Second order sucient optimality condition for a. nonlinear elliptic control problem. Eduardo Casas and Fredi Troltzsch and Andreas Unger

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1 Second order sucient optimality condition for a nonlinear elliptic control problem by Eduardo Casas and Fredi Troltzsch and Andreas Unger Technical University of Chemnitz wickau Faculty of Mathematics 97 Chemnitz Germany Introduction Optimal control problems governed by nonlinear elliptic partial dierential equations have already been considered by many authors. We refer only to the papers of Casas [], [3], to the book of Tiba [4], and to the references therein. Meanwhile, the existence of optimal controls and rst order necessary optimality conditions of the maximum{ principle type are well investigated. It is known that in the case of nonlinear equations the rst order conditions are not in general sucient for optimality. In this paper we are going to derive a second{order sucient optimality condition for a class of semilinear elliptic boundary control problems. For parabolic boundary control problems second{ order sucient conditions were established in the papers by Goldberg and Troltzsch [6]. It is more or less obvious that these conditions can be transferred by the same technique to elliptic problems. However, a comparison of the results in [6] with second order conditions for optimization problems in spaces of nite dimension reveals that the gap between the sucient optimality conditions in [6] and corresponding second order necessary optimality conditions is quite large. Taking into account the active set of optimal controls, this gap can be partially closed. The known diculty in the theory of sucient optimality conditions for extremal problems in function spaces is the so{called two{norm discrepancy. This problem was resolved, for instance, by Ioe [8] and Maurer []. In recent years further considerations have shown that some weaker sucient optimality conditions can be established for abstract optimization problems in Banach spaces and for optimal control problems governed by ordinary dierential equations. We refer, for instance, to Dontchev et al. [4] and Malanowski []. The main idea is to introduce a third norm taking into account the measure of positivity of some terms occuring in the variational inequality. We will use this idea to derive of a second{order sucient condition in the elliptic case.

2 Formulation of the optimal control problem We consider the optimal control problem to minimize F (w; u) '(w(x))dx (u(x); w(x))ds x () subject to and to the constraints on the control u 4w(x) w(x) in (x) b(w(x); u(x)) on u U ad fv L ()j < a v(x) b < a. e. on g: (4) U ad is a bounded, convex, and closed subset of L (). The partial dierential equation is considered in the following weak sense. Denition A function w W () is said to be a weak solution of the partial dierential equation (), (3), if for all v W () the equation is valid. @x i wv! dx To make this well dened we impose the following assumptions: b(w; u)vds x (5) (A) The function ' : IR! IR is of class C ;. The functions : IR IR! IR and b : IRIR! IR are twice continuously dierentiable with respect to both arguments. Furthermore, the function b is monotone decreasing and all partial derivatives of and b up to the second order are globally Lipschitz{continuous. This property of the function b guarantees the existence of a unique weak solution of the partial dierential equation (), (3) for each xed u U ad (cf. Kinderlehrer and Stampacchia [9]). (A) The set IR n is assumed to be a bounded domain with a sucently smooth boundary such that the weak solution of the partial dierential equation (), (3) can be shown to be an element of W p () with some xed p > n. Then known embedding theorems ensure that the solution is continuous on the closure of. (A3) Moreover, we suppose that the following estimate is valid for all admissible controls u ; u U ad and the corresponding solutions of the partial dierential equation w and w : kw w k W p () cku u k Lp() Cku u k L(): (6)

