AN A PRIORI ESTIMATE FOR DISCRETE APPROXIMATIONS IN

SIAM J. CONTROL AND OPTIMIZATIONVol. 34, No. 4, pp. 1315-1328, July 1996

() 1996 Society for Industrial and Applied Mathematics011

AN A PRIORI ESTIMATE FOR DISCRETE APPROXIMATIONSIN NONLINEAR OPTIMAL CONTROL*

ASEN L. DONTCHEVt

Abstract. We examine the convergence of an approximate discretization applied to the first-order optimalityconditions for a nonlinear optimal control problem with convex control constraints. Under an assumption of thecoercivity type we prove the existence of an optimal control for the original problem such that its L distance fromthe approximating sequence is proportional to the error. In a corollary we give conditions for Riemann integrabilityof the optimal control.

Key words, optimal control, discrete approximations, error estimates

AMS subject classifications. 49M25, 65K10

1. Introduction. In this paper we obtain an a priori error estimate for a discrete approx-imation provided by the Euler scheme to a nonlinear optimal control problem with convexcontrol constraints. We show that if the sequence of successive approximations satisfies acoercivity-type condition, then for a sufficiently fine approximation there exists an optimalcontrol, the L distance from which to the approximating sequence is proportional to theerror.

An excellent survey of earlier works on computational optimal control, including discreteapproximations, is contained in Polak [48]. Based on an unpublished work by J.-R Aubinand J.-L. Lions, Daniel [16] (see also his papers [14], [15]) developed an abstract approachfor proving convergence of discrete approximations of optimal control problems, parallelto the direct method of the calculus of variations (but formally independent of it; see also[55]). In a series of papers Cullum [11]-[ 13] showed value and solution convergence of Eulerapproximations, applying virtually the same idea. In 11 the problem is linear-convex, 12]considers problems of Mayer’s type that are linear in control, and [13] is an extension of12] for a state and control constrained problem. More recent results on convergence analysisof algorithms involving discrete approximations are contained in [26], [37], [46], [49]-[51 ],[61]. There is also a body of work in Russian focusing mainly on convergence of discreteapproximations combined with Tikhonov’s regularization; see [2], [51-[7], [28], [38], [54].

As typical in perturbation analysis of optimal control, the sequence of optimal valuesof perturbed (approximating) problems converges to the optimal value of the correspondingrelaxed problem (e.g., in the sense of Warga [59]); then the value convergence is equivalent tothe relaxability of the continuous problem. This was observed first by Mordukhovich [43]; seealso [44] and [25, Ch. 6]. In an earlier paper Cullum 13] proved that a sequence of solutions ofthe discretized problem converges to a solution of the relaxed problem; instead of relaxabilityshe used necessary conditions for optimality of the relaxed problem.

Compared with the extensive literature on computational optimal control, see the mono-graphs [30], [39], [47], [53], [54]; there are relatively few results available providing errorestimates for discrete approximations in optimal control. Hager [33] considered higher or-der schemes applied to unconstrained problems, obtaining error estimates under appropriatesmoothness assumptions. Related results are given in [52]. Error estimates for the Eulerscheme applied to state and control constrained convex optimal control problems were de-rived in [17]; see also [1], [40]-[42]. As usual in numerical analysis, the estimation of theerror provided by a discrete approximation is limited by the regularity of the solution. In a

*Received by the editors June 19, 1994; accepted for publication (in revised form) March 28, 1995. This researchwas supported by National Science Foundation grant DMS 9404431.

Mathematical Reviews, 416 Fourth Street, Ann Arbor, M148107.

1315

1316 ASEN L. DONTCHEV

recent paper [21 we consider a nonlinear optimal control problem obtaining an a posteriorierror estimate. More specifically, we show in [21 that, if an optimal solution of the contin-uous problem satisfies a coercivity-type condition, then for a sufficiently small mesh spacingh there exists a local minimizer of the corresponding discrete problem at a distance from theoptimal solution that is proportional to the averaged modulus of continuity r(u*; h) of theoptimal control u*. The only assumption for the regularity of the optimal control is Riemannintegrability; i.e., the optimal control may have infinitely many points of discontinuity (butmust be almost everywhere continuous). If the optimal control is of bounded variation, thenwe automatically obtain that the error is O (h).

