e r g o d i c t h e o r e m a n d c o n c l u d e t h a t f o r i = 1 ^ . . . , n , v ( A ) = P ; , a c o n s t a n t , f o r a l m o s t a l l A € . T h e c o l l e c t i o n
. . . , i^n } i s c a l l e d t h e a u x i l i a r y s p e c t r u m . T h e m a i n d i f f i c u l t y w i t h t h e a u x i l i a r y s p e c t r u m i s t h a t i t i s n o t c l e a r h o w o n e c a n r e l a t e t h e a u x i l i a r y s p e c t r u m w i t h t h e d y n a m i c a l p r o p e r t i e s o f t h e f l o w it. T h e s e s e e m s t o b e w o r t h f u r t h e r s t u d y .
5 . A N E W S P E C T R A L C O N C E P T A N D T H E F L O Q U E T P R O B L E M
I n [ 7 , 8 ] R . J . S a c k e r a n d G . R . S e l l i n t r o d u c e a s o m e w h a t d i f f e r e n t s p e c t r a l c o n c e p t b a s e d o n t h e d y n a m i c a l p r o p e r t i e s o f t h e l i n e a r s k e w
224 SELL
p r o d u c t f l o w 7Г = (ф ,с т ) o n X X Jii . W e s h a l l p r e s e n t t h e b a s i c f e a t u r e s o f t h i s t h e o r y h e r e . W e n o w a s s u m e t h a t jrf i s a n a l m o s t p e r i o d i c m i n i m a ls e t , i . e . i s t h e h u l l o f a n a l m o s t p e r i o d i c f u n c t i o n .
F o r e a c h r e a l n u m b e r X l e t
7г х ( х , А , т ) = ( е " Х тф ( x , A , t ) , A t )
T h i s t o o i s a l i n e a r s k e w p r o d u c t f l o w o n X X j ÿ " . N e x t d e f i n e t h e b o u n d e d s e t
â ë x = { ( х , А ) : II e " x t ф ( х , А Д ) II i s u n i f o r m l y b o u n d e d i n t }
t h e s t a b l e s e t
Sfx = { ( x , A ) : ü e~x t ф ( х , А Д ) | | - * 0 a s t - * + ° o }
a n d t h e u n s t a b l e s e t
<2/x = { ( x , A ) : D e ' X t ф ( х , А Д ) | | - > 0 a s t ->■ - o o }
A l s o d e f i n e t h e f i b e r s
3 § ¿ A ) = { x G X : ( x , A ) e SBJ
S^x(A) = { x G X : ( x , A ) G 5 ^ }
= { x G X : ( x , A ) G W J
T h e f i b e r s A ) , 5 ^ x ( A ) . a n d <%SX(A ) a r e l i n e a r s u b s p a c e s o f X .T h e f o l l o w i n g t h e o r e m i s p r o v e d i n S a c k e r a n d S e l l [ 6 , 1 0 ] .
T h e o r e m 1 . A n e c e s s a r y a n d s u f f i c i e n t c o n d i t i o n f o r
X X = £ x + W x ( W h i t n e y s u m ) ( 7 )
i s t h a t ¿ ^ x i s t r i v i a l , i . e . 3BX = { 0 } X F u r t h e r m o r e , i n t h i s c a s e t h e r e e x i s t s f o r e a c h A G Jii a p r o j e c t i o n P X ( A ) : X -► X s u c h t h a t P x ( A ) x i s j o i n t l y
c o n t i n u o u s x a n d A a n d
I Ф Х ( А Д ) Р Х ( А ) Ф ^ 1 ( А , 8 ) | à к e ~ “ ( t ’ s) , s s t(8)
| ® x ( A , t ) [ I - P x ( A ) ] « ' ¿ ( A , s ) | g K e ' a (s ‘ t ) , t s s
f o r p o s i t i v e c o n s t a n t s K a n d а. ( Н е г е Ф Х ( А Д ) = е ~ х ‘ ф ( А Д ) . )T h e m e a n i n g o f E q . ( 7 ) i s t h a t d i m 5 ^ ( A ) a n d d i m < ? /х( А ) a r e c o n s t a n t
o v e r л / , a n d t h a t & % A) a n d <2/x(A ) v a r y c o n t i n u o u s l y i n A , a n d t h a t
X = S ^ A ) + « ^ ( A )
IA E A -S M R -1 7 /5 1 225
f o r a l l A G T h a t i s 5 ^ a n d ^ a r e i n v a r i a n t s u b b u n d l e s f o r it, s e e R e f . t l O ] .
I n e q u a l i t y ( 8 ) s a y s t h a t e v e r y e q u a t i o n A G л/ a d m i t s a n e x p o n e n t i a l d i c h o t o m y . I t i s n o t h a r d t o s e e t h a t t h e r a n g e o f P ^ ( A ) i s p r e c i s e l y 5 ^ ( A ) a n d t h e n u l l s p a c e i s A ) . I n p a r t i c u l a r w e s e e t h a t t h e r a t e o f d e c a y i n 5 ^ a n d ^ 4 i s e x p o n e n t i a l , p r o v i d e d âë^ i s t r i v i a l .
W e n o w d e f i n e t h e r e s o l v e n t s e t p ( A ) t o b e t h e c o l l e c t i o n o f a l l X e R f o r w h i c h âB\ = { 0 } X J&. T h e c o m p l e m e n t о(л/) = R - p ( j a ? " ) i s c a l l e d t h e s p e c t r u m o f A . T h i s i n c l u d e s t h e c o n c e p t o f s p e c t r u m d i s c u s s e d i n s e c t i o n s 2 a n d 3 a s s p e c i a l c a s e s .
T h e f o l l o w i n g t h e o r e m i s p r o v e d i n S a c k e r a n d S e l l [ 7 , 8 ] .
T h e o r e m 2 . A s s u m e t h a t d i m X = n j 1 . T h e n t h e s p e c t r u m и(л/) i s n o n - e m p t y a n d c o m p a c t a n d c o n s i s t s o f t h e u n i o n o f к n o n - o v e r l a p p i n g c l o s e d i n t e r v a l s L a ¡ , b ¡ j , i = 1 , . . . , к w h e r e k g n . F u r t h e r m o r e , a s s o c i a t e d w i t h e a c h i n t e r v a l [ a ¡ , b ¡ ] t h e r e i s a n n o n - t r i v i a l i n v a r i a n t s u b b u n d l e > ' ' i w i t h t h e p r o p e r t y t h a t f o r e v e r y e > 0 t h e r e e x i s t p o s i t i v e n u m b e r s К a n d a s u c h t h a t f o r a l l ( x , A ) G o n e h a s
| | e - ( b i + e )t < p ( x , A , t ) | | § K | x | e ~ a t , t ê 0 ( 9 )
a n d
II e ' (a i ' 0 t c p ( x , A , t ) | | á K | x | e + a t , t s O ( 1 0 )
F i n a l l y o n e h a s
X X + . . . + ( W h i t n e y s u m )
A s u b s e t (У Ç X X ^ i s c a l l e d a s u b b u n d l e i f ( i ) f o r e a c h A G j a f t h e s e t ‘У ' ( А ) = ( x E X : ( x , A ) G ' X ' } i s a l i n e a r s u b s p a c e X , ( i i ) d i m < X " ( A ) i s c o n s t a n t o v e r a n d ( i i i ) <У(А) v a r i e s c o n t i n u o u s l y i n A .
N o t i c e h o w t h e l a s t t h e o r e m g e n e r a l i z e s b o t h a s p e c t s o f t h e s p e c t r a l t h e o r y f o r d i f f e r e n t i a l e q u a t i o n s w i t h c o n s t a n t c o e f f i c i e n t s . T h e d e c o m p o s i t i o n o f t h e p h a s e s p a c e i s r e p r e s e n t e d i n t e r m s o f t h e W h i t n e y s u m f o r X Х Л ' a n d t h e e x p o n e n t i a l g r o w t h r a t e i s g i v e n b y ( 9 ) a n d ( 1 0 ) . W e c a n s a y m o r e a b o u t t h e g r o w t h r a t e s b y c o m p a r i n g t h i s t h e o r y w i t h t h a t o f L y a p u n o v .
L e t u s n o w c o n s i d e r ( x , A ) G (x f 0 ) w h e r e <У[ i s g i v e n b y T h e o r e m 2 . L e t A ( x , A ) d e s i g n a t e a n y o f t h e f o u r L y a p u n o v - t y p e n u m b e r s . T h e n i t f o l l o w s f r o m t h e o r e m 2 t h a t
a ¡ s X ( x , A ) s b ¡
I n p a r t i c u l a r , i f t h e s p e c t r a l i n t e r v a l [ a ¡ , b j d e g e n e r a t e s t o a s i n g l e p o i n t { a ¡ } , i . e . a ¡ = b t , t h e n w e s e e t h a t a l l f o u r L y a p u n o v - t y p e n u m b e r s a r e t h e s a m e , a n d c o n s e q u e n t l y f o r a l l ( x , A ) G ( x f 0 ) o n e h a s
l i m ^ l o g | | c p ( x , A , T ) | | = l i m ¿ l o g || ф ( х , А , Т ) || = a ¡T —► +oo i. T -> -0O 1
226 SELL
F r o m o u r p o i n t o f v i e w t h e n t h e F l o q u e t p r o b l e m f o r a l m o s t p e r i o d i c
e q u a t i o n s r e d u c e s t o a s k i n g : D o t h e s p e c t r a l i n t e r v a l s [ a j , b ¿ ] d e g e n e r a t e t o p o i n t s ? W e m i g h t s a y t h e s p e c t r u m c o n t a i n s a c o n t i n u o u s s p e c t r u mi f i t c o n t a i n s a n o n - d e g e n e r a t e i n t e r v a l . T h e F l o q u e t p r o b l e m i s t h e n e q u i v a l e n t t o a s k i n g w h e t h e r t h e r e e x i s t s a n a l m o s t p e r i o d i c e q u a t i o n w i t h c o n t i n u o u s s p e c t r u m .
A p r e l i m i n a r y r e p o r t o f o u r i n v e s t i g a t i o n s o n t h i s p r o b l e m c a n b e f o u n d i n R e f . [ 7 ] .
N o t e a d d e d i n p r o o f : A n e x a m p l e d u e t o M i l l i o n & d i k o v s h o w s t h a t t h e r e i s a n a l m o s t p e r i o d i c l i n e a r e q u a t i o n w i t h c o n t i n u o u s s p e c t r u m , s e e R e f . [ 8 ] ,
R E F E R E N C E S
[ 1 ] CODDINGTON, E . , LEVINSON, N .. Theory o f Ordinary D ifferential Equations, McGraw H ill,New York (1955).
[ 2 ] LILLO, J. C . , A note on the continuity o f characteristic exponents, Proc. Nat. A cad . S c i. USA 46(1960) 247 -250 .
[ 3 ] LYAPUNOV, A. M . , Problèm e général de la stabilité du m ouvem ent, Annals o f Math Studies 17,Princeton Univ. (O riginally published in Russian in 1892).
[ 4 ] MILLIONSCIKOV, V . V . , A m etric theory o f linear systems o f d ifferential equations, D okl. Akad.Nauk SSSR 179 (1968) 20 -23 .
[5 ] MILLIONSCIKOV, V . V . , On the theory o f characteristic Lyapunov exponents, Mat. Zam etki 7 (1970) 503 -513 .[ 6] SACKER, R .J . , SELL, G .R . , Existence o f d ichotom ies and invariant splittings for linear differential
systems, I, J. D iff. Eqs, 15 (1974) 429 -45 8 .[ 7 ] SACKER, R .J ., SELL, G .R ., A spectral theory for linear alm ost period ic differential equations.
Preliminary report. Proc. C onf. D iff. E qs., Los Angeles (1974) (to appear).[ 8] SACKER, R .J . , SELL, G .R . , A spectral theory for linear alm ost period ic d ifferential equations, Int.
C on f. D iff. Equations, A ca d em ic Press, New York (1975) 698-708.[ 9 ] SELL, G. R ., T op o log ica l Dynam ics and Ordinary D ifferential Equations, Van-Nostrand Reinhold,
London (1971).[1 0 ] SELL, G. R ., Linear D ifferential Systems. Lecture Notes, University o f Minnesota (1974).
IA E A -S M R '1 7 /6 6
SOME EXAMPLES OF DYNAMICAL SYSTEMS
S . S H A H S H A H A N I
D e p a r t m e n t o f M a t h e m a t i c s a n d C o m p u t e r S c i e n c e ,
A r y a - M e h r U n i v e r s i t y o f T e c h n o l o g y ,
T e h r a n , I r a n
Abstract
SO M E EXAM PLES O F D Y N A M IC A L S Y S TE M S .
T w o generalizations of Van der Pol's equation are studied. T h e a im is to construct structurally stable planar systems with a number of closed orbits w hich are e x p lic itly defined, and for w hich inform ation
regarding the number and nature of the closed orbits m ay be read off the defining equation.
I n t h i s p a p e r w e s h a l l b e c o n c e r n e d w i t h t w o c l a s s e s o f t w o - d i m e n s i o n a l a u t o n o m o u s d i f f e r e n t i a l e q u a t i o n s ( A D E ) t h a t m a y s e r v e a s u s e f u l e x a m p l e s i n t h e q u a l i t a t i v e t h e o r y o f o r d i n a r y d i f f e r e n t i a l e q u a t i o n s . T h e s e A D E ( v e c t o r f i e l d s , d y n a m i c a l s y s t e m s ) h a v e t h r e e a d v a n t a g e s : ( 1 ) T h e y a r e r o o t e d i n n a t u r a l p r o b l e m s , ( 2 ) t h e y a r e e x p l i c i t l y d e f i n e d w i t h o u t b e i n g t r i v i a l — i n f a c t u n s o l v e d p r o b l e m s a r o u n d t h i s s t u d y m a y b e p o s e d , a n d ( 3 ) t h e y e x h i b i t " s t r u c t u r a l s t a b i l i t y " , a c o n c e p t t h a t w i l l b e g r a d u a l l y c l a r i f i e d . R o u g h l y s p e a k i n g , a s y s t e m i s s t r u c t u r a l l y s t a b l e i f b y a d d i n g a t e r m t o t h e e q u a t i o n w h i c h i s s m a l l i n b o t h m a g n i t u d e a n d d e r i v a t i v e , t h e q u a l i t a t i v e p i c t u r e i n t h e p h a s e p l a n e r e m a i n s t h e s a m e .
E x a m p l e s o f n o n - s t r u c t u r a l - s t a b i l i t y
C o n s i d e r a c l a s s i c a l m e c h a n i c a l s y s t e m o f o n e d e g r e e o f f r e e d o m w i t h p o t e n t i a l e n e r g y V ( x ) , i n w h i c h e n e r g y i s c o n s e r v e d . N e w t o n ' s l a w x + V ' ( x ) = 0 d e s c r i b e s t h e e v o l u t i o n o f t h e s y s t e m . L e t t i n g x = y , t h e s e c o n d - o r d e r O D E m a y b e r e g a r d e d a s a f i r s t - o r d e r s y s t e m i n t h e x y - p l a n e , n a m e l y
í¿ = y (1) l y = - V ' ( x )
T h e t o t a l i t y o f t h e t r a j e c t o r i e s f o r m t h e ' p h a s e - p o r t r a i t ' o f t h e s y s t e m . T h e ' e n e r g y ' E = V ( x ) + y 2 / 2 i s c o n s e r v e d , h e n c e t h e t r a j e c t o r i e s o f ( 1 ) a r e j u s t
t h e l e v e l c u r v e s o f E . T h e e q u i l i b r i u m p o i n t s o f ( 1 ) a r e p o i n t s ( x Q, 0 ) , w h e r e x 0 i s a c r i t i c a l p o i n t o f V . A n o n - d e g e n e r a t e c r i t i c a l p o i n t ( i . e . , w h e r e V " ( x 0 ) f 0 ) i s e i t h e r a m a x i m u m i n w h i c h c a s e a s a d d l e p o i n t i s o b t a i n e d ,
227
2 2 8 SHAHSHAH ANI
o r a m i n i m u m w h e r e i t c a n b e s h o w n ( s e e R e f . [ 1 ] ) t h a t a l l t r a j e c t o r i e s s u f f i c i e n t l y c l o s e t o t h e e q u i l i b r i u m a r e c l o s e d o r b i t s ( p e r i o d i c s o l u t i o n s ) .
T h e l a t t e r s i t u a t i o n c a u s e s a b r e a k d o w n o f s t r u c t u r a l - s t a b i l i t y s i n c e t h e a d d i t i o n o f a n y a m o u n t o f d i s s i p a t i o n w i l l t r a n s f o r m t h e c l u s t e r o f p e r i o d i c s o l u t i o n s ( a ' c e n t r e ' ) i n t o t h e d o m a i n o f a t t r a c t i o n o f a n a s y m p t o t i c a l l y s t a b l e e q u i l i b r i u m p o i n t ( a ' s i n k ' ) . T h u s , a s t h e e x p l i c i t e x a m p l e s i l l u s t r a t e , t h e t o t a l q u a l i t a t i v e p i c t u r e i s n o t p r e s e r v e d u n d e r p e r t u r b a t i o n .
( A ) H a r m o n i c o s c i l l a t o r
x + x = 0
H a r m o n i c o s c i l l a t o r w i t h d a m p i n g
x + e x + x = 0 , e > 0
M AX. M IN.
( B ) D u f f i n g ' s e q u a t i o n f o r a s o f t s p r i n g
x + x - fix5 = 0 , ¡x > 0
D u f f i n g ' s e q u a t i o n f o r a s o f t s p r i n g w i t h d a m p i n g
x + e x + x - / и х 3 = 0 , e > 0
IA E A -S M R -17 /6 6 2 2 9
DISSIPATION
I t c a n b e s h o w n t h a t s u f f i c i e n t l y s m a l l c h a n g e s i n t h e v e c t o r f i e l d s ( s m a l l e r t h a n t h e o r i g i n a l p e r t u r b a t i o n ) w i l l n o t c h a n g e t h e q u a l i t a t i v e p i c t u r e s o n t h e r i g h t s o t h a t t h e s e r e p r e s e n t s t r u c t u r a l l y s t a b l e s y s t e m s .
N o t e t h a t L y a p u n o v - t y p e a r g u m e n t s s h o w t h e l a c k o f e x i s t e n c e o f p e r i o d i c s o l u t i o n s i n m e c h a n i c a l s y s t e m s c o n t a i n i n g p o s i t i v e d i s s i p a t i o n ( s u c h a s t h e d a m p e d s y s t e m s ) . I n f a c t f o r m o s t o f t h e h i g h l y - s t u d i e d e x p l i c i t l y - d e f i n e d s t r u c t u r a l l y - s t a b l e A D E [ 2 ] , t h e r e i s e i t h e r a l a c k o f c l o s e d o r b i t s ( s e e e . g . , [ 3 ] a n d [ 4 ] ) o r a n i n f i n i t e l y d e n s e - d i s t r i b u t e d f a m i l y [ 2 ] . T h e m o s t f a m o u s e x c e p t i o n i s V a n d e r P o l ' s e q u a t i o n
x + /л(x :2 - 1 ) x + x = 0 w h i c h a r i s e s i n t h e s t u d y o f e l e c t r i c a l n e t w o r k s .T h e s e c r e t h e r e i s t h a t e n e r g y i s a l t e r n a t e l y d e c r e a s i n g a n d i n c r e a s i n g a l o n g a g i v e n t r a j e c t o r y , a n d , a s w e s h a l l s e e , a u n i q u e s t a b l e p e r i o d i c s o l u t i o n i s o b t a i n e d . B e f o r e c o n s i d e r i n g i t , h o w e v e r , w e m a k e s o m e e l e m e n t a r y r e m a r k s o n p l a n a r s y s t e m s a r i s i n g f r o m s e c o n d - o r d e r A D E i n o n e v a r i a b l e . W e s h a l l b e e x c l u s i v e l y c o n c e r n e d w i t h s u c h s y s t e m s .
C o n s i d e r t h e e q u a t i o n x + f ( x , x ) = 0 , f : s m o o t h , o r e q u i v a l e n t l y
( i ) A l l o r b i t s i n t h e u p p e r - ( r e s p e c t i v e l y l o w e r - ) h a l f - p l a n e m o v e f r o m l e f t t o r i g h t ( r e s p . r i g h t t o l e f t ) ; o r b i t s a r e v e r t i c a l a t t h e i r i n t e r s e c t i o n w i t h t h e x - a x i s . Z e r o s o f t h e v e c t o r f i e l d ( 2 ) o c c u r o n l y o n t h e x - a x i s .
( i i ) A t a z e r o P o f t h e v e c t o r f i e l d , i . e . , w h e r e у = 0 a n d f ( x , 0 ) = 0 , t h e H e s s i a n m a t r i x i s
(2)
T h e n :
H e n c e t h e c h a r a c t e r i s t i c e x p o n e n t s a r e t h e r o o t s o f t h e e q u a t i o n
230 SHAHSHAH ANI
T h u s t h e e q u i l i b r i u m p o i n t i s n o n - d e g e n e r a t e ( X f 0 ) i f a n d o n l y i f
A n e q u i l i b r i u m p o i n t i s c a l l e d h y p e r b o l i c i f t h e r e a l p a r t s o f t h e c h a r a c t e r i s t i c e x p o n e n t s a r e n o n - z e r o . H e r e t h i s h o l d s i f a n d o n l y i f
T h e s i g n i f i c a n c e o f h y p e r b o l i c i t y i s t h a t t h e l o c a l s i t u a t i o n a r o u n d a n e q u i l i b r i u m p o i n t i s s t r u c t u r a l l y - s t a b l e i f a n d o n l y i f t h e e q u i l i b r i u m p o i n t i s h y p e r b o l i c ' 1 5 ] . I n t h e h y p e r b o l i c c a s e , i f t h e t w o X ' s a r e r e a l a n d o f d i f f e r e n t s i g n s
o n e o b t a i n s a s a d d l e p o i n t ; a n d i f t h e r e a l p a r t s o f t h e X ' s a r e b o t h p o s i t i v e ( r e s p . n e g a t i v e ) , a s o u r c e ( r e s p . a s i n k ) a r e i n d i c a t e d . H e r e w e h a v e a s o u r c e i f
a n d a s i n k i f
SADDLE SOURCE SINK
L e t u s n o w r e t u r n t o V a n d e r P o l ' s e q u a t i o n ; i t c a n b e w r i t t e n a s
t h e s y s t e m
(3)
IA E A -S M R -1 7 /6 6 231
w h i c h i s o b t a i n e d f r o m t h e h a r m o n i c - o s c i l l a t o r r e l a t i o n x + x = 0 , w i t h e n e r g y E = \ ( x 2 + y 2 ) , b y a d d i n g t h e t e r m + ц (х2 - l ) x . T h e c h a n g e i n e n e r g y i s
E = Ê = x x + y y = -ц ( х 2 - l ) y 2
A n e a s y c o m p u t a t i o n s h o w s t h a t ( 0 , 0 ) i s a s o u r c e ( t h e o n l y e q u i l i b r i u mp o i n t , i n f a c t ) . T h u s a l l o r b i t s s t a r t i n g s u f f i c i e n t l y c l o s e t o 0 m o v e a w a yf r o m 0 .
A s t h e e n e r g y i s i n c r e a s i n g i n t h e v e r t i c a l s t r i p - 1 < x < 1 , a n d d e c r e a s i n g o u t s i d e , o r b i t s s t a r t i n g s u f f i c i e n t l y f a r a w a y f r o m t h e o r i g i n m o v e i n w a r d s , t o 0 ( t h i s c a n b e m a d e r i g o r o u s , s e e , e . g . , [ 6 ] ) .
I t f o l l o w s t h a t s o m e o r b i t m u s t c l o s e t o f o r m a p e r i o d i c s o l u t i o n ( w h i c h , i n t h i s c a s e , c a n b e s h o w n t o b e u n i q u e ) . I t i s k n o w n t h a t t h e b e h a v i o u r o f t r a j e c t o r i e s i n a ' t u b e ' a r o u n d a c l o s e d o r b i t o f a p l a n a r s y s t e m i s s t r u c t u r a l l y - s t a b l e i f a n d o n l y i f a l l o r b i t s s u f f i c i e n t l y c l o s e t o t h e c l o s e d o r b i t e x p o n e n t i a l l y a p p r o a c h o r r e c e d e r e l a t i v e t o t h e c l o s e d o r b i t [ 5 ] .I n t h i s c a s e t h e c l o s e d o r b i t i s c a l l e d h y p e r b o l i c . A t e s t o f t h i s f a c t i s t h e s i g n o f t h e c h a r a c t e r i s t i c e x p o n e n t
T
i fA = - J ( d i v X ) d t
о
w h e r e T i s t h e p e r i o d o f t h e o r b i t a n d X t h e v e c t o r f i e l d o f t h e s y s t e m . H y p e r b o l i c i t y i s e q u i v a l e n t t o A f 0 , w i t h A < 0 ( r e s p . A > 0 ) i n d i c a t i n g e x p o n e n t i a l a p p r o a c h ( r e s p . e x p o n e n t i a l r e c e s s i o n ) o f r a t e Л .
I t c a n b e s h o w n t h a t t h e c l o s e d o r b i t o f V a n d e r P o l ' s s y s t e m i s h y p e r b o l i c , a n d t h a t i n f a c t ( 3 ) i s o f g l o b a l s t r u c t u r a l s t a b i l i t y . T h u s s m a l l p e r t u r b a t i o n s o f ( 3 ) p r o d u c e s y s t e m s w i t h a u n i q u e e q u i l i b r i u m p o i n t ( a s o u r c e ) , a n a t t r a c t i n g c l o s e d o r b i t s u r r o u n d i n g i t , a n d e v e r y o r b i t ( e x c e p t t h e u n i q u e s o u r c e i t s e l f ) a p p r o a c h i n g t h e c l o s e d o r b i t .
T h e t w o c a s e s o f v e c t o r f i e l d s c o n s i d e r e d i n t h i s p a p e r b o t h g e n e r a l i z e V a n d e r P o l ' s e q u a t i o n . I n o n e c a s e , w e s t a r t a g a i n w i t h t h e h a r m o n i c o s c i l l a t o r x + x = 0 , a d d a t e r m m o r e g e n e r a l t h a n ^ ( x 2 - l ) x , a n d o b t a i n r e s u l t s r e g a r d i n g t h e n u m b e r a n d n a t u r e o f c l o s e d o r b i t s . I n t h e s e c o n d c a s e , w e e x p l o i t t h e s i m i l a r i t y b e t w e e n t h e t e r m ( x 2 - 1 ) a n d t h e p o t e n t i a l e n e r g y j x 2 o f V a n d e r P o l ' s s y s t e m t o o b t a i n a n o t h e r c l a s s o f s y s t e m s t h a t c a n b e t h o r o u g h l y s t u d i e d .
F i r s t c o n s i d e r t h e e q u a t i o n x + ^ W ( x ) x + x = 0 , w h e r e
W ( x ) = ( x 2 - a 2 ) . . . ( x z - a 2 )
0 < a 1 < . . . < a n ; p > 0
W e w i s h t o s t u d y t h e n u m b e r o f c l o s e d o r b i t s o f t h i s s y s t e m . I n p a r t i c u l a r , u n d e r w h a t c i r c u m s t a n c e s a r e t h e r e n d i s t i n c t p e r i o d i c s o l u t i o n s ? W h i l e i t d o e s n o t s e e m t h a t t h i s q u e s t i o n h a s a g e n e r a l a n d e l e g a n t a n s w e r , v a r i o u s s u f f i c i e n t c o n d i t i o n s m a y b e f o u n d . O u r a n s w e r s d e p e n d o n t h e s i z e o f ц.