3 For instance, suppose that is a bounded domain with a C {boundary. Using results of Groger [7], the following regularity result can be proved Lemma The weak solution w of the boundary value problem 4w w g on (8) is an element of W p (), provided the function g is an element of L p (). Let w i denote the solution of the equation (7), (8) corresponding to g i L p () for i ;. Then the estimate kw w k W p () ckg g k Lp() is valid, where c does not depend on the choice of g i. Proof: We will outline only the main idea of the proof. For the details we refer to the paper of Groger [7]. The domain is bounded and has a C {boundary. Therefore, it can be shown that the dierential operator represented by the equation (7), (8) is a continuous mapping from Wp () onto the dual space of W q (), pq. In the known way, any g L p() can be regarded as an element of (Wq ()). The statement of the lemma is now a conclusion of the properties of the dierential operator. The uniqueness of the solution in W () ensures also the invertibility of the dierential operator between Wp () and (Wq ()). We can now apply the Banach theorem to derive the continuity of the inverse operator. This lemma allows us to ensure the assumed regularity of the weak solution. We proceed in the following way. The unique solution w W () of the state equation (), (3) exists due to the assumptions on b and. The trace theorem shows that the trace wj is in W (). This space is embedded in L p () with some p > because of the embedding theorem. Owing to the Lipschitz property of b and to u L (), we can identify the function b(w(); u()) with a function g L p (). The regularity result ensures that the solution w belongs to Wp (). We repeat the considerations starting with p instead of. In this way the solution w of the state equation is seen to be an element of Wp () with some p > p. Repeating this bootstrapping argument suciently often, we construct a sequence fp i g, where p i > p i. It can be shown (see Eppler and Unger [5]) that there exists a constant c > such that p i > p i c for all i. So we are sure to arrive at p > n after a nite number of steps. The assumed estimate follows from the estimate in the regularity result and the monotonicity of the function b with respect to the rst variable. As a further corollary we have that the norm of all feasible states is uniformly bounded in Wp () since of U ad is bounded. If we assume in addition that the cost functional is convex and the function b is linear both with respect to the control u, the existence of a solution of the optimal control problem follows by standard methods (cf. for instance Eppler and Unger [5]). However, we shall not rely on this assumption. We just suppose that a xed reference control u is given. Let us introduce the Langrange{function for our optimal control problem in the following way: L(w; u; y) '(w)dx (w; u)ds x 3 wy i b(w; u)yds x : (9)

4 This function is well dened and twice continuously dierentiable with respect to the space C() \ W () L (). Standard considerations apply to derive the rst order necessary optimality conditions at u. They can be written in the equivalent form (adjoint equation) and L w (w ; u ; y ) L u (w ; u ; y )(u u ) 8u U ad (variational inequality). Subscripts denote associated partial derivatives. In detail, the adjoint equation reads 4y y ' w (w ) b w(w ; u )y w (w ; u ) on : () The variational inequality admitts the form ( u (w ; u ) b u (w ; u )y)(u u )ds x 8u U ad : () To simplify the notation we denote in the sequel the pair (w; u) by v. Denition The pair v (w; u) is called an admissible pair, if u belongs to U ad and w is the weak solution of (), (3) corresponding to u. Two norms of v corresponding to the spaces V W () L () and V Wp () L () are given by kvk (kwk W () kuk L ()) and kvk maxfkwk W p (); kuk L()g; respectively. The simplest second order sucient optimality condition which can be transferred to our problem from the parabolic case can be formulated as follows: Assume that for a pair v (w ; u ) the rst order necessary conditions are satised. Let y be the associated adjoint state. Suppose that the second order derivative of the Lagrange{function with respect to w and u fulls the estimate L vv (v ; y )[(h; ); (h; )] k(h; )k (3) for all (h; ), where uu with u U ad and h is the weak solution of the corresponding linearized equation 4h h b w(v )h b u (v ) on : (5) In this condition, the second order derivative of the Lagrange{function has to full the estimate with respect to all admissible controls u. From the theory of optimization in spaces of nite dimensions weaker second order sucient conditions are known. More precisely, the estimate of the type (3) has to be fullled only with respect to all u from some subset of the admissible set. Using ideas of Dontchev et al. [4] and Malanowski [] we will derive such a sucient condition in the next sections. 4

5 3 Motivation of a second order sucient optimality condition Let us consider the mathematical program to minimize subject to x IR k f(x) h(x) g(x) ; wheref, h, and g are smooth functions. For this problem the following second order sucient condition is known (cf., for instance, Spellucci [3]): Theorem Let x be a admissible point satisfying the rst order necessary optimality condition with associated Lagrange{multipliers and. The point x is a strong local minimizer, if an > exists such that z r f(x ) mx i for all z satisfying the conditions i r g i (x ) kx j j r h j (x ) A z z T z (6) and z T rh i (x ) ; (7) z T rg i (x ) ; if g i (x ) ; (8) z T rg i (x ) ; if i > : (9) Compairing this condition with the sucient condition (3) for the elliptic control problem we observe: The estimate (6) corresponds to the estimate (3). The equation (7) can be viewed as a representation of the partial dierential equation in the sucient condition for the control problem. The inequality (8) means that ~x x z is admissible with respect to the linearized inequality constraints for suciently small. In the context of the control problem this corresponds to restrictions on in the sucient condition for the elliptic control problem. However, there is no corresponding term for the condition (9). It is this additional condition, which we shall add to the sucient condition. Usually, necessary optimality conditions for optimal control problems have to be satised for all u from some control set U ad. To simplify the presentation, let this set be described by U ad fu L () j u(x) a.e. on g: Introducing formally a Lagrange{multiplier function with respect to the inequality constraints, the variational inequality reads ( u (w ; u ) b u (w ; u )y )(u u )ds x 5

