SECOND ORDER SUFFICIENT OPTIMALITY CONDITIONS FOR SEMILINEAR ELLIPTIC EQUATIONS

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1 SECOND ORDER SUFFICIENT OPTIMALITY CONDITIONS FOR SOME STATE{CONSTRAINED CONTROL PROBLEMS OF SEMILINEAR ELLIPTIC EQUATIONS EDUARDO CASAS y, FREDI TR OLTSCH z, AND ANDREAS UNGER z Abstract. This paper deals with a class of optimal control problems governed by elliptic equations with nonlinear boundary condition. The case of boundary control is studied. Pointwise constraints on the control and certain equality and set{constraints on the state are considered. Second order sucient conditions for local optimality of controls are established. Key words. Boundary control, semilinear elliptic equations, sucient optimality conditions, state constraints AMS subject classications. 49K20, 35J25 1. Introduction. In contrast to the optimal control of linear systems with convex objective, where rst order necessary optimality conditions are already sucient for optimality, higher order conditions such as second order sucient optimality conditions (SSC) should be employed to verify optimality for nonlinear systems. (SSC) have also proved to be useful for showing important properties of optimal control problems such as local uniqueness of optimal controls and their stability with respect to certain perturbations. Moreover, they may serve as assumption to guarantee the convergence of numerical methods in optimal control. In this respect, we refer to the general expositions by Maurer and owe [15] and Maurer [14] for dierent aspects of (SSC). The approximation of programming problems in Banach spaces is discussed in Alt [2]. Moreover, Alt [3], [4] has established a general convergence analysis for Lagrange{Newton methods in Banach spaces. Meanwhile, an extensive number of publications has been devoted to dierent aspects of (SSC) for control problems governed by ordinary dierential equations. In particular, the well known two{norm discrepancy has received a good deal of attention. We refer for instance to Ioe [13] and Maurer [14]. First investigations of (SSC) for control problems governed by partial dierential equations have been published by Goldberg and Troltzsch [11], [12] for the boundary control of parabolic equations with nonlinear boundary conditions. In [9], Casas, Troltzsch, and Unger have extended these ideas to elliptic boundary control problems with pointwise constraints on the control. Moreover, they tightened the gap between second order necessary and sucient optimality conditions. This was done by the consideration of sets of strongly active constraints according to Dontchev, Hager, Poore and Yang [10]. This technique is also related to rst order sucient optimality conditions introduced by Maurer and owe [15]. It should be mentioned that already four norms have to be used in this case (L 1 {norm for dierentiation, L 2 {norm to formulate (SSC), L 1 {norm for the rst order sucient optimality condition, and certain L p {norms to obtain optimal regularity results). This research was partially supported by European Union, under HCM Project number ER- BCHRXCT940471, and by Deutsche Forschungsgemeinschaft, under Project number Tr 302/1-2. The rst author was also supported by Direccion General de Investigacion Cientca y Tecnica (Spain) y Departamento de Matematica Aplicada y Ciencias de la Computacion, E.T.S.I. Industriales y de Telecomunicacion, Universidad de Cantabria, Santander, Spain. z Fakultat fur Mathematik, Technische Universitat Chemnitz-wickau, D Chemnitz, Germany 1

2 2 E. CASAS AND F. TR OLTSCH AND A. UNGER F. Bonnans [5] has shown that a very weak form of second order sucient conditions can be used to verify local optimality for a particular class of semilinear elliptic control problems with constraints on the control: If the second order derivative of the Lagrange function is a Legendre form, then it suces to have its positivity in all critical directions. In our paper, the results of [9] will be extended to additional constraints on the state. In this way, we continue the investigations by Casas and Troltzsch [8] on second order necessary conditions. We will also rely on general ideas of Maurer and owe [15] combining their approach with a detailed splitting technique. At the beginning, we aimed to establish second order sucient optimality conditions for boundary control problems governed by semilinear elliptic equations in domains of arbitrary dimension with general pointwise constraints on the control and the state. However, we soon recognized that pointwise state{constraints lead to essential and quite surprising diculties. To establish second order sucient optimality conditions for problems with pointwise state{constraints given on the whole domain, we had to restrict ourselves to two{dimensional domains with controls appearing linearly in the boundary condition. These obstacles might indicate some limits for the "traditional" type of (SSC) for control problems governed by PDEs. If pointwise state{constraints are imposed on compact subsets of the domain, while the other quantities are suciently smooth, then arbitrary dimensions can be treated without restrictions on the nonlinearities. In this case the adjoint state belongs to L 1 (). Moreover, we are able to avoid the assumption of linearity of the boundary condition with respect to the control by introducing some extended form of second order optimality conditions. 2. The optimal control problem. We consider the problem: Minimize the functional (2.1) F 0 (y; u) = f(x; y(x)) dx + g(x; y(x); u(x)) ds(x) subject to the equation of state (2.2) y(x) + y(x) = 0 y(x) = b(x; y(x); u(x)) on ; to the constraints on the state y (2.3) (2.4) F i (y) = 0; i = 1; : : :; m; E(y) 2 K; and to the constraints on the control u (2.5) u a (x) u(x) u b (x) a. e. on : In this setting, R n is a bounded domain with a Lipschitz boundary according to the denition by Necas [17]. Moreover, suciently smooth functions f : R! R and g; b : R 2! R are given. The is used for the derivative in the direction of the unit outward normal on. The functionals F i : C()! R, i = 1; : : :; m, are supposed to be twice continuously Frechet dierentiable, that is to be of class C 2. By E we denote a mapping of class C 2 from C() into a real Banach