2 3 2 SH AH SH AH AN l
C a s e 1 ( p » 0 ) . L e tX
F ( x ) = J W ( f ) d Ç
0
t h e n t h e c r i t i c a l p o i n t s o f F ( x ) a r e j u s t ± a , . . . , + a n , w h i c h a l t e r n a t ea s m a x i m a a n d m i n i m a . L e t m . = F í a . ) Ii 1 ' i ' i
T h e o r e m 1 C o n s i d e r t h e s y s t e m
x + f i W ( x ) x + x = 0 ( 4 )
F o r ц s u f f i c i e n t l y l a r g e , ( 4 ) h a s n c l o s e d o r b i t s p r o v i d e d m j < m 2 < . . . < m n .T h e p r o o f i s j u s t a n a p p l i c a t i o n o f t h e c l a s s i c a l i d e a s o f S t o k e r - F l a n d e r s
a n d L a S a l l e ( s e e , e . g . , [ 7 ] ) . O n e c o u l d a c t u a l l y u s e q u i t e g e n e r a l f u n c t i o n s r a t h e r t h a n t h e p o l y n o m i a l W ( x ) , p r o v i d e d t h a t t h e c o r r e s p o n d i n g c o n d i t i o n f o r t h e m j ' s i s s a t i s f i e d . T h e l i m i t i n g p o s i t i o n o f t h e p e r i o d i c s o l u t i o n s a s ц -» + oo i n t h e x z - p l a n e , w h e r e
z = F ( x ) + — x
a r e s h o w n i n t h e f i g u r e . T h e s o l i d l i n e s t o u c h i n g t h e h e i g h t s o f t h e c r i t i c a l p o i n t s t o g e t h e r w i t h p o r t i o n s o f t h e c u r v e z = F ( x ) f o r m t h e l i m i t i n g p o s i t i o n s o f t h e c l o s e d o r b i t s .
\
N\
\
C a s e 2 . I n t h e o t h e r e x t r e m e c a s e , w h e r e 0 < ц « 1 , t h e f o l l o w i n g i s t y p i c a l o f t h e t h e o r e m s t h a t g i v e s u f f i c i e n t c o n d i t i o n s f o r t h e e x i s t e n c e o f n c l o s e d o r b i t s . M u c h s h a r p e r t h e o r e m s c a n p r o b a b l y b e p r o v e d .
T h e o r e m 2 . I n t h e s y s t e m ( 4 ) , s u p p o s e a 9 > 6 a , , a , > 5 a 0 , a n d a k + i > 4 a k f o r k ê 3 . T h e n , i f 0 < ц « 1 , t h e s y s t e m h a s n d i s t i n c t p e r i o d i c s o l u t i o n s .
T h e p r o o f u s e s t h e m e t h o d o f s m a l l p a r a m e t e r s , w h i c h r e d u c e s t h e q u e s t i o n t o c o u n t i n g t h e n u m b e r o f p o s i t i v e z e r o s o f t h e r e a l - v a l u e d f u n c t i o n
+ Г
F I— F 1 G ( r - ) = J W ( 5 ) d ?
I A E A -S M R -1 7 /6 6 233
I f n o r e s t r i c t i o n o n t h e s p a c i n g o f t h e a ¡ ' s i s g i v e n , e a s y c o u n t e r e x a m p l e s s h o w t h a t n o c l o s e d o r b i t s m a y e x i s t .
W e n o w t u r n t o t h e s e c o n d g e n e r a l i z a t i o n o f t h e V a n d e r P o l e q u a t i o n . A g a i n c o n s i d e r i n g x + p ( x 2 - l ) x + x = 0 w e n o t e t h a t t h e p o t e n t i a l e n e r g y i s d e t e r m i n e d t o w i t h i n a n a d d i t i v e c o n s t a n t s o t h a t w e m a y w r i t e V ( x ) = i ( x 2 - ! ) • V ( x ) a n d p ( x 2 - 1 ) h a v e t h e s a m e z e r o s , a n d t h e i r c r i t i c a l p o i n t s o c c u r f o r t h e s a m e v a l u e o f x a n d i s o f t h e s a m e t y p e ( m i n i m u m ) f o r b o t h . I n g e n e r a l , t w o r e a l - v a l u e d f u n c t i o n s V ( x ) a n d W ( x ) a r e c o v a n i s h i n g a n d c o c r i t i c a l i f t h e i r z e r o s a n d c r i t i c a l p o i n t s
c o i n c i d e ( a n d a r e o f t h e s a m e t y p e ) .
T h e o r e m 3 . C o n s i d e r t h e s y s t e m d e f i n e d b y x + W ( x ) x + V 1 ( x ) = 0 w h e r eV a n d W a r e c o v a n i s h i n g a n d c o c r i t i c a l , t h e c r i t i c a l p o i n t s { c ; } a r e n o n d e g e n e r a t e , a n d W ( c ¡ ) f 0 f o r b o t h . H e n c e f o l l o w s :
( A ) T h e z e r o s o f t h e s y s t e m o c c u r a t ( c ¡ ( 0 ) a n d a r e o f 3 t y p e s : ( i ) s a d d l e s i f c ¡ i s a m a x i m u m , ( i i ) s o u r c e i f c . i s a . m i n i m u m w i t h W ( c . ) < 0 , a n d ( i i i ) s i n k s i f q i s a m i n i m u m w i t h W ( c t ) > 0 .
( B ) E v e r y c l o s e d o r b i t s u r r o u n d s a t l e a s t o n e s o u r c e a n d i s a h y p e r b o l i c a t t r a c t o r .
( C ) I f a n i n t e r v a l i s ' s u f f i c i e n t l y d o m i n a t e d f o r W ' , i t i s e n c l o s e d i n a c l o s e d o r b i t .
T h e l a s t s t a t e m e n t m e a n s t h e f o l l o w i n g : L e t a a n d b b e t w o c o n s e c u t i v e z e r o s o f W , w i t h W n e g a t i v e o n t h e o p e n i n t e r v a l ] a , b [ = I , a n d l e t C * a n d C ¥ b e t h e c r i t i c a l p o i n t s ( m a x i m a ) i m m e d i a t e l y t o t h e l e f t a n d t h e r i g h t o f I .
W e s a y I i s s u f f i c i e n t l y d o m i n a t e d i f :
( i ) J W ( x ) d x > 0 a n d J W ( x ) d x > 0 a r e b o t h s u f f i c i e n t l y l a r g e , a n d c * b
( i i ) c * - b a n d a - с * a r e s u f f i c i e n t l y l a r g e .
I n p a r t i c u l a r i f W h a s n o c r i t i c a l p o i n t s o u t s i d e I a n d
W ( x ) d x = J W ( x ) d x = + oo, t h e c o n d i t i o n i s a s s u m e d s a t i s f i e d ,
b
T h e o r e m 4 . L e t t h e s m o o t h f u n c t i o n V : И - * К s a t i s f y t h e f o l l o w i n g :
( i ) V h a s a f i n i t e n u m b e r o f c r i t i c a l p o i n t s , a l l n o n - d e g e n e r a t e ,( i i ) V / 0 a t t h e c r i t i c a l p o i n t s , a n d ( i i i ) V -> + æ a s x - * + oo. T h e n f o r ' a l m o s t a l l ' s m o o t h f u n c t i o n s W : И - * И c o v a n i s h i n g a n d c o c r i t i c a l w i t h V , t h e s y s t e m x + W ( x ) x + V ' ( x ) = 0 i s s t r u c t u r a l l y s t a b l e .
T h e " a l m o s t a l l " s t a t e m e n t m e a n s t h e f o l l o w i n g :
( 1 ) I f t h e c o n c l u s i o n h o l d s f o r s o m e W 0 i t h o l d s f o r a l l W s u f f i c i e n t l y C 1 - c l o s e t o W 0 , a n d
234 SH AH SH AH AN I
( 2 ) I f i t f a i l s f o r a p a r t i c u l a r W Q3 t h e r e i s a W a r b i t r a r i l y C 1 - c l o s e t o W Q f o r w h i c h i t h o l d s .F o r t h e p r o o f s o f t h e o r e m s ( 3 ) a n d ( 4 ) t h e r e a d e r i s r e f e r r e d t o R e f . 1 9 ] .
R E F E R E N C E S
[ 1 ] A R N O LD , V ., Ordinary Differential Equations, M I T Press (1973).
[2 ] SM ALE, S., 'Differentiable D yn am ical Systems', Bull. A .M .S . 73 (1967) 747.
[3 ] SM ALE, S ., 'Gradient D yn am ical Systems’ , Ann. M ath. 74 (1961) 109.
[4 ] S H A H S H A H A N I, S., ’Dissipative Systems on M anifolds', Inventiones m ath. 16 (1972) 177.
[5 ] M A R K U S , L ., 'Structurally-stable Differential Systems', Ann. Math. (2 ) 73 (1961) 1.[ 6 ] H IR SC H , М ., S M ALE, S., D ifferential Equations, D yn am ical Systems, and Linear Algebra, Academ ic
Press (1974).[ 7 ] L E F S C H E T Z , S., D ifferential Equations: G eom etric Th e o ry , Interscience, N .Y . (1957).
[ 8 ] S H A H S H A H A N I, S., 'A Note on Nonlinear Oscillations', to appear.
[ 9 ] S H A H S H A H A N I, S . , ’Lienard Systems’ , to appear.
I A E A -S M R -1 7 /5 3
REALIZATION THEORY OF LINEAR DYNAMICAL SYSTEMS
R.E. KALMANCenter for Mathematical System Theory, University of Florida,Gainesville, Florida,United States of America andProfessur fiir mathematische Systemtheorie, Eidgenossische Technische Hochschule,Z u ric h ,Switzerland
Abstract
REALIZATION THEORY OF LINEAR DYNAMICAL SYSTEMS.These lectures cover the classical theory o f realization o f constant linear dynam ical systems from
input/output data, with a brie f discussion o f the broader scien tific aspects o f the realization problem . The linear realization problem is treated both from the abstract point o f view (ca teg ory -th eoretic proof o f the uniqueness theorem o f m in im al realizations) as w ell as the app lied -m ath em atica l point o f view , in which the Hankel m atrix is used to obtain a com putational algorithm . Som e aspects o f the partial realization problem are also treated.
0 . O V E R V IE W
R e a l i z a t i o n t h e o r y i s t h e g e r m o f a n ( a s y e t ) u n d e v e l o p e d t h e o r y o f
m o d e l i n g . I t m a y b e a r g u e d ( b u t w e w i l l n o t d o s o h e r e ) t h a t m o d e r n
s c i e n c e i s i n a p r o c e s s o f r e o r g a n i z a t i o n , i n w h i c h t h e c l a s s i c a l
p a t t e r s o f i n v e s t i g a t i o n b a s e d o n t h e p r e c e p t s o f p h y s i c s ( e x p e r i m e n t ,
t h e o r y , v e r i f i c a t i o n ) a r e g r a d u a l l y b e i n g r e p l a c e d b y a d i f f e r e n t
m e t h o d o l o g y , w h i c h i s n o t d i r e c t e d t o w a r d t h e d i s s e c t i o n o f i s o l a t e d
p h e n o m e n a b u t a i m s t o c o m p r e h e n d c o m p l e x s y s t e m s a s a w h o l e . I n t h i s
n e w m e t h o d o l o g y , c o m p u t e r - a i d e d m o d e l i n g a n d s i m u l a t i o n t e n d t o r e p l a c e
( t o a c e r t a i n e x t e n t ) d i r e c t e x p e r i m e n t s . M a n y m o d e l e r s a n d s i m u l a t o r s
a r e , a l a s , u n a w a r e o f d e e p e r " l a w s " u n d e r l y i n g t h e p r a c t i c e o f t h e i r
c r a f t . I n f a c t , t h e s e " l a w s " a r e a s y e t b u t d i m l y a p p a r e n t i f w e w i s h
t o e n c o m p a s s d y n a m i c a l s y s t e m s i n t h e g r e a t e s t g e n e r a l i t y . B u t i n t h e
l i n e a r c a s e t h e s e s a m e " l a w s " h a v e b e e n b r o u g h t i n t o s h a r p f o c u s i n
t h e l a s t 1 5 y e a r s . S o o u r o b j e c t i v e h e r e w i l l b e t o g i v e a n a c c o u n t o f
t h e m o d e r n v i e w o f m o d e l i n g i n t h e l i n e a r c a s e .
235
236 KALMAN
N e i t h e r t h e f a c t s n o r t h e m a t h e m a t i c a l m a c h i n e r y u s e d f i t n e a t l y
i n t o t h e f r a m e w o r k o f ( c l a s s i c a l ) a p p l i e d m a t h e m a t i c s . T o a p p r e c i a t e
t h e r e s u l t s , t h e r e a d e r s h o u l d b e e q u i p p e d w i t h s o m e s y s t e m - t h e o r e t i c a l
i m a g i n a t i o n , s o m e t h i n g t h a t c a n b e a c q u i r e d b y p h y s i c a l o r l i t e r a r y
c o n t a c t w i t h s e r v o m e c h a n i s m s , c o n t r o l s y s t e m s , o p t i m i z a t i o n t h e o r y ,
c o m p u t e r s , i n f o r m a t i o n t h e o r y , e t c .
T h e m a i n r e f e r e n c e s w i l l b e K A L M A N , F A L B , a n d A R B I B [ 1 9 6 9 ] ,
K A L M A N [ 1 9 6 9 ] , E IL E N B E R G [ 1 9 7 ^ , C h a p t e r X V I ] , a n d K A L M A N [ t o a p p e a r ] .
W e s h a l l a l s o p o i n t o u t s o m e n o t e w o r t h y r e c e n t d e v e l o p m e n t s w h e r e v e r
a p p r o p r i a t e .
1 . IN T R O D U C T IO N
I n o r d e r t h a t m a t h e m a t i c s m a y c o n t r i b u t e t o t h e s t u d y á n d
u n d e r s t a n d i n g o f m o d e l s , i t i s n e c e s s a r y t o h a v e a p r e c i s e d e f i n i t i o n
o f a m o d e l a s a d y n a m i c a l s y s t e m .
T h e c l a s s i c a l ( a n d b y n o w o b s o l e t e ) i d e a o f a d y n a m i c a l s y s t e m ,
a s u s e d i n p u r e m a t h e m a t i c s , i s b u i l t o n t h e c o n c e p t s o f d i f f e r e n t i a l
e q u a t i o n s , o n e - p a r a m e t e r t r a n s f o r m a t i o n g r o u p s ( " f l o w " ) , e t c . A l l
t h i s i s a l a r g e i d e a l i z a t i o n a n d a s m a l l g e n e r a l i z a t i o n o f t h e m o t i o n
o f t h e p l a n e t s a s d e s c r i b e d b y N e w t o n ' s d i f f e r e n t i a l e q u a t i o n s . T h e
m o d e r n i d e a o f a d y n a m i c a l s y s t e m i s i n s p i r e d b y t h e c o m p u t e r a n d u s e s
t h e c o n c e p t s o f i n p u t s , s t a t e s , o u t p u t s , s t a t e t r a n s i t i o n s , e t c . ; a
c o m p u t e r w i t h o u t a n i n p u t o r ( a f o r t i o r i ) w i t h o u t a n o u t p u t i s o f
n e g l i g i b l e i n t e r e s t .
A n e s s e n t i a l i n s i g h t o f t h e m o d e r n t h e o r y i s t h a t w e m u s t
d i s t i n g u i s h b e t w e e n " i n t e r n a l " a n d " e x t e r n a l " d e s c r i p t i o n s o f
d y n a m i c a l s y s t e m s . T h e f i r s t t e r m r e f e r s , r o u g h l y , t o t h e a n a l y s i s
o f t h e s y s t e m v i a s o m e p o s t u l a t e d " s t a t e v a r i a b l e s " . T h e s e c o n d t e r m
r e f e r s , r o u g h l y , t o t h e a n a l y s i s o f t h e s y s t e m a s a " b l a c k b o x " , t h a t
i s , f r o m p u r e l y b e h a v i o r i s t i c p o i n t o f v i e w , w i t h o u t p o s t u l a t i n g a n y
i n t e r n a l m e c h a n i s m i n v o l v i n g s t a t e v a r i a b l e s .
H a v i n g b e e n g i v e n t h e " i n t e r n a l " d e s c r i p t i o n o f a s y s t e m , o b t a i n i n g
i t s " e x t e r n a l " d e s c r i p t i o n i s a t r i v i a l p r o b l e m . T h e d i f f i c u l t i e s
e n c o u n t e r e d h e r e a r e m e r e l y t h o s e o f s o l v i n g a d i f f e r e n t i a l o r
d i f f e r e n c e e q u a t i o n ; t h e s e p r o b l e m s m a y b e f a r f r o m e a s y i n t h e
s e n s e o f p u r e m a t h e m a t i c s b u t t h e y h a v e n o s y s t e m - t h e o r e t i c a l i n t e r e s t .
IA E A -S M R -1 7 /5 3 237
The converse problem, o f in d u cin g an " in t e r n a l" d e s c r ip t io n o f a
system from i t s g ive n "e x te rn a l" d e s c r ip t io n , i s v e ry f a r from
t r i v i a l ; t h is i s , in f a c t , the problem of r e a l iz a t io n . I r i t u it iv e l y
speaking , we may equate an " e x te rn a l" d e s c r ip t io n to a problem and
the d e s ire d " in t e r n a l" d e s c r ip t io n to a computer program which so lve s
th a t problem . In more p h y s ic a l term s, the "e x te rn a l" d e s c r ip t io n o f a
system re p re se n ts i t s g ive n (or d e sire d ) beh avio r w h ile the " in t e r n a l"
d e s c r ip t io n w i l l sup p ly a b lu e p r in t (or c i r c u i t diagram , or computer
program, or . . . ) w ith w hich the g iv e n beh avio r can somehow be " re a liz e d " in some concrete way w it h in the p re se n t-d a y l im it a t io n s
o f technology.
Note th a t a r e a l iz a t io n (such as a computer program c a lc u la t in g
the s o lu t io n o f a m athem atical problem) need not e x is t and, i f i t
does, need not be n e c e s s a r ily unique. Thus we are le d fu rth e r to the
problem o f f in d in g a " n a tu ra l" or "ca n o n ica l" r e a l iz a t io n which
depends o n ly on the g ive n b ehavior and i s fre e o f any other (hidden)
extraneous assum ptions. A s t r ik in g l y im portant r e s u lt o f r e a l iz a t io n
th e o ry i s th a t each b ehavior (b la c k box) possesses a unique n a tu ra l
r e a l iz a t io n (p re c is e statem ent l a t e r ! ) . T h is shows th a t th ere i s a
n a t u r a l b i je c t iv e correspondence between the " in t e r n a l" and "e x te rn a l"
d e s c r ip t io n s o f system . We have here a m athem atical setup o f g re a t
in t e r e s t to a l l s c ie n t is t s ; fo r example, t h is r e s u lt dem olishes the
famous " re d u c t io n is t " v s . " b e h a v io ris t" co n tro ve rsy in psycho lo gy;
the d is t in c t io n between "models" and "behavior" i s r e a l l y nothing more
than ju s t a f a i r l y t r i v i a l (m athem atical) isom orphism .
In f in it e -d im e n s io n a l r e a l iz a t io n theo ry i s not ye t developed even
in the l in e a r ca se . The d i f f i c u l t i e s are to p o lo g ic a l (how to g iv e a
system a n a tu ra l to p o lo g y ?) . So f a r i t has not been p o s s ib le to show
how fu n c t io n a l a n a ly s is can have a d ir e c t b e a rin g on the r e a l iz a t io n
problem . (Note added in J u ly 19 7 5 : In the meantime, the uniqueness
theorem fo r n a t u ra l r e a l iz a t io n s has been proven in a f a i r l y la rg e
v a r ie t y o f s it u a t io n s ; see MATSUO and KALMAN [19 7 6 ]. )
Le t us now o u t lin e the problems to be co nsidered in these le c t u r e s :
A. P re c is e d e f in it io n o f a system— o n ly re so lv e d in the
f in it e -d im e n s io n a l l in e a r case.
B. E x iste n c e o f a n a t u ra l (c a n o n ic a l) r e a l iz a t io n co nstructed
from and in 1 - 1 correspondence w ith a g ive n b e h a v io r. I f th ere i s a
r e a l iz a t io n , then th ere i s a c a n o n ic a l one.
2 3 8 K ALM AN
C. Uniqueness. A re any two n a t u r a l r e a l iz a t io n s isom orphic in
some sense?
D. How can we t e l l th a t a r e a l iz a t io n i s n a t u r a l? Reduction o f
n o ncanonical r e a l iz a t io n s . S tru ctu re o f an a r b it r a r y r e a l iz a t io n .
E . R e a liz a t io n s from p a r t ia l d a ta . ( " P a r t ia l" r e a l iz a t io n th e o ry .)
2 . DEFINITIONS
The c r i t i c a l assumption used throughout i s l in e a r i t y . Somewhat
le s s im portant i s the assum ption th a t a l l systems tre a te d here are
co nstant (have "constant c o e f f ic ie n t s " ) ; in other words, the d e fin in g
data do not change w ith t im e .
We f i r s t co n sid e r the case where the tim e se t i s T = R = r e a ls ;
systems w ith such tim e se t axe c a lle d co n tin u o u s-t im e . F o rm a lly , a
l in e a r , co n stan t, f in it e -d im e n s io n a l, c o n tin u o u s-t im e , d if f e r e n t ia l
dynam ical system E over the r e a ls i s g iv e n by the r e la t io n s
( 2 . 1 ) d x /d t = Fx + G u ( t ) ,
(2 .2 ) y ( t ) = Hx ( t ) ,
where u ( t ) £ U = r” = in p u t sp ace, y ( t ) £ Ï = Í = output space,
x ( t ) = s o lu t io n o f ( 2 . 1 ) , w ith v a lu e s in X = Rn = sta te space, and
t £ R. U, Y, and X are viewed as v e cto r spaces over the f i e l d R.
The system Z i s s p e c if ie d by g iv in g the m a trice s F , G, H (w ith
co nstant c o e f f ic ie n t s ) ; w ith some abuse o f language, i t i s convenient
to re g a rd Z as the t r ip l e (F , G, H ) . By d e f in it io n , we se t
dim £ = dim X = n.
The o ther in t e r e s t in g case i s th a t o f d is c r e t e -t ia e systems whose
tim e s e t i s T = Z = in t e g e r s . F o rm a lly , a l in e a r , co n stan t, f i n i t e
d im e n sio n al, d is c re te -t im e dynam ical system over a f ie l d к i s g iv e n
by the r e la t io n s
(2 .3 ) x ( t + 1 ) = F x (t ) + G u ( t ) ,
(2.U ) y ( t ) = H x ( t ) ,
where ( in analo gy w ith the preced in g) U = к™, X = k 11, Y = are
v e c to r spaces over k ; the data w hich s p e c if ie s a system i s a g ain a
t r ip l e o f m a tric e s (F , G, H ) . (But note th a t F con . an<l ^ d is c ^ave
IA E A -S M R -1 7 /5 3 2 3 9
v e r y d if f e r e n t p h y s ic a l s ig n if ic a n c e ; t h is i s im portant in a p p lic a t io n s
b ut la r g e ly ir r e le v a n t as f a r as the theo ry i s co ncerned.)
The d is c r e t e -t im e case i s in t e r e s t in g and co nvenient because we are
no lo n g e r r e s t r ic t e d in the ch o ice o f the f ie l d by the requirem ent o f b e in g a b le to w rit e a d if f e r e n t ia l equation fo r the e v o lu tio n o f
sta te ( " d if f e r e n t ia l" system s). H is t o r ic a l ly , the fo llo w in g cases
have been co nsid e re d :
к = R, Ç (complex numbers), befo re 1950.
= f in i t e f ie l d , co nsidered beginning about 1955, o f in t e r e s t in computer sc ie n ce ( " lin e a r s e q u e n tia l sy ste m s").
= a r b it r a r y f i e l d ( in f in i t e or not, a r b it r a r y c h a r a c t e r is t ic ) , t h is i s the p re se n t "standard" th eo ry.
= commutative r in g ( e s p e c ia l ly a H o etherian in t e g r a l domain),a f t e r 1 9 7 1 ; see ROUCHALEAU, WYMAN) and KALMAN [ 1 9 7 1 ] .
= noncommutative r in g s , c u rre n t re se a rch a t the Center fo r M athem atical System Theory at the U n iv e r s it y o f F lo r id a .
I t i s im portant to emphasize the fa c t t h a t , from a t h e o r e t ic a l
p o in t o f v iew , o n ly the " a lg e b ra ic p ro p e rtie s" a sso c ia te d w ith the
m a tr ic e s (F , G, H) p la y a r o le (see below ). Thus there i s no r e a l
d iffe re n c e between co n tinu o u s-tim e and d is c r e t e -t im e system s; t h is was
not c le a r ly r e a liz e d befo re a lg e b ra ic system th e o ry began in the 1950' s .
Henceforth к w i l l be an a r b it r a r y (commutative) f i e l d .
We now tu rn to e x p la in in g what i s to be meant by " a lg e b ra ic
p ro p e rtie s " o f the t r ip l e (F , G, H ) . Two system s S and Ê are
e q u iv a le n t i f f there e x is t s a square m a trix T , w ith det T =/ 0 (an
element o f the g e n e ra l l in e a r group GL(kn)) such th a t the r e la t io n s
( 2 .5 ) FT = TF, G = TG, HT = H
h o ld . (These r e la t io n s can be e a s i ly found or v e r i f ie d as b e in g due to
the a c tio n o f T on X by X w Tx = x, i . e . , a change o f b a s is in
X .) S ince e q u iv a le n t systems have the same b e h a v io r when viewed as
b la c k boxes, i t i s co nvenient to th in k o f a system not as a s in g le
t r ip l e but as an eq u ivale n ce c la s s under ( 2 . 5 ) . We denote the e q u i
v a le n ce c la s s o f £ under the a c t io n o f G L(kn) b y [ £ ] . I t should
be borne in mind, o f co urse, th a t [z ] i s a much le s s elem entary
m athem atical o b je c t than a t r ip l e o f m a trice s (see a ls o l a t e r ) .
240 K ALM AN
The p receding d is c u s s io n was concerned w ith d e f in in g a system in th
in t e r n a l sense; we s h a l l now d e fin e a system in the e x te rn a l sense.
G iven Z (e it h e r co n tinu o u s-tim e or d is c r e te -t im e ) and the
assum ptions t = 0 and x (0) = 0, the fo llo w in g form ulas are v a l id :
(2 .6 ) y ( t ) = / A (t - t)u(t)cIt, A (t) = He 4 T = R;0
( 2 .7 ) y ( t ) = E A u ( t ) , A, = HFt_ 1G, T = Z.t > x - 0 1
F o rm a lly we have then: a l in e a r , co nstant ( f in i t e or in f in i t e
d im e n sio n a l, co ntinu o u s-tim e or d is c re te -t im e ) dynam ical system S
over к (" e x te rn a l" sense) i s an in f in i t e sequence o f m a trice s
(A^, А^, . . . ) , each m a tr ix being p x m w ith elements over k . For
d is c r e t e -t im e system s, t h is d e f in it io n i s suggested by the fa c t th a t
fo rm ula ( 2 .7 ) re q u ire s ju s t such a sequence fo r i t s s p e c if ic a t io n . For
co n tin u o u s-tim e systems (k = R ), the d e f in it io n fo llo w s from theF to b se rv a tio n th a t e i s a r e a l a n a ly t ic fu n c tio n o f t and th e re fo re
can be s p e c if ie d by g iv in g as data the c o e f f ic ie n t m a trice s of' i t s
power s e r ie s . (U n fo rtu n a te ly , the th eo ry o f in f in it e -d im e n s io n a l
a n a ly t ic l in e a r systems i s not ye t developed.)
