On continuous dependence of controls upon parameters in fixed-time free-terminal-state optimal control

Automatiea, Vol. 6, pp. 289-295. Pergamon Press, 1970. Printed in Great Bntmn.

On Continuous Dependence of Controls upon Parameters in Fixed-Time Free-Terminal-State Optimal Control* Sur la d6pendance continue des commandes par rapport aux param&res dans la commande optimale h temps fixe et 6tat final libre Uber die stetige Abhangigkeit der Steuerungen von Parametern bei optimalen Prozessen mit fester Zeit und freiem Ende der Trajektorie O nertpepbIBHOfi 3aBttCI4MOCTHynpaBzterlVlfi OT napaMeTpoB B OrITI4Ma2IbHOM ynpaB.rIennH

C IIOCTO~IHHbIM B p e M e n e M

I,I C B O S O ~ H b I M o r o n q a T e J I b n b I M

COCTO~IHHeM

A. J. K O R S A K ~ f

Continuous dependence of optimal controls upon perturbing parameters in the control law, objective function, or initial state o f a control problem is derivable for certain topologies on a f~mily of problems and its set o f controls. 1. INTRODUCTION

Summary--Let ,~a be the class of optimal control problems having given fixed dimensions for states and controls, time interval [to, tl], and control region, and assume each problem P in ~ is characterized by its initial state, control law, and integral objective functional:

J=

LEa" ~ BE a class o f o p t i m a l c o n t r o l p r o b l e m s P, characterized b y some c o n d i t i o n s on the c o n t r o l laws, initial a n d t e r m i n a l sets, objective functionals, a n d c o n t r o l regions o f the p r o b l e m s P c # . I f suitable topologies are defined on ~ a n d on the set o f all c o n t r o l s o f p r o b l e m s P e ~ , one w o u l d expect the following kind o f result to h o l d : Let a p r o b l e m P having an o p t i m a l c o n t r o l u be p e r t u r b e d to a p r o b l e m P e ~ . T h e n a n y o p t i m a l c o n t r o l o f P, i f one exists, is a p e r t u r b a t i o n o f the o p t i m a l c o n t r o l o f P. This p a p e r asserts such c o n t i n u o u s dependence o f o p t i m a l c o n t r o l s u p o n the o p t i m a l c o n t r o l p r o b l e m , u n d e r certain c o n d i t i o n s before p e r t u r b a t i o n , for a specific class 9 ' o f p r o b l e m s , n a m e l y those possessing a single initial point, a b o u n d e d c o n t r o l region, a free terminal state, a n d a fixed time interval. First, a general sufficient c o n d i t i o n is verified, a n d then it is shown that n o r m a l , linear c o n t r o l law p r o b l e m s in ~ with b o u n d e d convex p o l y h e d r a for their c o n t r o l region d o satisfy this general condition. Consequently, it is shown t h a t linear n o r m a l p r o b l e m s in ~ , when p e r t u r b e d , are " c o r r e c t l y p o s e d , " b u t in a m o r e general sense t h a n KIRILLOVA'S* [1], as further discussed in Section 3 for time o p t i m a l control, since a n y non-linear p e r t u r b a t i o n in ~ is allowed here, whereas K i r i l l o v a ' s p e r t u r b a t i o n s r e m a i n e d linear. The t o p o l o g y on ~ here consists o f a weak C ° function t o p o l o g y on the c o n t r o l law function a n d the euclidean metric o n initial points. T h e

["f°[x(t), u(t), tl dt

dto

Perturbations /3 of problems P in ~ are considered with respect to a certain topology r on ~ and the/-1 metric on the space of control functions, or controls. Essential uniqueness of an optimal control is introduced, and it is shown that any optimal control n*, for a perturbation ]5 of a problem P having an essentially unique optimal control u*, depends continuously upon/~, that is, with respect to J and the LI metric; i.e., given ~>0, there is a neighbourhood N of P in ~ such that PeN=*.d(u*, ~*) < e, where d represents the L1 metric. Also, it is shown that the optimal control for problems in ~ with a normal ** linear control law and a bounded convex polyhedron for the control region is essentially unique. We thus obtain a theorem that generalizes the type of result obtained by Kirillova in Izvestia VUZ, Matematika, No. 4 (5), 1958, pp. 113-126, which has been translated by L. W. Neustadt in SIAM Journal on Control, Vol. I, No. 2, wherein Kirillova considered time-optimal instead of fixed-time problems but allowed only linear perturbations of a linear problem. The theorems here are also related to some more recent works, which are referenced. * Received 29 January 1969; revised 31 July 1969. The original version of this paper was presented at the IFAC Symposium on System Sensitivity and Adaptivity which was held in Dubrovnik, Yugoslavia during August 1968. It was recommended for publication in revised form by associate editor P. Kokotovic. This research was supported by the United States Office of Naval Research under Contract Nonr 222(88). This paper is part of a dissertation submitted in partial satisfaction of the requirements for the Ph.D. degree in mathematics at the University of California (September 1966). t Research Mathematician, Information and Control Laboratory, Stanford Research Institute, Menlo Park, California, U.S.A.