6 for all u from the whole space L (). Thus, the left factor under the integral sign has to vanish identically on. In this way, we can identify with u (w ; u ) b u (w ; u )y in some sense. The formal multiplier function plays in this case the same rule as the vector in theorem. A suitable interpretation of the condition (9) on z is now given in the case of a optimal control problem by u(x) u (x) if u (w (x); u (x)) b u (w (x); u (x))y (x) > : In the next section we will modify this intuitive additional condition and prove the suciency of the corresponding second{order condition. 4 Second{order sucient optimality condition In what follows let v (w ; u ) be admissible for the optimal control problem. We do not assume that v achieves the global minimum of the optimal control problem. Suppose that the rst order necessary optimality conditions are fullled at v with the associated adjoint state y. This is the optimality system: (variational inequality), (adjoint equation), and ( u (v ) b u (v )y )(u u )ds x 8u U ad () 4y y ' w (v b w(v )y w (v ) on 4w b(w ; u ) on (state equation). It is well known that () holds i f u (v (x)) b u (v (x))y (x)g(u(x) u (x)) almost everywhere on for all u() U ad. We dene a set of positivity: Denition 3 For " > the set of positivity " denotes the measurable subset of such that for all u U ad holds for almost all x ". f u (v (x)) b u (v (x))y (x)g(u(x) u (x)) "ju(x) u (x)j () Remark: Owing to the assumptions on, the adjoint state y is continuous on. Thus, the function u (v (x))b u (v (x))y (x) is measurable on, and the denition make sense. It is possible that the set of positivity is of measure zero. 6

7 Corollary The estimate " f u (v (x)) b u (v (x))y (x)g(u(x) u (x))ds x "ku u k L (") holds true for all u U ad. Before we start to discuss second order sucient optimality conditions we derive some useful technical results. At rst we remind of the continuity of the adjoint state y. Therefore, the assumptions on the functions standing in the objective and in the partial dierential equations allows us to prove the following statements. Lemma The second order derivative of the Lagrange{function at (v ; y ) with respect to v fulls the estimates and jl vv (v ; y )[(h ; ); (h ; )]j ck(h ; )k k(h ; )k () jl vv (v ; y )[(h; ); (h; )] L vv (v; y )[(h; ); (h; )]j ckv v k k(h; )k (3) for all (h; ); (h i ; i ) W () L () and all v W p () L ().. Proof: The second order derivative of the Lagrange{function is given by: L vv (v ; y )[(h ; ); (h ; )] ' ww (w )h h dx ( ww (v )h h wu (v )h uw (v ) h uu (v ) )ds x (b ww (v )h h b wu (v )h b uw (v ) h b uu (v ) ) y ds x : The control u is bounded in the sense of L (), and the state w is bounded in the sense of C(). Therefore, all second order derivatives under the integral sign are uniformly bounded in the sense of L (). The adjoint state y is continuous on. For that reason we are able to estimate jl vv (v ; y )[(h ; ); (h ; )]j To prove the second estimate we consider h h dx (h h h h )ds x (h h h h )ds x A ck(h ; )k k(h ; )k : L vv (v ; y )[(h; ); (h; )] L vv (v; y )[(h; ); (h; )] ' ww (w )h dx ( ww (v )h wu (v )h uu (v ) )ds x (b ww (v )h b wu (v )h b uu (v ) )y ds x 7