The. fo llo w in g d e f in it io n i s now w e ll-m o tiv a te d : we say th a t Z
r e a l iz e s S i f f i n f i n i t e l y many r e la t io n s
(2 .8 ) A = HFt - 1 G, t = 1 , 2 , . . .
are s a t is f ie d . In the converse ca se , the r ig h t-h a n d s id e o f (2 .8 ) i s
g iv e n and the le ft -h a n d s id e g iv e s the e x te rn a l d e s c r ip t io n (" b la ck
box") o f Z , which we w rit e as S^,. Thus Z q r e a l iz e s Sq i f f
Sq = Sj-q . We s h a l l a ls o c a l l S the in p u t /o utp ut map.
Remark 1 . Z ~ Z im p lie s S , = Sg. Thus .S i s a constant
fu n c tio n on [ £ ] .
Remark 2 . I f S has a f in it e -d im e n s io n a l r e a l iz a t io n of
dim ension n, then the f i r s t 2n terms o f S u n iq u e ly determine a l l
the rem ain ing terms o f S. The problem, o f co urse, i s to f in d t h is n
from the data S.
I t i s c le a r th a t the A _, t = 1 , 2, . . . are abso lute a lg e b ra ic
in v a r ia n t s o f Z (th a t i s , p o lyn o m ia ls in the elements o f F , G, H
w hich are in v a r ia n t under the a c t io n ( 2 . 5 ) ) . There are a ls o other
in v a r ia n t s . The most im portant o f these are rank R and ran k 0,
where
IA E A -S M R -1 7 /5 3 241
R = [G , FG, . . . ]
and
0 =
(N o tice th a t R i s m u lt ip lie d on the l e f t by T under the a c tio n
(2 . 5) , hence ran k R i s in v a r ia n t ; s im i la r ly , 0 i s m u lt ip lie d on
the r ig h t by T under the a c t io n ( 2 . 5 ) and so rank 0 i s a ls o
in v a r ia n t . )
These in v a r ia n t s were d isco ve red in c o n tro l th.eory (see KALMAN
[ i 9 6 0 ] ) befo re the development o f the a lg e b ra ic th e o ry ; in f a c t , the
fo llo w in g sy ste m -th e o re tic d e f in it io n s were the o r ig in o f much o f th a t
th e o ry .
We say th a t Z i s (co m pletely) reachab le i f f rank R , = dim £ .
T h is i s e q u iv a le n t to r e q u ir in g th a t every sta te | £ Xv be reachab le
from 0, th a t i s , fo r each 5 there i s a f i n i t e sequence o f in p u ts
u (0) , u ( l ) , . . . , u (r^ ) such th a t t h is sequence a p p lie d to (2 . 5)
" t ra n s fe rs " the i n i t i a l s ta te x (0) = 0 to the f i n a l sta te
x ( r £ + l ) = i . C le a r ly i f T. i s reacha b le so are a l l members o f [Е ] .
S im i la r ly , we say th a t £ i s (com pletely) o b servable i f f
ran k 0 , = dim Z . T h is i s e q u iv a le n t to the statem ent th a t e very i n i t i a l
s ta te o f Xj, can be re co n stru cte d by l in e a r o p e ra tio n s on the output
v a lu e s ( y ( t ) , 0 < t < r^ } assuming th a t the in p u ts u ( t ) are a l l
zero (or are known). I f one member o f [z] i s o b se rva b le , so are a l l
o t h e rs .
Remark 3 . In the e a r l ie r papers by the a utho r, the c o n d it io n
rank R = dim E was c a lle d (complete) c o n t r o l la b i l i t y (e ve ry i n i t i a l
s ta te f 6 Xj, can be taken to 0 by a s u it a b le f i n i t e inp u t sequence).
T h is p ro p e rty i s e q u iv a le n t to complete r e a c h a b il it y in the co ntinu o u s
tim e case, but i s somewhat weaker in the d is c r e t e -t im e ca se. From the
242 K ALM AN
a lg e b ra ic p o in t o f view , the n o tio n o f complete r e a c h a b il it y i s the
a p p ro p ria te one s in c e i t works e q u a lly w e ll in both ca se s.
We can now sta te the main d e f in it io n : £ i s c a n o n ic a l i f f i t i s
both re a ch a b le and o b se rva b le ; we say a ls o th a t £ i s a n a t u r a l (or
c a n o n ic a l, or ir r e d u c ib le , or . . . ) r e a l iz a t io n o f S i f f £ r e a l iz e s
S and £ i s c a n o n ic a l. S ince any £ i s a r e a l iz a t io n o f some S
(fo r example, o f i t s own S jJ , i t makes sense to sa y th a t £ i s
c a n o n ic a l i f f i t i s a n a t u ra l r e a l iz a t io n o f i t s own in p u t/o u tp u t map.
3 . UNIQUENESS THEOREM FOR NATURAL REALIZATIONS
The te rm in o lo g y "c a n o n ica l" comes o f course from a lg e b ra ; i t i s
d i r e c t ly r e la te d to the "c a n o n ic a l f a c t o r iz a t io n " o f a l in e a r map
f : X -> Y as the sequence
( 3 .1 ) f : X — r-» X /k e r f --------------- im f r—? Yv onto ' isomorphism 1 -1
In t h is f a c t o r iz a t io n i t i s customary to suppress the isom orphism .
I t fo llo w s (although i t i s seldom po in ted out) th a t in the f a c t o r iz a t io n
o f a l in e a r map f
( 3 .2 ) f : X onto 1 -1
the space Z i s unique up to v e cto r space isom orphism . The g e n e r a li
z a t io n o f t h is r e s u lt from l in e a r maps to l in e a r systems le a d s to the
a u th o r's v e ry b a s ic uniqueness theorem fo r n a t u ra l r e a l iz a t io n s .
We s h a l l now g iv e a modern fo rm u la tio n and p ro o f o f t h is r e s u lt .
The main problem i s to o b ta in system isomorphism as the system e q u iv a
le n c e intro d u ced above. More a lg e b r a ic a l ly , the problem i s to d e fin e a
ca te g o ry o f r e a l iz a t io n s o f a f ix e d S so th a t isom orphism in t h is
ca te g o ry i s the same as system eq u ivalence ; then i t w i l l fo llo w th a t the
n a t u ra l r e a l iz a t io n o f a g iv e n S i s unique w it h in (catego ry)
isom orphism .
To d e fin e t h is categ o ry we co n sid e r dynam ical systems £ o f
g re a t g e n e r a lit y (not m erely l in e a r ) . We assume o n ly th a t
( i ) the sta te r e s u lt in g from an in p u t sequence <0 £ fl in
system £ i s denoted by [cd] . ;
IA E A -S M R -1 7 /5 3 243
( i i ) the c la s s of a l l s ta te s in £ w hich alw ays g iv e the
same output when sub je cted to the same input i s denoted by
W ith these co nventio ns, i t fo llo w s th a t £ and £ have the same
e x te rn a l d e s c r ip t io n s i f and o n ly i f = ( [cu]-}g fo r a l l cu f. SI.
The categ o ry R ealg o f a l l r e a l iz a t io n s o f a f ix e d b la c k box S
has fo r o b je c ts a l l dynam ical systems £ (w ith in a c e r t a in c la s s ,
s p e c if ie d as co nvenient) w hich r e a l iz e S and as morphisms Mor (£ , £)
any r e la t io n cp: X , -» X^ such th a t
( 3 - 3 ) M z - [ [ c o ] £ ] ¿
(That i s , the r e la t io n cp i s f ix e d on the reacha b le sta te o f X^ but
i s a r b it r a r y e lsew h ere.) Our p receding remark shows th a t t h is d e f in it io n
o f a morphism i s consonant w ith the requirem ent th a t S = S£.
I t i s im portant to note th a t in t h is d e f in it io n the morphisms are
d e fin e d v ia r e la t io n s on s t a t e s , not maps o f s t a t e s . T h is i s v e ry often
the case in modern a lg e b ra ic automata th eo ry (see EILEN3ERG (19 7^ ,
Chapters I and X V I] ) .
We should not le a v e the im p re ssio n th a t categ o ry th eo ry i s being
used h e re . A l l th a t t h is in v o lv e d i s a search fo r a s u it a b le d e f in i
t io n o f the morphisms in a categ o ry whose o b je c ts are g iv e n in an
obvious way. U n lik e in c l a s s ic a l a lg e b ra ic s tru c tu re s (groups, r in g s ,
v e c to r sp aces, . . . ) the d e f in it io n o f morphisms i s not a t a l l obvious in
the p resen t co ntext. I t i s even co n ce iva b le th a t s e v e r a l d if f e r e n t
morphisms can be d efined (see a ls o below ); the p re se n t d e f in it io n i s
adopted because o f the theorem we w ish to prove.
UNIQUENESS THEOREM FOR NATURAL REALIZATIONS ( I 962) . Any two
c a n o n ic a l o b je c ts in R e a l . are iso m o rp h ic.
For the p ro o f, a number o f sim ple lemmas should be noted. They
fo llo w im m ediately from the d e f in it io n s .
LEMMA 0 . For any p a ir o f o b je c ts £ and £ in R e a l^ there
e x is t morphisms cp: £ -> Z and \|r: £ -» £ ; in f a c t , cp (the in v e rse
r e la t io n ) i s a morphism l ik e 1);.
LEMMA 1 . I f £ i s reacha b le ( i . e . , ш € (!)) then
any morphism cp: Z -> £ i s a unique complete r e la t io n ( i . e . , everywhere
u n iq u e ly d efined by (3 - 1 ) ) .
244 K ALM AN
LEMMA 2 . I f Z i s observable ( i . e . , every se t = x )
then any morphism cp: Z -» Z i s a p a r t ia l map.
From these o b se rv a tio n s, we get the key r e s u lt :
MAIN LEMMA. I f Z i s reachab le and Z i s observable then
the morphism cp: Z -> Z i s a unique map.
P ro of o f the Theorem. Suppose th a t Z* and Z** are both
c a n o n ic a l o b je c ts . Consider the morphism
z * — 2 b > - z * * - A > z * .
By the Main Lemma, cp, are unique maps. A gain by the Main Lemma,
th ere are uniaue morphisms 1„ „ : Z* -» Z* and l-.* .. : Z** -> Z** (theyL * " * '
must be the id e n t it y morphisms s in c e the id e n t it y map X -» X i s always a
morphism Z -* Z ) . Consequently фср = l^,* and cplr = 1(. w hich i s
ju s t the d e f in it io n o f an isomorphism in a ca te g o ry. □
So f a r the uniqueness theorem was sta te d fo r v e ry g e n e ra l c la s s e s
o f system s. By r e s t r ic t in g to l in e a r system s, we o b ta in a somewhat
stro ng er
COROLLARY. C o nsid er the categ ory L in r e a l , o f l in e a r r e a l iz a
t io n s o f a f ix e d l in e a r b la c k box S, whose morphisms are l in e a r
r e la t io n s s a t is f y in g ( 3 -3 ) . Then any two c a n o n ic a l systems Z* and
Z*~* are l in e a r l y iso m o rp hic, th a t i s . t h e ir s ta te spaces X* and X**
are isom orphic as v e cto r spaces, and t h e ir d e fin in g m a trice s are
re la te d as in ( 2 .5 ) .
What i s somewhat s u r p r is in g about t h is r e s u lt i s th a t isomorphisms
in L in re a l g tu rn out to be v e c to r-sp a c e isom orphism s, but the morphisms
o f systems are NOT ve cto r-sp a ce morphisms ( l in e a r m aps). T h is shows
th a t the d e f in it io n o f a category o f systems i s a ra th e r d e lic a t e
m a tte r. (At le a s t in L in r e a lg , the m ystery can be illu m in a t e d some
what by re d e f in in g morphisms as fo llo w s : cp: Z -> S i s any l in e a r map
T : X , -» X # (not n e c e s s a r ily in v e r t ib le ) such th a t (2 .5 ) hoJ.ds.
Note th a t the e x iste n c e o f any l in e a r map T im p lie s th a t S , = Sg:
НГС = HFTG = HTFG = HFG. In t h is categ o ry Lemma 0 i s not tru e and in
g e n e ra l morphisms do not e x is t ; however, the Main Lemma rem ains
c o rre c t and the p ro o f o f the uniqueness theorem i s e x a c t ly as b e fo re .)
IA E A -S M R -1 7 /5 3 245
The p ro o f o f the e x is te n c e o f a t le a s t one n a t u r a l r e a l iz a t io n ££* in
the ca te g o ry L in r e a l^ w i l l be g iv e n below. By the uniqueness theorem,
a l l these r e a l iz a t io n s are the same w it h in system e q u iv a le n c e . I t i s
in t e r e s t in g to examine the " lo c a t io n " o f t h is s in g le system [£*]
w it h in the ca te g o ry. By the Main Lemma, th ere i s a unique morphism
from any re a c h a b le [£ ] to [ £ * ] ; so [E*] i s the te rm in a l o b je c t
fo r the subcategory o f reacha b le r e a l iz a t io n s . A ls o by the Main Lemma,
th e re i s a unique morphism from [£*] to any observable [ £ ] ; so
[£*] i s the i n i t i a l o b je c t fo r the subcategory o f observable r e a l iz a
t io n s . Between systems w hich are n e ith e r re ach a b le nor o b servable th ere
are many morphisms, but t h is circum stance does not seem to r e v e a l any
in t e r e s t in g sy ste m -th e o re tic f a c t s .
The preced in g d is c u s s io n o f the uniqueness theorem re q u ire d the
assum ption o f co nstant system s. T h is i s not a t a l l e s s e n t ia l. In the
l in e a r nonconstant case, a p ro o f o f the theorem i s g iv e n in KALMAN,
FALB, and ARBIB [1969» Appendix IO C ]; t h is p ro o f was the o r ig in a l
ro u te by w hich the theorem was d isco ve re d .
1+. CONSTRUCTION OF A NATURAL REALIZATION
Up to now our d is c u s s io n i s d e f ic ie n t ; we have not y e t s e t t le d the
c r u c ia l p o in t : What p ro p e rt ie s o f S guarantee the e x iste n ce o f at
le a s t one f in it e -d im e n s io n a l r e a l iz a t io n ? From t h is the e x iste n ce o f
a unique (w ith in system e q u ivale n ce ) n a t u ra l r e a l iz a t io n w i l l a ls o
fo llo w q u ite e a s i ly .
To answer these q u e stio n s ( in the l in e a r case o n ly !) , we fo rm a lly
d e fin e the beh avio r o f a l in e a r dynam ical system as the in f i n i t e Hahkel m a tr ix B(s) formed from the elements o f the e x te rn a l d e s c r ip t io n s :
№
The r e a l iz a t io n problem, in t u it iv e ly , i s the f a c t o r iz a t io n o f A
in t o th ree p ie c e s . I t i s e a s ie r , however, to d e a l w ith a tw o -fo ld
f a c t o r iz a t io n as in ( 3 . 2 ) , the two fa c to rs e x p re ssin g the in p u t and
A 1 A 2 A 3
A 2 А з \
А з \ A 5
246 K ALM AN
output p ro p e rt ie s o f a system . Such c o n s id e ra tio n s le d to the
d is c o v e ry o f the
MAIN LEMMA. I f S i s r e a l iz a b le by a f in it e -d im e n s io n a l
system E then B(S) =
P ro o f. Combine the d e f in it io n (2 .8 ) o f the r e a l iz a t io n problem
w ith the d e f in it io n ( U .l) o f the b eh avio r m a tr ix . □
CONSEQUENCE 1 . I f S i s r e a l iz e d by a c a n o n ic a l system E*
then dim E* < rank B ( S ) .
P ro o f. "E* c a n o n ic a l" means th a t ran k 0 , = rank R . = dim E* =
= n. Thus 0 and R . each co n ta in at le a s t one n o n sin g u la r n X n
su b m atrix , sa y , Ô and R. I t fo llo w s from the Main Lemma th a t B (S j
co n ta in s OR as a subm atrix and o b v io u s ly rank OR = n. □
CONSEQUENCE 2 . I f S i s r e a l iz e d by any system Z then
rank B(S) < dim E and the in e q u a lit y i s s t r i c t u n le ss Z i s c a n o n ic a l.
P ro o f. Use the Main Lemma and r e c a l l a l in e a r a lg e b ra ic fa c t :
rank B(S) < min [ra n k 0^,, ran k R }.
The ran ks o f 0 , and R , are bounded by dim Z and can a t t a in t h is
bound i f and o n ly i f Z i s observable and r e a c h a b le .. □
CONSEQUENCE 3 - For any r e a l iz a t io n Z o f S and any c a n o n ic a l
r e a l iz a t io n Z* o f S, dim Z* < dim Z and the in e q u a lit y i s s t r ic t
u n le ss Z i s a ls o c a n o n ic a l.
T h is shows th a t a c a n o n ic a l r e a l iz a t io n o f S, i f i t e x is t s , i s
n e c e s s a r ily m in im a l, th a t i s , a r e a l iz a t io n having the le a s t dim ension
in the c la s s o f a l l r e a l iz a t io n s o f S. Moreover, i f c a n o n ic a l
r e a l iz a t io n s e x is t , then a m in im al r e a l iz a t io n must be n e c e s s a r ily
c a n o n ic a l. Hence e ve ryth in g reduces to p ro ving the e x iste n c e o f a
c a n o n ic a l r e a l iz a t io n . Assuming t h is i s done, we have a ls o the
PROPOSITION. A r e a l iz a t io n o f S i s c a n o n ic a l i f and o n ly i f
i t i s m in im a l.
The e x iste n c e q u e stio n i s most e a s i ly s e t t le d by f in d in g a
r e a l iz a t io n Z_ whose dim ension i s rank B (S ) ; by Consequence 2 o f.D
IA E A -S M R -1 7 /5 3 247
the Main Lemma, Zg i s then o b v io u s ly c a n o n ic a l (and m in im a l)! T h is
r e a l iz a t io n i s co nstructed by a procedure emphasized e s p e c ia l ly by
ROUCHALEAU (c a . 19 7 0 ).
CONSTRUCTION OF A REALIZATION FROM B (S ) . The co n d it io n
"rank B(S) = n" i s e a u iv a le n t to sa y in g th a t the к -v e c to r space V_tígenerated by the ( in f in it e ) columns o f B(S) has dim ension n. So
! e t Хз := Vg.
The next q u estio n i s how to d e fin e the l in e a r map F ■ X_ X_.tí tí tíWe note f i r s t th a t the d ata S adm it a n a t u r a l s h if t operator
cr. At t-> A . Now B(S) i s d e fin ed in ju s t such a way th a t the a c t io n
o f cr on the columns e^ o f B(S) i s e s p e c ia l ly sim ple to e v a lu a te :
e^ >-> cre means "drop the f i r s t component o f the ( in f in it e ) v e cto r e / '
In view o f t h is r u le , the assignm ent e^ »-» cre fo r an a r b it r a r y subset
( e . , j £ J) o f l in e a r l y independent columns o f B(S) im p lie s the0
same assignm ent e^ >-> cre^ fo r a l l other i ^ J . Hence the map
F „ : e. t->cre. i s w e ll d e fin e d .В i i
Now d e fin e G^: U -» Xg by
" a ;
u u;
fu r t h e r , d e fin e H^: X^ i-> У by x h> Hx = f i r s t p components o f x .
I t i s now easy to v e r i f y th a t the r e a l iz a t io n co n d it io n (2 .8 )
i s met fo r a l l t = 1 , 2 , . . . . Thus E_ = (F,,, G_, H„) r e a l iz e stí tí tí tíS and dim £_ = ran k B (S ). tí
By the axiom o f ch o ic e , the dim ension o f a m in im al r e a l iz a t io n
i s alw ays w e ll d e fin e d ; i f the c la s s o f ( f in it e -d im e n s io n a l) .m in im a l
r e a l iz a t io n s i s empty, we sa y the dim ension i s ° ° . So i t i s meaning
f u l to w r it e : dim S := dim o f a m in im al r e a l iz a t io n o f S. □
Our c o n s id e ra tio n s can now be summarized as
FIR ST MAIN THEOREM (1965) . dim S = rank B (S ).
2 4 8 K ALM AN
In c o n stru c t in g the r e a l iz a t io n F , G, H were g iv e n asJd
a b s t ra c t l in e a r maps ra t h e r than as m a t r ic e s . The p re se n ta tio n o f X
as the columns space o f B(S) i s m a th e m a tica lly e le g a n t, but fo r
p r a c t ic a l purposes i t i s not app ealing to work w ith v e cto rs o f in f in i t e
le n g th . Much more concrete r e a l iz a t io n form ulas can be obtained by
co n s tru c t in g X as a space o f n -tu p le s (fo r example, by co n sid e rin g
o n ly n components in the columns o f B ( S ) ) . In f a c t , i t appears
th a t e very method which a llo w s num erica l computation o f rank B(s) can
be extended, w ith v e ry l i t t l e t h e o r e t ic a l e f f o r t , to a method fo r
o b ta in in g a r e a l iz a t io n .
We s h a l l now p re se n t some " r e a liz a t io n theorem s", th a t i s , theorems
in w hich a method fo r computing the ran k can be used, v ia a lg e b ra ic
fo rm ulas (not a lg o r it h m s ), fo r the co n stru c t io n o f a r e a l iz a t io n .
The c o n d it io n ra n k B(S) = n i s e q u iv a le n t to the e x iste n ce o f an
n x n n o n sin g u la r subm atrix Ф o f B (S ). G iven Ф, l e t Г be the
n x m subm atrix o f the f i r s t m columns o f B(S) whose rows are the
same as those o f Ф; s im i la r ly , l e t Л be the p x m subm atrix o f
the f i r s t p rows o f B(S) whose columns are the same as those o f Ф.
C a l l B ( s )ilc upper le f t -h a n d I x к b lo c k subm atrix o f the
in f i n i t e m a tr ix B ( S ) . Note th a t B ( S ) ^ depends o n ly on the data
A , . . . . A, . With these co nventio ns, we have the1 k + ¿ - l ’
SILVERMAN REALIZATION THEOREM (1966) . Suppose B(S) as w e ll
as B(S)„,_ have rank n, and l e t Ф be an n x n n o n sin g u lar- 1 - 1
subm atrix o f B (S )^ ^ . Then (ф сгф, Ф Г, л) i s a n a t u ra l r e a l iz a t io n
o f S.
P ro o f. As in the a b s tra c t c o n s tru c t io n o f the p re v io u s s e c t io n ,
d e fin e F : k n -» k n by сгф = i F (where a : A^ At + 1 ) > s in c e Ф i s
n o n s in g u la r, c l e a r l y F := Ф "'сгф.
Let Ф be a l l the in f in i t e columns o f B(S) in w hich Ф i s00
co n ta in e d . By the h yp o th e sis th a t ran k B(S) = n i t fo llo w s th a t
стФ = Ф F . S ince Л i s a subm atrix o f Ф' , c le a r ly сгЛ = AF. I t e r a -CO 00 CO
t in g cr and u s in g the s p e c ia l p a tte rn in w hich the A are enteredt t
in t o B (S ) , i t fo llo w s th a t а Л = AF . D efine H := Л.
Now d e fin e G by Г = $G, th a t i s , G := Ф 1 Г.
5. CONCRETE COMPUTATION OF NATURAL REALIZATIONS
IA E A -S M R -1 7 /5 3 249
Note th a t Г = o G. From the d e f in it io n s i t fo llo w s th a t 00 00
AG = A , . s in c e Л i s a subm atrix o f Ф whose rows are the same as1 oothose o f A as a subm atrix o f П ». T h is v e r i f ie s (2 .8 ) fo r t = 1 .
t -1 t -1In the same way, A^ = cr AG = AF G fo r t = 2 , 3 , ••• and so
a l l co n d it io n s (2 . 8) are v e r if ie d . □
Remark 1 . The co n stru c tio n ju s t g iv e n i s a g e n e r a liz a t io n o f
c l a s s ic a l form ulas u s in g determ in an ts, as g iv e n in the w ell-kncw n
textbook o f PÓLYA and SZEGÓ. I t was n o ticed by F L IE S S [19 7 2 ] th a t the
same c o n stru c t io n w i l l work whenever S ( s t i l l w ith elements in a
f ie l d k) has as the index se t o f the sequence (A^) not Z
but m erely a monoid M; then the (ц, v ) -t h element o f B(S) i s
d e fin e d s im p ly as А о у - In t h is way many known r e s u lt s o f automata
th eo ry can be deduced from u n iv e r s a l p ro p e rtie s o f H ankel m a tr ic e s ;
fo r example, the w ell-know n SCHUTZEHBERGER [1961] theorem may be viewed
as a v a r ia n t on the r e a l iz a t io n methods d is c u s s e s in t h is s e c t io n .
Remark 2 . Note th a t the co n stru c t io n o f the SILVERMAN r e a l i
z a t io n i s e s s e n t ia l ly a l o c a l procedure w hich i s c a r r ie d out over an
a lg e b ra ic neighborhood det Ф 4 0 (o f some u n d e rly in g a lg e b ra ic
v a r ie t y ; see S e ctio n 6) . The c o n stru c t io n y ie ld s a c e r t a in "ca n o n ica l"
form whose elements may be viewed as the lo c a l co o rd in a te s d e s c r ib in g
the system over the neighborhood ce t Ф / 0 in the space o f in p u t -
output maps S .
The data re q u ire d fo r the c o n stru c t io n o f a SILVERMAN r e a l iz a t io n
i s o f course f in i t e and i s co ntained (a t le a s t ) in the m a trix
B(S)£ k + 1 . The c o n d it io n th a t rank B(S) = rank B ( s )^k i s an
a b s tra c t h yp o th e sis which i s re q u ire d to assure th a t the r e a l iz a t io n
c o n d it io n s h old fo r i n f i n i t e l y many t . O b v io u sly i t i s d e s ira b le
to m odify the SILVERMAN theorem in such a way th a t i t i s a p p lic a b le
to f i n i t e data w ithout any a b s tra c t assum ptions. T h is co uld be done
b y exam ining more f in e ly the p ro o f o f the theorem. We p re fe r in ste a d
to c a r ry out t h is d is c u s s io n v ia the
ZEIGER REALIZATION THEOREM (1968) . Suppose B(S) as w e ll
as B (S )¿ k have rank n. Suppose a ls o th a t s ( s )¿!c = pQ> each o f
f u l l rank n. Let F be the uniaue s o lu t io n o f PFQ = crB(S)^k - Then
(F, Qx , Px ) i s a n a t u ra l r e a l iz a t io n o f S, where P^ and are
the f i r s t b lo ck s in P and Q regarded as b lo c k m a tr ic e s .
250 K ALM AN
P ro o f. S ince P has f u l l ra n k , there e x is t s a m a tr ix Lp
o f 0 's and l ' s such th a t det L pP / 0 (L p p ic k s out a se t o f
l in e a r l y independent rows from P ) . S im i la r ly , th e re i s a m a trix
Rq o f 0 ’ s and l ' s such that, det QR^ ^ 0 . Consequently, d e fin e
( 5 .1 ) F := ( L p P r ^ p a B Î S ^ R ^ Q R ç ) - 1 .