* Note that Kirillova considered the entirely different time-optimal control problem.

**See Section 8 for a definition. 289

290

A.J. KORSAK

topology on controls here consists of the L~ metric on the space of admissible controls. 2. PRACTICAL SIGNIFICANCE The significance of the results obtained here for practical computation lies in situations where numerical errors in computing optimal controls may be interpreted as arising from a fixed perturbation of the equations of the control problem. An example of this is where quantization of states and controls appears, such as in any digital computer application, or when the problem is made discrete in time. This fact will be demonstrated in section 6. Our Theorem 3 shows that, under such practical perturbations, any optimal control obtained for a linear normal problem in N must be close to the true optimal control, in the L 1 norm. Hence, they are also close in the L 2 norm, since lu] is bounded. Practically speaking, what this states is that the true and perturbed optimal controls have nearly the same energy over any time interval. If a more stringent topology on ~ were imposed, it may be possible to derive a closer relation between the true and perturbed optimal controls, but no results in this direction are known to the author. 3. RELATED HISTORY The results obtained here generalize the type of result in the paper of KIRILLOVA [1], but for a different class of problems, in which it is shown that small linear perturbations of linear normal time optimal control problems lead to small perturbations of the optimal control. Kirillova's result was proved in a simpler way by LEE and MARKUS in Ref. [2], Theorem 22, p. 148, but only for autonomous linear control laws. Also, JANE CULLUM [3, 4] has proved some related theorems. In Ref. [3], she has derived a type of continuityunder-perturbation result for a more general class than here or in Refs. [1] and [2]. However, her "continuity" is in a weaker sense, as follows: she shows that whenever a sequence of optimal control problems Pc converges to Po in her class ~ , and under certain restrictions on Po, then for any sequence of admissible collections of control, trajectory, and initial or terminal states of the P,'s, there is a subsequence, which converges to an admissible collection for Po, and the convergence of the controls is weak L~ convergence. Her Theorem 3 in Ref. [4] extends this to weak L2 convergence of the original sequence, if Po has a unique optimal control and has a linear control law and objective integrand f ° as indicated in the summary. In our case, the convergence is strong L1. It should be remarked here that the latter type of continuity result is the only one of practical

significance in numerical computation. Mrs. Cullum obtains another result in Refs. [3] and [4] for less stringent conditions on her class ~ that a subsequence of collections, of the trajectories, and initial, terminal states, converge to a collection for Po, without reference to the controls. In addition, in Ref. [4] she derives a slightly more general conclusion than Kirillova's for time-optimal control, but she still requires that the perturbed problem be linear. To our knowledge, no results of our strong type of convergence under non-linear perturbations have been obtained for another class of optimal control problems. 4. OTHER EFFECTS OF PERTURBATIONS Our investigations of certain "natural" topologies, and specific examples of problems, have indicated that one can always perturb a problem for which there exists an optimal control to obtain one for which there does not [5]. One is thus discouraged from attempting to prove the invariance of existence of an optimal control under perturbation, at least for the general types of perturbations considered here. In Refs. [1], [2] and [4], this difficulty does not arise, since the perturbed problem is linear. Hence the existence of an optimal control for a perturbed problem shall be assumed herein wherever required. One may very well ask if there is a more precise qualitative relation between optimal controls of the original and perturbed problems, assuming that the latter exists. For example, one may expect that a problem having a "bang-bang" optimal control, when perturbed, will continue to have a similar such control, i.e. one with nearby switching times. This is not the case for the general perturbations considered here [5], however, under more restricted perturbations this appears to be the case, and we hope to publish this result in the near future. Various results have been published concerning the rate of change of the objective functional's value at the optimal control with respect to changes in parameters introduced into the law of motion or initial and terminal conditions [6-9]. This appears in the literature under the name of "sensitivity analysis." 5. FIXED-TIME FREE-TERMINAL-STATE OPTIMAL CONTROL PROBLEMS The class ~ of problems treated here is as described in ReL [10]. Essentially the same class is considered by Pontryagin et al. in Ref. [11], Chapter l, para. 8.B. Briefly, the restrictions are as follows: (1) The initial set So consists of a single point Xo, and the terminal set S~ is the entire state space E".