8 ' ww (w)h dx ( ww (v)h wu (v)h uu (v) )ds x (b ww (v)h b wu (v)h b uu (v) )y ds x (' ww (w ) ' ww (w))h dx (( ww (v ) ww (v))h ( wu (v ) wu (v))h ( uu (v ) uu (v)) )ds x ((b ww (v ) b ww (v))h (b wu (v ) b wu (v))h (b uu (v ) b uu (v)) )y ds x The second order derivatives of ',, and b are assumed to be Lipschitz{continuous. Therefore, jl vv (v ; y )[(h; ); (h; )] L vv (v; y )[(h; ); (h; )]j jw wjh dx (jv vjh jv vj jhj jj jv vj )ds x (jv vjh jv vj jhj jj jv vj )y ds x A Now the second estimate follows immediately. The next results are concerned with properties of remainder terms. We will use the following notations: r b (v ; v) b(v) b(v ) b w (v )(w w ) b u (v )(u u ) (rst order remainder term) r L (v ; y ; v) L(v; y ) L(v ; y ) L v (v v ) L vv(v ; y )[v v ; v v ] Lemma 3 The estimate is valid for all v W p () L (). (second order remainder term) kr b (v ; v)k L () kv vk ckv v k (4) Proof: We have kr(v b ; v)k L () (b(v) b(v ) b v (v )(v v )) b v (v s(v v ))(v v )ds b v (v )(v v ) (b v (v s(v v )) b v (v )))(v v )ds 8 A A ds x ds x

9 @ cjs(v v )jjv v jds A by the Lipschitz property of b v c jv v j 4 ds x ckv v k C()L() ckv v k jv v j ds x : ds x jv v j ds x So we nish with kr b (v ; ~v)k ck~v v k k~v v k : Lemma 4 The second order remainder term r L (v ; y ; v) satises the estimate Proof: We have r L (v ; y ; v) jr L (v ; y ; v)j kv v k ckv v k : (5) L(v; y ) L(v ; y ) L v (v ; y )(v v ) L vv(v ; y )[v v ; v v ] L v (v s(v v ); y )(v v )ds L v (v ; y )(v v ) L vv(v ; y )[v v ; v v s L vv (v (v v ); y )[v v ; v v ]d L(v ; y )(v v ) A ds L v (v ; y )(v v ) L vv(v ; y )[v v ; v v ] s L vv (v (v v ); y )[v v ; v v ]d ds L vv(v ; y )[v v ; v v ] s L vv (v (v v ); y )[v v ; v v ] L vv (v ; y )[v v ; v v ]d ds Invoking the previous lemma we conclude jr L (v ; y ; v)j c s kv v k kv v k d ds ckv v k kv v k : 9

10 Now we are able to establish a strenghened second order sucient optimality condition. Theorem Let v (w ; u ) be admissible for the optimal control problem and full the rst order necessary optimality condition with associated adjoint state y. Choose " > and the corresponding set of positivity ". Suppose the existence of a > such that L vv (v ; y )[(h; z); (h; z)] k(h; z)k (6) holds for all z u u, where u U ad and u(x) u (x) for almost all x ", and h is the solution of the linearized equation Then there exist > and > such that for all v U ad with kv v k : 4h h b w(v )h b u (v )z on : (8) F (v) F (v ) kv v k (9) Remark: The solution h of the linearized equation (7), (8) exists, as b w (v) is nonpositive. Proof: Let v (u; w) be an admissible pair. Suppose that v fullls the assumptions of the theorem. We have F (v) L(v; y ) since v is admissible. By means of a Taylor{expansion of the Lagrange{function at v with respect to v we get L(v; y ) L(v ; y ) L u (v ; y )(u u ) L w (v ; y )(w w )) The admissibility of v yields L vv(v ; y )[v v ; v v ] r L (v ; y ; v): L(v ; y ) F (v ): Moreover, the necessary optimality conditions are fullled at v with adjoint state y. Thus, the third term is equal to zero. On the other hand, the second term is nonnegative due to the variational inequality (). However, we get even more. The denition of " yields: L u (v ; y )(u u ) ( u (v ) b u (v )y )(u u )ds x ( u (v ) b u (v )y )(u u )ds x " "ku u k L ("):