T h is shows th a t i f a m a trix F s a t is f y in g PFQ = B(S) e x is t s , thenК
i t i s n e c e s s a r ily unique and g iv e n by (5 - 1 )•
Next i t must be shown th a t F as ju s t d e fin e d i s a c t u a l ly a
s o lu t io n . Here e n ters the h yp o th e sis th a t the ra n k o f the in f in i t e
b e h a v io r m a trix B(S) i s n. More e x p l ic i t ly , the r e la t io n s
(5 .2 ) ran k B (S )i k = ran k B(S)^ k + 1 ,
(5 - 3) ran k B (S )i k = ran k B(s)¿+1 k ,
im p ly the e x iste n c e o f m a trice s Z and Z„ such th a tr I
B ( S W = B < s V r ’
B ( s ) i k = zi B ^ k -
Now check th a t
PFQ = P i L p P j ' ^ B O ) ^ ^ (Q J^)- ^ ,
= P a p P f ^ p P Q Z ^ Q R ^ Q ,
= Р« У У % ГЧ= Z ^ Q R ^ Q ,
= a B (S )i k .
(Note th a t t h is p ro o f re v o lv e s around d if f e r e n t re p re se n ta t io n s o f
the s h if t o perator a .)
I t rem ains to v e r i f y the r e a l iz a t io n co n d it io n s ( 2 .8 ) . F i r s t
d e fin e
crP := Z^P and Q := QZ^.
Then PFQ = ctB(S) . = (crP)Q = P(crQ) ; s in c e P and Q have f u l l rank X/ кFQ = crQ and PF = crP. Consequently FQ. = Qi + 1 > i = . . . , k - 1 ,
IA E A -S M R -1 7 /5 3 251
and P jF = P .j+ i, j = 1 , I - 1 . C le a r ly , A1 = ЪУ in d u c tio n ,
A . = P_Fi + «3“1 Q ; i = 1 , . . . , k, j = 1 , . . . , ¿ , and i + j < к + г . i + 0 i l —
T h is v e r i f ie s the r e a l iz a t io n co n d it io n s up to and in c lu d in g t = к + I .
To v e r i f y the c o n d it io n s fo r t > к + I re q u ire s a separate
argument. Let к* > к and I* > I . S ince rank B'(S) = rank B(S)
= n, c le a r ly a ls o ran k B (S )^ #k# = n. I t i s easy to see th a t the
re p re s e n ta t io n B(S) = PQ can be extended to the re p re se n ta t io n Z к
г к
B(SW [Q Q],
u s in g s u it a b ly chosen m a tric e s P and Q. The ex ten sio n o f the
re p re se n ta t io n has no e f f e c t on L p and RQ, hence the r e a l iz a t io n
rem ains as befo re but now the preced in g d is c u s s io n proves the r e a l iz a
t io n c o n d it io n fo r a l l 0 < t < k* + £*. S ince k* and i * are
a r b it r a r y , (2 . 8) i s proved fo r a l l t . □
Remark 3 - The ZEIGER r e a l iz a t io n i s m otivated by " f in it iz in g "
the Main Lemma in S e ctio n b. Thus B ( S ) ^ = PQ i s the analog o f
B(S) = Oj-Ryj P i s ju s t a f i n i t e p a rt o f 0 , and Q i s a f in i t e
p a rt o f R ., provided E i s taken to be the ZEIGER r e a l iz a t io n .
The p ro o f o f ZEIG ER's theorem co nta in s in i t a stro ng er r e s u lt
w hich we can s ta te as the
SECOND MAIN THEOREM ( I 968) . Let SR+i = (A , . . . , A ¿+ ¿ )
be any f i n i t e sequence o f p x m m a trice s over a f i e l d k . Let
E = (F , P^, Q^) be as in ZEIGER1 s theorem. Then t r e a l iz e s
i f and o n ly i f co n d it io n s (5 .2 ) and (5 - 3) h o ld .
COROLLARY. I f ( 5 - 2) and ( 5 - 3) h o ld , the p a r t ia l sequence
S^+ has one and o n ly one co n tin u a tio n to an in f in i t e sequence such
th a t rank B(S) = rank B ( S ) , , , and t h is c o n tin u a t io n i s g iv e n by-----------------------------------------------------A^ = HF G fo r a l l t > к + Í,, where (F , G, H) i s the ZEIGER
r e a l iz a t io n .
Note th a t the ZEIGER r e a l iz a t io n i s computed on the b a s is o f the
data in hut no more. In other words, the a b s tra c t argument
in the l a s t p a rt o f the p ro o f o f ZEIGER's theorem has no b e a rin g on
the com putation o f the r e a l iz a t io n , but i s s im p ly a means o f u sin g the
a b stra c t assum ption ran k B(S) = rank B(S)^k to assure th a t the system
252 K ALM AN
computed from f i n i t e data i s in fa c t a r e a l iz a t io n o f S s a t is f y in g
the i n f i n i t e l y many co n d it io n s ( 2 . 8 ) .
When the b la c k box i s g iver, m erely as a f i n i t e sequence S^, we
speak o f a p a r t ia l r e a l iz a t io n problem o f order N. The deeper p a rt
o f r e a l iz a t io n th e o ry i s concerned w ith the p ro p e rt ie s o f p a r t ia l
r e a l iz a t io n s . For example, the Second Main Theorem does not say th a t
th e re are no r e a l iz a t io n s o f S ' i f e it h e r (5 -2 ) or ( 5 -3 ) f a i l s to
h o ld , but m erely th a t ZEIG ER 's r e a l iz a t io n (whose com putation v ia ( 5 - 1 )
re q u ir e s o n ly rank B ( S ) ^ = n) w i l l not w ork; under these circu m
stan ces there w i l l e x is t many p a r t ia l r e a l iz a t io n s , l in e a r l y p a ra
m etrized by c e r t a in a r b it r a r y co n stan ts, w hich must be computed by a
d if f e r e n t method. An e a r ly r e s u lt in p a r t ia l r e a l iz a t io n th e o ry (see
KALMAN [1969, Theorem ( 9 - 7 ) 1 i s the fo llo w in g
THIRD MAIN THEOREM (1968) . Let SN = A , . . . , A^ be any
p a r t ia l sequence over a f ie l d k . Then there e x is t s an in te g e r
n = n(S„,) such th a t .N -------------------------
( i ) E v e ry co n tin u a tio n o f to an i n f in i t e sequence has
dim ension > n.
( i i ) There i s a co n tin u a t io n of S^ to an i n f in i t e sequence
o f dim ension e x a c t ly n.
( i i i ) The fa m ily o f in f in i t e c o n tin u a tio n s o f S o f dim ension
n i s a l in e a r fa m ily depending on param eters ;
th a t i s , t h is fa m ily i s a b s t r a c t ly isom orphic w ith k^.
Note th a t t h is theorem does not in v o lv e any c o n d it io n s on S^!
Even more s u r p r is in g i s the fo llo w in g r e s u lt :
FOURTH MAIN THEOREM (19 7 2 ) . For any in f i n i t e sequence
A^, A2 , . . . the co n d it io n s (5 -2 ) and (5 .3) occur i n f i n i t e l y o fte n .
The l a s t two theorems and t h e ir appurtenant m athem atical m achinery
a llo w us to have a complete overview o f the p a r t ia l r e a l iz a t io n problem,
th a t i s , to c o n c re t iz e the a b s t ra c t f in it e n e s s c o n d it io n s used in the
SILVERMAN and ZEIGER r e a l iz a t io n theorems. The r a t h e r su b tle d e t a ils
o f t h is are w e ll beyond these le c t u r e s and we r e f e r the read e r to
KALMAN [to app ear].
IA E A -S M R -1 7 /5 3 253
(a) "Homotopy Problem" (BROCKETT). I t i s w e ll known th a t the se t
o f r e a l symmetric n o n sin g u la r n x n m a trice s has e x a c t ly as many
connected components as s ig n a tu re c la s s e s , th a t i s , on each connected
component the s ig n a tu re i s co nstant and unconnected components have
d if f e r e n t s ig n a t u re . Thus l in e a r , s c a la r systems (m = p = 1 ) can be
c l a s s i f ie d a cco rding to the s ig n a tu re o f t h e ir b e h a v io r m a tr ic e s . The
s ig n if ic a n c e o f t h is f a c t i s not y e t c le a r , altho ugh i t i s known th a t
the s ig n a tu re c la s s (n, 0) , corresponding to Bnn( s ) p o s it iv e
d e f in it e , c o n s is t s o f the s o -c a lle d "RC p o s it iv e r e a l" systems w hich
are fundam ental in c l a s s ic a l network s y n th e s is . I t i s not even known
what the topology o f the components i s , as a fu n c t io n o f the s ig n a tu re
c la s s . T h is i s an area where a lg e b ra ic topology i s in co ntact w ith
a lg e b ra ic l in e a r system th e o ry . I t would be in t e r e s t in g to explo re
these problems fu rth e r and e s p e c ia l ly to remove the r e s t r ic t io n to " s c a la r " system s. V e ry l i t t l e has been done in t h is area s in c e the
famous r e s u lt o f FROBENIUS (19th century) th a t the s ig n a tu re o f a
square m a tr ix may be computed from the s ig n s o f the p r in c ip a l m inors.
(b) R e la t io n s w ith a lg e b ra ic geometry. The fa m ily o f a l l systems
£ S : dim S = f ix e d ) i s undoubtedly some s o rt o f a lg e b ra ic v a r ie t y , but
t h is v a r ie t y has not been e x p l i c i t l y computed, l e t alone stu d ie d . I t
i s , o f co urse, an open subset o f the a f f in e a lg e b ra ic se t (S :
( S : ran k S < n + 1) g iv e n by the v a n is h in g o f a l l m inors o f s iz e
(n + 1 ) x (n + l ) o f B(S) and the nonvanishing o f a t le a s t one minor
o f s iz e n x n. See F i r s t Main Theorem. The much s im p le r case o f
p a ir s (F , G) = re a c h a b le , n = f ix e d , modulo the a c t io n ( 2 . 5) tu rns
out to be a v e ry n ic e q u a s i-p r o je c t iv e v a r ie t y , w hich i s everywhere
l o c a l l y isom orphic to see KALMAN [197U ]. T h is v a r ie t y
may be viewed as an in t e r e s t in g g e n e ra liz a t io n o f the c la s s ic a l
GRASSMANN v a r ie t ie s .
(c) R e la t io n s w ith a lg e b ra ic to p o lo gy. I t i s tem pting to guess
(c o n je ctu re may be too stro ng a term here) th a t the a lg e b ra ic methods
o f homology th e o ry , or perhaps even the concept o f homology i t s e l f ,
has something to do w ith the p a r t ia l r e a l iz a t io n problem . In f a c t , the
stu d y o f the homology o f the GRASSMANN v a r ie t y (the s o -c a lle d SCHUBERT
v a r ie t ie s ) i s q u ite s im i la r to the techniques used in p a r t ia l r e a l i
z a t io n th e o ry .
6. SOME OPEN PROBLEMS
254 K ALM AN
(d) R e a liz a t io n o f l in e a r systems over a r ig g . The f i r s t steps
here are due to ROUCHALEAU, WYMAN, and KALMAN [19 7 1] and ROUCHALEAU
and WYMAN [197М ; see a ls o EILENBERG [197*b Chapter X V I] . There i s no
q uestio n th a t the study o f these problems re q u ire s a much more
s o p h is t ic a te d m athem atical m achinery than th a t used in these le c t u r e s .
So t h is w i l l be a k in d o f p ro ving ground where the a r t i l l e r y o f modern
a lg e b ra can prove i t s re le va n ce to system th e o ry .
(e) R e a liz a t io n o f m u lt i l in e a r system s. Such a system i s one
whose in p u t/o u tp u t map i s m u lt i l in e a r ; t h is i s o f co urse a ra th e r
s p e c ia l c la s s o f n o nlin e a r system s. The f i r s t steps toward a r e a l i
z a tio n th e o ry were taken in KAIMAN [1968] ; the problem i s one o f
s u r p r is in g d i f f i c u l t y . Systems o f t h is type can be r e a l iz e d by in t e r
connecting b lo ck s o f l in e a r systems v ia r e a l-t im e m u lt ip l ie r s . One o f
the co nceptual d i f f i c u l t i e s i s due to the f a c t th a t G L(kn ) i s no lo n g e r the re le v a n t group; the sta te v a r ia b le s can be transform ed
s e p a ra te ly w it h in each l in e a r b lo ck as w e ll as " t ra n s fe rre d ” from
one such b lo ck to ano ther. I t appear c e r t a in (197*0 th a t a
d e f in it iv e th eo ry o f such systems w i l l re q u ire f a i r l y advanced
methods from a lg e b ra ic geometry.
7 . FUTURE RESEARCH
I t i s w orthw hile to spend a few m inutes in r e f le c t in g over the
o r ig in s o f the r e s u lt s o f l in e a r r e a l iz a t io n th e o ry . As in a l l
a lg e b ra ic q u e stio n s, the c r u c ia l r o le i s p layed by f in it e n e s s
co n d it io n s ; t h is i s , in f a c t , the main content o f the Main
Theorems o f S e ctio n s 5 - 6 . In a ra th e r s im ila r way, the s o -c a lle d
Laplace transform s in l in e a r system th eo ry (see KAIÍ-1AN, FALB, and
ARBIB [ 1969, Chapter 10 , S e ctio n 9]) u s e fu l m a in ly because they
are u s u a l ly r a t io n a l , th a t i s , s p e c if ia b le by g iv in g a -f in it e number
o f c o e f f ic ie n t s . These two cases o f " f in it e n e s s " a re not a t a l l
a c c id e n t a l; they are c lo s e ly r e la te d to each other and th e y are r e a l ly
ju s t s p e c ia l cases o f a famous c o n je c tu re o f HILBERT (189О and l a t e r ) :
The r in g o f in v a r ia n t s o f a tra n sfo rm a tio n group i s f i n i t e l yg en erated .
HILBERT proved t h is c o n je c tu re fo r the g e n e ra l l in e a r group G L(kn) ,
w hich i s p r e c is e ly what l in e a r system th e o ry i s concerned w ith . The
in t u it iv e content o f H ILBERT's c o n je c tu re in the g e n e ra l case i s
IAEA -S M R -1 7 / 53 255
s im p ly th a t the e x te rn a l d e s c r ip t io n (= r in g o f in v a r ia n t s ) o f any
dynam ical system , th a t i s , a system w ith any sta te -tra n s fo rm a tio n
group, i s f i n i t e l y generated. The e a r ly success o f the Laplace
transfo rm method and the la t e r su ccess o f a lg e b ra ic l in e a r system
th e o ry stem from the f a c t th a t HILBERT guaranteed in advance the
c r u c ia l f in it e n e s s p ro p e rty whenever the group was GL(kn ) .
We may expect a s im ila r g e n e ra l approach to work a ls o in the
n o n lin e a r c a se ; there are a lre a d y in d ic a t io n s in th a t d ir e c t io n in the
th eo ry o f m u lt i l in e a r systems (see KALMAN [1968] ) . But i t i s w e ll to
remember th a t HILBERT'S c o n je c tu re i s not tru e in g e n e ra l (see
DIEUDOHNÉ and CARRELL [19 7 1) P - k l ] ) and th a t the re se a rc h on n o n lin e a r dynam ical systems from t h is p o in t o f view has ju s t b a r e ly begun.
N onetheless, i t i s not an exag geration to say th a t HILBERT'S (as
y e t unproven) theorem on the f in it e n e s s o f in v a r ia n t s should be
re co g nize d , alm ost one hundred ye a rs la t e r , as one o f the c o rn e r
stones o f system th eo ry.
8 . REFERENCES
J . A . DIEUDONNÉ and J . B. CARRELL
[1971] In v a r ia n t th e o ry , o ld and new, Academic P re s s .
S. EILENBERG
[197!+] Automata, lan g u ag es, and m achines, V o l. A, Academic P re ss .
M. FLIESS
[1972] "Sur c e rta in e s f a m ille s de s é r ie s fo rm e lle s " , thèse d o cto rat d ' é t a t , P a r is V I I .
R . E . KALMAN
[ i 960] "On the g e n e ra l th eo ry o f c o n tro l system s", Pro c. 1 s tIFAC Congress on Autom atic C o n tro l, Moscow, B utterw orths, London, V o l. 1 , U81-14 9 2.
[1968] "Raspoznavanie obrazov p o l i l in e in y m i m ashinam i", Pro c.IFAC Conf. on A dap tive Systems, Erevan, USSR, pp. 7 -3 0 , Iz d a t e l's t v o "Nauka", Moskva, 1 9 7 1 . E n g lis h t r a n s la t io n (w ith a nn o tatio ns) to appear in C o n tro l and C yb e rn e tics (Warsaw), 19 76 .
[1969] "Lectures on c o n t r o l l a b i l i t y and o b s e r v a b il it y " , Pro c.CIME Summer School, Cremonese, Roma, pp. l - l i j - 9 .
256 K ALM AN
[ I 97U] "A lg e b ra ic -g e o m e tric d e s c r ip t io n o f the c la s s o f l in e a rsystems o f co nstant dim ension", Pro c. 8th Annual P rin ce to n Conf. on In fo rm a tio n S cien ces and Systems, pp. 1 8 9 -1 9 1 -
[to appear] R e a liz a t io n th e o ry : a lg e b ra ic methods and problem s.
R. E . KAIMAN, P. L . FALB, and M. A. ARBIB
[1969] To p ics in m athem atical system th e o ry , M cG ra w -H ill.
T . MATSUO and R. E . KAIMAN
[1976] "N a tu ra l r e a l iz a t io n s of in f in it e -d im e n s io n a l l in e a r systems" (to a pp ear).
Y . ROUCHAIEAU and B. F . WYMAN
[19 7^ ] " L in e a r dynam cial systems over in t e g r a l domains", J . Comp. System S c i . , £ : 1 2 9 -1 !t2 .
Y . ROUCHALEAU, B. F . WYMAN, and R. E . KALMAN
[1971] "A lg e b ra ic stru c tu re o f l in e a r dynam ical system s. I I I .R e a liz a t io n th eo ry over a commutative r in g " , Pro c. Nat. Acad. S c i . (USA), б £: 31м Л -3^0б.
M. P. SCHUTZENBERGER
[1961] "On the d e f in it io n o f a f a m ily o f automata", In fo rm atio n and C o n tro l, 2 ^ 5 -2 7 0 .
IA E A -S M R -1 7 /8 1
BASIC EQUATION OF INPUT-OUTPUT MODELS AND SOME RELATED TOPICS
M. RIBARIC Institute Jogef Stefan,University of Ljubljana,Ljubljana, Yugoslavia
Abstract
BASIC EQUATION OF INPUT-OUTPUT MODELS AND SOME RELATED TOPICS.For a given physical system composed of N sub-systems, one m ight like to explain its properties and
behaviour in terms of the properties of its constituent parts. W hen the properties of sub-systems are given in the
form of input-output relations, it turns out that the system of equations whose solutions describe the behaviour
of the system can always be put into a standardized form which w ill be called the Basic Equation. Some
applications w il l be g iven.
N1. Introduction. Suppose we have a physical system, say U T , composed of N parts
k=1 K N(subsystems), say T 's,and we would like to explain the properties and behaviour of U T
K 1 K in terms of the properties of its constituent parts T , к = 1,2,..., N. In many different casesthe properties of subsystems TK are given in the form of input—output relations (i.e. wheneverparts T are considered as black—boxes), and then it turns out that the system of equa-
K Ntions whose solutions describe the behaviour of the system U T can always be put into a
K=1 Kcertain kind of standarized form we call the Basic Equation. In what follows we will show how. As an application of the Basic Equation we will show how some qualitative properties of the whole system depend on the corresponding properties of its constituentparts. Further, we will show how in the case of causal parts we can describe the system NU TK by time—evolution operators forming a semi—group.
к = 1
2. Input—output,black—box description of parts TK. We assume that parts TKare characterizedby input—output type descriptions as follows: For any part T* there is given an additive, zero- element containing set, say l¡níí«of all permissible inputs, say i. ; so that
' ' ¡ п К ‘' Г п К е , т к i m P l i e S ]\ п к +УШ е ] Ш - K = 1 ' 2 ............ N <2 Л )
Now, to any permissible input ¡in e l¡nK there corresponds a unique element, say ioutK = ‘outK inK belonging to a set, say loutK of all possible outputs of T . This assumption is mathematically equivalent to the existence of an operator, say AK (in general non-linear), and an element, say qKe*outK.such that for any pair of the permissible input ijnK and the corresponding output ¡outK the following relation is true
'o u tK - Л с^ '¡ПК + ^K №-2)
with
\ l ¡ ¡ n „ = o ! = o (2.3)
257
258 RIBA RIC
so that qK is the output of TK corresponding to the zero input ijn(t = 0. We may call operator AK the response, scattering or albedo operator of TK; and together with the set I. of all possible inputs and the zero—response qKt it contains all the pertinent information about the part (subsystem) TK we are interested in. So to say, TK '-e-' is mathematically the set of all possible corresponding ordered pairs (i¡nK, iout(»inK)); cf- Rïbariô 1973, Ch. XXII. for a relationship between an input—output description of TK and the corresponding field theory, see also Scott 1974.
Considering identical parts TK (i.e. such as have the same linK's,AK's and q 's) they may still differ in their inputs which determine the physical situation each part TK is in. Considering part TK let us describe the particular physical situation TK is actually in by the pair
(¡¡n K> w 1 formed by the corresponding input of T, and the associated output of T . Analogously, let us describe the particular physical situation of N parts TK by a 2N dimensional vector, say
*u *'in1> *out1 * *out/i' "■ ' 'inN* 'outN (2.4)
which is an element of the additive 1, zero—element containing set N
I = 2 $ (I. ®l t ) (2.5)u K¡¡_ in К OUtK
i = P i i =P i k = 1 2 N (2.6)‘¡ПК in К II / OUtK OUtK 'u f * 1 ' ... v '
so that we have the following identity
Viuelu (2.7)
indicating that F¡ní('s and Poutf£'s are in essence of the nature of projections. Further, we defineP .
u i n
and
so that
and
(2.8)
Kin uout'u — ^Kout uout’u — PKOUt ¡u, viuelu anc* K~ b2,...,N (2.9)
P u i n ' u + P u o u . ' u = i u v i u e l u ( 2 . 1 0 )
P2. £= P . p2 = p P P =P Р = П /О 111. u r n u i n ) u o u t u o u t ) u i n u o u t u o u t u i n \¿.\\)
3. Constituent assumptions (definition) of the system ÜT . Now let us specify how we N k=1K Neffect a system U T from a collection of N parts T i.e. how the system U T is Com
ic— 1 K K=1posed from parts T . In our case it is the interaction between parts TK which effects and
(2. 1 ) Addition of any two elements of I is defined component—wise.
IA E A -S M R -1 7 /8 1 259
distinguishes the system U T from a mere collection of N parts, and we make the fol-k=1 K
lowing two assumptions about this interaction.(i) We assume that the input i. to any part TK is determined uniquely and additively byoutputs i of all parts TK as follows: There are N2 connection operators, say CKK* mapp- ing outputs ioutK, o f T , 's into inputs o f T (i.e. CKK,loutK,C I ¡rJ s0 that
V K - 1 . 2 ......N (3.1)
and
С к к ' ^ ‘o u t / i' 1 + 'out* * 2 ^ ' « W * W l * + C KK'^outK'2^
V ioutK'1 *'outK'2 6 i ..... N (3 .2)
N
About this assumption we note:(a) We allow also for the case when the output of TK contributes to the input of TK ;(b) The assumption in (3.1) that outputs of different TK/s contribute additively to the input of T does not restrict the applicability of the considered^model, since in a case where it isnot so, we can always enlarge the considered system U T by adding an additional subsystem such that assumption (3.1) would be true for its input and which would make outputs of other parts contribute in the required non-linear fashion to the input of TK ; e.g., when i. = CK { Puout ¡u I with CK being a non-additive operator, then we may introduce an additional part, say TKC,by defining its input as ¡¡n(tc = puout¡u* its output as
■out.cc - [С к* W * " С к * ° П + C í 0 í ' and then assumin9 that ‘¡n« = ‘outKC*
(c) For the same reasons also assumption (3.2) does not impose any restriction on the applicability of the considered model.
(d) It is possible of course that we have defined inputs and outputs of parts TK so that contrary to (3.1) some combination of outputs ¡outK and inputs ijnK additively determines the inputs of individual parts. It seems, however, that in such a case quantities designated as inputs and outputs should better be called otherwise, since it is conceptually desirable that the output of one part should be the input to its adjacent neighbour and vice versa. Often one chooses such a description of parts TK and of their interactions that either the output of one part TK is directly the input to one or more other parts T ?or that this is true for redefined inputs and outputs 5-(ioutK + ¡inKJ and ^'out« “ ‘¡тЛ respectively. Within the theory of electrical networks one designates the first approach as the scattering formalism and the second one as impedance or admittance formalism, see Youla, Castriota and Carlin 1959; Newcomb 1966; Kuh and Rohrer 1967; Zemanian 1968. Note that in the first case any connection operator Ckk, is in essence either a projection operator or it equals zero when the output of TK> does not contribute to the input of T . Shekel 1974 considers computation of connection operators Скк> (collected in the form of a junction matrix) associated with scattering formalism in the case when interconnections between parts T of the
N Kcomposite system и TK are originally given in terms of Kirchoff's laws appropriate toadmittance formalism.
N(ii) We assume that the outputs ioutK and inputs ¡¡nK of parts TK of the system U TKare related by the same relations (2 .2 ) as they have been before composing T's into N. *system U T .
K = 1 K
260 RIBARIC
NSo we assume that by composing parts TK into U T we do not change their reflectionк=1 к Nproperties A s and q s; each individual composed system U T that can be
к к к=1 x
effected by the same parts T differs only in corresponding connection operators С * butN KKnot in associated A„ s and q s. In effect we assume that parts T's of U T
k=1 Kinteract solely by means of ioutK's, which of course does notrestrict the applicability of the model since the definition of what is considered under ijnK's and ioutK's is not restrictive. Following Bellman and Kalaba 1959 we may call this assump* tion a localization principle, cf. also Ribarifi 1973, § V.b.2.
. Basic Equation. The question arises whether we can put the preceding constitutive assumptionsN(3.1), (3.2) and (2.2) of the system U T into a compact form of an equivalent single
K=1 N equation. The answer is positive and the result we call the Basic Equation of U TK . We proceed as follows. First we note that 2N constitutive assumptions (3.1) and (2.2) are equivalent to relations
A W + Ч ,. ■•-» I4 - 1»
¡ou«« = Pou,< ¡u and W = P¡n* ¡u - K = 1 ■ 2......N <4 2 >
since using relations (4.2) we can retreive from (4.1) the assumptions (3.1) and (2.2), whereas(4.1) and (4.2) are implied by (3.1), (2.2), (2.4) and (2.6). Now using relations (4.2) weNeliminate iout(/ s and ijnK's from (4.1), obtaining the Basic Equation of U T :
k=1 K
¡U = <C u + V 1 ^ 1 + % <4 -3>
where operators Cu and Au are defined as follows:
NZ
k'=1(4.4)
and
P¡ n / » ¡U5D ' V(í = 1 ’ 2 ....N a n d V i uelu (4.5)
q = (...; 0, q ;...) (4.6)
This equation together with relations (4.2) is equivalent to the 2N constituent assumptions(3.1) and (2.2). Note that
Cuiol = 0 , AUI о! = 0 (4.7)
andCu ■ Ри Л = CuPuout , AU = PuoutAu = AuPu|n (4.8)
by (4.4), (4.5), (2.6), (2.7), (2.8) and (2.9).