On continuous dependence of controls upon parameters (2) The objective functional is J e ( u ) = c x , ( t l ) , where c is a constant vector in E ~ and x , is the solution to (1) with the boundary condition x,(to)= Xo, when the feasible control u is used. (Note: all integrable admissible* controls are feasible* in this case.) It is well known that Jp(u) formulated in this way is equivalent to an integral objective function as stated in the Summary. (3) There is only one time interval / = [ t o, t~] used for all controls of all control problems in our class N. (4) U is compact. In Ref. [10] it is assumed that f ( x , u, t)in the law of motion

Yc=f(x, u, t)

(1)

is continuous in all variables and twice differentiable in x and t. In Ref. [11] it needs to be only once differentiable in x, t and continuous in x, u, t. Such assumptions are needed for applying the maximum principle. We need the assumptions of both Ref. [11] and Ref. [10] for our Theorems 2 and 3, although for Theorem 1 it suffices to assume that f i s only continuous in x, u, t and Lipschitz in x. We shall represent a problem P as described above by the triple (f, Xo, c). 6. THE TOPOLOGY ON The basis for the topology T on ~ at P is given by the sets

Ne(P)=

k

(ii)

final time tl, amounts also to regarding u as piecewise constant on the same intervals, in practical computation. This limits the family of admissible controls, strictly speaking; however, the discretized problem can be regarded as the continuous one, with the same set of admissible controls but with the f i ' s piecewise constant, since any two controls whose integrals over all intervals [tk_ 1, tk] are equal are equivalent in the piecewise c o n s t a n t f problem, provided that the set U is convex, so that averaging the ui's over the intervals It k_ 1, tk] will yield admissible controls. Thus, discretizing the problem amounts to perturbing f i n (1) to f, where f ( x , u, t ) = f ( x , u, tk), with tk < t < t k + 1. Clearly, by the uniform continuity of f on X~ x U × I, where (i) is satisfied when the intervals [tk- 1, tk] are sufficiently small; thus PeN~(P) when P = (f, x o, c) while P = (f, Xo, c). As for quantization of states and controls, the effect of this in practical computation is to have xi's, u~'s all integer multiples of some basic numbers, while the actual "quantization" is really taking place only on the ui's and the functions f i ( x , u, t) in (1); i.e. whenever the computer evaluates f i ( x , u, t), it truncates the function, although some computers take the nearest multiple of the basic number, while only discrete values of u~'s are allowed. The first effect is clearly a perturbation t o f a s per (i) above; the latter effect is a restriction on the admissible controls which can be interpreted as a perturbation f ( x , u, t) to f ( x , u, t) being constant on any interval u~
(xl Ilxll

II o-Xotl< , Ila-cll<

This is the so-called "weak" C ° function topology (cf. Ref. [12]) in cartesian product with E 2n with its usual topology. It will now be demonstrated that quantization of state and control variables, as well as discretization with respect to time, as described specifically in Ref. [4], for example, leads to perturbations in the sense of this topology z. Discretizing, as Jane Cullum points out in Ref. [4], for example, clearly amounts to perturbing ihe component functions f t in (1) to being piecewise constant over intervals t k_ 1 ~ t< t k, thus, using discrete variables x~, UR ~ for the components of x, u indexed by "time" k = 0 . . . . . K, where tK is our * Admissible controls are measurable functions mapping [to, td into the control region U; feasible controls are admissible controls which iesult in a trajectory lying in the constraint sets at all times, which in this case are the entire space except at to.

291

J

leads to satisfaction of (i) above when the quantization intervals are sufficiently small. 7. THE METRIC ON THE SET OF CONTROLS The usual metric on the space L~ of measurable functions on a dosed interval I will be assumed. Let u be these functions with range in U. 8. ESSENTIAL UNIQUENESS OF AN OPTIMAL CONTROL Definition. An optimal control u* for P will be called an essentially unique optimal control for P if d(um, u * ) ~ 0 as m--*~ for every sequence of controls {Um}~=l with Jv(um)~inf{Jv(u)lueql}. The geometric significance of essential uniqueness is closely related to normality, assumed here to be definedt as in Ref. [2], or, equivalently, the t This is quite distinct from the classical definition of "normal arcs" in the calculus of variations, as in Ref. [15].