11 Continuing, we estimate F (v) F (v ) L u (v ; y )(u u ) L w (v ; y )(w w )) L vv(v ; y )[v v ; v v ] r L (v ; y ; v) F (v ) "ku u k L (") r L (v ; y ; v) L vv(v ; y )[v v ; v v ]: Next, we discuss the second derivative of the Lagrange{function. exploit (6). To this aim, a new control ~u is introduced by Here, we intend to ~u(x) ( u (x) on " u(x) on n " Let ~w be the state corresponding to the control ~u, i. e. b(w; ~u): Let ~v denote the pair ( ~w; ~u). Inserting these notations, we obtain by means of (): L vv (v ; y )[v v ; v v ] L vv (v ; y )[v ~v ~v v ; v ~v ~v v ] L vv (v ; y )[v ~v; v ~v] L vv (v ; y )[~v v ; ~v v ] L vv (v ; y )[~v v ; v ~v] L vv (v ; y )[~v v ; ~v v ] ck~v vk ~ck~v vk k~v v k : Moreover, we introduce w t as the solution of the linearized state equation at v : 4w t w b(v ) b w (v )(w t w ) b u (v )(~u u ): Setting v t (w t ; ~u) and invoking (), we continue by L vv (v ; y )[~v v ; ~v v ] L vv (v ; y )[~v v t v t v ; ~v v t v t v ] L vv (v ; y )[~v v t ; ~v v t ] L vv (v ; y )[v t v ; v t v ] L vv (v ; y )[v t v ; ~v v t ] L vv (v ; y )[v t v ; v t v ] ck~v v t k ~ck~v v t k kv t v k : (3) For the rst term of (3) the estimate (6) applies. To handle the second and the third term we consider rst the dierence ~v v t. By denition we have k~v v t k (k ~w w t k W k~u () ~uk L () k ~w w t k W (), and ~w w t fulls the equation 4( ~w w t ) ( ~w w t ~w w t b( ~w; ~u) b(w ; u ) b w (w ; u )(w t w ) b u (w ; u )(~u u ):

12 For the right hand side of the boundary condition we get b( ~w; ~u) b(w ; u ) b w (w ; u )(w t w ) b u (w ; u )(~u u ) b( ~w; ~u) b(w ; u ) b w (w ; u )(w t ~w ~w w ) b u (w ; u )(~u u ) b( ~w; ~u) b(w ; u ) b w (w ; u )( ~w w ) b u (w ; u )(~u u ) b w (w ; u )( ~w w t ) r b (v ; ~v) b w (w ; u )( ~w w t ): Thus the boundary condition is equivalent to Lemma ~w w t b w (w ; u )( ~w w t ) r b (v ; ~v): k~v v t k k ~w w t k W () ckr b (v ; ~v)k : (3) Inserting (3) we continue the estimation of (3) by L vv (v ; y )[~v v ; ~v v ] L vv (v ; y )[v t v ; v t v ] ck~v v t k ~ck~v v t k kv t v k kv t v k ckr b (v ; ~v)k ~ckr b (v ; ~v)k kv t v k kv t ~v ~v v k ckr b (v ; ~v)k ~ckr b (v ; ~v)k kv t ~v ~v v k k~v v k kv t ~vk k~v v k ckr b (v ; ~v)k ~ckr b (v ; ~v)k kv t ~vk ~ckr b (v ; ~v)k k~v v k k~v v k ckr(v b ; ~v)k ~ckr(v b ; ~v)k k~v v k! k~v v k ckrb(v ; ~v)k ~ckrb(v ; ~v)k : k~v v k k~v v k By (4), L vv (v ; y )[~v v ; ~v v ] k~v v k ck~v v k ~ck~v v k If v is suciently close to v with respect to the L {norm, then k~v v k is suciently close to zero, too. Therefore, the term in the brackets is greater or equal to ~ > for all admissible v such that kv v k. In this way we conclude that L vv (v ; y )[~v v ; ~v v ] ~k~v v k (3) holds for all admissible v with kv v k and all corresponding pairs ~v. Further, L vv (v ; y )[v v ; v v ] L vv (v ; y )[~v v ; ~v v ] ck~v vk ~ck~v vk k~v v k ~k~v v k ck~v vk ~ck~v vk k~v v k ~k~v v v v k ck~v vk ~ck~v vk k~v v v v k ~kv v k ~kv v k k~v vk ck~v vk ~ck~v vk kv v k ~kv v k ck~v vk ~ck~v vk kv v k