Using operators Pu¡n and Puout and taking account of (4.8) and (2.11), we can reduce the Basic Equation (4.3) to the following equivalent form:
(4.9)
«W u » Cu(Puout'u) (4.10)
IA E A -S M R -1 7 /8 1 2 6 1
(P i ) = A Í C (P i )i + q (4.11)' uout V u ( u ' uout u n 4 u
indicating that on determining outputs (Puout>u) of all parts TK we can obtain associated inputs (Pujniu) of all TK's by a mere substitution. When (4.3) has a unique solution for any qu then the same is true for (4.11)/ and vice versa, so that
[I - (Сц + AU)] - 1 = <I+CU)[I - A uCur ‘ (4.12)
For some additional modifications of the Basic Equation see Ribarié 1973, Sec. XVII.a.
and
5. An example. The preceding derivation of the Basic Equation (4.3)fbeing of very great genera- lityr suggests that it should be possible to put any nonlinear equation, say
Ф(£.а) = 0 (5.1)
£ being the unknown variable and a a parameter, into a form such as (4.3)? say
x - Ax = y (5.2)
And indeed, on defining
x = g , y = ( m ° > ) a n d A x = ( * № . » » - 0 ( 0 . 0 ) ^ ,5 .3 ,
we can verify by inspection that equations (5.1) and (5.2) are equivalent, showing that any nonlinear equation can be put into a form such as (5.2).
6 . Input and output of a system Ü T . The question presents itself as to how are we toK = 1 K
define within the given conceptual framework the exterior of a system and the associatednotions of input and output of a system U T . Conceptually we can consider any part,
N•1 1 Ksay AM as an exterior of the system U T of the remaining N — 1 parts T ,whenever TwI’* К К J It
is such that
V W J - O Viuelu <6-1)
since in such a case the output
‘outN = (6 .2 )
of TN as well as the corresponding contributions C^q^ to inputs of other parts TK?к s 1 » 2.N - 1, do not depend on the properties A 's, q 's, к = 1, 2,..., N —1 of Darts of theM —1 к к • rsubsystem U TK . So to speak the exterior is such a part whose behaviour (i.e. whose output) isnot influenced by the properties of other parts, and does not interact with them. So it makes
N—1sense to define qM as the input i. of the subsystem м T , i.e.14 mu к
262 RIBARlC
N - 1and input ¡¡nN to TN as the output of U TK, i.e.
'ou,u s i - ; s k n ¿ c KK' W u- 0- - » <6-4»
N- 1In this sense we can consider ¡outu as the response of U TK to its input ¡¡nu .
N N7. Properties of the system U T . Having a system U T how do we study its properties?
k ~ 1 K N k= 1What we usually do with a system U T of N parts T is to consider a set of all
N k = 1 K Ksuch systems, say U T . , ¡ *1 ,2 .... whose parts have the same response operators as parts
N к- l K)T of U T , i.e.K K = 1 K
A . = A , j = 1, 2 ........ к = 1, 2, ... N (7.1)К J K f » * i •
but whose zero-responses q . may be different from q 'sofT 's. To put it mathematically,we k j n к кstudy the properties of a system U TK by considering solutions iu of the Basic Equation (4.3) corresponding to some given set, say lq of zero-responses of TK's, and then study ‘u = iu<qu> as a function of sources quel In particular, I may consist only of all possiblezero responses qN of the exterior TN, i.e. of all possible inputs ijnu to the systemN- 1U T ; in this case we study the dependence of solutions i of the Basic Equation k=1 K
i = (A + С ) Í = q. * + i « q. . = q — j (7.2)u u u ' и 4 m t jnu 4 »nt Hu ‘ inu
on the input iinu of the subsystem V T(<; in particular, we may study dependence of ou,put 'outu of TK on its inPu* ¡¡nu-
So immediately there are three basic problems we have to solve:(i) Which of the sources 4uelp are physically possible, i.e. for which quelq does theBasic Equation (4.3) have a solution ¡uelu ? Were it the case that for no q el would the Basic
N u QEquation have a solution, then the system TK as defined is not physically possible.
N N(ii) Is the description iu of the physical situation of U T (i.e. the properties of U T )K“ 1 K K- 1 Kuniquely determined by the given reflection properties A 's and qK's of parts T's, i.e.
is the solution ioelu of the Basic Equation (4.3) unique?(iii) Given a certain source que1u find the corresponding i
In paragraphs 9 and 13 we will consider two cases where any Цие\ц is physically possibleand where the corresponding reflection properties of parts JK uniquely determine the physical situation i ; for some additional results cf. Desoer and Wu 1970; Flanders 1971, § V; Sandberg 1971; Abdullah and Tokad 1972; Ribariâ 1973; Secs. V.c. and VI.a; Singh and Liu 1973; Zemanian 1974, 1974.
When lu is an infinite-dimensional linear space, then the Basic Equation is essentially equivalent to an infinite system of coupled equations, and one may ask whether it is possible to approximate its solution i by a solution of seme finite—dimensional system.
¡¡nu = ' <6-31
N-1(6 . 4 We could as well define input to U T as L =(...;(1 - 5 M)C MqM,0;...).
^ ^ К inU К IN К IN IN
IAEA -S M R -1 7 /8 1 2 6 3
i.e., is there a sequence of arbitrarily accurate, approximate finite—dimensional models (Basic NEquations) of our system U T , e.g. see Flanders 1971, 9 VI; Ribaric 1973, Ch.XVI; and
K = 1 KZemanian 1974.
N NSince to study system U T means to study i of U T as a function of q e I , in the
к= 1 K u к=1 K u 4^time-invariant case we may conduct resonance experiments by taking a harmonic time—dependent input i . el , study the dependence of i as a function of the harmonic frequencyIMU q 7 N — 1°f ijnu and consider how the resonances of U TK depend on the properties of its parts TK#Or more generally, we may study how the analytic properties of the Laplace transform of the
Nunit pulse response of U depend on the analytic properties of the Laplace transformsof the unit pulse response of T 's, cf. e.g. Desoer and Lam 1972; Ribaric 1973, Ch.VI. andSec. VIII.d. Furthermore, we may be looking for subsets, say linyC lu,such that
q u e 'in v im P ,ieS ¡u e l m v -
i.e., we are looking for a subset |jnv С lu such that when quelu has the property ofbelonging to I. , then the corresponding i has the same property. When parts T of N u N KU T have the property that ¡¡nKcP¡nK l¡nu implies ¡outKeP I and U T is suchк- 1 N «=1
that queljnv implies ¡uel¡nv, then we can say that U TK inherits property ljny fromits constituent parts T , cf. paragraphs 8 and 11 .We may also study the stability (i.e. structural stability as well as asymptotic behaviour
Nof I (t) as t °°) of the composite system U T in terms of properties of its con-U к = 1 K
stituent parts, and in particular inheritance of some specific kind of stability from partsT by Ü T , cf. e.g. Desoer and Wu 1970; Ribaric 1973 § V.c.28, Chs.VI. and VIII;
K k = 1 K
Cook 1974; Grujic 1974; Grujió and Siljak 1974; Michel 1974.
When the geometrical structure of the interconnections between parts TK of compoundNU T which is described by connection operators Cut/. gives some information about
k=1 K Nproperties of U T one studies it using topological methods, cf. e.g. Seshu and Reed 1961; Harary,Norman and Cartwright 1965; Sellers 1970.
NWhen studying (computing) properties of a compound system U T consisting
к = 1 Kof many parts, i.e. when N is large, then we may either study (compute) properties of NU T directly in terms of properties of its parts T * or we may take a hierarchical ap-K~ 1 K K
proach by grouping N parts TK into, say, Ng subsystems say
T k a i u i T « - *и Л = ы к п ы к . = 0 i f k * k '\<ik k=1 k = 1
i k\k being the set of indices к of parts TK of subsystem Tk? study (compute) first properties of each individual subsystem 17 in terms of its parts T « ке\к\. , and then expressN K kj кэ(compute) properties of U Tk In terms of properties of its Ns subsystems Tk*?e.g., it is customary in analyzing general nonlinear RCL networks to analyze them in terms of three one- element—kind of (sub—)networks.Many times it turns out that only the hierarchical approach is possible (especially so when approximations have to be made for computational reasons) and its success depends on the mannerin which subsystems T. have been chosen, i.e. how we have torn apart and interconnectedN .,again the original system U T . For some relevant information see e.g. Kron 1953, Mesarovic,
k * 1 K
264 RIBA RIÓ
Macko and Takahara 1970; Ribarió 1973; § V.c.20., §Vll.c.7, See. IX.d; §XVII.c.3; Singh and Liu 1973; Nicholson 1974; Rosenbrock and Pugh 1974.
Suppose that Basic Equation (7.2) has a unique solution for any PoutN ¡¡nu e*outN
N-1 ,nu k=1U T . In analogy with (2.2) and (2.3) this correspondence determines uniquely both the
k=1 n-1 N-1response operator, say A( U Tw) and zero—input response, say q( U T^hof the system N-1U T as followsK = 1 K
K= 1 K=1
Wu = AiV-rj I i inu! + 4 < V T.) . A ( V t )|0|=0 (7.3)
The basic task of any system engineering is the problem of synthesis and decomposition which we may pose as follows:
N-1 N-1i) Given are the response operator A( U T ) and the zero-input response q( U T )N-1 k=1 к k=1 *of U T but the number N—1 of its parts T may not be specified.K = 1 K Kii) Given are M parts T ? к = 1,2,..., M, reflection properties of each TK depending onparameters, say a and к , so A„-A (a ) and q„ = q„(k );к к к к к к к к
iii) Given are M2 connection operators, each of them depending on some parameter, say y KK. so that CK
iv) Chosing N—1 parts T such that the associated Basic Equation (7.2) has a unique so*K N-1lution for any i. .determine the corresponding response operator A( U T ,a, k, y) and inu ’ n-1 N-1 *=1 Kzero—input response q( U T , a, k, 7 ) of U T , where <*=(....<* ,...), к = (..., к ...)
к =1 K k=1 K * ' Kand 7 = are parameters of N- 1 chosen parts T of U T ;
K K=1
v) Determine N—1, a, к and 7 so that
A(n-1 n-1 n - i t , .7 ~ »U T , a, k. 7 ) * A( U TJ and q( U T ,<x, k; 7 ) = q( U TJK = 1 X' K = 1 « - 1
cf. e.g. Newcomb 1966; Dewilde, Belevitch and Newcomb 1971; Voula 1971; Levan 1972; Ribariè 1973, § VIII.a.2; Chua and Lam 1974.
In connection with the synthesis of compound Û T let us note the significantK = 1
fact that it is quite possible that there is an essential qualitative difference between the N - 1 N - 1reflection properties A( U T ) of composite system U T and the reflection properties 4 = 1 K k = 1 K N - 1
Av of its parts T , e.g. it is possible to build a composed system ü T able to supportК К K = 1 к
self—sustaining oscillations from parts T which are not able to do so, cf. also Ribariè 1973;Table (VIII.b.5). On the other hand certain qualitative properties of parts T are always
N - 1 Kinherited, i.e. exhibited by compound U T , e.g.; (i) when all parts T are causal so is N - к=1 K *
compound U T , cf. §9; Ribaric 1973, V.c.5:K = 1 K
(ii) when reflection properties A of all T ‘s are linear and the Basic Equation (7.2) hasK N— 1 n - 1
always a unique solution, then the reflection properties A( U T ) of U T are linearK=1 K K=1 K
too;(iii) when reflection properties A of all T 's are time—invariant (i.e. they do not change
K K N - 1 N - 1in time), then the same is true also of reflection properties A( U T ) of U Tkjcf. (8.5), (10.10) and (10.11).Furthermore, on composing a part Ti together with a lossless part T2 it turns out that certain qualitative properties of Tt are inherited by TxUTî/ cf. Rozzi and Van Heuven 1974 and the literature therein.
IA E A -S M R -1 7 /6 1 265
8. Invariances and conditional conservation laws. Let BK's be linear operators such that
В I. C l . and В I . С I к = 1,2 N (8.1)К 1ПК in К К OUtK OUtK ' ' ' '
and let all A K's and CKK''s be BK-in va ria n t, i.e., let
(8 .2 )
Then it fo llows from (4.4) and (4.5) that
B (A u + Cu) = (A u + CU)B (8.3)
where В is defined as follows:
Biu s ( -■ BKP¡nK¡u,BKPou,K¡u.- -> Viuelu (8-4>
When the Basic Equation (4.3) has a unique solution fo r any q ue lq UBIq , then the inverse
operator [I — (Au + C J ] ' 1 may be defined as [I — (Au + Си) ] " \ = ¡ц , where ¡u = ¡u (qu)
is the unique solution of the Basic Equation (4.3) w ith rhs(4.3) = qu , and we have
— <AU + Cu) ] - ‘ = [ I - ( A u +C u ) ] - ‘ В (8.5)
i.e., when iu is the solution o f the Basic Equation corresponding to rhs(4.3) = qu , then B¡u
is the solution of the Basic Equation corresponding to rhs(4.3) = BquI a fact which can be
verified by m ultip ly ing the Basic Equation by В and taking in to account (8.3). In particular,
implies(8 .6 )
by (8.5); to w it , when q u is B— invariant, then the corresponding physical situation i is
also В-in va ria n t. For an example o f В-in va ria n t reflection properties see (10.10).
Let us now consider the case when fo r any qu^Ço u tl u the Basic Equation (4.3)
has a unique solution i which is given e xp lic itly by the fo llow ing relations:
'u “ (A u + c u > m 4 ! <8.7)
where m is a given integer,
<Au + ‘V n K 1 SqU f0r n=0 <8’8)and
<A u + Cu > n 4 ! = < A u + C u>l ' A u + Cu 'n . , * q n î î+ qu fo r n - 1 . 2 .... m (8.9)
When both A u and Cu are linear operators, then the right-hand side of (8.7) turns out to
be equal to the m - th partial sum o f a Neumann series. Now suppose that A U1 Cu, an additive
subset |ц С I and q ¡n, e lu are such that
■u e IÛ im PlieS (А и + СиИ 'и , + Ч т , е |и <8 ' 10»
266 RIBARlC
then
(8.11)
by (8.7), (8.8), (8.9) and the add itiv ity o f l \ Relations (8.10) can be interpreted as a conditional
conservation law o f the fo llow ing kind: (i) if the input ¡¡nK to a part TK belongs to
P i ' , then the corresponding ou tput i„ , , ,„ o f T belongs to P „,,,„ l', and (ii) when thein К Ц UUXK К O U IK u
output ioutK, o f a part T^, belongs to PoutJ Ú , then the induced inputs CKf£,¡outff, to all
parts TK belong to P¡nKly's. And relation (8.11) tells us the sense In which the system
U T inherits such a conditional conservation law. Now, suppose we have a linear operator P suchK= 1 K
that
a result which is again a kind of conditional conservation law, w ith relations (8.12) andN
(8.13) Indicating the sense In which system U T inherits this property from its Conic— 1 K
stituent parts. For some additional results about inheritance see Ribarlô 1973, Sec.lll.c.,
§ V.c.5, § V .c.11, § V.c.12, § V.c.25, § V.c.26, § X X I.b .4 ; Zemanian 1974.
9. The case o f causality w ith delay. Suppose that »ue lu are functions o f tim e, i.e. iu = ¡u (t)
is a mapping of an in fin ite time interval, say [0 , °°) in to some additive, zero— element con
taining set, say I . Let us denote by [ t i . t j ) also the characteristic function o f the interval
[ t b t 2) С [0, *>), i.e.,
We say that reflection properties A K o f T^'s are causal w ith delay tq > 0 and the con
nection operators CKK. are causal if f fo r any pair i'. i¡ñK e *¡nK = anc* anY Te&« °°)
we have
For some sufficient conditions fo r causality, and additional defin itions o f causality and
the ir properties see Youla, Castriota and Carlin 1959; Sandberg 1965; Winslow and Saeks
1972; Ribarià 1974, § lll.a .10 , § lll.a .11 , § lll.a .13 , § XXI.a.2; De Santis 1974, 1974 and the
bibliography therein. In particular, when A Ki i is causal, then
(8 .12 )
Then
(8.13)
by (8.8), (8.9) and (8 .12 ); in particular
Pqu = qu implies Piu u (8.14)
t ^ [ t i , t i )
t e [ t , , t 2 )(9.1)
(9.2)
and
(9.3)
[0 , t) A Kj [ r , °») i |nKI = 0 V T > 0 and ¡¡nKe l¡n|t (9.4)
IA E A -S M R -1 7 /8 1 267
Further, when A is causal then fo r any r > 0 we have the fo llow ing relation:
= A r K¡ [ r ' “ » ¡¡n J + РГК (9.5)
where
A T J t T’ °°) 'in* ! = [X, A K Í [0 , ’ ■ ) ' ! „ « + [7, ~ ) ¡ ¡n(í| - [ r , “ ) A Kí [ 0 , r ) i inJ (9.6)
and
Якг = [т, °°) qK + [ T ,~ ) A f [ 0 , T ) ¡ ¡nK| (9.7)
Suppose we start observing part T K at the tim e instant t = r by applying d iffe ren t inputs ,
[ t , °°) ijn wtaking care that during the tim e interval [0 , t ) inputs to T^are the same, so that
we are observing all the tim e the same part, say T ([0, r ) i . ). Then A tk is the reflection
operator o f T K([0, r ) i jnK ) and qT/< its zero— response, and relations (9.6) and (9.7) tell
us how the history (the past) [0, t ) ¡ . o f TK influences its fu ture behaviour.
NWhen system TK is such that reflection operators A K o f its parts J K are
causal w ith delay r 0 > 0 and ail associated connection operators Скк> are causal, then
the solution i o f the Basic Equation (4.9), (4.10), (4.11) w ith in any fin ite time interval
[0, t0 ), to e(0, °°], i.e. [0, t 0) ¡u is given by the fo llow ing re la tion:
[ 0 . t „ ) iu = [0 , to ) < l+ C u)(A uCu) J q u l e l u (9.8)
where
<A UCU>0 = ' and <A uCu > n 4 ! = ,AuCuH<A|]Cu>n_ 1 i q u ll + qu (9.9)
n = 1 ,2 ,...,m, and m is any integer > t 0 / r 0 ; this solution is the only one, i.e. [0, to ) iu is
unique, see R ibaric 1973 [A .X II .1.7]. So we have shown tha t in the case o f causality w ith
a non— zero delay the reflection properties A K's and qK's o f T K's and the connection opera
tors CKi(,'s determine uniquely the properties o f the system T^ w ith in any fin ite tim e—
interval [0 , t 0 ). Furthermore, a result analogous to (8.13) implies tha t the operator
[0, t 0 ) (I + Cu)((A uCu)m — I) is causal w ith delay r 0 , showing the sense in which also the
system U T is causal. For some additional results concerning the connection betweenK = 1 Kcausality and the unique solvability o f the Basic Equation see Vidyasagar 1972; Ribarié 1973,
Secs.V.c. and V II I .b; De Santis 1974 and the references thereof.
10. State o f a system U T whose parts are causal. Suppose A 's and С /s are causal i.e.K— 1 K KK * •let them satisfy relations (9.2) and (9.3) w ith Tq ^ 0, and w ith I consisting o f mappings
o f [0 , °°) in to lUJ. Then m ultip ly ing the Basic Equation (4.3) by [0 , r ) and [ t, t 0), t0 > r > 0,
we can decompose i t in to tw o equivalent equations
[ ° , T ) iu - [0 ,7 ) (A u + Cu) { [ 0 , r ) i u ) = [0 ,T )q u (10.1)
and
It, to) iu - [т. to ) (A u + C J ([0 , t ) ¡u + [r , to) ¡u i = [ r , t „ ) q u (10.2)
26 8 RIBAR IÍ
which have to be solved consecutively. When the Basic Equation (4.3) has a unique solution
fo r any qu e lq , then the inverse operators [0 , to ) [I - [0 , t 0)(A u + С u)]~ ‘ [0, t 0) ,
[0, r ) [ l - [0, r ) ( A u + Cu) ] _1 [0, t ) and [ t , to) [I - [ t, t0 ) (Au + Cu) ] ' ‘ exist V t» > 0
and are related as fo llow s:
[0. to ) [ I — [0. t 0 ) (A u + Cu) ] - X [0, t 0 ) = [ 0 , r ) [ l - [ 0 , T ) ( A u + Cu) ] - 1 [0, r) +
[r , to) [I - [T, to) (Au + Cu) ] - ‘ j [ r , to) +[0, r ) [ l - [0 , r ) ( A u + C J ] ’ 1 [0, r ) j (10.3)
as can be verified by inspection on applying identity (10.3) to qu , and taking account o f
(10.1), (10.2) and [ 0 , t ) + [ t , t0 ) = [0 , t„ ) .N
In what fo llows we assume that the system U T consists o f part T.. which N -1 K=1 K
is exterior o f the subsystem U T (cf. s 6). Suppose we are considering this system at the t im e -K = 1 K
instant t = r ; then considering relation (10.3) together w ith an equivalent relation
[ 0 , t o ï i u (t) = [0 f T )iu (t) + [T ï t o ) [ l - [ r ft o ï ( A u + Cu ) ] - I i[T fto H q i n t + i u in ) + [0 f THu (t)J (10.4)
(c f.(7.2)), we can interpret (10.3) as follows: The firs t term of (10.3) is relating past causes
[0. r ) iujn w ith the past events [0 , r ) iu (t), whereas the second term is relating past events
[0 , t) ¡ (t) i and present and fu tu re causes [t , t0 ) ¡u ¡n w ith the present and fu tu re events
[r , t 0 ) iu (t). The second term shows that past events and fu ture causes interact so that the
future behavior [ r , to ) ¡ (t) o f the system U T is influenced by past events [0 , r ) i„ ( t ) .U «=1 K N
This result suggests tha t the state o f the considered system U T at the tim e— instantK = 1
t = t is determined by the sum o f past events [0, r ) i u and fu tu re causes [ t, t0 ) ¡uin . And
indeed, on defining fo r any re ( - ° ° , ° ° ) and ¡це lu the sh ift-opera to r
, > _ f 0 V t i [— г , t 0 — t ) , te ( - <*>,“ >)
(T- r ' „ « И - ] j u(t + T) V t e [ - T , t o - T ) (10.5)
we can verify that the quantity
stu ( T ) 5 T . r l ° ' T ) i u ( t ) + T 7 M ° > ¡u¡n< t>( 10.6 )
has the essential property o f a state o f a system that its value at, say, t = r ' uniquely
determines its value at any later instant, say t = т " > t ' as fo llows :
stu (r"> = E [ t ' , t " ) stu (r ') (10.7)
where
E ( t \ t " ) ... s T ^ „ [t \ t " ) [ l - [ f . r " ) (A u + Cu ) ] - * I [0 , r " )T r , ... + [ t ' , T " )q ¡n t l
+ Т т „ [0 , t■') TT. ... + T .„ [г", о») T , ... (10.8)
¡s the tim e— evolution operator which satisfies the fo llow ing analogue to the Chapman—
Kolmogorov equation
Е(т', т т ) = E (т", т т ) E (t ', t ") (10.9)
IA E A -S M R -17 /8 1 26 9
as can be verified by inspection. So introducing by (10.6) the state st ( t ) o f the system N NU T we can cast any physical process w ith in U T described by causal input— output
к=1 к к—1 K
relations in to the form o f a Markov process where any past situation influences any fu ture
one only indirectly through the present one; cf. also De Santis 1973 fo r connection be
tween causality and state o f a system. The preceding considerations poin t o u t that in our
case causality and the unique solvability o f the Basic Equation are basic ingredients fo r theN
existence o f the state description o f U T see Saeks 1970; M iller 1971; Bryant and Tow 1972;K= 1 кГIkeda and Kodama 1973; Singh and Liu 1973 fo r some related questions.
For some additional results about the relationship between properties o f input— output descrip
tions and state description see Willems 1971; Goknar 1972; Ribari6 1973, Ch. X X ; Kalman's
lectures on realization theory.
When the reflection properties o f parts T^ as well as the associated connection
operators С K, are tim e— invariant, i.e. when 1
q^ = о, Т А к = A kTt and TTCKK' = c Kií'TT V 7 > 0 , к , к ' = 1, 2......N - 1 , (10.10)
then
T r < A u + C u> ■ <A u + C u > T r
and one can infer from the defin ition (10.8) o f the tim e-evo lu tion operator that
E ( t \ t " ) = E ( t0 — (t " — t'>, t 0 ) V Г < r " « t0 e (0 ,« ) (10.12)
indicating that in the tim e— invariant case the tim e-evolution operator E ( r \ r " ) is actually
a function o f the difference r " — r ' as one may be led to expect by physical in tu ition ; this
would not be so had we in defining state stu (r) by (10.6) om itted the sh ift operator T_T.
11. Yield o f a system. Suppose we have a possibly nonlinear mapping © o f sets l jnK's and
*out/c*s *nt0 an additive, zero-element containing set, say l0 , i.e.
0:|т К" ' 0 and 'ou.K^'e
Let us assume tha t the connection operators CKK, are such that
- ? QSC.i -I VK-1,2.....N (11.2)
and
■ 0 l i o » « J V K - - 1 . 2 ...... N (11.3)K=1
Considering ® í ¡ ¡nl(! 's and ® í¡outK! 's as amounts o f some © -q u a n tity in ¡¡nK's and ¡outK's,
respectively, we can interpret relations (11.2) and (11.3) as species o f conservations postulates,
since (11.2) tells us tha t the amount o f © -q u a n tity received by part TK equals the sum of
contributions supplied by outputs o f all parts, and relation (11.3) tells us that the total amount
o f © -q u a n tity supplied by part TR to all parts o f the system equals the amount o f © -
quantity lost by T . These tw o assumptionsdo not place any restriction on the subsequent
(10. 4 Note tha t in the case considered part TN is the exterior whose A N = 0 and q N = u¡n
270 RIBARlC
results, since were the connection operators CKK, not such,we could introduce additional
parts and new connection operators such tha t the assumptions (11.2) and (11.3) would be
true. On defining © — yield o f the part TK as fo llows
Y<0.V w > ,11-4>
we infer from (11.2) and (11.3) t h a t1
V<®.TK.iinK) = 0 - (11-51K = 1
N -1In particular, when T w is an exterior o f the subsystem U T (i.e. when A N = 0 ) , then
N N - 1 *=1¡t makes sense to define the yield o f the subsystem U T as
k=1 K
Y < ® - Nu! T K ' ¡inu> = - Y < ® 'T N ' ¡¡nN> <11 '6 >K= 1
and then relation (11.5) reads as fo llows:
Y <® .NuA - ¡¡nul = V < ® / V i n « > <11-71к = 1 k = 1
N -1I.e. © — yield o f the system U T equals the sum o f © — yields o f its N— 1 parts T . Whenk=1 Kl 0 is also partia lly ordered, then we can define the maximal and the minimal yield of system
TK by
Y (0 , TK t Max) = sup Y (© iT K? iinK) (11.8)
'¡пке hn/<and
Y ^ T ^ M i n ) = in f Y (0 ?TKfi jnK) (11.9)
'¡пк€ пк
respectively, and relation (11.7) implies
N -1 N -1Y (©, U T } Max) < £ Y (©. T • max) (11.10)K=1 к K—1 K
and
N - 1 N -1Y (ô , U T , min) > 2 Y (© ,T ,m in ) (11.11)
K = 1 * K=1 K
N -1So when all parts T o f U T are stric tly © — passive, i.e. whenк k=1 K
Y (0 , TK, Max) < 0 V k = 1 , 2 , . . . , N - 1 (11.12)
then
y ^ u r K. Max)<o (11 13)
( I I . 1) Throughout this paragraph we ta c it ly assume that the corresponding Basic Equation
does have a solution, i.e. that i¡n tí's and ¡outK's do exist.
so tha t the system U T composed o f stric tly © — passive parts T is also stric tly © — passive, i.i k=1 K K
strict ©— passivity is an inheritable property. For some additional results see Ribaric 1973,
Sec.V.d; §X X I.b .6 ; § XX I.b.7 ; § XX l.b .9 .