292

A . J . KORSAK

general position hypothesis of Pontryagin, et al. [11], of a linear control law. However, it is a bit more stringent. Normality implies that the attainable set K(tl) interacts the support hyperplane with normal c, the cost vector, at a unique point.* Essential uniqueness of the optimal control, on the other hand, appears to require that the attainable set have a proper tangent cone, i.e. one that does not degenerate to a half space in any hyperplane, at the contact point of the cost hyperplane, although we have not been able to show this is sufficient for essential uniqueness. It turns out, as our Theorem 2 shows, that normality does, in fact, imply essential uniqueness. In Ref. [5] a counter example is given where the optimal control is unique but not essentially unique, for the above reason. This example is as follows: Denote the 2-vector by (x, y) and let u be a scalor control, with a > 0 and let Po be

5c= 1 -- ay 2

I - [ 0 , 1"]

ee,t

f

u=[--l,

J.(u)=x(O. Clearly the control ,~.(t) = c o s

n~t

yields the trajectory

x,(t)=t, y,(t) =

sin nnt

nig

,

0_
which is obviously optimal for Pc, and uniquely so but the u,(t)'s do not converge in L I to the optimal control Uo(t)-O for Po. Note that this example satisfies the conditions of convergence in the topology ~.

Let x~ denote the trajectory for problem Q obtained by using the control v.

So(U)=XO). Clearly the sequence of controls for n = 1, 2, . . . 2m 2"

2m-2 2m-1 --1, ~ t < - 2" 2"

m = l , 2 .....

2n-1

0, t--1 is an optimizing sequence, but its L~ norm is 1, while the obviously unique optimal control is Uo(t) - 0 . Thus d(u,, ao)-=l, and Uo is not essentially unique. Note, however, that the u,'s do converge to Uo weakly in L 1, which they should, since this example, when trivially phrased in a nonfixed time context, fits the conditions of Jane Cullum's Theorem 2 in Refs. [3] and [4]. Note also that the attainable set here does not have a "corner" at (1, 0), which is the optimal terminal point, but is "smoothly round" there. It can be easily shown that the above problem is not "correctly formulated." Let us take a sequence of problems P~. with e = 1In as follows: * See Theorem 3, p. 76, Ref. [2].

2

9. STATEMENT OF RESULTS

x(0) = y ( 0 ) = 0

a,(t) =

'

y = ll

~=u Po

2m - 1 1, - - _ < t < - 2"

I=[0, l]

~ = 1 - a ( y-sinnrt mrx)/

Theorem I. Suppose Pg~ has an essentially unique optimal control u* and x,e is uniformly bounded for all u in q/. Let {P,,}~=I be any sequence of problems in ~ converging to P with respect to the topology z, and having optimal * respectively. Then d(u*, u * ) o 0 as controls urn,

m---~oo.

Theorem 2. Suppose that the f in the equation of motion (1) takes the formf(x, u, t) = A(t)x + B( t)u, in matrix notation, where x, u are column vectors, and A(t), B(t) are matrix functions on 1 satisfying the so-called general position hypothesis relative to U as described in Ref. [ll], p. 116, pp. 182-183. Suppose also that the compact set U is a convex polyhedron. Then there exists an essentially unique optimal control u* for the problem P = ( f , x o, e).

Note. When f is linear with continuous matrix functions as in Theorem 2, it follows from the fact U is compact that the "forcing functions" B(t)u(t) are uniformly bounded for all ueU. Hence the "variation of constants" formula, e.g. Ref. 13, pp. 74-75, yields uniformly bounded trajectories x e over all u in 0?/, and so Theorems 1 and 2 can be combined as follows: Theorem 3. Let the fixed-time free-terminal-state optimal control problem P e ~ have a law of motion of the form 5¢= A ( t ) x + B ( t ) u ,

On continuous dependence of controls upon parameters where A(t), B(t) satisfy the general position hypothesis relative to the range of the controls, which is a compact, convex polyhedron. Then:

293

From Lemma I it follows that lim

IsP,,,(u*)- Sp(u*)] =o,

(4)

~1"-* O0

(1) a unique optimal control u* exists for P

lim IJe~(u*)-dv(u*)l = 0 .