In a series of papers Hager [32], [34]-[36] studied approximations to dual problems inoptimal control. Dual problems can be defined in various ways, depending on the assumedspaces for the variables and the constraints. By choosing appropriate spaces one may obtaindual problems that are tractable numerically. The dual approach is exceptionally efficient whenone deals with entropy-like minimization problems; see, e.g., [4]. In [19], [20] we applieddual methods to a class of best approximation problems with applications in computer-aidedgeometric design.

Another approach is based on the dynamic programming formulation related to theHamilton-Jacobi equation. The Hamilton-Jacobi equation typically has a nonsmooth (vis-cosity) solution whose computation requires special procedures. An overview of a priori errorestimates for the approximation of optimal control problems by discrete models is given in[9]; see also [3], [8], [10], [29], [31].

Related to the subject of the present paper is the work on discrete approximations ofdifferential inclusions; for a survey see [24]. In a recent paper [23] we approximated theHausdorff distance between the sets of solutions to a controlled boundary value problem andits discrete approximation. For other results in this direction, including higher order schemesand error estimates for the reachable set, see [56]-[58], [60].

Although discrete approximations are primarily intended for computations, they are alsoa useful tool for obtaining purely theoretical results. The idea of using broken lines to proveexistence of solutions to differential equations goes back to Euler. Apparently one can usevarious approximations, depending on the purpose of the study. For instance, one can de-rive necessary optimality conditions from necessary conditions for a discretized problem bypassing to a limit when the step size goes to zero; for a recent paper using this approach see[45]. Depending on the kind of approximation one may obtain various necessary or sufficientoptimality conditions.

In 2 we present our error estimate. A proof of this result is given in 3 and is based onan abstract lemma of inverse mapping type for a sequence of set-valued maps. As a corollarywe obtain that if the problem has a unique optimal control and a certain coercivity conditionis fulfilled for any regular partition, then the optimal control is Riemann integrable. Section 4contains a discussion of numerical results.

2. The error estimate. We consider the nonlinear optimal control problem

(1) minimize g(x(t), u(t))dt

subject to

2(t) f(x(t), u(t)) for a.e. 6 [0, 1], x(0) a;

u(t) U for a.e. [0, 1],u LC,x Wl’,where x(t) Rn, U is a closed and convex set in Rm, g R x Rm -+ R, f R x R -+ Rn,and a is a fixed vector in Rn. Throughout L is the space of essentially bounded vector

DISCRETE APPROXIMATIONS IN OPTIMAL CONTROL 1317

functions in [0, 1] with the norm u II.o= ess sup{lu(t)l 0 < < and W1, is the spaceof functions with values in R that are Lipschitz continuous on [0, 1 equipped with the normx IIw,o=ll x I1o + I1oo. We denote by I" any norm in R and by the superscript r

the transposition.Assuming that the functions f and g are continuously differentiable, we write the first-

order conditions (Pontryagin maximum principle) as a variational inequality of the form

(2) 2(t) f(x(t), u(t)), x(O) a;

(3) (t) -VxH(x(t), u(t), X(t)), X(1) 0;

(4) VuH(x(t), u(t), X(t)) N(U; u(t)),

for a.e. 6 [0, 1], where X is the adjoint variable, H is the Hamiltonian, H(x, u, X)g (x, u) + Xrf(x, u), VxH is the derivative of H with respect to x, and N(U; u) is the normalcone to the set U at the point u; that is,

{y 6 Rm yr (v u) < 0 for each v 6 U}N(U; u)