12. Tellegen's theorem. First we w ill derive a special form o f relations (11.4) and (11.5)
suitable fo r inferring a generalized Tellegen's theorem. Througout this paragraph we
assume that the connection operators » are tim e— invariant and memory less 1/ i.e.
we are considering a tim e— dependent case where i = ¡u (t), te [0 ,00) , assuming tha t for
lA E A -SM R -17/61
N - 1
any iue lu we have
(CkA u« ' ) « = C „ . w ( t ) v t > 0 . K X = 1 ,2 ..................................................... N (12.1)
So relations (3.1) read
¡ Í „ ,M • ¿ C . / W W V t > 0 , K.K- = 1,2.......N (12.2)
Let A be a linear operator, acting on i. , (t)'s and L ll t„ (t) 's as functions o f t > 01ПК OUTK
(e.g. Л * d /d t) , such that it commutes w ith connection operators, i.e.
(Aiin>> ■ f ,Скк'(Ли«')(‘> vt > 0, « = 1,2.N (12.3)K = 1
Further, fo r any t > 0 let i*nK and £ utK(t) be linear operators (possibly functionals)
on Pjn#( I and PoutK lus such tha t they satisfy the fo llow ing relations adjoint to
( 12.2 ) :
SutK'W “ ! , v t > 0 . * ' - 1 . 2 ..... N (12.4)
And let Л ' be a linear operator acting on ¡£,K(t),. i*ut (t) as functions o f t > 0, such
that
(Л' ¡o *u ,> > = J, (Л'С.) w Си. V t > 0. к- = 1,2.......N (12.5)
On m ultip ly ing relation (12.3) by ( A 'i j * K) (t), summing over к = 1 ,2 , ...,N and takingin to acount (12.5) we get the fo llow ing special case o f relations (11.4), (11.5):
J [ ( A ' ¡ * > ) ^ ¡ ¡rJ ( t ) - (A 'i0* t > ) ( A i 0ut> ) ] = 0 V t > 0 (12.6)
Now fo r any к = 1,2,..., N let sets P_11#J lie and P. I ie be equivalent real H ilbertu u i k us <пк us
spaces. When fo r any set o f A '-transform ed inputs A 'iL „ and outputs A 'i ' , thatIn К OUtK
satisfy (12.3) the connection operators CKK, are such that by scalar product ( , ) to
A 'i'. 's and A 'i ' 's associated linear functionals■ПК OUTK
(Л 'С )< *> s ((Л Т о и , > Ь ) (12.7)
and
s (И п > > ' ) , к = 1.2... N (12.8)
(12. ! ) For reasons analogous to those given in § 3 this assumption about CKK/s
does not restrict the applicability o f the subsequent results.
271
2 72 RIBAR lë
satisfy relations (12.5), then we get from (12.6) the fo llow ing form o f generalized
Tellegen's theorem:
J / K uJ W ’ K kH = J , ( k u „ < H t> . ( A - r J ( t ) ) V t > 0 - (12.9)
cf. Kishi and Kida 1968; Penfield, Spence and Duiken 1970. And when
( 12.10)
then combining (12.3) and (12.10) we get
| , [ ( ( л и < ‘ >. - ((A i outK) W . K u J w ' » = 0 V * ( 1 2 . 11 )
a result equivalent to (11.4); (11.5) w ith 0 ... = (A ..., A ...) .
Let us recapitulate the assumptions made in deriving Tellegen's theorem (i.e. rela
tions (12.9)):
(i) We assumed that N inputs ¡;„ „ ( t ) and N outputs irtll* „ (t) satisfy N connection rela-1ПК OUT к
tions (3.1) which impose N restrictions on the ir choice. We are however free to choose,
e.g. N outputs ioutK(t) arbitrarily and compute the corresponding N inputs ¡¡nK (t) by
relations (3.1),
(ii) We assumed operator A acting on inputs i in (t) and outputs i + (t) as functionsin re outk
o f t > 0 (e.g. d ifferentiating them) to be such that also the transformed A i jnK(t) 's and
A ¡ou,K(t)'s satify N connection relations (3.1).
(iii) Connection operators CKK»'s are assumed to satisfy relations (12.5), (12.7) and
(12.8) fo r any set o f inputs A '¡ ¡nK(t) and outputs A 'i^ utK(t) satisfying N connection
relations (3.1) and not necessarily the same Basic Equation as A¡nl<(t) 's and A ioutK (t) 's.
Consequently Tellegen's theorem (12.9) as well as relations (11.4) and (11.5), (12.6) and
(12.11) are based on properties o f connection operators С , which are essentially deter-K N
mined by the geometrical structure o f the composite system U T and do not dependN k » 1 K
on scattering properties A 's and q 's o f parts T o f U T .к к к кThough the structures o f relations (11.4) and (11.5), (12.6) and (12.11), and
(12.9) are analogous there are essential conceptional differences between the generalized
form o f Tellegen's theorem (12.9) and, say relation (11.5), because relation (11.5) is
essentially a conservation theorem, whereas Tellegen's theorem is usually interpreted as
an algebraic identity between tw o d iffe ren t sets o f inputs and outputs which do notN
neccesarily belong to the same system U T . This is due to the fact that deriving
Tellegen's theorem from (12.6) we impose on a set o f functionals acting on inputs and
outputs an additional assumption that they may be interpreted as a possible matched
set o f inputs and outputs consistent w ith connection equations (3.1).
13. Linearization. Throughout this paragraph we w ill assume that !u ¡s Banach space and that
the response operators к = 1,2,..., N. o f parts T„ o f U T„ are continuously' K K = 1 K
Frechet d ifferentiab le, at least In the v ic in ity o f, say iuoe lu , so that there is r > 0 such
that fo r any ¡u eSr = \ ¡u e lu : II¡u — ¡uo II < r, r > 0} there is a bounded linear operator,
say d A u(¡u , countinuously depending on a parameter ¡u e lu in the norm topology of
I so that
(13.1)
IAE A-SM R -17 /8 1 273
“ A j i ' + i " ! - A J ¡ ' ! - d A u( ¡ - . g » / l l ¡ " » - 0 as l i " l - 0 V i ' , i " e S r (13.2)
When A K is a linear operator, then PoutKc*A K(i'u, i" ) = A KP¡nK¡" . We can consider
the Frechet derivative PoutKС*АЦ(¡^ , )as the linear approximation to the nonlinear
response operator A KPjnK in-the v ic in ity o f the po in t P¡nK¡u, ¡'u eSr | since P ^ ^ d A ^ i^ , i" ) ,
depending linearly on P¡nK¡" , gives the approximate change of ou tpu t A j PjnK¡úÍ + QK of
T K corresponding to change P¡nK¡" o f input P¡n|t iÿ. In what fo llows we w ill give a few
examples of how we can infer from properties o f linearizations d A K(i[11 )o f the reflec
tion operators A K the properties o f A K's themselves; fo r some related ideas see Liu,
Saeks and Leake 1971. First we note that when any linearization Pout„ d A u(i'u, ) o f
A I P. I is passive, then also A K¡ PjnK | is passive; namely, when
IIPou,)id A u<iu' *й>“ < “Pin* ~'"J V ¡'u, *йе1и, ttlen
IIPo u , A l ¡u ! » = I I / Pou.Kd A u4 . ¡u>del1 < / <lPoü«Kd A u (^u H u > lld e « l l P ^ i J :
fo r an analogous result see Sandberg 1965.
Suppose that ¡UQ = iuo(quo* is a solution o f the Basic Equation (4.3} w ith
rhs(4.3) = q uo> i.e. let
q = ¡ — (C + A ) í i \ (13.3)Mu o u o 4 u u ' u o *
Then we may look upon the solution, say (¡u — >u0) j o f the associated linearized Basic
Equation
<¡u - ¡u o > e - ( Cu + d A u<¡uo: i>( iu - ¡uo>e = % ~ Ч и о <1 3 4 >
as an approxim ation to (i (q ) — iuo), where ¡u (qu) is a solution o f the Basic Equation
(4.3) fo r rhs(4.3) = qu > provided it exists. Now, suppose that the linearized Basic Equa
tion (13.4) does have a unique solution, say ( iy - ¡u0)g(qu - qu0) fo r any (qu - q uo)e lu i
so that by the bounded inverse theorem the inverse operator
[ l - ( C u - d A u (iu01 J ] - 4 q u - q uo) = (iu - i u0)e (qu - q u0) V q u| q ^ e l , , (13.5)
is a bounded linear operator. Then we might ask ourselves whether this fact implies the
existence o f a solution, say i u(qu) o f the Basic Equation (4.3) also fo r qu ¥ quo , and
what is the relation between ¡u (qu) and the solution (¡u ~ ¡ u0)j2 o f the linearized Basic
Equation (13.4). The answer to these questions is provided by the fo llow ing local im pli
c it - fu n c t io n theorem, cf. Schwartz 1969, (1.20), (1.21).
Theorem. When A u is continuously Frechet differentiable in the v ic in ity Sr o f
‘uoe *u* 'uo 's a solut'on ° f ^ e Basic Equation (4.3)>and the inverse [I — (Cu + d A u(iu0j i)
exists, then there are positive constants r¡ and r„ such that [l-(c + dA (i « ))l-1'uo Quo u u uexists as a bounded linear operator fo r any i e l J i — i ЛИ < r . , and such tha t fo r r 1 и u7 u и о iuo 'any qu ^ lu, llqu — qu o H < rq , the corresponding Basic Equation (4.3) has only one so
lu tion, say i i q , , ) such tha t Hi (q ) - i (q J i l < r¡ and such thatu u u u u u o uo
HiUí4C)-iuK»)“ [iuK , ” iu(4Ú)lfill/ll4u- (l i l|->0 as 4 “ ,K °
'^q ‘ , q ” el № llq’ — q D ^ r n llq" — q II < г И3.6)4 u ' 4 u u*? 4 u H u o q u o r 4 u 4 u o q uo
and such that
27 4 RIBARIC
where
(13.7)
Consequently, when fo r some ¡uo(quo) the corresponding linearized Basic Equation
(13.4) is solvable fo r any Qu - Q uoe lui then any iuelu in the v ic in ity o f ¡u0 is a locally
unique solution o f the Basic Equation (4.3) which also has a locally unique solution
fo r any que lu in the v ic in ity o f quo There ¡s also a global version o f this result, cf.
Schwartz 1969, (1.22).
Theorem. When fo r any iuoe lu the operator A u is continuously Frechet dif-
ferntiable, the inverse [I — (Cu +c*Au (iuot J]"1 exists and
then the corresponding Basic Equation (4.3) has a unique solution fo r any que lu .
In particular, in view o f (4.12) we have such a case when
For some related results see Vidav 1970; Fujisava and Kuh 1971; Sathe 1972;
Wu and Desoer 1972; Ikeda and Kodama 1973.
Result (13 .9) is an example o f many existence theorems o f system theory based on the as
sumption o f some kind o f passivity, cf. Desoer and Wu 1970; Holtzman 1970; Sandberg and
Wilson 1971; Anderson 1972; Sandberg 1972; Wu 1974.
The preceding theorems give tw o examples of the case where linearizability o f the reflection N -r1
properties A K o f parts TK o f U T is inherited by the reflection properties
A ( U T ) o f U Tu , cf. also Ikeda and Kodama 1973.K=1 K K=1 KLet us give an additional example o f how we can infer from linearized reflection
properties d A K(P jn J u o i ) o f T^'s the corresponding properties o f the composite sy
stem. Suppose A u satisfies conditions o f the preceding theorem, and its Frechet deriva
tive together w ith Cu are s e lf-a d jo in t operators, then the Frechet derivative of the inverse
[I — (C + A ) ) -1 is a se lf-a d jo in t operator by the preceding tw o theorems; about further u u N
properties of reciprocal systems U TK whose d A u(iuoj )+ Сц is selfadjoint see Brayton
1971; Chua and Lam 1973, 1974; and the literature therein.
Abdullah, K. and Y.Tokad: On the existence o f mathematical models for multiterminal НСГ networks. IEEE Trans.Circuit Theory, CT-19, PP*U19-U2 U ( 19T2 ).
sup H [ l - ( C u + d A u(iuo, ) ) r ‘ l l <¡..„el.,
(13.8)
sup HdAu(iuo, Cu )ll < 1 (13.9)
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I A E A -S M R -1 7 /5 9
SUPPORT FUNCTIONS AND THE INTEGRATION OF SET-VALUED MAPPINGS
G.S. GOODMAN Istituto Matematico,Florence, Italy
Abstract
SUPPORT FUNCTIONS AND THE INTEGRATION OF SET-VALUED MAPPINGS.1. Definition of support functions; 2 . Characteristic properties of support functions; 3 . Examples of
support functions; 4. Directional derivatives of support functions; 5. Further properties o f h (y* ; x1*);6. Extremal faces and their support functions; 7. The fundamental equations in Rn; 8. Consequences o f the fundamental equations in Rn; 9. Extreme points; 10. Theorem of Kudô; 11. Comments on Kudo's theorem.
1. DEFINITION OF SUPPORT FUNCTIONS
Let X be a (locally convex) vector space over the reals and X* its dual space. If К is a subset of X, we define the support function of K, hK, by means of the formula
hK (x*) = supx e K <x, x* >, x* G X*
If К is (weakly) unbounded, then for some x* hK will assume +00 as a value, while hK(x*) = - 00 if and only if К is empty (and then hK = - 00).
Support functions were introduced in the last century by Minkowski in order to study the properties of convex sets in R3 . Further developments have been treated in the works of Bonne sen and Fenchel and Hormander.
2. CHARACTERISTIC PROPERTIES OF SUPPORT FUNCTIONS
It follow at once from the definition that hK is sublinear, that is, it satisfies
(1) h(Xx*) = Xh(x*) for X S 0 (positive homogeneity)(2) h(x*+y*) s h(x*) + h(y*) (subadditivity)
and, as a supremum of weak* continuous functionals on X*, hK must be weak* lower semicontinuous.
Hormander (Ark. Mat. ¿(1954) 181) showed, conversely, that these properties characterize the support functions of non-empty closed convex sets.
EXERCISE. Show that (1) and (2) imply that h is a convex function, i .e . h(ax* + ( l -а) y*) S ah(x*) + ( l -а) h(y*), when 0 S a g 1.EXERCISE. Show conversely that any convex function which is positively homogeneous is subadditive.
281
282 G O O D M A N
Example 1. К = {x}. Then hK(x*) = <x, x* > and hK is a linear functional. The converse is also true if the elements of X* separate the points of X, i. e. if, given any pair x f у inX, there is some x* such that <(x, x*^> f ( y, x (This is usually assumed in dealing with locally convex spaces, since it assures that they are Hausdorff.EXERCISE. Show that if h is sublinear and h (-x*) = ~h(x*) for each x*, then h is linear, i. e. h (x*) = <(x, x* > for some x 6 X.
Example 2. X a real Banach space, К = {x | ||x|| s 1} (the unit ball). Then
hK(x*) = supxeK<x, x*> = ||x*||
Hence hK is simply the norm in X*.
Example 3. X = Rn and К is the unit hypercube, i. e.
3. E X A M P L E S O F S U P P O R T F U N C T I O N S
n
i=l
where the x¿ are the components of x relative to an orthonormal basis. Then X * = X and
n
h(x) = ^ |xi I i=i
(This is a special case of z)
Example 4. X = Rn, К = {x | x 4 ê 0, i = 1, . . . , n}. Then
r 0 if x e -K = {x I Xi SO, i = 1, . . . , n}h K<x ) = 1
+ 00 otherwise.
Example 5. К a closed linear subspace of X = Rn. Then
r 0 if x G К hK(x) = j
+ 00 otherwise.
Observation. Note that in each of these examples, К is a closed convex set, i .e . cl coK = K, where cl = closure and coK , the convex hull of K, is the set of all finite convex combinations of elements of K.
QUERY: Is it true in general that hK = hclcoK?
IA E A -S M R -1 7 /5 9 283
The sublinearity of a function h implies a number of interesting consequences. One concerns the existence of directional derivatives
4 . D I R E C T I O N A L D E R I V A T I V E S O F S U P P O R T F U N C T I O N S
. / -m t h(y*+Xx*) - h(y*)h (y-jx '-) = l im x io -^ --------r-1------ u - 1
THEOREM. If h is sublinear and finite-valued, then h(y*jx*) exists for аП y*, x*.
The proof is based upon the following important PROPOSITION. Under the assumptions of the theorem, there holds
h(y* +Xx*) - h(y*) s h(y* +цх*) - h(y*)A “ M
whenever 0 < X s ц.Proof of the Proposition. -By sub additivity,
h(/iy* +Xjnx*) = h(Xy* + [ju-X]y*] + X/jx*) S h(Xy* + Xjux*) + h([/u-X]y*)
Since ju —X g 0, we can use the positive homogeneity of h to get
A* My* +Xx*) - juh(y*) s Xh(y*+¡ux*) - Xh(y*)
Dividing both sides by X/j now gives the desired inequality.
Proof of the Theorem. The Proposition tells us that the difference quotients are monotone decreasing when X I 0. To show that a finite limit exists, it is thus enough to show that they are bounded from below. Now, by subadditivity,
h(y*) = h(y* + Xx* - Xx*) S h(y*+ Xx*) + h(-Xx*)
so that
- h(- x*) = ■ h(rI> Ax*) x > QA A
and this gives a bound from below.
COROLLARY
h(y*;x*) â h(y* +x*) -h(y*)
P r o o f . S e t ¡л = 1 i n t h e P r o p o s i t i o n a n d l e t X * 0 .
5. FURTHER PROPERTIES OF h(y*; x*)
PROPOSITION. h(y*;x*) has the following properties: it is
(1) positively homogeneous in x*;(2) subadditive in x*;(3) h(y*;x*) g h(x*) for all x*;(4) h ¿y * ; x*) = h(y*;x*), if/n > 0;(5) h(y*, x*;x*) = lim X|0 > x + )—^(у > x ) exists for each z*.A
Proof of (1). Suppose /л > 0. Then
h(y*;„x*) = l i m ц ГЫУ- + У ) -h (y * r L л/и
But Х/u i 0 when X J- 0, heneé the limit at right is precisely juh(y*; x*)
Proof of (2). By subadditivity,
2h(y* + I (x* + x|)) g h(y* + Xx*) + h(y* + Xx|)
Thus
h (y- + j (xi~ + x j)) - hK(y-) h(y* +X xt) _ h(y*) h(y* + Xx%) - h(y*)Х/2 " X X
Sending X-l 0 then gives
h(y*; x* + x f) g h(y*; x*) + h(y*; x*)
Proof of (3). This follows immediately from
My* + Xx*) g h(y*) + Xh(x*); X > 0
For, dividing by X gives
h(y* + Xx*) - h(y*) sA
and sending X 4 0 yields (3).
Proof of (4). We have, , л . , , л, h(y* + — x*) - h(y*)
h(py*; x*) = lim u o ---- 1-------1 -t o ..) = iim xw ----------^ j- ------------- =h(y*,x
Proof of (5). This follows from (1) and (2), and the reasoning of the preceding section.
2 8 4 GOODMAN
IA E A -S M R -1 7 /5 9 285
It turns out that the directional derivatives hK(y*; x*) have an interestinggeometrical interpretation (at least when X = Rn).
Let К С X = Rn be compact and convex. For any vector y* G x* we definethe exposed face (or supporting set) Ky* as
Ky* = {x € К I < x ,y *> = hK(y*)}
Then Ky* is a non-empty compact convex subset of К of co-dim ension 1.Thus Ky* possesses a support function hKy#(x*).
THEOREM
6. E X T R E M A L F A C E S A N D T H E I R S U P P O R T F U N C T I O N S
hKy*<x*) = hK<y*;x*)
Proof of hKy* s hK(y*; x*). When X > 0, we have, by definition,
If now we restrict the supremum to x G Ky*, we have
< x ,y*> = const. = hK(y*) = supxeK <x, y*>
and since Ky* С К we get
hK(y* + Xx*) - hK(y*) X
<x,y* + Xx*> - < x ,y*> X
s u P x e K y * <\ х » x ' У Ц су* (x )
x
Sending XiO now gives the desired inequality.Actually, equality holds — this will be proved later on. We remark now only
286 G O O D M A N
that one approach to such a proof would be to establish that, for all X > 0 sufficiently small,
s u Px eKy* < x > x * > = s u Px е к < x > У* + X x * >
i .e . X is a Lagrange multiplier which converts the constrained maximization problem into a free one (see Fig. 1).
EXERCISE. Show that such a X > 0 need not always exist.
7. THE FUNDAMENTAL EQUATIONS IN Rn
Let h(x*), x* € Rn = X* be subline ar. Then, if yjj', . . . , yjjj form a basis, we can take the successive directional derivatives h(y*;x*), h(y|, y|; x*), . . . , My*» • • •» y^-ij x*) form the basic system of equations
where, for j = 1, we mean h(y£).
THEOREM (Bonnesen and Fenchel; Kud'ô).The foregoing system of equations possesses a unique solution y, and у
satisfies
<y, x*> S h(x*)
for all x* G Rn
Proof. The existence and uniqueness of the solution follows at once from the fact that the y j , . . . , y* form a basis. It thus remains to prove the inequality.
Let x* be an arbitrary vector in X* = Rn. It has an expansion of the form
x* = c ^ f + . . . + Cj y f + . . . + c ny*
relative to the basis у*, . . • ,У*. Substituting into the basic equations, weobtain, by the linearity,
<У.У*> = h(y*, . . . « y j ^ y * ) j = 1, . . . n ,
П n
We must thus prove that
n n
d )j=i j=i
F or sim plicity, we shall first assume the cj f 0 for j = 1, . . . , n.
We start with the j = nth term in the left side of (*). If c n ê 0, then
cn h(yf j • • • » У n - i> Уп ) = h(yî> спУ'п)
by positive homogeneity. If c n < 0, then - c n = | cn | and subadditivity in the final variable gives
- h ( y * , . . . , y ; . 1;y ; ) ' п(уТ>• • • >-Уц-!; Уц)
Multiplying both sides by | cj then yields, by positive homogeneity,
° nh (y i , . . . , y*_i ; y„) = h(yí» • • • > y¡í-i ; с пУ n)
and thus this inequality holds regardless of the sign of c n.Next we observe that, since Cj.j f 0
h ( y ï , . . . Jy ] .1; c jyn) = h (y * ,... ,y | _ 2, I c j . J y r ^ j c ^ p , j = 2 , ----- n (2)
Now let j = n and use the fact that
h(yf, . . . ,y*_2, I c ^ ly*^ ; cny*) S h(y*, . . . , y*_2 ; | c ^ ¡y*^ + C[iy*)
- h (y j------,У гЙ-2;|с п-1 |Уп‘-1 ) (3)
The term at left dominates the final term cnh(y!J, . . . , y*_1 ; y*) in the sum (1), while the penultimate term in (1), c^ h ty if, . . . , y^_2; y*_i )» is just1
(sgm Ch.j) h(y*, . . . , y * .2; I cn. 1 ly*^ ) (4)
There are two cases to consider.(a) сп-! ê 0. Then ¡Cn.j | = c n.j, and, adding the foregoing term to
both sides of (* *) gives
cn-i h<y p • • • - Уп-2 ; Уп-i ) + cnh(yï> • • • - y ; - r Уп-i )
s h (y 11 . . . . Уп- 2 ; Сп-1Уп-1 + с дУп) (5)(b) Сц.! < 0. Then Cn.1 = - Icn_1[ and
h(yf------.У^-2; I c n .J y ;.! + c ny*)
= h < y î ...........Л - г > с п-1У п - 1 + с п Уп + 2 l c n - i l y n - i >
s h (yf, . . . , y* . 2 ; c ^ y * . ! + cny*) + 2 h(y*. . . . , y*_2 ; | c n.1 ly*^ )
Make this substitution in the right side of (3). The 2 will cancel with the last term in (3). Adding (4) to both sides of (3) will give an additional c^ ce lla tion , since s g m c„.i = -1. Hence (5) emerges once again.
IAE A-SM R-17 /5 9 2 8 7
1 By sgm cn_ i is meant cn_]/| cn_ i l , unless c , ^ = 0, when sgm cn- i = 0.
28 8 G O O D M A N
We have thus established that
n n-2
Y Cjh(yp . . . , y|.i; y p á Y, С)Ь(У1.........УХ- i ; У1)j=i j=l
+ h(y i.........Уп-2; сп-1Уп-1 + спу£)
that is, we have reduced the order of differentiation by one. It is clear that we can continue in this fashion, until Cj.1 = 0 for some j > 1, which we have so far excluded. But in this case, we can use the fact that
h(yf, . • •, y f .i ; y f + . . . + cny*) g h(yf, . . . ,y f_ 2 ; Cjy j + . . . + c ny*)
and just skip the с^.х term.
8. CONSEQUENCES OF THE FUNDAMENTAL EQUATIONS IN Rn
In the foregoing section we have only required that the function h(x*) besublinear - that is, we have not associated it with any convex set K.Suppose now we set
К = П {x I < x ,x*> S h(x*)} (1)x * e R n
Then K, as the intersection of closed half-spaces, is closed and convex.The foregoing theorem now implies that К is non-empty. Moreover, h(x*) is the support function of K. For, if x* is given, we can always find a basis in which y j = x*, and then the first equation tells us that the corresponding solution у satisfies
< y ,x*> = h(x*)
Since y G К, this shows that
s u Px x * У = h <x *)
The same argument works for every x*, and thus the assertion is proved. We have thus established all but the boundedness assertion in the followingTHEOREM. (Minkowski). There is a 1-1 correspondence between bounded closed convex sets in X = Rn and the sublinear, real-valued functions on X* = Rn.To see that K, defined above, must be bounded, let x G K. Then it is well known (Hahn-Banach.1 ) that
IIх II = SUP] x*l s i < x . x * >
Hence, if we can show that h(x*) S M for some M, for all x* such that Il x* I s 1, then it will follow that К is bounded. Let y'ij, . . . , y* be a basis.
IA E A -SM R -17/59 289
Set y* + j = -y*> j = 1, . . . , n. Then every x* with |x* || s 1 will possess an expansion
2n
x* = £ a jy f j=i
in which 0 S aj S 1 for all j. Thus
2n 2n
h(x*) = h ( ^ æ y*) S ^ h(y.) = M j=i j=l
/where M is independent of x*, as required. /EXERCISE. Show that h is continuous at the origin, and thus everywhere.
Another set of consequences arises when we replace h in thefundamental equations by hK, where К is a compact convex set in Rn . Then the inequality < y, х*У S hK(x*) for all x* tells us that y G К. For, if not, there would be a hyperplane which strictly separates x from K. Such a hyperplane has the equation
<( x, x* У - d = 0
for some x* and some real d, and the separation condition means that
<(x, x*> - d < 0 for x G К and < y, x*)> - d > 0
(Geometrically, К belongs to one of the open half-spaces bounded by thehyperplane, and x belongs to the other. ) Thus
suPx e к <x. x*> - d s 0
(Actually, inequality holds, since the supremum is assumed by some x G K, but we do not need this. ) Consequently,
hK(x*) < <y, x*>
which is a contradiction. Hence y G К.In fact, y G Ky|, since у also satisfies the equation
< y .y f> = hK(yr)
This allows us to complete the proof of the assertion that hKy*(x*) = hK(yif;x*). Recall that we had shown that S holds. We thus need to find an y G Ky* such that
<У.х*> = hK(y*; x*)
Take a basis y | ,. . . , y* in which y f = y* and y f = x*. Then the solution у of the fundamental equations will belong to Ky* and the foregoing equation becom es just the second fundamental equation.