(2) for any sequence of optimal control problems Pie~ convergent to P in the topology z above, and having optimal controls u~, although these may not necessarily exist no matter how "close" P~ is to P, the u*'s converge to u* strongly in the L1 metric.

From the optimality of um and u for Pm and P, respectively, we have

Remark. The existence result here is not new. The ideas in our proof may be found in Ref. [2], Chapter 2. Our contribution lies in showing that the unique optimum is essentially unique. The existence proof is included, however, because certain results in it are needed in the other part.

Combining (4) and (5) we obtain Jp(u*m).--~Jp(u*) as m ~ o v , as was claimed. But this combined with the assumption d(u*, u*) > e contradicts the essential uniqueness of u*. This completes the proof of Theorem 1.

m---~ 0 0

Jp.(U*.) Jp(U*).

Proof of Theorem 2. Let ~b*: I~E" be the unique absolutely continuous adjoint vector function for P, satisfying

10. P R O O F S O F T H E O R E M S

Proof of Theorem 1. Suppose the theorem is false. Then there must be an e > 0, and a problem Pm in every N1/m(P) for which an optimal control u* exists with the property d(u*, u*)>8, m = 1, 2 . . . . . We claim that {U*m},= 1 is an optimizing sequence for P, i.e. Jp(u*)~Jp(u*) as m ~ Lemma 1. If P,,~N~/m(P), r e = l , 2 . . . . . then Je,(u)-Je(u)~O uniformly in ueq,¢ as m ~ o o . Proof.

de*_

a'(t)¢*

¢*(t,)=-c

dt (cf. Ref. [10]). We shall first prove that there exists a u*eql for which the maximum principle

H[x*(t), d/*(t), u*(t), t-] = sup n[x*(t), Ip*(t), u, t] a.e. on I

IJj. (u)- Jp(u) I_ Ile.-cll'llx~
+(llell + l/re)Ix,

H(x, ~b, u, t)= ~b.f(x, u, t).

(tl)-x,(tl)],

• Pn, "P IIx~ (t)-x.(t)ll<-Ilf.,(xX'~(o, u(t), t)

-f(xeu'(t), u(t), t)[[ + IIf(x~.m(t), u(t), t) -f(x~(t), u(t), t)[I _<1 +Kl[xp.(t)_x~(t)ll ' (2) where K is a Lipschitz constant for f. Denoting the inside of the norm in (2) by y(t), and using the relation Y

(6)

is satisfied, where

where R is the radius of a sphere containing the trajectories xue of P. It therefore suffices to show that x.e m (t)-xu(t) e converges to 0 uniformly for all u ~ and tel as m ~ ~ . We have

_

(5)

,

Since U is compact, the supremum in (6) is always attained on L Also, ~k* depends only on the matrix function .4 and the vector c. Thus, we can define a control u* on I by means of (6). Now the general position hypothesis implies that any such function u* is well defined by (6) uniquely a.e. o n / , and is, in fact, piecewise constant, taking values at the vertices of U. This has been proved by Pontyragin etal. in Ref. [11], Theorem 9, p. 117, in connection with a time-optimal problem, however, their proof is readily seen to be independent of the nature of the objective functional, and is not affected by making the time interval fixed. Therefore u* so defined is certainly measurable, hence admissible. Furthermore, Rozonoer proved in Ref. [10] that the maximum principle is necessary and sufficient for optimality in the case where f takes the form

f(x, u, t) = A(t)x + dp(u).

we obtain

Ily(t)tl ~ t 1-t°eK', m

which proves the Lemma.

(3)

It is clear that Rozonoer's argument holds for the extended case

f(x, u, t)= A(Ox + ok(u, t),

294

A . J . KORSAK

for any q~ continuous on U × L and it is therefore true in our case. Thus, the u* we have defined by (6) is an optimal control for P. We now proceed to show that u* is essentially unique. Let S be the finite set of points in I at which u* is not well defined by (6). Define co*(t), t d - S , to be the smallest angle ranging in (rr/2, ~r] which is attained, since U is a closed, finite polyhedron, between all allowable increments v in the control u* at t and the vector B'(t)~[;*(t), where the prime is used to denote the transpose of a matrix. That is, 09*0) satisfies cosog*(O=

v'B'(t)~*(t)

sup

I[ 1111B'0) '*(t)[I

, tel-S.