13 Now we are able to complete the estimation of the cost functional. We have F (v) F (v ) "ku u k L (") r L (v ; y ; v) L vv(v ; y )[v v ; v v ] F (v ) "ku u k L (") r L (v ; y ; v) ~kv v k ck~v vk ~ck~v vk kv v k F (v ) "ku u k L (") r L (v ; y ; v) ~kv v k ck~v vk ~ck~v vk kv v k : The next term under consideration is k~v vk kv v k. Youngs inequality implies k~v vk kv v k k~v vk kv v k for all >. Moreover, we have by construction k~v vk ck~u uk ; where the term on the right hand side is uniformly bounded. estimation with Thus, we continue the F (v) F (v ) "ku u k L (") r L (v ; y ; v) ~kv v k ck~v vk ~ck~v vk kv v k F (v ) "ku u k L (") r L (v ; y ; v) ~kv v k ck~u uk ~c k~u uk kv v k F (v ) "ku u k L (") c k~u uk! ~ kv v k ~c jrl (v ; y ; v)j kv v k The general assumptions imply by (5) that jr L (v ; y ; v)j ckv v k kv v k : Therefore, for kv v k 3 and suciently small, the term in the brackets is greater or equal to > for all v with kv v k 3. Hence F (v) F (v ) "ku u k L (") c k~u uk! ~ kv v k ~c jrl(v ; y ; v)j kv v k F (v ) "ku u k L (") c k~u uk kv v k ~u diers from u only on ", where ~u is equal to u. Therefore the norm k~u uk is equal to ku uk L (") 3ku u k L ("). Now we take 3 suciently small such that 3

14 " c( ) 3 >. Then we are allowed to omit the terms with ku u k L (") and arrive at F (v) F (v ) kv v k (33) for all admissible v with kv v k 3. Re{formulating this result we arrive at Theorem 3 Suppose that the assumptions of Theorem hold true. Then u is a locally optimal control in the sense of L (). 5 Remarks The results of the proceding sections ensure local optimality of the control u in a suciently small neighbourhood in L (). If jumps of u cannot be excluded, all functions in this neighbourhood must have jumps at the same position. This is too strong for many applications. Aiming to weaken this, we consider now the optimal control problem under stronger assumptions. We shall show that in this case the assumptions of Theorem are sucient for local optimality in the sense of L p (), where p < is suciently large (cf. section ). To do so we suppose the following additional properties of and b : The functions and b full the assumptions (A), (A), and (A3) from section and have the form: (A4) b(w; u) b (w) b (w)u (34) (w; u) (w) (w)u u ; (35) where L () with (x) almost everywhere on. We assume that the functions b i and i and all of their derivatives up to the second order are Lipschitz continuous. The crucial point in the proof of Theorem were the estimates of the rst and second order remainder terms of b and the Lagrange{function L. In general, these estimates are valid only with respect to the norm in W p () L (). For that reason Theorem 3 states only local optimality in the sense of L (). The additional assumptions (34) and (35) allow us to improve the estimates of the remainder terms. We set for convenience kvk p p k(w; u)k p p kwk p W p () kukp L p() : Lemma 5 The rst order remainder term of b fullls the estimate for all v (w; u) W p () L (). kr b (v ; v)k L () ckv v k kv v k p (36) Proof: We have kr b (v ; v)k L () (b(v) b(v ) b v (v )(v v )) ds fb v (v s(v v ) b v (v )g(v v )ds A ds x : (37) 4

15 Making use of the special structure of b we can rewrite the integrand at almost all x in the following way jb v (v (x) s(v(x) v (x)) b v (v (x))g(v(x) v (x))j jfb w (w (x) s(w(x) w (x))) b w (w (x))g(w(x) w (x)) fb w (w (x) s(w(x) w (x)))(u (x) s(u(x) u (x))) b w (w (x))u (x)g (w(x) w (x)) fb (w (x) s(w(x) w (x))) b (w (x))g(u(x) u (x)))j: The functions b and b and the corresponding derivatives are assumed to be Lipschitz{ continuous. Using u L (), we can estimate jb v (v (x) s(v(x) v (x)) b v (v (x))g(v(x) v (x))j c (js(w(x) w (x))(w(x) w (x))j sj(u(x) u (x))(w(x) w (x))j ju (x)jjs(w(x) w (x))(w(x) w (x))j) c (js(w(x) w (x))(w(x) w (x))j sj(u(x) u (x))(w(x) w (x))j) : Inserting this estimate in (37) we get kr b (v ; v)k L () c c fjs(w(x) w (x))(w(x) w (x))j sj(u(x) u (x))(w(x) w (x))jgds fj(w(x) w (x))j j(u(x) u (x))(w(x) w (x))jg ds x (jw w j 4 jw w j 3 ju u j jw w j ju u j )ds x ckw w k C() ckw w k W p ()kv v k ckv v k p kv v k (jw w j jw w jju u j ju u j )ds x The statement of the Lemma is now a simple conclusion. To prove estimates of the second order remainder term of the Lagrange{ function we have used in the general case the Lipschitz{argument of Lemma. This general result cannot be transfered to our special case replacing the {norm by the p{norm. However, we are able to show Lemma 6 The estimate jl vv (v (vv ); y )[vv ; vv ]L vv (v ; y )[vv ; vv ]j ckvv k p kvv k (38) holds for all admissible pairs v W p () L () and (; ). Proof: By denition we have L vv (v ; y )[v v ; v v ] A ds x 5