2 9 0 G O O D M A N
We can iterate this reasoning. If y j , __ ,y * is a basis, Ky* is compactand convex, and thus possesses a non-empty exposed face in the direction y|:
K y^ yg = {x 6 Ky* I <x, x*> = hRy* (a*)}
we know that hK * (x*) = hK(yj; x*). Now, applying just this reasoning to Ky:| in place of К will yield
h Ky * y * (x * ) = М У * > у 1 ; х * )
Continuing in this way, we get the following THEOREM. Let y*, . . . , y* be a basis in R n and К a non-empty compact convex subset. Then
hKy*...... = V y * « • • • ’ y f ; x*> for j = 1, . , . , n
9. EXTREME POINTS
The foregoing theorem allows us to recognize that the solutions of the fundamental equations are precisely the extreme points of K, where К is given by (1) of the previous section.
This follows at once from Olech's characterization of the extreme points of К as the lexicographic maxima relative to the bases of X* = Rn. That is, y G К is an extreme point of К if and only if there is a basis y*, . . . , y* such that any point x G K, x f y, satisfies, for some 1 S к S n.
r<x, y f> = <y, y ]> for j < k, and
<x, y p < <У,У]> for j = к
Now this means that
< У » У р = suPx e K < x» y f > = h K&V>
<У.У!!> = supx6Ky* < x ,y*> = hKy* (y*),
<y,y*> = supx eKy*.....yg.^^yn > = Чу*.....yg./Уп)The previous theorem tells us that
hKyf...... = hK( y ï - - - > y f - i ; y î )
for each j. Hence у is precisely a solution of the fundamental equations.Since, for any set K, hclcoK (x*) = hK(x*), we have thus proved the
following result!
IA E A -SM R -17/59 291
THEOREM. (Kudo). Let К be a bounded set in Rn. Then the extreme points of c lc o K are precisely the solutions of the system of equations
where, for j = 1, is meant hK(y!£).
COROLLARY. A compact convex set in Rn has extreme points.
10. THEOREM OF KUDO2
Suppose that for each t G [ 0, 1], K(t) E R“ is a compact convex set with support function hK(tj (x*) such that
(a) supxeK(t) [Ix У S M(t), where M(t) € L j [ 0 , 1](b) hK (x * ) is measurable in t for each fixed x*.
о
for i = 1, . . . , n.(c) Each extreme point e of I is the integral of a function whose value
at t is the extreme point of K(t), relative to the same directions as
P roof. Let us first observe that hK(t) satisfies the inequality
<y.y*> = Vyî* — y*-i;yj). j = i-• • ->n
Let1 1
0 0
Then(a) I is a non-empty, compact convex set.(b) The support function h^x*) of I satisfies
1
о
and if y j , - • • j Уд is a basis, then
1
М У*.........
lh K(t)(x *) I - м № IIх * II fo r a11 x '
2 Nat. Sci. Reports, Ochanomizu University, 4 , N o.2 (1953), 151-163.
292 GOODMAN
This follows from Schwarz's inequality <(x,x*^ g ||x|| ||x* ||, for then
h K(t) ( x * ) = s u P x € K (t ) < x > x * > = SUP X e K(t) I I х II I I х " ' H á M W I I х * II
Hence
- h K(t)(-x*) ê - M(t) ||-x*|| = - M(t) ||x*||
and, by the subadditivity of h and the fact that h(0) = 0, there follows
h K < t > ( x * ) g - h K ( t ) ( " x * ) й I I х * II
To establish that I is non-empty, we must prove that a measurable selection x(t) exists. This follows from general theorems (e. g. Kuratowski and Ryll-Nardzewski, A general theorem on selectors, Bull. Acad. Pol.Sci. 13 (1965), 397), but a direct proof yields m ore. Following Kudo, we take a basis y * , . . . , y* ; and define x(t) as the solution of the system of equations
<x, y*>= hK(t)(yp
< x ,y*> = hK(t)(y* ;y f)
< x ,y *> = hK(t)( y * , . . . ;y * )
The measurability of h K tj(x>:‘ ) for each x* implies the measurability of hK(t)(yyj; x*), which, in turn, implies the measurability of hK t) (y*, y|; x*) etc. Thus each term at the right is measurable; consequently, the solution x(t) of the system is a measurable function of t. We know from our previous discussion that x(t) €K (t) for each t; hence, by hypothesis (a), ||x(t) || â M(t) for each t,which im plies that x(t) is integrable. Hence I is non-empty.
To see that I is bounded, suppose that x € I. Then
x(t) I x(t) У й / M(t) < oo
sup x e I M(t) < 00
i. e. I is bounded.We have not proved that I is closed, although we could, by arguing in
terms of weak convergence (see below). We shall establish that I is closed by proving, with Kudo, that I is convex and contains the extreme points of its closure, cl I (cf. the contribution of Olech in Vol. 1). The support function of cl I is
h j ( x * ) = s u p x e I < x , x * >
IAE A-SM R-17 /5 9 293
and the extreme points of cl I are the solutions of the equations
< x ,y *> = h jiyp
В
- < x ,y*> = hj(y*, — ,y*)
where y:j‘ , . . . , y* form a basis. Now, for any x G I and any x*
j. i
< x , x * > = y < x { t ) , x 5'*> = J h K ( t ) ( x * )
о о
for some selection x(t). Hence
lh^x*) = supxeI <x, x*> S J hK(t)(xi!)
To see that equality holds, note that we can, in the system of equations A, set y* = x*, and find vectors y|, . . . , y*, such that x*, y*, . . . , y* form a basis. Solving the system for x(t), we shall then have
<x(t), x* > = hK(t)(x*)
so that, integrating, we shall get
J.
< Х , Х * > = У ' hK{t)(x*)
wherel
x = J x(t) £ I0
Hencei
h,(x*) = supx6I <x, x*> s / hK(t)(x*)
and therefore equality holds. Thus
lhi(x*) = f hK(t)(x*)
0
29 4 G O O D M A N
We can now prove that for any y* and x*
1Ь1(У* ;х * )= У ' hK(t)(y*;x*)
0
For, when X > 0,i
hi(yj' + Xx*) - ht(y*) = Г hKm(y*+ Xx*) - hKm(y*) X J X
and when X -» 0 the left side tends to hj(yj‘ , x*) and the integrand at the right tends to hK(t)(y*; x*). It is thus just a matter of justifying the passage to the limit under the integral sign. Now, when X S i , the difference quotient is dominated by
h K(t) ( y ï + x *> “ h K(c) ( y f ) s M ( t > [ IIУ 1 + x * Il + Il y î I I ] s M ( t ) t 2 II у * II + IIх " I I ]
On the other hand, as we saw earlier
-h K(t)(-x*) s 4m <y| + Xx^ - Ь кг^ у! ) , X > 0
and this leads to an estimate from below by - M(t) ||x* ||. Since M(t) € L].[0, 1], the existence of these estimates allows us to apply the Dominated Convergence theorem of Lebesgue, and thereby to justify the passage to the limit under the integral sign. We thus arrive at the integral representation of hj(y’j; x*).
The estimate
1 hK(tj (у*; x*) I s MW [ 2 lly'i II+ IIх* II:K(t)
permits us to repeat this reasoning and thereby to establish the successive formulas
1hj(y*, — , y f; x*) = / hK(t)(y j.........y*;x*)
оfor i = 1, . . . , n.
Now suppose x is an extreme point of cl I relative to the basis y*, . . . , y * . Then x satisfies the system of equations B, and therefore
i l< x ,y ’{ > = / hK(t)&V = / < x ( t ) , y * >
0 01 1
<X, y|> = J hK(t) (y*; y*) = J <x(t), y* >
IA E A -SM R -17/59 2 9 5
where x(t) is the solution, for each t, of the system A. But, for eachi = 1, . . . , n
l l J < x(t),y*> = < J x(t),y*>о 0
so thatl
<x,yf> = < J x(t).yf>0
and this means that, relative to the basis y* . . . , y*, the vectors x andl i nJ x(t) have the same co-ordinates. Hence
о
and this proves the assertion (c) of the theorem. It also shows that the extreme points of cl I belong to I and since, trivially, I is convex, I must be closed, hence compact.
11. COMMENTS ON Kudo's THEOREM
1. Closedness of I. As observed by K ellerer (Bemerkungen zu einem Satz von H. Richter, Arch. Mat. 15 (1964), 204-207), the closedness of I can be proved by a weak convergence argument. We can sketch a direct proof along these lines, as follows.
Suppose that x G cl I. Then there is a sequence of points xr G I, r = 1, 2 ,. . . such that x ^ x a s r - * » , To each xr there is associated a function x r(t) assuming values a. e. in K(t), such that x r is the integral of x r(t) from 0 to 1.
Lett
FrM = J xr(u) du, r = 1, 2, . . .о
Then the fact that ||xr(t)|| M(t) 6 1^ [0 ,1 ] implies that t i l
||Fr(t)|| g J ||X[(u)|| s J ||Xl(u)|[ S J M(u) du0 0 0
so that the functions Fr(t) are uniformly bounded. Similar estimates showt
that the Fr (t) are absolutely continuous (equally in r), since / M(u) du is.
296 G O O D M A N
By A rzelà 's Theorem, there is a subsequence of r 's such that the corre sponding Fr(t)'s converge uniformly on [0, 1] to a function F(t). As the uniform lim it of equi-absolutely continuous functions, F(t) is absolutely continuous. Hence its derivative x(t) = F'(t) exists a .e . Accordingly, setting x(t) = 0 at those t where F '(t) does not exist, we have
lx = F (1) = J x(t) dt
0
It remains to show that x(t) GK(t) a .e . Observe that if x* is given, then, on any interval [u, v ], 0 u < v á 1, we have
v
<Fr (v) - Fr (u), x*> = <jT xt(t) dt, x*> u
V v
= f <xr(t). x*> dt S J hK(t)(x*)dtu u
Hence, sending r oo through a suitable subsequence, we getv
<F(v) - F(u), x*> â J hK(t)(x*) dt u
Now, if t is avalué at which F '(t) exists and at which the fundamental theorem of calculus holds for the integral, we can surround t by a sequence of intervals [u, v ], u < t < v, such that v - u ->• 0. Dividing both sides of the foregoing inequality by v - u and passing to the limit then gives
<x(t), x*> S h K(t)(x*)
Accordingly, this inequality holds, for fixed x*, at almost all t, where the exceptional set of t may depend upon x*. Still, for a countable dense set of x*, the inequality will hold simultaneously for all t outside a set of measure zero. But this im plies that, outside of this set, the inequality holds for all x*, since both sides are continuous in x*. Hence, x(t) 6 K(t) a .e . , as required.
IA E A -S M R -1 7 /6 0
TOPICS IN SET-VALUED INTEGRATION
R. PALLU DE LA BARRIERE University o f Paris,Paris, France
Abstract
TOPICS IN SET-VALUED INTEGRATION.In set-valued integration — where the integrand or measure is set-valued instead o f single-valued —
Strassen’ s theorem is one o f the most important results. It is interpreted and considered in connection with other problems and results, in particular with the Radon-Nikodym theorem. A new proof is dealt with. The problem of the conditional expectation o f a set-valued function is discussed.
INTRODUCTION
Set-valued integration is a natural extension of the classical theory of integration where the integrand or the measure is allowed to become setvalued instead of single-valued. Such an extension is a useful tool in many fields of applied mathematics such as the theory of control, statistics, mathematical economy. One of the most important results of the theory of set-valued integration is Strassen's theorem which ensures the equivalence of two natural definitions of the integral of a convex compact-valued function. Recently Valadier [l] has given an abstract formulation of this theorem from which the most concrete versions may be derived. In this paper we want to give a broad interpretation of Strassen's theorem which connects this theorem with other problems or results, in particular the Radon- Nikodym theorem, and to supply a new proof announced in R ef.[l], This interpretation may be extended to other problems and we discuss one of them, namely the concrete interpretation of the conditional expectation of a setvalued function (i.e. a set-valued random variable). The incitement to study conditional expectation of set-valued random variables com es from the theoretical interest of the problem (see,for example [2]) and also from the application to the stochastic control (see for example [3]).
1. CLASSICAL DEFINITIONS AND RESULTS
1.1. Sublinear functions
Let F be a real vector space, F* the algebraic dual of F.A real function f is said to be sublinear if
f(X x )= X f(x ) V Xs 0, x € F
f ( x 1 + x 2) a f ( x 1) + f ( x 2) V X j , x 2 £ F
297
29 8 PALLU DE LA BARRIERE
(i.e. if it is convex and positively homogeneous). For every sublinear function f we denote by Vf the set defined by
Vf = {g e F* I g(y) S f(y), V yG F }
i.e . the set of all linear functions dominated by f. This set is the subdifferential Sf(0) of f at 0. It is convex and compact for the cr(F*,F) topology (i.e. the topology of simple convergence on F). By virtue of the Hahn-Banach theorem, if f is sublinear, we have
f(y) = sup {g(y)| gG Vf}
More precisely, if Fj is a subspace of F and if g iGV(f|Fi), then gj may be extended into a element of Vf.
Let H be an ordered vector space. A function f of F into H is said to be sublinear if
f(Xx)=Xf(x) VAêO, xG F
f(x1+x2)g f(x 1)+ f(x2) VXj, x2€ F
and we put
V f = { g ë ^ (F,H )|g(y)sf(y), V y G F }
If H is completely reticulated (i.e. every m ajorized/m inorized subset has a super/low er bound) then the formal generalization of the Hahn-Banach theorem remains valid:
g(y) = sup {f(y)| f € Vg}
Let E and F be a dual pair of vector spaces (in other words, a pair of vector spaces in duality) and С a non-empty convex, ct(E ,F )-compact subset of E. We denote by 6*(. |c) the function from F into IR defined as follows:
6*(y I c) = sup {< x ,y> I xG С }
This function is called the support function of C. It is a sublinear function. The mapping C ->6*(.|C) is injective. Conversely, if f is a sublinear real function defined on F, then f is the support function of V f (for the duality between F and F* ). A necessary and sufficient condition for Vf to be included in E is that f is continuous for the Mackay topology t (F ,E ), (see section 1.5).
1.2. Set-valued measures
Let (T, ? ) be a measurable space (i.e. T is a set and Я? is a ст-algebra of subsets of T) and E a vector space. A set-valued measure (or multimeasure) on (T, (&) is a mapping A->M(A) from T into the set @ (E) of subsets of E) such that М(ф) ={0} and such that a convenient property of "cr-additivity" is satisfied. A ст-additivity property is a property of the
IA E A -S M R -1 7 /6 0 299
form: If A i 'ÿ ’ and {Aj} is a denumerable partition of A with A jG ^, then M(A) is in some sense the sum of the M(A¡ ).
In this paper we are only concerned with set-valued measures with non-empty convex <r(E, F)-com pact subsets of E (E and F being a dual pair of vector spaces), and the ст-additivity property used is the so-called "weak ст-additivity" defined as follows:
{Aj} ^ -p a rtition of A = > 6 * (у | M(A)) 6*(y | М(А;)), V y G Fi
This property implies that for every finite 'if-partition {AJ of A we have
M ( A ) = ^ M ( A j )i
Note that, if M is a set-valued measure in this sense, then 6*(y|M(.)) is a measure for every y'GF.
A measure m defined on (T,W) is said to be a section of the set-valued measure M, if m(A) £M (A), VAS ^ . The set of all sections of M will be denoted by 5^ .
1.3. Set-valued functions
Let (T, be a measure space with ц ê 0, ст-finite, and E,F a dualpair of vector spaces. A set-valued function Г from T into E (i.e. a function from T into the subsets of E) with non-empty convex, cr(E,F)-compact values (briefly with c .c . values) is said to be scalarly measurable (resp. scalarly integrable) if the function ê*(y|r(.)) is measurable (resp.integrable) for every y 6 F . Two scalarly measurable set-valued functions Га , Г2 from T into E with c .c . values are said to be scalarly equal almost everywhere if for every у 6 F there exists a negligible set Ny С T such that
«*(y|r1(t))=ô*(y|r2(t» VtG Ny
Let Г be a scalarly integrable set-valued function from T into F* with c .c . values and A é ? . The convex cr(F*,F)-compact subset of F* is called weak integral of Г on A with respect to m; it is denoted by w/Г/и or w/r(t)iu(dt) and defined as follows:
J Гц = Iх G F * I ^X,y^ sJ 6*^y |г М м № ), Vy G F A ' A
ô*(y| w J Гц) = J ó*(y |r (t))M(dt), Vy G F
3 0 0 PALLU DE LA BARRIERE
Let (T, , p) be a measure space with ц g О, а -finite. A lifting of5^"(Т, Я!, ц) is a mapping р of SC“ (T, c€, ц) into itself such that
1 . 4 . L i f t i n g
(1) p(f) = f a.e.(2) f = g a.e. ==> p(f) = p(g)(3) p(l) =1(4) fë 0 a.e. = » p(f)3 0(5) p linear(6) p(f.g) = p(f) • p(g)
Following Ionescu-Tulcea [4], such a lifting always exists if (T, ’W, ц) is a complete measure space. A function f such that f = p(f) is said to be p-compatible.
1.5. Special properties of vector spaces
A topological space E is said to be a Suslin space if it is a Hausdorff space and if there exists a continuous mapping from a complete separable m etric space onto E.
If E,F is a dual pair of vector spaces, t (F,E) denotes the Mac key topology on F, i.e. the finest locally convex topology such that the topological dual of F is E. This topology is the topology of uniform convergence on convex a(E ,F)-com pact subsets of E.
A locally convex vector space E is said to be sem i-reflexive if it is algebraically equal to its bidual E" (i.e. the dual of E' endowed with the topology of uniform convergence on bounded subsets of E), in other words, if t (F,E) is equal to the topology of uniform convergence on bounded subsets of E.
A locally convex vector space E is said to be barrelled if every lower semi-continuous sem i-norm on E is continuous.
A topological vector space E is said to have the (SCG) property if every linear mapping u from E into an arbitrary Banach space having the property: u ix^-’-O.if xn-*0 and и(х„) has a limit, is continuous (see R ef.[12]).
2. GENERAL ABSTRACT RESULTS
(T .^ .m ) denotes a measure space with /л s 0, ст-finite. F denotes a vector space.
Theorem 2.1.1
We consider the three following spaces:
(1) the space of all classes (modulo the a.e. scalar equality) of scalarly integrable functions from T into F*,
(2) the space of all measures defined on eg with values in F* (the а -additivity is taken in the weak sense) which are absolutely continuous with respect to ц.
(3) the space of all linear maps from F into L1.
IA E A -S M R -1 7 /6 0 301
There, if f, m, ф, respectively, are elements of these three spaces, each of the following relations is a consequence of the two others:
(i) m (A)=w J * ( w h e r e f^ G f)A
i.e.
<m (A ),y>=y<ft,(t),y>^(dt) V y G FA
(ii) <Иу) = j « A . ) , y> V yG F
where j is the canonical mapping from i^1 (ц) to L/fju); we shall also write this relation as follows:
Ф(y) = < f( .) .y » Vy 6 F
(iii) <m (A ),yy {ф(у))^ц V y G F A
where
( ф ( у ) ) ^ < Е ф ( у)i.e .
ф(у) = ^ <m (.),y>
These three relations define a commutative diagram of isomorphisms between the considered spaces.
Proof
The first part needs only routine checking. Thereafter we proceed as follows: assume f given, then the relation (i) defines m and the map f^ m is injective.
With given f, relation (ii) defines cp and the map f->cp is injective;Relation (iii) defines an isomorphism between the second and the third
space.It remains to prove that the map f->m defined by (i) is surjective. This
is a consequence of the following theorem.
Theorem 2.1.2
(Abstract Radon-Nikodym theorem for m easures.)F or every vector measure with values in F* which is absolutely
continuous with respect to /л, there exists a scalarly integrable function from T into F* such that
m(A) = w j ffi VAG <if A
30 2 PALLU DE LA BARRIERE
Let f-^f^ be a right inverse of the canonical mapping from â f” to L°° (i.e. j( f^ ) =f^, V f We get the result by defining f as follows:
P r o o f
<f(t),y> = (t)
Theorem 2.2 Consider the three following spaces:
(1) the space of all classes (modulo the scalarly a.e. equality) of scalarly integrable set~valued functions from T into F* with non1 empty, convex, cr(F* ,F )-compact values (briefly with c .c . values')
(2) the space of all set-valued measured M defined on with convex a(F *,F )-com pact values (briefly with c .c . values), which are absolutely continuous with respect to ц (i.e. satisfy the following implication: ц{А)=0 =>• M(A) = 0.
(3) the space of all sublinear functions from F into L1.
These spaces are endowed with two composition laws: addition and multiplication by positive real numbers, and they are "abstract convex cones". Then, if Г , M, cp are respectively elements of these three spaces, each of the following relations is a consequence of the two others:
A
where Г is an element of Г
i.e.
A
(ii) cp(y) =j(ô*(y|r (.)), V y e F
A
whereц
(ф(у )) is an element of cp(y)
ф ( у ) = ^ 5 * ( y I M ( . ) )
Relation (iii) defines an isomorphism between the second and the third spaces.
IAEA-SMR-И /б О 3 0 3
Relations (i) and (ii) define isomorphisms between the first space and a subspace of the second space and between the first space and a subspace of the third. A ll these isomorphisms form a commutative diagram. They extend the isom orphism s defined by theorem 2.1.1. They are order-conserving if we endow the three spaces considered with the following orders:
С f 2 <=» ô* (y |ï} (t)) S 6*(y |r2 (t) for all у and all t, except the t that belong to a negligible set depending on у
M jC M 2 « M j(A )C M 2(A) V A €<r
Ф1 S ф2 Ф^у)-Ф2(у) V yG F
Proof
The proof is sim ilar to the proof of theorem 2.1. The lack of an abstract Radon-Nikodym theorem for set-valued measures makes the result formally weaker.
Definition
If M and Г satisfy (i), we say that Г is a weak Radon-Nikodym derivative of M with respect to ц, or that M is a set-valued measure, having Г as a density, with respect to ц.
Let Г be a scalarly measurable set-valued function from T into F* with c .c . values. A scalarly measurable function f from T into F* is called a pseudo-section of Г if for all y € F , one has
<f(t),y> S 6*(y |r(t))
except for the t that belong to a negligible set Ny depending on y. The set of all pseudo-sections will be denoted Ьу^5^(Г), and the quotient space of 0>5Р(Г) by the a.e. scalar equality is denoted by P S (r ). Every section of Г is a pseudo-section but the converse is generally false.
Theorem 2.3. (Main isomorphism theorem)
Let Г be a scalarly integrable set-function from T into F* with c .c . values. Let M be the set-valued measure and cp the function from F into L1 associated with Г by theorem 2.2.
Then, between the sets PS(T ), 5£¡ and Уф, there exists a commutative diagram of isomorphisms such that f,m,i// correspond to each other by these isom orphism s if and only if two of the three following relations (and therefore also the third) are satisfied:
(f И denotes an element of f)
(i) m(A)=wy^ f^M V A £ ?A
(ii) ф(y) = < f(.),y> v y 6 F
(iii) ф(у) = ^ <m (.),y> Vy e F
Proof
Immediate consequence of theorem 2.1 and 2.2.
3 0 4 PALLU DE LA BARRIERE
2.4.
Theorem 2.4.1. (Abstract Strassen theorem)
Let Г be a scalarly integrable set~valued function from T into F* with c .c . values. For every j e w JГр, there exists an f 6 ) such thati =w/f/u.
Proof
By means of the main isomorphism theorem (2.3) we see that it is equivalent to the proof of the existence of an m e S y such that m(T) =| or the existence of а ф £ Vcp such that иоф = Ç, where u is the linear functional h -/h p on L1. The proof will therefore be achieved by .applying one of the following theorems.
Theorem 2.4.2
Let M be a set-valued measure from ^ into F* with c .c . values.For every f e M(T), there exists m e 5 , such that f =m(T).For the proof see Pallu de la Barrière [5]. A m ore general theorem
may be found in Coste [6] without any convexity assumption on the values of M.
Theorem 2.4.3.
Let Ф be a sublinear function from F into iJ and u a positive functional on L1. Then
V (и»ф) = UoVcp
(i.e. it is equal to {иоф|<// G Vcp})
F irst proof (adapted from Valadier [1])
The inclusion u0VcpC V(uocp) is trivial.Let L^be the space L 1 endowed with a iL ^ L ” ), and 5^(F,Lg) the space
of all linear mappings from F into Lj, endowed with the topology inducedby (LI f .
IA E A -S M R -1 7 /6 0 30 5
Then Уф is a relatively compact subset of this space because for every y E F, we have
{ф(у)\ф£ V p } = [ - < р ( - у ) , ф ( у ) ]
and the right-hand side is compact for crfL^L” ). M oreover, Уф is closed in Я?% (F, Lo) because the positive cone of l}a is closed. Finally, Уф is a convex compact subset of ¿fs(F , L*). As the mapping ф ^и°ф from £^(F,Lq) into F* is continuous, иоУф is a convex compact subset of F*.
F or every y € F and every фЕ Уф, we have ф (у ) S ф(у ). Furthermore, for every y€ F, there exists ai//€:Vp such that ф (у) = ф(у ). Therefore, for every y € F and every ф E Уф, we have
(и» Ф)(y) s (иоф) (у)
and for every y €= F, there exists а фЕУф such that
(udp)(y) = (иоф) (у)
This means that
(и-фНу) = max {(uo^) (у)\ф& Уф}
or that V(uoФ) is the closed convex hull of шУp. Recalling that иоУ// is closed and convex, we get the desired equality.
Second proof
This proof is adapted from the proof of a sim ilar result given by Konig [10]. The theorem is elementary if the a-algebra W is finite because this means that У(фх + ... +ФП) = Vpj + ... +^ФП.
Suppose we have ¡¡u =1, и =IE and denote by IEe the conditional expectation with respect to a finite subalgebra <-0 of : 1ЕЯ is a linear mapping ofL 1(<if ) on to L For every f E У(1Еоф), there exists a linear mapping фаof F into L^JJO such that фа Е VflE^o?) and IEo Фа = f . We have
Фа ( y ) e [ - (œ ÆOf)(-y ), (Ша of )(y )]
For a given y, the right-hand side remains inacr (L1, L°°)-compact subset Ky (this is true for both bounds of the interval) and the set-valued function
3] is upper semi-continuous. Put К = П К' . This is a compact subset of (Lj)F. yeF У
Consider the directed set of finite subalgebras and ^ an ultrafilter finer than the section filter. Let ф =1исп.фа . As we haveiMy) = lim Фа (у),ф is linear. Furthermore, for every h £ L “ , we have
y).h> = lim <<// (y),h>&Let л/0 be a fixed finite subalgebra of <3?. For h e 1.Г(J&0) and
we get
<Фа <у)> h> s <№а Ф{y),h> =<^(y),h>
30 6 PALLU DE LA BARRIERE
and therefore фа (y)scp(y) and, taking the limit,
Ф(у) s ф(у)
i.e.