(7)

4*'(t)B(t)v(t)dt" to

dp(Um)- Je(u*) >- -.Ito~V t~*'(t)B(t)[um(t)- u*(t)]dt • 0

-k 2/

h(O)

lu~(t)-u*(t)ldt,

where

{tell(rc/2)(l + O) < co*(t) < ~ }, for 0 < 0 < 1. We claim that each 1o is measurable. Recalling that u*(t) is piecewise constant, let I ~, i = 1. . . . , K denote the open intervals into which 1 is subdivided by S, and it should be noted that S consists of a finite number K of j u m p points of u* and zero's of B'O*. The expression being maximized on the R.H.S. of (7) is continuous in t, and the allowable set of v's remains constant over each I ~. Thus o9" is continuous on each I ~. Since Io is the set where to* =co*It_s,

h(O) =inflln'(t)~,,(t)ll # 0 ( l u l -< 4Kllull, u s ~ ) • teFa

Since p ( I ) - #(IomF~) < 26,

d(u~, u*)<.(zo~r,[u,,(t)- u*(t)ldt

+ 2M6,

where M = sup

lul < ~

ue~d

Hence, given 5>0, if we choose 6=5/4M and N sufficiently large so that

Jl,(u,,) -Jp(u*)< h(O)" (sin 0/2) "e/(2 x K ) , for any m_> N, then d(um,U*) < (5/2) + (e/2) = e for m > N. This completes the p r o o f of Theorem 2, and therefore of Theorem 3.

it is measurable. Since

l=Su~

f

tl

Here it should be noted that in the nonlinear case, the adjoint function ~ , depends upon the control. Given a sequence {u,,)~= 1 of admissible controls for which Jp(u,,)--,Je(u*) as m ~ , we must show that d(u,,, u*)--+0 as m - - ~ . Let 6 > 0 be given and choose 05(0, 1] with tl(Io) >/~(I) - 6. This is possible on account of (8). Let Fa be the set remaining when open intervals of width 6/K > 0 centered on the K points of S are removed from I. Hence,

> (sln-'~ --r-_ f

Next, define Io~1 as the set

o9"-1{[(n/2)(1 +0), n]},

Jp(u + v) - Je(u) = -

w Io~,

LO~(O, l)

_]

~(s)=0,

11. CONCLUSIONS

and Io, ~Io2, we have lim #(Io) = p( l) ,

(8)

0~0

where /~ is Lebesgue measure. We now use the following remarkable formula derived by ROZONOER

[10]:

J,.(u + v) - J~.(u) = -

{H[O.(t), x.(t), . ( 0 to

t]-n[o.(O, x.(O, u(t), t]}dt, which holds for arbitrary~" increment function v, i.e., +v(t)eU, tel In the present linear function above, we get

u5~1 and allowable one for which u(t) case, where f is the a unique 4 " and so

Rozonoer assumes all controls are piecewise continuous. However, this formula holds for measurable controls, on account of the Lebesgue monotone convergence theorem (Ref. [14], p. 186) and the density of piecewise continuous functions in the set of measurable functions.

The continuity of dependence of an optimal control upon the optimal control problem has been derived, under certain conditions. Under these conditions we have shown that one is assured, as in result of KIRILLOVA [1], that a given problem is "correctly formulated," using the English translation of the terminology of Kirillova. That is, it is not liable to produce drastic changes in its optimal control when small perturbations in the formulation of the problem occur, the amount of change being measured with respect to the L1 metric on the family of control functions. It has been demonstrated that a form of such perturbations arises when states and controls are quantized or a control problem is made discrete in time, for a typical numerical computational method•

Acknowledgement--The author wishes to express his gratitude to Prof. S. P. Diliberto for his suggestion of the problem and supelvision of this research as part of a Ph.D. thesis, and also to the IFAC reviewing committees for their constructive ciiticisms and suggestions pertaining to the revisions in this paper.