0

Suppose that to solve problem (2)-(4) we use a discrete approximation provided by theEuler scheme. More specifically, let N be a natural number; let h 1/N be the mesh spacing;let ti ih; and let xi, ui, Xi denote approximations of x(t), u(t), and X(t), respectively, at

ti. The Euler scheme applied to the variational inequality (2)-(4) results in a finite-dimensionalvariational inequality. There are various numerical techniques for solving such problems; inthis paper we shall not discuss this topic. We suppose that, after applying a numerical procedurewith certain accuracy, a sequence of vectors (xN, UN, XN) C= RNn X RNm X RNn is availablethat satisfy the following approximation to (2)-(4):

(5) Xi+l Xi + hf(xi, Ui) -1" h3, xo a;

(6) Xi Xi+I -- hVxH(xi, lli, Xi+I) -- ho/N, XN O"N"

(7) VuH(xi, bli, Xi+I) At- KiN N(U; lli) O, N 1.

Here the vector/N (3/N,/.N, ,O"N,x/N),3/N Rn,0.N, E Rn,tc/N E Rm,i 0,1,...,N-1, cr N G Rn represents the accuracy ofthe algorithm used for solving the discretized problem.We denote by xv (.) and XN (.) the piecewise linear and continuous extensions over [0, 1] ofxN and XN, respectively, and by uv(.) the piecewise constant extension of uN over [0, 1],which is continuous from the right across the grid points ti.

in our previous paper with W. Hager [21] we showed that if a given optimal control u*and the corresponding state x* and adjoint variable X* for the continuous problem (1) satisfycertain conditions, then there exists a solution of the discretized problem that is "close" to(u*, x*, X*). In this paper we are concerned with the converse estimate; namely, we considerthe distance from a given sequence of solutions of the approximating problem (5)-(7) to theset of solutions of the continuous problem (1). More precisely, in Theorem 1 we give ananswer to the question of under what conditions on a sequence of successive approximationsxN, uN, XN there exists an optimal control u* of the original problem (1) such that its Ldistance from uu (.) is proportional to the step size h and to the error eN.

THEOREM 1. Assume that the sequence (xiN uy, L/N) is contained in a compact set 2( CRn x Rm x Rn for all O, 1 N andfor all N and that thefunctions f and g are twice


continuously differentiable in an open set containing 2(. Let there exist a constant ot > 0 suchthatfor every sufficiently large N

(8) fo fo[x (t)Qu(t)x(t) + 2xr (t)Su(t)u(t) + u7 (t)Ru(t)u(t)]dt > ot lu(t)12dt

for all x(.) E wl’2, u(.) E L2, x(0) --O, 2(t) Au(t)x(t) + BN(t)u(t),u(t) U- Ua.e. in [0, 1], where AN Vxf(XN, uN), BN Vuf(XN, uN), RN V2uuH(xN, uN, )N),SN V2xuH(xN, uN, )N), QN V2xxH(xN, uN, ,,N). Then there exist positive constants c,and an integer N* such thatfor every N > N* for which

max I/NI(9)0<i<N-1

there exist an isolated local minimizer (X*N (’), U*N (’)) of the continuous problem (1) and a

corresponding adjoint variable L*u (.) such that

X*N (’) XN (’) IIw,. + ,,N (.) ,N (.) IIw’.

(10) + u*N (’) uN (’) IIL__< c(h + max0<i<N-11/N I).

In the statement of the theorem we use the uniform grid in [0, 1], i.e. with a constant stepsize h. The same result holds for any regular partition of [0, 1] in which the maximal step sizegoes to zero. From Theorem 1 we obtain the following corollary.

COROLLARY 1. Let problem (1) have no more than one optimal solution, let

lim max I/NI 0,N cx O N-1

and letfor every regularpartition of[0, 1] in N intervals the sequence (XN, UN, N) obtained

from (5)-(7) satisfy coercivity condition (8). Then there exists a (unique) optimal controlfor(1), which is Riemann integrable.

