Necessary and sufficient conditions for optimal

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d2l(t,s,Z) = jf*S{t-s)A-1. S{s-r)
Proceedings of the Royal Society of Edinburgh, 124A, 211-251,1994

Necessary and sufficient conditions for optimal controls in viscous flow problems H. O. Fattorini* and S. S. Sritharan| Department of Mathematics, University of California-Los Angeles, Los Angeles, California 90024-1555, U.S.A. (MS received 10 February 1992. Revised MS received 30 November 1992) A class of optimal control problems in viscous flow is studied. Main results are the Pontryagin maximum principle and the verification theorem for the Hamilton-Jacobi-Bellman equation characterising the feedback problem. The maximum principle is established by two quite different methods.

1. Introduction

Optimal control theory of viscous flow has several applications in engineering science. In [20], a fundamental optimal control problem in exterior hydrodynamics was studied. In that paper, the task of accelerating an obstacle from rest to a given speed in a given time, minimising the energy expenditure, was considered. In [13] a unified formulation of optimal control problems in viscous hydrodynamics, covering wind tunnel flow, flow inside containers and exterior hydrodynamics, was considered. Both of these papers were concerned with proving existence theorems for optimal control. The present paper is a sequel to these papers. In this paper we establish the following two fundamental steps in the optimal control of viscous incompressible flow: (i) the Pontryagin maximum principle to obtain the necessary conditions; (ii) an analysis of the feedback problem using the infinite dimensional HamiltonJacobi-Bellman equations. In Section 2, we consider a nonlinear evolution equation in a Hilbert space with a certain type of cost functional. The form of this system represents several control problems in fluid mechanics. The major theorems of this paper are stated in Section 2 and proved in later sections. This section also contains the hypotheses on various operators and, as shown in [13], these are in fact satisfied for the specific flow control problems. In Section 3 we consider the task of computing the optimal control. The Pontryagin maximum principle we prove in this section provides the necessary conditions for such computations, in the form of an adjoint backward linear evolution problem and a variational inequality which states that a certain Hamiltonian takes its maximum value at the optimal control. The maximum principle proved in this section is * Supported by the NSF grant DMS-90011793. t Mailing address: Code 574, NCCOSC, San Diego, CA 92152-5000, U.S.A.; supported by the ONR-URI Grant No-N00014-91-J-4037.

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powerful, in the sense that it accommodates a very general target condition. Ekeland's variational principle plays a key role in this proof. In Sections 4, 5 and 6 we elaborate the dynamic programming concept introduced to fluid mechanics in [19]. In Section 4 we consider the feedback problem. The value function is defined as the minimum value of the cost. It is shown that the value function is locally Lipschitz. An important result proved in this context is that the value function is a viscosity solution (in the sense of Crandall and Lions [5, 6]) of the Hamilton-Jacobi-Bellman equation associated with our control problem. In Section 5 we use the method in [3, 4] to provide another proof for the Pontryagin maximum principle. In this proof, however, no target set is included. Finally, we establish the verification theorem for the Hamilton-Jacobi-Bellman equation in Section 6. This theorem provides the mathematical resolution of the feedback control problem for the Navier-Stokes equations. Some of the results of this paper were announced in [21]. 2. Unified mathematical formulation and main theorems As in [13], we consider the general control system: te(T,T),

(2.1) (2.2)

in a Hilbert space H. The operator si satisfies the following hypothesis: HYPOTHESIS 2.1. si is self-adjoint and positive: (siy,y)H^0,

VyeD(si).

The origin belongs to the resolvent set of si. For a ^ 0, we denote by Ha the space D(si"). This space is a Hilbert space equipped with its natural inner product (y, z)a = (s/"y, si*z)H. The inner product (-,-)« corresponds to the norm ||j>||a= I I ^ ^ H H - For oc^O, Ha is the closure of H under the norm ||-|| a . The nonlinear term ^T(-) is unbounded and satisfies the following hypothesis: 2.2. There exists p, 0 ^ /? < 1/2 such that the map JT(-):Hi-+H-f is continuous and locally bounded and has a Frechet derivative [DyjV](y) which is continuous and locally bounded (as an S£(H^,H.^-valued function).

HYPOTHESIS

The control U(t) takes values in a Hilbert space F. The linear operator 3$ satisfies: HYPOTHESIS 2.3. J 1 e Z£(F; H).

We shall assume that controls {/(•) belong to L 2 (T, T;F). By definition, solutions (or trajectories) of the initial value problem (2.1)-(2.2) in an interval x rg t ^ T are H±-valued functions y(t) continuous in the norm of H± and satisfying S(t-r)®U(r)dr, (2.3)

Optimal controls in viscous flow problems

213

The following result is useful in the justification of (2.3) and other integral equations of this paper. LEMMA 2.4 ([13,16,18]). Let si be self-adjoint and non-negative definite, and let S{t) = exp {—tsi) be the analytic semigroup generated by — si. Then: (I) for any CeH, si^S{-% e L2(0, oo; H) and ll^(o,=o:fl) = - ^ l l f l U .