On continuous dependence of controls upon parameters REFERENCES [1] F. M. KIRILLOVA: O n the correctness of the formulation

[2]

[3] [4]

[5] [6] [7] [8]

[9] [10]

[11]

[12] [13] [14] [15]

of an optimal control problem. J. Soc. Ind. Appl. Math., Series A (Control), 1, No. 2 (1963). E. B. LEE and L. MARKUS: Foundations of Optimal Control Theory. John Wiley, New York (1967). JANE CULLUM: Perturbations of optimal control problems. SIAMJ. Control 4, No. 3 (August 1966). JANE CULLUM: Perturbations and approximations of continuous optimal control problems. Mathematical Theory of Control (Ed. A. V. BALAKRISHMANand L. W. NEUSTADT). Academic Press, New York (1967). A. J. KORSAK: Perturbed optimal control problems, Ph.D. Thesis, University of California, Berkeley, California (June 1966). P. DORATO: On sensitivity in optimal control systems. IRE Trans. Aut. Control AC-8, No. 3 (July 1963). H. J. KELLEY: Guidance theory and extremal fields. IRE Trans. Aut. Control AC-7, No. 5 (October 1962). W. R. PERKINS and J. B. CRuz, JR.: The Parameter Variation Problem in State Feedback Cont.ol Systems, Report R-182, Coordinated Sciences Laboratory, University of Illinois, Urbana, Illinois (October 1963). R. A. ROHR~R and M. SOBRAL, JR. : Sensitivity and Optimal System Design. Coordinated Sciences Laboratory, University of Illinois, Urbama, Illinois. L. T. ROZONOER: The L. S. Pontryagin maximum principle in the theory of optimal systems I. Avtomatika i Telemekh. 20, No. 10 (English Transl. Automn remote Control 20, No. 10). L. S. PONTRYAGIN, V. G. BALTYANSKI, R. V. GAMKRELmZE and E. F. MISCHENKO: The Mathematical Theory of Optimal Processes. Interscience, John Wiley, New York (1962). M. W. HmSCH: Notes on differential topology, University of California, Berkeley, Califomia (1962). E. A. CODDINGTON and N. LEVINSON: Theory of Ordinary Differential Equations, McGraw-Hill, New York (1955). M. E. MUNROE: Introduction to Measure and Integration. Addison-Wesley, Reading, Mass. (1953). G. A. BLISS: Lectures on the Calculus of Variations, para. 76-7, pp. 210-9). University of Chicago Press, Chicago (1946).

R ~ u m f - - S o i e n t ~ la classe des probl~mes de commande optimale ayant des dimensions fixes donn6es pour 6tats et les commandes, un intervalle de temps (to, tl) et un domaine de commande et admettons que chaque probl~me P dans est caract6ris6 par son 6tat initial, la loi de commande et la fonctionnelle int6grale objective:

J=

u(t), t]dt

probl~mes ~ temps fixe mais admettait seulement des perturbations lin6aires d'un probl~me lin6aire. Les theor~mes sont ~galement rapport6s ici ~t certains travaux plus recents, qui sont mentiorm6s en ref6rence. Zusammenfasstmg--Sei .~ die Klasse von Problemen der optimalen Steuerung, bei denen die Dimensionen des Zustands- und des Steuervektors, das Zeitintervall (to, t 0 und der Steuerbereich fest vorgegeben sind, und es soll vorausgesetzt werden, dab jedes Problem P in ,~ durch seinen Anfangszustand, das Steuergesetz und das Giitefunktional

J= C |t~ f ° [ x ( t ) , u(t), t]dt J to

charakterisiert wird. StSrungen p von Problemen P in ~ werden mit Riicksicht auf eine gewisse Topologie r auf ~ und die Metrik L~ auf dem Raum der Steuerfunktionen oder Steuerungen betrachtet. Die wesentliche Eindeutigkeit einer optimalen Steuerung wird eingeftihrt, und es wird gezeigt, dab irgendeine optimale Steurung ~* ftir eine S t 0 r u n g p eines Problems P, das eine wesentlich eindeutige optimale Steuerung u* hat, stetig von p abh/ingt unter Berticksichtigung von J und der Metrik L1; d.h. bei gegebenem e > 0 gibt es eine solche Nachbarsehaft N von P in ~ , dal3 PeN~d(u*, t~*)< e wobei d die Metrik L~ darstellt. Augerdem wird gezeigt, dab die optimale Steuerung far Probleme in g~ mit einem normalen linearen Steuergesetz and einem beschr~inkten konvezen Polyeder als Steuergebiet wesentlich eindeutig ist. Wir erhalten also ein Theorem, das den Typ eines Ergebnisses verallgemeinert, das von Kirillova in Izvestija VUZ Matematika, No. 4 (5), 1958, S. 113-126 (tibersetzt von L.W. Neustadt in SIAM Journal on Control, Vol. 1, No. 2) erhalten wurde, wobei Kirillova anstatt der Probleme mit fester Zeit zeitoptimale Probleme betrachtete, aber nur lineare StSrungen eines linearen Problems zulielL Die Theoreme werden hiernach auf einige modernere Arbeiten bezogen, auf die verwiesen wird. ee31oMe---I-IycTI~ ~ - - x n a c c npo6~IeM OnTmvlan~noro ynpaBnCHH.q n M e t o ~ n x nOCTO~HH~Ie ~aHn~ie pa3Mepbl ~IJIfl COexOgHmt~ ~ ~Ina ynpaBaenm~, HHTepBaJI BpeMeHn (to, tl) H