Proof. Theorem 1 implies that problem (1) has a solution (x*, u*) that is unique andsatisfies (10). Then for every regular partition {ti U

i=0’

maxo<i<N-lSUPti<_t<t,+lu*(t) -u*(ti)l -+ 0 as N --+

which implies Riemann integrability of u*.The coercivity condition (8) is a stability-type condition which implies that the inverse

of the linearization of (5)-(7) is Lipschitzian around the reference sequence uniformly inN. For instance, it holds when the Hamiltonian is convex in x and strongly convex in uin a neighborhood of a solution and the sequence of successive approximations is in thisneighborhood. Practically, during computations one may check whether there exists a constantc > 0 such that

for all N, x, u, and t, which is a computationally available procedure. The consistency con-dition (9) means that the error of solving the discretized problems is to be sufficiently smallwhen increasing the number N. Note that a finer discretization results in higher dimensions ofthe discrete problem and, in turn, is more likely to lead to accumulation of various errors ac-companying the computations. The practical validity of the error model adopted in Theoremmay vary from problem to problem and from computer to computer.


3. Proof of Theorem 1. The proof is based on the following abstract inverse-function-type result for a sequence of set-valued maps.

LEMMA 1. Let (X, p) be a complete metric space, let Y be a linear normed space, andlet Ba (x) be the closed ball centered at x with radius a. Let N be a sequence in X, let (R)N bea sequence ofset-valued mapsfrom X to Y with 0 (U(U)for all N, and let )U X --+ Ybe a sequence offunctions. Suppose thatfor each N there exists afunction qU Y --+ X withthefollowing properties.

(A1) u kilN(O) and there exists a constant > 0 such thatfor every N

N(Y) (R)vl(y) for every y B#(O).

(A2) There exists a constant , > 0 such that kiiN is Lipschitz in B(O) with constant

for every N.(A3) For every > 0 there exists ot > 0 such thatfor every N

)N(U) --(N(I)) I1 p(u, o)

whenever u, v e B(N), N 1, 2Thenfor every c > y there exists A > 0 such thatfor every Nfor which

N(N) A

there exists v e (()N N)-1 (0) satisfying

(11) P(b#/, ’N) --. C )N(N) II.

Moreover, if (R)1 is single-valued near O, then there exists exactly one v with the aboveproperties.

ProofofLemma 1. Let c > y and let > 0 be so small that

(12) 0 < ’e < and < c.1 -e,

Let A > 0 be such that

(13) A< min{ fl c}ce+l’c

and let

(14) rN --c qbN(N) II-Define the function

dPN(X

Let N be such that N(N) I1_< A, and letx e BrN(N). Using (A3), (13), and (14) we have

N(x) I1--<11 qu(x) --N(N) + N(N) II--< 6rN -k- A <_ A(1 q-ce) _</3.

Then from (A2), (A3), (12), and (14) we obtain

P(N, dPN(X)) P(qN(O), klIN()N(X))) y )N(x)

Y )N(x) --(U(U) -[" N(N) 11 ’rN -b ’rN/C < rN.


Hence, (I)N maps BrN (N) into itself. Moreover, if x’, x" BrN (N), then

p(dPN(X’), di)N(Xtt)) p(tllN((/)N(X’)) klIN((/)N(Xtt)))

<-- / CN(x’)- CN(Xit) II_< p(x’, x") < p(x’, x").

Thus PN is a contraction from Bru (N) to Bru (N). By the contraction mapping principle forevery N there exists 6 Bru (U) such that / U(). The definition of U yields that

6 ((R)U CU) -1 (0). If (R)V is locally single-valued near 0, then is the unique fixedpoint of U in BrN (U). This completes the proof of Lemma 1. [3

ProofofTheorem 1. We apply Lemma with the specifications

X { (x, ,k, u) 6 W’ W1’ x L, x(0) a}, Y L Rn,

Jc f (x, u) (JC N) .ql_ AN(X XN) -[- BN(U UN)

+ VxH(x, u, ) ( N) Aru(k N) QN(X xN) SN(U UN)