16 ' ww (w )(w w ) dx ( ww (v )(w w ) wu (v )(w w )(u u ) uu (v )(u u ) ds x (b ww (v )(w w ) b wu (v )(w w )(u u ) b uu (v )(u u ) ) y ds x : (34) and (35) imply: L vv (v ; y )[v v ; v v ] ' ww (w )(w w ) dx (( ww (w ) ww (w )u )(w w ) w (w )(w w )(u u ) (u u ) )ds x y (b ww (w )(w w ) b ww (w )u (w w ) b wu (w )(w w )(u u ))ds x Analogously we have with w w (w w ) and u u (u u ) L vv (v ; y )[v v ; v v ] ' ww (w )(w w ) dx (( ww (w ) ww (w )u )(w w ) w (w )(w w )(u u ) (u u ) )ds x y (b ww (w )(w w ) b ww (w )u (w w ) b wu (w )(w w )(u u ))ds x Using the Lipschitz{continuity of the partial derivatives of ',, and b, we can estimate jl vv (v ; y )[v v ; v v ] L vv (v ; y )[v v ; v v ]j c c (jw w j(w w ) ju u j(w w ) jw w jj(u u )(w w )j)ds x jw w j(w w ) dx (jw w j(w w ) ju u j(w w ) jw w jj(u u )(w w )j)ds x jw w j(w w ) dx ckw w k jw w jjw w jdx ckv v k p kv v k : ((w w ) ju u jjw w j j(u u )(w w )j)ds x A 6

17 In particular, the embedding result kwk C() ckwk W p () was used. Therefore, the statement of the Lemma holds true. The next statement is an immediate consequence. Lemma 7 The second order remainder term r L (v ; y ; v) satises the estimate for all admissible v W p () L (). jr L (v ; y ; v)j ckv v k p kv v k (39) Proof: The proof is along the lines of the proof of Lemma 4. We only have to use Lemma 6 instead of Lemma for the Lipschitz{estimate. Summarizing up we arrive at Theorem 4 Let the general assumptions (A), (A), and (A3) and the assumption (A4) of this section be fullled. Suppose further that the assumptions of Theorem are satised. Then u is locally optimal in the sense of L p (). Proof: The proof of local optimality of u in the sense ofl () was mainly based on Wp (){regularity of the state w, on dierentiability of the nonlinearities in C()L (), and on the estimates of the remainder terms of b and L. Wp (){regularity is preserved for controls u L p (). Due to the strengthened assumptions on the appearance of u, the dierentiability is guaranteed in C() L p (), too. Therefore, to prove the statement of Theorem 4, we can proceed along the lines of the proof of Theorem. The only dierence is the use of estimates according to Lemma 5 and Lemma 7 instead of Lemma 3 and Lemma 4. References [] Robert A. Adams. Sobolev Spaces, volume 65 of Pure and Applied Mathematics. Academic Press Inc., 978. [] E. Casas. Pontryagin's Principle for Optimal Control Problems Governed by Semilinear Elliptic Equations, volume 8 of International Series of Numerical Mathematics, pages 97{4. Birkhauser Verlag Basel, 994. [3] E. Casas and J. Yong. Maximum principle for state-constrained optimal control problems governed by quasilinear elliptic equations. Dierential and Integral equations, 8():{8, January 995. [4] A. L. Dontchev, W. W. Hager, A. B. Poore, and B. Yang. Optimality, stability, and convergence in nonlinear control. Appl. Math. Optim., 3(3):97{36, May/June 995. [5] K. Eppler and A. Unger. Boundary control of semilinear elliptic equations { existence of optimal solutions. to appear, 995. [6] H. Goldberg and F. Troltzsch. Second{order sucient optimality conditions for a class of nonlinear parabolic boundary control problems. SIAM J. Control and Optimization, 3(4):7{5, July

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