o

5. 6.

L. V. Kantorovich and R. Sh. Rubinshtein, "On a function space in certain extremal prob- lems," Dokl. Akad. Nauk SSSR, !15, No. 6 (1957). P. Meyer, Probability and Potentials [Russian translation], Mir, Moscow (1973). V. M. Zolotarev, "Ideal metrics in the problem of approximation of distributions of sums of independent random variables," Teor. Veroyatn. Primen., 22, No. 3 (1977).

CONDITIONS OF STABILITY OF A LINEAR SYSTEM

UNDER FREQUENCY FLUCTUATIONS

Yu. N. Kamnev

i. In various practical problems we encounter the case that a linear oscillatory system

with low friction loses its stability due to the presence of small random frequency fluctua-

tions.

If the frequency is a sum of a constant component e and a stationary random process,

then such a linear system can be adequately described by the equation

2+k2+(~+~(0) 2x=0

If the random disturbances ~(t) are small compared to ~, then

(~+ ~ (t))~-- ~ + 2 ~ (t).

If, moreover, the process ~(t) has a small correlation interval, then it can be replaced

by white noise W, and it will be necessary to study the stability of the solution of the

stochastic equation

2 + k 2 + (~2 + 2 ~ ' (0) X=O. (1)

Equation (i) can be more correctly written in the form of a system of stochastic Ito

equations

dX~ (t) : ~-u (t) dr, d X 2 (t) = - - (kX2 -~ ~X~) d t - - 2~X~dW/ (t), (2)

where W(t) is a standard Wiener process.

General results obtained by methods of stability analysis of such systems have been ob-

tained in [i].

The stability of system (2) has been analyzed in [2, 3] with the aid of the methods de-

scribed in [i]. However, in [2, 3] the authors are using a circumventing procedure for find-

ing an invariant measure of the corresponding Markov process on the circle, by assuming that

the presence of a singularity in the diffusion coefficient precludes the possibility of solv-

ing the Fokker--Planck equation [2, p. 415].

The principal aim of this note is to find a method of determination of necessary and

sufficient conditions of stability for system (2) that is based on solving the Fokker--Planck

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei -- Trudy Seminara, pp. 57-61, 1983.

0090-4104/86/3206-0619512.50 �9 1986 Plenum Publishing Corporation 619

equation for the corresponding process on a circle. It seems to us that such a procedure of

stability analysis is not only of methodical interest, because in the multidimensional case

the procedure used in [2, 3] is inapplicable. Moreover, in contrast to [2, 3] we analyze the

asymptotic region of stability of Eq. (I) for ~-+0.

2. At first let us briefly recall the resultg of [I] that will be used by us. Let X(0

be a Markov process that takes its values in a k-dlmenslonal Euclidean space E g, and that

satisfies the equation

l

d x (t) = B x (0 dt + ~ a , x (t) du;', (t). (3)

Here B and er are constant matrices of dimension ~ and ~ M l , respectively, and the

~ ( t ) are independent standard Wiener processes.

Let us denote !

X

r ~ l

Q (~) = (B)~, ~) 1 + g Sp A (i)-- (A (~.))~ ~,)

~the symbol * denotes transposition).

x ( t ) ~ The process A(t)=!x(O I is a Markov process on the surface of the sphere S={[kI~---I}. If

it has a unique finite invariant measure with a density ~(~), then the necessary and suffi-

cient condition of stability of system (3) will in probability have the form

j" Q (k) p, (~,) d~, < 0. (4) S

In general the function i~(X) satisfies the Fokker--Planck--Kolmogorov equation which in

the two-dimenslonal case~A (t) ~- (cos ~ (t), sin~ (t))* has the following form (see [i, p. 277 ] ) :

where

Hence we obtain

drp (t) -----r (~ (t)) d t + ~ (~ (t)) d W (t),

l

t--I

(q~) _- - -~* (q0) B~, (q~) + ~ * (rp) A (~, (q~)) X (rp)

d~

�9 0 (rp)= --co-- sinq0 cos q~ (k + 4~2 cos2 q~) #~ (cp) =4~2 cos4 rp

Q (;L) = - - k sin 2 q~ + 2a 2 (cos 2 q~- 2 sin ~ q~ cos 2 q0.

(5)

The diffusion $2(~) vanishes for ~=4-~ However, at these points the drift coef-

ficient ~(~) of the process ~(t) is equal to ~o. Therefore, we shall have the first of the

620

possibilities indicated on p. 279 of [i], i.e., the process ~(t) is ergodic. We shall show

that irrespective of a singularity in the diffusion coefficient, it is possible to find the

density of an invariant measure of this process by solving the Fokker-Planck-Kolmogorov equa-

tion

1 d ' d (,2 (~) ~)_ ~ (.(rt~) = o 2

under t h e a d d i t i o n a l c o n d i t i o n s of n o r m a l i z a t i o n

(6)

2~

~" ~, (9) a ~ = : 0

and of continuity on the circle, i.e.~

v, (o) = ~ (2n). The general solution of Eq. (6) is

(~) = C, (W (~) ~2 (~))-, + C2 (W (~o) ,2 (~))-, j" w (u) au, 0

where (see [i]),

l[/(o~,=exPl--2!~dco}=co~eXp{2JK,(stgs~-{-r176176 �9

the interval [

The only combination of constants C: and C2 for which the function ~(~) is integrable in

yields the expression

-y (7)

2<~<~, ~(~+~=~(~).