--VuH(x u, I) 21- Vu HN -- RN(U uN) 2t- STN(X XN) -JI- Ov( AN) Ku

(T N

{N(

( N) AN(X XN) BN(U UN)( fN) 21 ATN() .N)

__QN(X xN) ql_ SN(U uN)

--VuHN RN(U UN) STN(X XN) B() N) -4y KN .qt_ N(U; u)

,k(1) --O"N

where VuHN(t) VuH(xN(ti),uN(t),)N(ti+I)) for [ti, ti+l) and N,r]N, KN areassumed piecewise constant and continuous from the right across the time grid. Clearly,0 (U(U).

Let (p, q, r) 6 L, s Rn, and denote y (p, q, r, s). Then 6 (R)v (y) if and onlyif (x,), u) satisfies the relations

(15) Jc ANX + BNU + p + OlN, x(O) a;

(16) --ArN QNX SNU + q + jN, )(1) S .ql_

(17) RNu + Svx + BfvL + r + yg e N(U; u)

for a.e. e [0, 1], where OlN jN ANXN BNtIN, 1N fN -Jr- ATN)N + QNxN +SNttN, YN VuHN RNuN S;xN B)N teN. We show first that there exists afunction tPu(y) (XN(y), IN(Y), UN(y)) e (R)v(y) that satisfies conditions (A1), (A2) ofLemma 1.

Under the coercivity condition (8) the system (15)-(17) is equivalent to the linear-quadratic problem of minimizing

(18) --X(1)T (s+crN)-FO.5 [xT QNx+uTRNu-F2xT SNu--(q+flN)Tx+(r+yN)Tu]dt

subject to

(19) Yc ANX + BNU -Jr- p + OlN, u(t) U for a.e. 6 [0, 1], x(0) a.


Let wN(p) be the solution of

ANW "+" p + OIN, w(O) a.

Changing the state variable x z + tON(p) we obtain the following problem equivalent to(18): minimize

(20)

-z(1)z (s + O"N) -- 0.5 [ZT QNZ 2r- u T RNU -+- 2Z SNU

--(q + N 2QNWN(p))Tz -+- (r + ’U + 2SvWN(p))Tu]dt

subject to

(21) , ANZ 2r- BNU, u(t) U, a.e. 6 [0, 1], z(0) 0.

Let N L2 L2 be the linear and bounded input-state map for system (21); that is,

(NU)(t) N(t, V)BN(V)u()d,

where N is the fundamental matrix solution to ANZ. Note that the conjugate map is

(rNX)(t) BN(t)r dPN(T, t)rxQ:)dz,

and, since the components of AN and BN are bounded in L, there exists a constant C

independent of N such that the operator norm satisfies

(22) N I1 C1.

Denote

CN[S](t) --Bv(t)N(1, t) (s + O’N)

I)N(Y) Tu(--q fiN -- 2QNtON(p)) -- r + ?’U + 2SvWN(p) + CN(S),

(23) 7ZN CrNQNCN + 2NSN + RN;

and let {u 6 L2 u(t) U for a.e. 6 [0, 1]}. Then problem (20) can be rewritten as

(24) minimize 0.5(u, "[’N u) 2t- (VN(y), U) subject to u 6 Q,

where (., .) is the L2 scalar product. The set is a closed and convex subset of the space L2

and the coercivity condition (8) is equivalent to

(25) (u, "Nu " Ol u 112 for all u 6 fa- .It is a standard observation that, under (25), for every y 6 L x Rn there exists a uniquesolution uN(y) of the problem (24). Thus there exist an unique optimal state xN(y) and anunique optimal adjoint variable )vN(y); that is, the function qN(Y) (XN(y),)’,N(Y), UN(y))is well defined for all y 6 L x Rn. Hence assumption (A1) of Lemma 1 holds. To show that


qN(Y) is Lipschitzian we apply a well-known argument; see, e.g., [18, Ch. 2]. For any" satisfyy’, y" 6 Y the corresponding solutions uN, uu

(’]N u’ 2c- VN(y’), U" U’) >_ O,

(’JN ut’ -- VN(y"), U’ U") > O.