фЕ Уф
We finally have
ТЕоф = lim IEoii' =fgr «
We conclude that for every f E IEocp, there exists а ф E Уф such that Ç = 1 Е о ф .
That means that ^(ЕоФ)С 1Е<,У//. Since the converse inclusion is trivial, the proof is achieved.
3. CONCRETE RESULTS ON INTEGRALS AND RADON-NIKODYMDERIVATIVES
Suppose that E, F are a dual pair of spaces: that means that E is a subspace of F* and separates (the points of) F. In order to obtain more concrete results we assume that, for instance, a set-valued function and a setvalued measure have values in E and try to find a hypothesis ensuring that our notions such as weak integrals, R.N. derivatives, have also their values in E. F irst'w e give some results concerning the weak integral.
Theorem 3.1.1 (Fundamental rule)
If Г and Fj are two scalarly integrable set-functions from T into E with c .c . values such that T(t) С FjÇt) a.e. and /Ц цСЕ. then JrjuCE.
If К is a convex o(E ,F )-com pact set and g is an integrable positive function such that T(t) С g(t)K, then we have w /l> C E (more precisely w /r^CK /g/u).
Theorem 3.1.2
If E is a Suslin sem i-reflexive space and F = E ', then, for every scalarly integrable set-valued function Г with non-empty convex ст(Е, E' ) - compact values,
i.e . the set-valued measure with Г as a density has its values in E.
Example
A
Proof (see Valadier [7,8])
IA E A -S M R -1 7 /6 0 307
If we assume that the theorem is known for a single-valued function Г, we can obtain a generalization to a set-valued function in the following way: We write
Ф(у )=J ô*(y|r(t))M(dt)A
Let y0 be an element of F and f a measurable section of the facet of T(t) in the direction yQ.
Then
<f(t),y0> = 6* (y0|r(t))
M oreover, f is scalarly integrable and w /f^ С E. With
ф(у) = J <f(t),y> /J (dt)
we have
ф(у0) = ф(у0)and
ф(у)«Ф(у) V y 6 F
i.e . ф£ЭФ(у0).The function ф is continuous for the strong topology on E' and Ф is lower
semi-continuous for the same topology. As E' is barrelled for the strong topology, Ф is continuous and therefore УФ CE.
The following result is a generalization of the well-known theorem by Gelfand and Dunford (see R ef.[12]).
Theorem 3.1.3
Suppose F is endowed with a locally convex topology having the (SCG) property and let E = F ', then for every scalarly integrable set-valued function from T into E, with c .c . values, we have
A
Proof
This is an immediate consequence of the following theorem. We let ф (у ) =/б*(у|г (t))M(dt); then we have ф( у) =/ф(у)й where ф is the sublinear function associated with Г by theorem 2.2; since ф is continuous, ф is also continuous and УФС F.
•Jt /x C E , V A £ ?
3 0 8 PALLU DE LA BARRIERE
If F is endowed with a locally convex topology having the (SCG)-property, then for every scalarly integrable set-valued function from T into F' with c .c . values, the function cp from F into L1 defined by
<p(y) = j(6*(y |r (.))
is continuous, as also all elements of Уф.
Proof
F or the sake of sim plicity, use the same symbol to denote an element of â f1 and its class in L 1. Leti^S Уф. Consider a sequence {yn} converging to 0. Then
Ф(yn) e [-ф("Уп)> ф(у„)]
ф(уп) (t) =6*(y„ lr (t)) -* 0
ф(-уп) (t) — o
So if ф(yn) has a limit in L1, it is necessarily equal to zero and ф is continuous.
Moreover, we have
№(у)|0ЕУф} =[-ф(-у),ф(у)]
and this set is bounded for every y e F. Therefore Уф is bounded in LfF.L1 ) for the topology of simple convergence and thus it is equicontinuous (every space having the (SCG)-property is barrelled).
F or every e> 0 there exists a neighbourhood 9^of 0 such that
у е У ■» Il ^ (y ) Il s e ViZ/ е У ф
and consequently
y € f => II Ф(у)|| s e
Example. Theorems 3.1.3 and 3.1.4 may be applied if F is a Fréchet space or if F is the dual of a reflexive Fréchet space.
In order to obtain a concrete version of the abstract Strassen theorem it is useful to be in possession of conditions ensuring that a pseudo section is scalarly a.e. equal to a section.
Theorem 3.2.1 (Valadier, R ef.[l])
Let E, F be a dual pair of vector spaces. We suppose that a denumerable subset exists in F , which separates the points of E (more briefly .E is separated
T h e o r e m 3 . 1 . 4
by a denumerable subset of F).
IA E A -S M R -1 7 /6 0 3 0 9
Let Г be a scalarly integrable set-valued function from T into E with c .c . values, and suppose that
wJF/^CE, V A e '? ’A
(i.e. M (A )C E , VAG<if, M being the set-valued measure, Г is a density).Then every pseudo-section of Г is scalarly a.e. equal to a section
(i.e. P S (r ) = S (r )).
Proof
Let {л,,} be a sequence of elements of F which separates E. Let F0 be the vector space over Q generated by {rij and F0the vector space over ]R generated by {r¡n} . Let f G ^ ^ i r ). As F0 is denumerable, there exists a negligible set N such that, for t$ N , we have
< f(t),y > Sб* (y I Г (t)), V y 6 F 0
and this inequality remains valid by continuity for у € F0 .The linear functional y-* <Cf(t),у defined on F0 may be extended in a
linear functional m ajorized by 5*(y(F(t)) and therefore of the form y-^fOtJ.y^ with f(t)G Г (t). One has
f(t) = {xG F (t)|<x, r)n> = <f(t), rin>}
and (see Valadier [7,8]) this ensures the scalar measurability of Г. We have
w I f ц G M(A), w J Ï/lî e M(A) A A
and
<wj fju,y> = < w jf^ ,y > VyG F0 A A
<f(t),y> = <f(t),y> V yG F 0
and this relation im plies that
w J ffJ = wjTai a A
and, equivalently, that f and ? are scalarly a.e. equal. As a corollary we get the following concrete version of the Strassen theorem.
Theorem 3.2.2 (Strassen theorem, concrete version)
Let E ,F be a dual pair of vector spaces. Suppose E is separated by a
3 1 0 PALLU DE LA BARRIERE
denumerable subset of F. Let Г be a scalarly integrable set-valued function from T into E with c .c . values. We suppose
This theorem has a m ore general version which we quote without proof.
Theorem 3.2.3 (Valadier [1])
Let E ,F be a dual pair of vector spaces. Suppose E is separated by a denumerable set of F. Let Г be a scalarly integrable set-valued function from T into E with c .c . values. We suppose that w J fp € E for every f£
By use of a lifting of it is possible to obtain "good" versions of the derivative of a set-valued measure. Throughout this section, p will denote a lifting of .
Definition
Let Г be a set-valued function from T into F* such that fi*(y |г (.))6£^°°
for every y £ F . Then Г is said to be p- compatible if and only if 5*(y |r(.)) is p-compatible for every y £ F .
Theorem 3.3.1 (Weak Radon-Nikodym theorem)
Let M be a set-valued measure from into F* with c .c . values absolutely continuous with respect to Ц. and such that
A
Then for every | e w jl> , there exists an f6 SÇ , such that
Then
^ 6 * (y | M (.))G L ”
There exists a unique scalarly integrable function from T into F* with c .c . values, p-compatible, such that
I A E A -S M R -1 7 /6 0 311
The mapping M - Г preserves inclusion. The "density" Г will be denoted by d* M / d|U.
Proof
If v is a measure such that v is absolutely continuous with respect to ju. we denote by d*y/dju the p-compatible R.N.-differentiation of v with respect to ц.
The proof is completed if we define Г by the following relation:
6*(ylr(t)) = ïïü 6*(ylM(J
To get a m ore concrete version of the preceding theorem, it is necessary to make some assumptions ensuring that if M has values in E C F * , then d#M/dix also has values in E.
Theorem 3.3.2 (The Radon-Nikodym theorem for set-valued measures with c .c . values, concrete version)
Let E , F be a dual pair of vector spaces. Suppose ц is finite.Let M be a set-valued measure from into E with c .c . values.A convex p(E,F)~compact subset К of E is assumed to exist such that
M (A )C m(A)K
Then d* M/dM is well defined and we have
J # 1\ IT
■(t)CK, v t e T
Proof
Define the set-valued measure Mj by Mj(A) = ju(A)K. Then
dAi
and from MC Mj, we deduce
V , £ T
The following theorem gives sufficient conditions for the majoration hypothesis in theorem 3.3.2 to be satisfied.
Theorem 3.3.3 (Valadier, unpublished)
Suppose F is endowed with a locally convex topology such that F is
3 1 2 PALLU DE LA BARRIERE
barrelled and put E =F '. Suppose ц is finite.Let M be a set-valued measure from into E with c .c . values such that
^ 6*(y| M (.))G L°° , V y 6 F
Then there exists a convex o (E ,F )-compact subset of E such that M (A )C m(A)K * -
Proof
If
K = ^ w | u { ^ | A G ^ j ц(А)>о|
then К is a convex closed subset of E. We have
and
6*(y|K)=sup |б*(у|М^-)) | А 6 ^ , м(А)>о|
Therefore К is bounded for the topology of simple convergence and therefore it is equicontinuous (since F is barrelled). As E = F ',K is cr(E,F)-compact and we have by construction M(A)C;u(A)K.
4. GENERAL RESULTS ON CONDITIONAL EXPECTATION
Notation: from now on, ц is a probability measure, .jS'denotes a complete sub-a-algebra of <3f and ц is the restriction of й toja'. For every h 6 L1^ ) , IEÆh denotes the element of L1^ ) , whose elements are conditional expectations ofh with respect toj®'.
Definition 4.1
Let Г be a scalarly integrable set-valued function from Çf into F* with c .c . (i.e. non-empty convex o(F* ,F)-com pact) values. We call a (weak) conditional expectation any л/- scalarly measurable and ц 1а- scalarly integrable
s et ~ value d function 7 from T into F * , with c .c . values, such that one of the
IA E A -S M R -1 7 /6 0 3 13
following equivalent properties is satisfied:
(D ô*(y\ y(.))eœ a ô*(y\r(.)) v y e F
( 2 ) w j 7 M = w j r / u V A G J * '
A A
Condition (1) may be written as follows: ф =1Ел °ф where ф is the sublinear function from F into ) defined by ф(у) = j(0*(y |r(.)) ) and ф is thesublinear function from F into L 1(j 2 ) defined by ф{у) = j(ê* (у | т(.))).
Let M be the set-valued measure having Г as a density. Condition (2) can also be formulated as follows : 7 is a weak density with respect to of the restriction Mi of M to ja .
I C t
Let us check the equivalence of (1) and (2).
(1) *=> j ô * t y \ y ( . ) ) n = j 6*(y\r( .) )u VA б Д y G FA A .
<=* (y I wJ' 7 /л ) = ô* (y | w J Гм ) VA Ejà, y G FA A
*=* (2)
If 7 is a conditional expectation of Г with respect to J&, then also 7 ' is one, if and only if 7 ' is scalarly a.e. \xa -equal to 7 .
We denote by Е а Г the set (which may be empty) of all conditional expectations of Г with respect tom/.
Remarks
I. If Г is single-valued then 1ЕЛГ is non-empty (apply the abstract Radon-Nikodym theorem for measures).
II. If Г is set-valued, 1ЕЛГ ф ф, 7 е1Ея Г and f G 3PSf\V), then I E ^ fG ^ ’W . Let us discuss the existence of a conditional expectation.
Theorem 4.1.1
Suppose 6 * (y |r(.)G ¿f °° for every y G F. Then 1ЕЛГ ф ф. For every lifting p of 1ЕЛГ contains a p -compatible set-valued function whichwill be denoted by IE^F. The mapping Г-»1Е*Г conserves inclusion.
Proof
Use the weak Radon-Nikodym theorem for М|л and М|я and
* d#M |«= -A-----~
Let us now suppose that E,F is a dual pair of vector spaces. We want to find a conditional expectation of Г with values in E (instead of F*) when Г has values in E.
Theorem 4.2.1
If Г is a scalarly measurable set-valued function with c .c . values such thatr(t) CK, where К is a convex ct(E ,F )-compact subset of E, then (Ë jT )(t )C K .
Proof
S e tr i (t)=K, V t S T ; then ( lE ^ K t) =K, V t € T and (IE*r)(t) С (ffi£q)(t), Vt GT .
The two following theorems are generalizations of theorem 4.1.2.
Theorem 4.2.2
I fr (t )C g (t ) К where gE££1( r&’) and К is a c .c . subset of E, then ДС^Г фф and there exists а such that y(t)C h(t)K with h £IEa g.
Proof
One has
wJ'ï'V С A A A
We can apply theorem 3.3.2 and obtain a density y jof Mja with respect to the measure Ьм|я with 7 a(t) £ K , Vt 6 T. Then 7 =h7a is a density of ц|e with respect to and this density satisfies the relation 7 (t)Ch(t)K .
Theorem 4.2.3
If there exists an jz/-partition {T; } of_T and for every i a c .c . subset K, of E exists such that r(t) CK¡ VtG T¡ , then IE^F is not empty and contains an element 7 such that 7 (t) e Kj, Vt ST¡ .
Proof
The proof is completed when a routine checking is performed.
3 1 4 PALLU DE LA BARRIERE
5. GENERALIZED STRASSEN THEOREM
Theorem 5.1.1
Let Г be a scalarly integrable set-valued function from T into F* withc .c . values, M the set-valued measure with Г as a density with respect to д , and cp the sublinear function from F into L1 associated to Г and M (i.e.
Ф(У) = d*(y| M(.))
К / gt! =K / hfi. , V A e ^
IAEA-SM R-17/60 31 5
Letja^be a sub-algebra of and suppose ШС^Г^ф. Let
Vcpj
The horizontal arrows are the isomorphisms defined by the main isomorphism theorem (theorem 2.3). The vertical arrows denote the following mappings:
1ЕЛ: the mapping which associates to every class of pseudo-sections of Г the class of a conditional expectation of one of its m em bers, \a : the operation of restriction to JÍ, IE ■>: the mapping ф -*IEa °ф.
Let о,т,ф be elements of P S (r ),,9 ^ , Vcp which correspond to each other by the main isomorphism theorem, i.e.
we have to prove that а1,т 1,ф1 correspond to each other by the second-line isom orphism s, i.e.
(ctj is an element of ctx). But this is an immediate consequence of the equivalent definitions of the conditional expectation.
Theorem 5.1.2
Let Г be a scalarly integrable set-valued function from T into F* with c .c . values. Suppose that IE^r фф and let 7 g 1Е^Г.
Then for every h€&â*(y ) , there exists a g e ^ ( r ) such that h€IE Æg.
Proof
a - Ф( У) =j(5*(y |ст(.)))
where ст is an element of ст. When we denote
CTj =JEæct, m¡ = т|я, фг =Ша оф
'¿'i (У ) = j (5* (У I Ух ( - » ) V y e F
31 6 PALLU DE LA BARRIERE
The theorem states that the first vertical arrow in the commutative diagram described in theorem 5.1 is a surjection. It is equivalent to the surjectivity of the two other vertical arrows. So it is equivalent to the following theorem and is a consequence of theorem 5.1.4.
Theorem 5.1.3
Let cpbe a sublinear function from the vector space F into L1( ).Let J Í be a sub-cr-algebra of <«f. Then
V(IECT°cp) =Ея оУф
(i.e. it is equal to {IEe oip \ip e Уф} )
P r o o f
Proof
As previously mentioned, this theorem follows from the next one.A direct proof, which is a generalization of the first proof of theorem 2.3.4, will be given in Ref.[5a],
Theorem 5.1.4 (Pallu de la Barriere)
Let M be a set-valued measure defined on W with convex cr(F*,F)-compact values. Let be a sub-p-algebra of and m 1g . Then there exists m such that т|^ = т г 'a
Proof
See Pallu de la Barrière [5]. It is also possible to drop the assumption of convexity by applying the more general results of Coste [6 ].
5.2
We want to obtain m ore concrete versions of the generalized Strassen theorem ensuring that a section of 1ЕЯГ is in fact a conditional expectation of a section of Г. F or this purpose we may introduce assumptions such that every pseudo-section is scalarly a.e. equal to a section. That is what is done in the following theorem. Theorem 5.2.2 is m ore sophisticated.
Theorem 5.2.1 (Generalized Strassen theorem, first concrete version)
Let E,F be a dual pair of vector spaces and suppose that E is separated by a denumerable subset of F. Suppose Г is a scalarly integrable set-valued function from T into E with c .c . values. Suppose that, M being the set-valued measure with Г as a density, we have M(A) CE, VA € .
Suppose ДЦ„Г фф and take 7 £ ffi^r. Then for every section g of 7 there exists a section f of Г such that h 6 Е ЛГ.
Theorem 5.2.2 (Generalized Strassen theorem, second concrete version)
Let E,F be a dual pair of vector spaces and suppose that E is separated by a denumerable subset of F. Suppose Г is a scalarly integrable set-valued function from T into E with c .c . values. Suppose in addition that 1Е„Г j=<¡> and that for every f S ^ , lE ^f has a version with values in E. Take 7 e IE дГ.Then for every section g of 7 , there exists a section f of Г such that gGIE^f.
Proof
There exists anf G ^ ^ r ) such that gGIE^f, i.e.
<Я(.),у>е1Ел < « .) ,у > V y s F
For every y G F there exists a negligible set Ny such that
<f(t),y> S6*(y|r(t)) Vt GNy
Let {rjn} be a sequence of elements of F which separates E.Let F0 be the vector space over Q generated by {rin} and F0the vector
space over IR generated by {r)n} . There exists a negligible set N such that for t G N we have
<f(t), y> s 6*(y j Г (t )) V у GF0
and this inequality remains valid by continuity for у GF0 . The linear functional y-* <Cf(t),у defined on F0 may be extended in a linear functional m ajorized by 6 *(y |r(t)) and therefore of the form y^<Jf(t),y^ with f(t) GT(t). We have
f f t j ^ x e r i t l ^ x , ^ ) =<f(t),r)n>}
and (see Valadier [7,8]) this ensures the scalar measurability of f.We have
IE^<f(.),y> = IE^<f(.),y> = <g(.),y>
Let g be a version of IE^f with values in E. Then
< !(-). У> = ' < g ( . ) » y > VyGF0
and,in particular,
<g(t). nn> = <g(t), nn>
for every rj and for t not belonging to a negligible set N which may be assumed to be independent of n.
We conclude that g(t) =g(t) V t GN and the proof is achieved.
IA E A -S M R -1 7 /6 0 3 1 7
P r o o f
A p p l y t h e o r e m s 5 . 1 . 2 a n d 3 . 2 . 1 .
Remark
Theorems 5.2.1 and 5.2.2 are sim ilar to previous results by Van Cutsem, Refs [13, 14].
3 1 8 PALLU DE LA BARRIERE
R E F E R E N C E S
[13 VALADIER, M. f Sur le théorème de Strassen, C .R . Paris 278 (1974), 17; and Séminaire d’ analyse convexe, Montpellier № 4 (1974).
И NEVEU, J. , Convergence presque sûre des martingales multivoques, Ann. Inst. Poincaré, Sêr. B, J3 1 (1972).
[3] BISMUT, J. M ., Intégrales convexes et probabilités, J. Math. Anal. Appl. 42 3 (1973); Thèse de Doctorat, Paris (1973).
[4] IONESCU, A . , TULCEA, C . , Topics in the Theory o f Lifting, Springer (1969).[5] PALLU DE LA BARRIERE, R ,, Quelques propriétés des multi-mesures, Séminaire d'analyse convexe,
Montpellier No. 11(1973).[5a] Sur l ’ espérance conditionnelle d’ un processus aléatoire souslinëaire, Publ. Statist. IRMA, Grenoble (1975).[6] COSTE, A. , On multivalued additive set functions, to appear in Ark. Mat.[7] VALADIER, М ., Contributions à l ’ analyse convexe, Thèse de Doctorat, Paris, 1970 published in [8] and [9]).[8] VALADIER, М ., Multi-applications mesurables à valeurs convexes compactes, J. Math. Pures Appl.
50 (1971) 265.[9] VALADIER, M . , Sous diffrentiabilité des fonctions convexes à valeurs dans un espace vectoriel ordonné,
Math. Scand. 30 (1972) 65.[10] KONIG, H ., Sublineare Funktionalen, Ark. Math. 23 (1972) 500.[11] VALADIER, M ., Espérance conditionnelle d'un convexe fermé aléatoire, C.R. Paris 273 (1971).[12] BOURBAKI, N. , Intégration, Paris
[13] VAN CUTSEM, B ., Eléments aléatoires à valeurs convexes compactes, Thèse de Doctorat, Grenoble (1971); (partially published in [16] and [17]).
[14] VAN CUTSEM, B ., Espérance conditionnelle d’une multi-application à valeurs convexes compactes,C .R. Paris 269 , 212.
[15] VAN CUTSEM, B ., Martingales de multi-applications à valeurs convexes compactes, C.R. Paris,269, 429.
[16] STRASSEN, V . , The existence o f probability measures with given marginals, Ann. Math. Stat. 36 (1965) 423.
SECRETARIAT OF SEMINAR
ORGANIZING
R. Conti
L. Markus
C. Olech
EDITOR
J.W. Weil
COMMITTEE
Mathematics Institute "U .D ini",University of Florence,Italy
Mathematics Institute,University of Warwick,Coventry, United KingdomandSchool of Mathematics, University of Minnesota, Minneapolis, USA
Institute of Mathematics,Polish Academy of Sciences,Warsaw, Poland
Division of Publications, IAEA, Vienna, Austria
319
The following conversion table is provided for the convenience o f readers and to encourage the use o f Sf units.
FACTORS FOR CONVERTING UNITS TO SI SYSTEM EQUIVALENTS*
SI base units are the metre (m), kilogram (kg), second (s)f ampere (A), kelvin (К), candela (cd) and mole (mol).[F or further in form ation, see International Standards ISO 1000 (1973), and ISO 31/0 (1974) and its several parts]
Multiply to obtain
Mass
pound mass (avoirdupois) 1 Ibm 4.536 X 10"1 kgounce mass (avoirdupois) 1 ozm * 2.835 X 101 gton (long) (= 2240 Ibm) 1 ton = 1.016 X 103 kgton (short) (= 2000 Ibm) 1 short ton = 9.072 X 102 kgtonne (= metric ton) 1 t = 1.00 X 103 kg
Length
statute mile 1 mile = 1.609 X 10° kmyard 1 yd = 9.144 X 1 0 '1 mfoo t 1 f t = 3.048 X 10“ ' minch 1 in = 2.54 X 10“ 2 mmil (= 10“ 3 in) 1 mil = 2.54 X 10~2 mm
Area
hectare 1 ha = 1.00 X 104 m2(statute m ile)2 1 mile2 = 2.590 X 10° km 2acre 1 acre = 4.047 X 103 m2yard2 1 yd 2 = 8.361 X 10~’ m2fo o t2 1 f t 2 = 9.290 X 10~2 m2inch2 1 in2 = 6.452 X 102 mm2
Volume
yard3 1 yd3 * 7.646 X 1 0 '1 m3fo o t3 1 f t 3 = 2.832 X 1 0 '2 m3inch3 1 in3 = 1.639 X 104 mm3gallon (Brit, o r Imp.) 1 gal (Brit) = 4.546 X 10-3 m3gallon (US liquid) 1 gal (US) = 3.785 X 10-3 m3 ’litre 1 I = 1.00 X IQ '3 m3
Force
dyne 1 dyn = 1.00 X 10"5 Nkilogram force 1 kgf = 9.807 X 10° Npoundal 1 pdl = 1.383 X 10"' Npound force (avoirdupois) 1 Ib f = 4.448 X 10° Nounce force (avoirdupois) 1 ozf = 2.780 X 10_1 N
Power
British thermal unit/second 1 Btu/s = 1.054 X 103 Wcalorie/second 1 cal/s = 4.184 X 10° Wfoot-pound force/second 1 ft - lb f /s = 1.356 X 10° Whorsepower (electric) 1 hp * = 7.46 X 102 Whorsepower (metric) (= ps) 1 ps = 7.355 X 102 Whorsepower (5 5 0 ft- lb f/s ) 1 hp 7.457 X 102 W
* Factors are given exactly or to a maximum o f 4 significant figures
M u l t i p l y b y t o o b ta i n
Density
pound mass/inch3 1 lb m /in 3 2.768 X 104 kg/m 3pound mass/foot3 1 lb m /f t3 = 1.602 X 101 kg/m 3
Energy
British thermal un it 1 Btu = 1.054 X 103 Jcalorie 1 cal = 4.184 X 10° Jelectron-volt 1 eV 1.602 X 10"19 Jerg 1 erg = 1.00 X 10*7 Jfoot-pound force 1 f t - lb f = 1.356 X 10° Jkilow att-hour 1 kW-h = 3.60 X 106 J
Pressure
newtons/metre2 1 N /m 2 = 1.00 Paatmosphere3 1 atm = 1.013 X 105 Pabar 1 bar = 1.00 X 10s Pacentimetres o f mercury {0°C) 1 cmHg = 1.333 X 103 Padyne/centimetre2 1 dyn/cm 2 = 1.00 X 10_1 Pafeet o f water (4°C) 1 f tH 20 = 2.989 X 103 Painches o f mercury (0°C) 1 inHg = 3.386 X 103 Painches of water (4°C) 1 inH 20 = 2.491 X 102 Pakilogram force/centimetre2 1 kgf/cm 2 = 9.807 X 104 Papound fo rce /foo t2 1 I b f / f t2 = 4.788 X 101 Papound force/inch2 (= psi)b 1 1 b f/in 2 = 6.895 X 103 Pato rr (o°C) (= mmHg) 1 torr = 1.333 X 102 Pa
Velocity, acceleration
inch/second 1 in/s = 2.54 X 101 mm/sfoot/second (= fps) 1 ft/s = 3.048 X 10-1 m/sfoot/m inute 1 ft/m in = 5.08 X 10“3 m/s
m ile/hour (= mph) 1 mile/h4.470 X 10_1 1.609 X 10°
m/skm/h
knot \ knot = 1.852 X 10° km /hfree fa ll, standard (= g) = 9.807 X 10° m/s2foot/second2 1 ft/s 2 = 3.048 X 1 0 '1 m/s2
Temperature, thermal conductivity, energy/area■ time
Fahrenheit, degrees — 32 °F — 3 2 l 5 f cRankine °R [ 9 t к1 B tu - in /ft2 -s- °F = 5.189 X 102 W /m -K1 B tu /ft-s - °F 6.226 X 101 W /m -K1 cal/cm-s-°C = 4.184 X 102 W /m -K1 B tu /ft2 -s = 1.135 X 104 W/m21 cal/cm2 ■ min = 6.973 X 102 W/m2
Miscellaneous
fo o t3/second 1 f t 3/s = 2.832 X 10“ 2 m3/sfo o t3/m inute 1 f t 3/m in = 4.719 X 1 0 '4 m3/srad rad = 1.00 X 10-2 J/kgroentgen R = 2.580 X 10-4 C/kgcurie Ci = 3.70 X 1010 disintegration/s
a atm abs: atmospheres absolute; atm (g): atmospheres gauge.
b lb f/ in 2 (g) lb f/ in 2 abs
(= psig): gauge pressure;(= psia): absolute pressure.
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IN TE R N A TIO N A L SUBJECT GROUP:ATOMIC ENERGY AGENCY Physics/V IE N N A, 1976 Theoretical Physics