Formula (7) coincides with the expression obtained in [2, 3] in a more complicated way. From

(7) and (4) we obtain the necessary and sufficient condition of stability of Eq. (2).

It has the form

2

d 1 S ,[2a2 (cos 2 ~--2 sin2 ~ cos2 ~)--k sin 2 ~I ~ eo--~ exp {--1(8 tg~ + ~ tg2~ q_~ ig~)} X

2

) < f 1 - ! co a k 2 (8)

The number of parameters of the problem can be diminished by introducing the reduced

friction coefficient k: and the reduced spectral noise density a:2 , i.e.,

/~t ~ _~k ; ~ t 0"12 ~_~- - - .

621

It is also expedient to effect a change of variables

t g ~ = x , t g u = y .

Thus the trivial solution of Eq. (2) will be stable in probability in the large if and only

if the parameters kl and ~i 5 are such that

-V tvTx)+l dv<O. �9 3 (9)

--oo --oo

T h i s c o n d i t i o n c o i n c i d e s w i t h t h e c o n d i t i o n ( 3 . 5 ) - ( 3 . 6 ) i n [2] i f we c o r r e c t t h e e r r o r i n

formula (3.6) of [2].

3. The condition (9) is not convenient for applications. It is possible, of course, to

study the dependence of the critical noise density al ~ on the friction coefficient kl with

the aid of a computer, as was done in [2, 3]. However, in the case which is of greater in-

terest for applications, i.e., when ~15<<I and k1<<l, it is possible also to carry out an

analytic study.

By effecting in the inner integral a change of variables Z=-- Z- we can reduce condi-

tion (9) to the form

co

i { ( r } f I- 5 l--x' klx'~ z__ x2+l__2xz + z5 + k~ ~zei U~-fi), l+xt jdx exp e , a-~- ( z S - - z x ) d z < O . (10)

It follows from (i0) that the boundary of the stability region for a1!-+O lies in a neighbor-

hood of the straight line kI-----71~i 2 for a positive y1>O. For convenience let us denote ~15=~

and write this boundary in the form k1=--ylS-~y282-~y383 @o(s 3) ; thenwe try to find the constants

Yl, 75, ?3, by equating to zero the left-hand side of (i0). Hence we obtain the equation

i(2_O-x,) (v,+v,8+~,,'+o(,'))x,)dxX \(1 + x*) t 1 + x*

=o

• S + tz - + + 0

+ ~r 2 (2 5 - zx ) + o (,5)} d z = O. ( l l )

By e v a l u a t i n g t h e i n n e r i n t e g r a l i n (11) by t h e L a p l a c e method ( w i t h t h e a i d o f Theorem

I.i of [4], p. 35), and by substituting the result into (ll), we obtain Y1 =I, Y2 =0, Ya=~ 1.

41 Hence kl -~-" S --~- S8 "~ -]- 0 (S3)"

By returning to the original notation, we obtain

41 a ~ ~k = ~ + -~ gv + o (~6) (a -+ 0). (12)

Thus, in the case of a low noise intensity the critical (bifurcation) value of the fric-

tion coefficient can be approximately obtained by the formula

622

64 0 2 .

In conclusion, let us note that the assumption of a small noise Correlation interval of

the process ~(t) can apparently be dropped for ~i -+0, since in this case we can use the aver-

aging principle for Markov processes [5].

LITERATURE CITED

i. R.Z. Khastminskii, Stability of Systems of Differential Equations under Random Disturb- ances of Their Parameters [in Russian], Nauka, Moscow (1969).

2. F. Kozin and S. Prodromou, "Necessary and sufficient conditions for almost sure sample stability of linear Ito equations," SlAM J. Appl. Math., 21, 413-424 (1971).

3. R.R. Mitchell and F. Kozin, "Sample stability of second-order linear differential equa- tions with widehand noise coefficients," SlAM J. Appl. Math., 27, 571-605 (1974).

4. M.V. Fedoryuk, The Saddle-Point Method [in Russian], Nauka, Moscow (1977). 5. R.Z. Khas'minskii, "On the operation of a self-oscillatory system under weak noise,"

Prikl. Mat. Mekh., 27, No. 4, 683-688 (1963).

ASYMPTOTIC PROPERTIES OF PARAMETER ESTIMATES

OF GENERALIZED AUTOREGRESSION SCHEME IN THE UNSTABLE CASE

O. V. Lepskii

In this paper we consider the asymptotic properties of parameter estimates in a model

described by the equation

L(O)F=~, (1) where the vectors Yr=(vl ..... , yG) and ~=(~i ..... ~n) are defined on the probability space

(~, ~, P) and they take their values in R I. The components of the vector ~ are pairwise

independent, i.e., E~m~O, 00C@~@ l, o0~S~ I, E~m2=~m(Oo, ~o), Eto~O,

li ~r i = j

L(Oo)= (0o) ~ i = j + l otherwi~.

Such schemes are resulting, for example, from discrete observations of objects described

by stochastic differential equations, and from models of autoregression type in the case of

missed or nonequidistant observations. Such a scheme differs from the conventional scheme by

the dependence of the dispersion matrix of the vector ~ on the unknown parameter 00 �9 Let

m-- I n--1

7=! m=0

The behavior of the process (i) is determined in many respects by the quantities Lm and M n.

In this paper we shall consider the case that these quantities are increasing in a certain

Translated from Problemy Ustoichivosti Stokhasticheskikh Modelei -- Trudy Seminar a, pp. 61-71, 1983.

0090-4104/86/3206~0623512.50 �9 1986 Plenum Publishing Corporation 623