Combining these two inequalities with (25) we obtain that

1(26) u’ "N- /AN IILz- VN(y’)- VN(y")I1.

Using (22) and the boundedness in L of the matrices AN, BN, QN, SN we conclude that forevery y Y there exists a unique optimal control UN(y) of problem (18) that is Lipschitzian iny from L to L2 with a Lipschitz constant independent of N. The Gronwall lemma and (22)applied to the state and adjoint equations yield that the optimal state xN(y) and the optimaladjoint variable )N(Y) are Lipschitzian with respect to y from L x R to W1’2 uniformly inN. Furthermore, the coercivity condition (8) implies that for some c1 > 0,

uTRN(t)u >_ CllUl2 for a.e. 6 [0, 1],

whenever u 6 U U; see [27]. By repeating the argument in deriving (26) for optimalitycondition (17), we obtain that the optimal control UN(y) is Lipschitzian in y from L toLc uniformly in N. The Gronwall lemma applied to the state and adjoint equations yieldsthat XN(y) and )N(Y) are Lipschitzian in y from L x Rn to W1, uniformly in N. Hence,condition (A2) of Lemma 1 is satisfied.

Let us show that 4N satisfies (A3) at N(’) (xN (’),)N (.), UN (.)) uniformly in N as afunction from W’ x W1, x L x R to Lc. Denote (x, u) and consider the firstcomponent of bN. From the continuity of the derivative Vf it follows that for any > 0 thereexists c > 0 such that for every ( with ( fin IIL <_ ,

vf(() Vf((N IILo .If ’1, ’2 Bot((N), then

f(l)- f(2)- Vf(N)(’I- 2)IILoo ’- ’2

The proof of (A3) for the remaining components of qN is completely analogous.Thus we can apply Lemma 1, obtaining that there exist constants c and A such that if

U(U) < A, then there exists , 6 ((R)U --4U)-1 (0) satisfying (11). It remains to estimate4N(N) I[- The relations (5)-(7) imply that U X and satisfies

Jc(t) f (x(ti), u(t)) + 3N (t);

.(t) -VxH(x(ti), u(t), )(ti+l)) + 0N(t), 3(1)

VuH(x(ti), u(t), )(ti+l)) + teN(t) - N(U; u(t))

for [ti, ti+l), 0, N 1. Applying the Gronwall lemma to the adjoint equationand using the assumption that xN (.), uN (.) are bounded in L and f, g and their derivativesare continuous, we obtain that )N (.) is bounded in L. Hence, the sequence of the derivatives(kN (.),)N (.)) is bounded in Lo. Then


CN(N) I1 max0<i<N-1 suPti<t<_ti+ {[l f(xN (ti), lgN (t)) f (xN (t), lgN (t))l

+ ]VxH(xN(ti), uN(t), )vm(ti+l)) VxH(XN(t), uN(t), )vN(t))[

+ IVuH(xN (ti), blN (t), .U (ti+l)) VuHN (x(t), lg

N (/), AN (t)) l]

+ IE/I + Irgl}

< c2h + max0<i<N-11qN I,

where c2 is a constant independent of N. Take 3 > 0 and N* such that c2/N* + 3 < A.From Lemma 1, if N > N* and maxo<i<N-11qNI _< , then there exist x*N, u*N, and ).Nsatisfying the estimate (10) and the maximum principle (2)-(4). The last step of the proof is toshow that (x*N, u*N) is an isolated local solution of (1). Clearly, the values of the derivativesof f and H at (x*N u*N ,k’N), denoted A* * * *

N, BN, QN, SN, RN, are close in L to the valuesof the corresponding matrix functions AN, BN, QN, SN, RN defined in the statement of thetheorem. Then the operator Rv defined as in (23), where AN, BN, QN, SN, RN are replacedby Av, Bv, Qv, Sv, Rv, satisfies (25). It is known that (8) (or, equivalently, (25)), togetherwith the maximum principle (2)-(4), is a second-order sufficient condition for an isolated localminimum; see, e.g., [21, App. 1 ]. The proof is complete.