(2-4)

Let g(-) e L2(a, b; H) be given. If we define the function y(t)= then: (II)

S(t-r)g(r)dr,

a^t^b,

(2.5)

y(-)eC{H,fc:J?^>Rbe given maps and Y a subset of H. Find U e Ji such as to: PROBLEM

minimise fc(U),

UeJt,

subject to/([/) e y.

(3.1) (3.2)

In the application we have in mind, f(U) is defined as the final value of the state: f(U)=y(T,z;C,U) and fc(U) is taken as the cost functional: Let us first recall certain basic tools of nonsmooth analysis. DEFINITION 3.2 (Contingent cone). Let Y be an arbitrary subset of a Hilbert space H. Given y e H, the contingent cone to Y at y is the set KY(y) defined as follows: w e KY(y) if and only if there exists a sequence {Xk} of positive numbers with Xk-+§ and a sequence {yk} c F such that

Equivalently, w e KY(y) if and only if there exists a sequence {Xk} as above and there exists a sequence {wk} w and y + kkwk e Y. Ky{y) is a closed and in general nonconvex cone. DEFINITION 3.3 (Clarke tangent cone). Let Y be an arbitrary subset of a Hilbert space H. Given y e H, the Clarke tangent cone to Y at y is the set CY(y) which consists of all w € H such that, for every sequence {kk} of positive numbers and every sequence

Optimal controls in viscous flow problems

217

{jj} c y such that Ak->0, yk->y, there exists a sequence {yk} oo.

Equivalently, w> e Cy(j>) for every sequence {Xk} and {j 4 } w and yk + Xkwk e F. It can be shown that CY(y) is convex and closed (see [1, p. 407]). It is evident from the definitions that CY(y) . n->

QO

I

n—* oo

I

Our proof of the maximum principle will be based on the following theorem of Kuhn-Tucker type. The assumptions are the following, where (£m denotes the minimum of (3.1) subject to (3.2). (a) The metric space Ji is complete. (b) The function f:Ji^H (respectively fc:Ji->R) is denned in D{f) 0. (c) The target set Y is closed. 3.6. Let U be a solution of (3.1)—(3.2). Then there exists a sequence {&„} c / ? + , „—>0, a sequence {Un} cz Ji, a sequence {y") c Y such that

THEOREM

?>n

(3.7)

and a sequence {(fin, zn)} c R x H satisfying Hn^O,

||(^,zn)||i{XH=l,

such that, for every (S", 3") e conv do(fc,f)(U")

(3.8)

and every w" e KY(y") we have: (3.9)

where conv denotes closed convex hull. Moroever, for every weak limit point (/z, z) of the sequence {(/xn, zn)} a R x H, we have H^O, and for every (Ec, S) e lim inf^^ conv

zeNY(f(U))

(3.10)

do(fc,f)(U")

LiEc + (z,Z)H^O.

(3.11)

Finally, assume that there exists p > 0 and a compact set Qc H such that

0 {[n(conv3(/ c , /)(£/"))] -KY(y«)^B(0,p)+

Q}

(3.12)

1 = 1

contains an interior point in H, where Yl denotes the canonical projection from R x H into H. Then (n,z)^ 0.

Optimal controls in viscous flow problems

219

Proof. Given an arbitrary sequence {en} of positive numbers tending to zero, we define for each n the real-valued function fSn by 9H(U, y) = {Pe(U, w" andy" + lnkwnk e Y(see Definition 3.2). In (3.17), we set y = y" + lnkwnk and V = \J\, where {[/£} is the sequence used in the Definition (3.3)-(3.4) of variation. Thus, using the expression (3.13) for ^ ( - , •), we obtain {[max (0,fe(Ul) ~%m + e j ] 2 + \\f(U"k)~yn \\f(Un)~yn\\2H}i-et{d(U"k,U") n

-

+ lnk\\Wnk\\H}.

n

n

Note that from (3.3) we have d(U k, U ) ^ Xnk. Now let (E c, a") e Dividing by Xnk and letting k-> oo, we obtain (3.9) with

(3.18) do(fc,f)(Un).

(^,zn) = (yn,xn)/\\(yn,xn)\\RxH,

(3.19)

(yn, xn) = {max (0,/ c (£/") - ^m + en),f(U»)-y"}.

(3.20)

Obviously (3.9) extends to conv do(fc,f)(U"). Inequality (3.11) is obtained from (3.9) setting w" = 0 and taking limits as n —• oo. We note here that in this limiting argument we have used the fact that zn—>z weakly and H"—>3 strongly implies (zn, E")H->(z, E)H. The first condition (3.10) is obvious, thus it only remains to prove the second, namely that z eNY(f(U)). Let we CY(f{U)) and let {Xn}cR+ be such that lj\\(7n, xJW^O asn^oo.