o6naexb ynpaBnermfl H ~onyCTnM qTO Iea~41a~ npo6neMa P B ~ onpe~IeneHa ce HCXO~IH~IM COCTOgHneM, 3aKOHOM ynpaB~IeHHfl n O~eKTHBHIaIM nHTerpan~HbIM ~yHKI~OHa~OM:

/'tl J = J t o f ° [ x ( t ) ' u(t), t]dt CTarba paccMarpnBaeT Bo36y~r.glemI~ p npo6aeM P B ~ no oTnomenmo K Heroropo~ Tono.uornn ~ Ha ~ r~ r pa3MepHOCTH L 1 B npocTpaHcTBe ynpaBnamxunx dl)ynxun~ )tan ynpaBYteHKi~.

L'article consid~re les perturbations /~ des probl~mes P dans ~ par rapport ~ une certaine topologie r sur ~ et le scalaire L~ sur l'espace des fonctions de commande ou commandes. L'univocit6 essentielle d'une commande optimale est introduite et il est montr6 que toute commande optimale ~*, pour une perturbation p du probl~me P ayant une eommande optimale u* essentiellement univoque, d6pend continuellement de p , c'est h dire de J e t du scalaire L~; ainsi, 6tant donn6 e > 0 , il y a u n e proximit6 N de P dans telle que PeN~d(u*, a*) < e, ou d repr6sente les ealaire L1. I1 est 6galement montr6 que la commande optimale pour les problemes dans ~ avec uneloi de commande lin6aire normale et un poly6dre convexe limit6 pour le domaine de commande est essentiellement univoque. Nous obtenons ainsi un theor~me qui g6n6ralise le type de r6sultat obtenu par Kirillova dans Izvestia VUZ, Matematika, n ° 4(5), 1958, pp. 113-126, qui a 6t6 traduit par L.W. Neustadt darts SIAM Journal on Control, vol. 1, n °. 2, oh Kirillova consid6rait des probl/~mes optimaux dams le temps au lieu de

295

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Mam,noro ynpaBaeHns n nora3umaeTcs qTO aCaKOe onTm~are,Hoe ynpaB~Ierme ~*, ~aa~ BO36y',~eHVOI ~ npo6ner~,i P aMexomero B ropHe O~HO3Ha~'HoeonTnMaamnoe yripaaneHHe u*, HenpepunnO 3anncnT OT ~, T.e. OT J rx OT pa3Mepnoexn L1; TaxnM o6paaoM, 3a~aBaa e > 0 , aMeeTca 6a~a3OCTb N r P B , ~ TaroBaa "ITO PeN~d(u*, ~*) < e, r~e d npe~exaBngeT co6o1~ pa3MepnocTb L1. Tar)re noKa3biBaeTca ~TO onTnMa.rll,noe ynpaBneHne n c BI,IrHyTI,IM orpami~eHHbtM ~¢OHTypOM B raqecTBe o6nacTn ynpaBneHa~ B ropHe O~tHOaHa,Ino. M u nonyuaeM TarnM o6paaoM TeopeMy, roTopa~ o6o6maeT Trin pe3ym, TaTa nony~euHoro KapnnnoBol~ B Haaecrnax BY3, MaTeMaTnra, rio. 4(5), 1958, exp. 113-126, rOTOpb~ 6I,ta nepeBe~eH H.B. He~J-UTa~TOM B CtlAM ~xopna_rm on KonTpom,, TOM 1, no. 2, r~e KHpr[nnoaa paccMaTpmmna npo6nesa, x oirrnMa.rmm,ie no BpeMenH BM~rO npo6neM c nOCTOaHrmIM BpeMeHeM no ,tIonycta.rta nmm, Jrmiei~H~e Boa6y:~c~enn~ ~ i ~ o ~ npo6neMu. TeopeMH TaK)Ke CB~taaHhI 3~CC1~C 6OYlee HOBhlM]~pa6oTaMn, pc~pentu~

roTop/~IX npnBo~.qTC~l.