4. A computational example. If the final state in problem (1) is fixed, e.g. x (1) b,then using Lemma 1 one can easily prove a result completely analogous to that in Theorem 1on the additional assumption that there exists c > 0 such that the ball Bc(b) is in the reachableset of the linearized system (15) for all N (for a related analysis see [22]). As an illustrationwe consider the problem

subject to

minimize 0.5 u(t)2dt

31 X2,

-2 /g,

Xl (0) 0, Xl (2) 5/6,

X2(0) 0, Xl (2) 1/2,

U >0, [0,2].

The optimal control is a nonsmooth function in time, that is,

1-t for [0, 1),u*(t)

0 for [1, 2];

and the first-order optimality conditions result in a boundary-value problem with nonsmoothright-hand side

3 X2

)2 max{0, )v2},

1 --0,


2.4

2.3

2.2

2.1

1.9

1.8

1.7

1.6

10a

10- 10" h

FIG. 1. The dependence ofthe relative accuracy on h for 10-6.

(27)

Xl (0) 0, Xl (2) 5/6,

x2(0) 0, x1(2) 1/2,

We applied the Euler discretization scheme to the latter problem and solved the resulting finite-dimensional problem by the shooting method with a quadratic penalty. The unconstrainedoptimization problem is solved with the help of the BFGS code of MATLAB on HP Apolloworkstation 715/50.

It turns out that even for the simple linear-quadratic problem considered one observeseffects in the line of the theoretical analysis. Figure 1 shows the dependence of the relativeaccuracy u* (.)-uN (.) I1 ! h on the mesh spacing h when the tolerance in the stopping test ise 10-6. We see that this relative accuracy is bounded, hence the convergence rate is O (h) asproved theoretically. Note that the computational time for h 5 x 10-4 was 1.9542 x 104 sec.In Fig. 2 the same dependence is obtained for the tolerance e 10-3. in this case, forh < 5 x 10-3 the relative accuracy significantly increases; that is, for a tolerance comparablewith the mesh spacing, there is no first-order convergence. Figure 3 shows the exact and theapproximate optimal controls for e 10-6 and h 0.05.

An area for future research is error analysis of higher order approximations (e.g., Runge-Kutta schemes) applied to constrained optimal control problems. We did some computationsfor problem (27) with the standard second-order Runge-Kutta scheme. The dependence ofthe relative accuracy on h is given in Fig. 4; it indicates O(h2) convergence.

In the author’s opinion the abstract result in Lemma can be applied to variationalproblems with integral state and control constraints, as well as with systems governed byabstract (functional or partial) differential equations. The main steps in such an analysis willbe to prove the consistency and stability of the approximation considered. A direct applicationof the abstract scheme may fail for stiff systems where, in general, the stability condition fails.


10

10

10

10-1

10"’ 103

10-2

10 10h

FIG. 2. The dependence of the relative accuracy on h for e 10-3

FIG. 3. The exact and approximate optimal controlsfor h 0.05 and 10-6.

The presence of state constraints considerably complicates the analysis of nonlinear opti-mal control problems. The main difficulty here is to find a compromise between the require-ments for differentiability of the functions involved and the coercivity conditions guaranteeingstability. An O(h) estimate for the L2 norm of the error for the optimal control was obtainedin 17] for a convex problem with linear inequality state and control constraints. We do notknow whether such an estimate holds with a stronger norm, for example, in L.


100

0_2

103031 102 101 h 10

FIG. 4. The dependence of the relative accuracy on h for the second-order Runge-Kutta methodfor e 10-6.

Acknowledgment. The author thanks Ilya Kolmanovsky for his help in computations.

REFERENCES

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