Maximum Principle for State-Constrained Optimal Control Problems Governed by Quasilinear Elliptic Equations* by

In this paper, the authors study an optimal control problem for quasilinear elliptic PDEs with pointwise state constraints. Weak and strong optimality conditions of Pontryagin maximum principle type are derived. In proving these results, we penalized the state constraints and respectively use the Ekeland variational principle and an exact penalization method. Keywords. quasilinear elliptic equations, optimal control, Pontryagin's principle, state constraints. AMS(MOS) subject classi cations. 49K20, 35J65, 35J85 x1. Introduction. In this paper, our aim is to prove Pontryagin's principle for pointwise state-constrained optimal control problems governed by very general quasilinear elliptic equations. The control is distributed and takes values in a bounded subset, not necessarily convex, of some Euclidean space. The cost functional is Lagrange type. Standard results of optimal control problems for linear elliptic equations with convex control set and convex functional can be found in [16]. In [1,5], the results were extended to linear or semilinear equations with state constraints. In the framework of semilinear elliptic equations, the Pontryagin type principle was rst proved in [2] for problems without * This work was completed while the authors were visiting the IMA, University of Minnesota, USA, and partially supported by the IMA. The rst author was also partially supported by Direcci on General de Investigaci on Cient ca y T ecnica (Madrid) and the second by the NSF of China under Grant 19131050 and the Fok Ying Tung Education Foundation. 1 Departamento de Matem atica Aplicada y Ciencias de la Computaci on, Universidad de Cantabria, 39007-Santander, Spain. 2 Department of Mathematics, Fudan University, Shanghai 200433, China. 1 state constraints; later, in [3,4] and [21], di erent approaches were used to deal with the state-constrained case. In this paper, we improve the techniques of [3,4] and [21] so that the extension of the results to the quasilinear equations is possible. We remove the weak stability assumption made in [4] for the weak version of Pontryagin principle (see x4). We also obtain the strong version of Pontryagin principle, which was not carried out in [21]. As in [3,4], to prove this strong principle, we assume a stability condition for optimal cost functional with respect to small perturbations of the feasible state set. This leads an exact penalization of the state constraint. The penalty functional used here is di erent from that in [3,4], which allows us to shorten the proof. Let us mention some other papers related to the present one. In [6], optimal control of quasilinear elliptic equations without state constraints was considered; and for the evolution case of nite and in nite dimensions, see [10,13,18], and the references cited therein. This paper is organized as follows. In x2, optimal control problem is formulated and the state equation is studied. x3 is devoted to the derivation of the variation along given feasible pairs, which is needed to deal with the case of a not necessarily convex control set. The approach followed in this section is based on the method used in [12]. In xx4 and 5, we obtain the weak and strong Pontryagin maximum principles. x2. Formulation of the Problem. This section is devoted to a formulation of the control problem which will be studied in this paper. Our state equation is as follows: (2:1) 8< : r a(x;ry(x)) = f(x; y(x); u(x)); in ; y @ = 0: In what follows, we always assume that is a bounded region in lR with a C boundary @ , for some > 0 and U is a bounded measurable set in some Euclidean space. We use j j as the norm of vectors in Euclidean spaces or of matrices, which can be identi ed from the context. Also, we let h ; i be the inner products or duality in possibly di erent spaces. For any measurable set S lR, we use jSj to denote the Lebesgue measure of the set S. We make the following assumptions. 