(3.21)

Since \\(yn, xn)\\ = ^n(U",y") ^ e n ->0, we must have Xn^>-0 as well, thus by Definition 3.3 of C y (/((/)) we can pick a sequence {wn} in H such that w"->w strongly and y" + lnw" e Y. We then have from (3.17) that

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That is, {||(yn, xn) ||2 - 2Xn(f(U") - y \ w»)H + A2 || w" | | 2 } *

= II (y n ,*JII{l - 2 ( ^ / | | (yn,xn)\\)(zn, "")„ + % II w" 117II (?„,*,,) II2}* = I (y». x») I {1 - a . / I (?», *„) I )(z«. W")H + o(XJ ^ ^ ( l / - , /•) - e* {XJ ||fa,,x j ||} || (yn, xn) || || w" ||. Subtracting || (?„,*„) II (which is equal to co, we deduce that

Since w e C y (/(t/)) is arbitrary, we deduce from Definition 3.4 that z e Afy(/(l/)). As before, in this limiting argument we have used the fact that zn->z weakly and w" -»w strongly implies (zn, w")H->(z, w)H. It only remains to show that (3.12) guarantees that (/x, z ) ^ 0 . This will be a consequence of the following result: 3.7. Let E be a Hilbert space. Let {An} be a sequence of sets in E and P cz E a compact set such that the set

LEMMA

A= f\{com(An)

+ P}

(3.22)

contains an interior point. Let {zn} be a sequence such that l|z.ll£ = l

(3-23)

and (zn,y)E^en^0,

VjeAn.

(3.24)

Then every weakly convergent subsequence of {zn} has a nonzero limit. Proof. Let {zn} be a subsequence (denoted by the same symbol) convergent to zero. Let x be an interior point of A and let B(x, S), S > 0 be a ball contained in A. For each n there exists yn e conv (AJ and pneP with x+

8zn=yn+pn.

By compactness, we may assume that {/>„} is strongly convergent. Since (3.24) extends from An to conv (An), we have (zn,x)E = (zn,yn)E-3

||z n ||1 + (zn,pn)E^sn-S

+ (zn,pn)E.

Since zn is weakly convergent to zero and pn is strongly convergent, we should have il m J ( z n./'n)£ = 0We thus conclude that \im(zn,x)E^

-5.

This contradicts the fact that {zn} is weakly convergent to zero.



Completion of the proof of Theorem 3.6. Assume that (3.12) contains a ball B(x, 3).

Optimal controls in viscous flow problems

221

Set £ = R x H, P = [0, 1] x Q, and

If (3.12) contains an interior point in H, it is then evident that the intersection (3.22) will contain an interior point in E as well. We apply Lemma 3.7 to the sequence { — (nn,zn)}. The second condition in (3.8) implies (3.23); as for (3.24) it is (3.9). Remark 3.8. We note here that in the above arguments concerning the condition (3.12), the compact set Q does not seem to play any role. In fact this set can be omitted in condition (3.12) if the remaining set AR =

£ \ { t n ( c o n v dW /)(U"M - KY(y")nB(0, p)}

contains an interior point. However, there are special cases such as the exterior hydrodynamics problem described in [13] where the target set is in the form F = X x {lT} with XL(t) e S^(Ha; H*_ p) is continuous (in the uniform operator topology of operators).

Hypothesis 3.9 is equivalent to requiring that for each t e [T, T], stf ~pL(t)srf~* be a linear bounded operator in H and that the function t^>stf~l3L(t)3?~a e J5?(/7; H) be continuous in the uniform operator topology. The results on (3.25)-(3.26) will be applied to the operator L(t) = [_DyjV](y(t, x; £; C/)), where y(t, z; (; U) is a trajectory of the nonlinear system (2.1); here we use Hypothesis 2.2 and take a = \. We shall also treat the adjoint 'final value problem' dsz{s)-(^ + L(s)*)z(s) = h(s), O^s^T, z(T) = z0. x

l)

(3.27) (3.28)

ll

Hypothesis 3.9 implies that stf~ L(s)*stf~ = (s/~ L(s)£/~°")* is a continuous function in the uniform topology of operators, so that L(s)* also satisfies Hypothesis 3.9 with a and )3 switched. We only treat the case a == \ below, since the case a. < \ is simpler; the necessary modifications will be pointed out later. In the computation of the variations, we only need to consider z0 = 0 in (3.25)-(3.26); however, in the adjoint initial value problem we have no information

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on the final condition z 0 , thus we have to consider the general case £ e H. To unify the two cases, we shall only assume that z 0 e H in (3.26). 3.10. Let A(-) e L 2 (T, T; H) be given. Then the system (3.25)-(3.26), or, rather, the integral equation

LEMMA

ft z(t) = S{t-x)z0-

rt

s/l)S(t-a)sf-l)L(