2 (A1) The function a : lR ! lR is continuous. For each x 2 , a(x; ) is di erentiable and a ( ; ) is continuous (we use as the dummyargument forry). Moreover there exist constants > 1, 0 < 1, > 0 and > 0, such that for all x; b x 2 , ; 2 lR, (2:2) ( + j j) j j h a (x; ) ; i ( + j j) j j; (2:3) ja(x; ) a(b x; )j (1 + j j) 1jx b xj : (A2) The function f : lR U ! lR has the following properties: f( ; y; u) is measurable on , f(x; ; u) is in C(lR) with f(x; ; ) and fy(x; ; ) being continuous on lR U . Moreover, (2:4) fy(x; y; u) 0; 8(x; y; u) 2 lR U; and for any R > 0, there exists an MR > 0, such that (2:5) jf(x; y; u)j + jfy(x; y; u)j MR; 8(x; u) 2 U; jyj R: Next, we set U = fu : ! U u is measurable g: Any element u 2 U is referred to as a control. In what follows, we will denote by C0( ) the set of all continuous function on which vanish on @ and by C ( ) the set of all continuously di erentiable functions on for which the rst order partial derivatives are Holder continuous with the exponent 2 (0; 1). Now, we state the following basic result. Proposition 2.1. Let (A1){(A2) hold. Then, for any u 2 U , there exists a unique y y( ;u) 2 C ( )TC0( ) solving (2.1) for some 2 (0;minf ; g). Furthermore, there exists a constant C > 0, independent of u 2 U , such that (2:6) ky( ;u)kC1; ( ) C; 8u 2 U : Sketch of the Proof. First of all, we truncate f : For any m > 0, let (2:7) fm(x; y; u) = >>>< >>>: f(x; y; u); if jyj m; f(x; m;u); if y < m; f(x;m; u); if y > m: 3 Then, we consider the following truncated problem: (2:8) 8< : r a(x;ry(x)) = fm(x; y(x); u(x)); in ; y @ = 0: By [15], we know that (2.8) admits a unique solution ym 2 W 1; 0 ( ). Then, as in [19], we are able to show that there exists a constant C > 0 independent of m and u 2 U , such that (2:9) kym( ;u)kL1( ) C; 8m> 0; u 2 U : Consequently, for m > C, we obtain that ym = y is a solution of (2.1). Thus, by [14], we obtain that in fact this y is in C ( ), for some 2 (0;minf ; g) and the estimate (2.6) holds. Finally, the uniqueness follows immediately from the coercivity of the operator (see (2.2) and (2.4)). In what follows, any pair (y; u) 2 (C ( )TC0( )) U satisfying (2.1) is called a feasible pair and we refer to the corresponding y and u as feasible state and control, respectively. Clearly, under (A1){(A2), U coincides with the set of all feasible controls and for each feasible control u 2 U there corresponds a unique feasible state. Now, we let f : lR U ! lR be a given function. We make the following assumption on this function: (A3) The function f( ; y; u) is measurable on , f(x; ; u) is in C(lR) with f(x; ; ) and f y (x; ; ) being continuous on lR U . Furthermore, for any R > 0, there exists a function 'R 2 L( ), such that (2:10) jf(x; y; u)j + jf y (x; y; u)j 'R(x); 8(x; u) 2 U; jyj R: It is easy to see that under (A1){(A3), for any u 2 U , the following functional is well-de ned: (2:11) J(u) = Z f(x; y(x); u(x))dx: This functional is referred to as the cost functional. Next, we introduce another map g : lR! lR. We assume the following: 4 (A4) The map g is continuous, gy( ; ) exists and is also continuous on lR. Moreover we assume that (2:12) g(x; 0) = 0; 8x 2 : The assumption (2.12) can be relaxed; see Remark 4.2. >From above, we know that under (A1){(A2), for any u 2 U , the corresponding feasible state y is in C ( ). Thus, we may talk about the state constraint of form (2:13) g(x; y(x)) ; 8x 2 ; where > 0 is given. Of course, for any given u 2 U , the corresponding state y does not necessarily satisfy the constraint (2.13). We refer to any feasible pair (y; u) satisfying (2.13) as an admissible pair and the corresponding y and u as admissible state and control, respectively. We denote the set of all admissible controls by U , indicating the dependence on by the subscript. Now, our optimal control problem can be stated as follows: Problem (P ). Under (A1){(A4), nd a control u 2 U , such that (2:14) J( u) = inf u2U J(u): Any admissible control u satisfying (2.14) is called an optimal control, the corresponding state y is called an optimal state and the pair ( y; u) is referred to as an optimal pair. x3. Variation along Given Feasible Pairs. In deriving necessary conditions for optimal pairs, one needs to make certain perturbations for the control and the corresponding variations of the state and the cost functional need to be determined. This section is devoted to such a determination. We note that since the control domain is not necessarily convex, the perturbation of the control is restricted to be of \spike" type. This causes the computation somewhat technical. Our basic idea here is taken from [12] and [13,21]. 5 For any feasible pair (y; u), we de ne (3:1) >>>< >>>: aij(x) = ai; j (x;ry(x)); 1 i; j n; a0(x) = fy(x; y(x); u(x)); c(x) = f y (x; y(x); u(x)); and given u 2 U , (3:2) ( h(x) = f(x; y(x); v(x)) f(x; y(x); u(x)); h(x) = f(x; y(x); v(x)) f(x; y(x); u(x)): Set