3 SUFFICIENT SECOND ORDER OPTIMALITY CONDITIONS 3 space. K is a non-empty convex closed set, and u a, u b :! R are functions of L 1 () satisfying u a (x) u b (x) on. The control u is looked upon the control space U = L 1 (), while the state y is dened as weak solution of (2.2) in the state space C() \ H 1 () = Y, that is (2.6) (ryrv + yv) dx = b(; y; u)v ds 8v 2 H 1 (): We endow Y with the norm kyk Y = kyk C() + kyk H 1 (). The following assumptions are imposed on the given quantities: (A1) For each xed x 2 or, respectively, the functions f = f(x; y), g = g(x; y; u), and b = b(x; y; u) are of class C 2 with respect to (y; u). For xed (y; u), they are Lebesgue measurable with respect to x 2 or x 2, respectively. Throughout the paper, partial derivatives are indicated by associated subscripts. For instance, b yu stands 2 b=@y@u. By b 0 (x; y; u) and b 00 (x; y; u) we denote the gradient and the Hessian matrix of b with respect to (y; u): b 0 (x; y; u) = by (x; y; u) b u (x; y; u) ; b 00 (x; y; u) = byy (x; y; u) b yu (x; y; u) b uy (x; y; u) b uu (x; y; u) jb 0 j and jb 00 j are dened by adding the absolute values of all entries. In the next assumption, xed parameters p > n 1 and s, r are used, which depend on n. For the possible (minimal) choice of s and r we refer to the discussion of regularity in (3.13). Roughly speaking, we have yj 2 L s () and y 2 L r () in the linearized system (2.2), if u 2 L 2 (). As usual, s 0 and r 0 denote conjugate numbers. For instance, s 0 is dened by 1=s 0 + 1=s = 1. (A2) For all M > 0 there are constants C M > 0, functions M f 2 L (r=2)0 (), M;1 g 2 L (s=2)0 (), M;2 g 2 L 2(s=2)0 (), M;3 g 2 L 1 (), and a continuous, monotone increasing function 2 C(R + [ f0g) with (0) = 0 such that: (i) (2.7) b y (x; y; u) 0 a. e. x 2 ; 8(y; u) 2 R 2 ; b(; 0; 0) 2 L p (), for a p > n 1, jb 0 (x; y; u)j + jb 00 (x; y; u)j C M, jb 00 (x; y 1 ; u 1 ) b 00 (x; y 2 ; u 2 )j C M (jy 1 y 2 j + ju 1 u 2 j) for almost all x 2 and all jyj; juj; jy i j; ju i j M, i = 1; 2. (ii) f(; 0) 2 L 1 (); f y (; 0) 2 L r0 (); f yy (; 0) 2 L (r=2)0 () jf yy (x; y 1 ) f yy (x; y 2 )j M f (x) (jy 1 y 2 j) for all x 2 ; jy i j M, i = 1; 2. (iii) g(; 0; 0) 2 L 1 (); g y (; 0; 0) 2 L s0 (); g u (; 0; 0) 2 L 2 (); g yy (; 0; 0) 2 L (s=2)0 (); g yu (; 0; 0) 2 L 2(s=2)0 (); g uu (; 0; 0) 2 L 1 () (here, stands for x) jg yy (x; y 1 ; u 1 ) g yy (x; y 2 ; u 2 )j M;1 g (x)(jy 1 y 2 j + ju 1 u 2 j) jg yu (x; y 1 ; u 1 ) g yu (x; y 2 ; u 2 )j M;2 g (x)(jy 1 y 2 j + ju 1 u 2 j) jg uu (x; y 1 ; u 1 ) g uu (x; y 2 ; u 2 )j M;3 g (x)(jy 1 y 2 j + ju 1 u 2 j) for almost all x 2 and all jy i j M; ju i j M. ;

4 4 E. CASAS AND F. TR OLTSCH AND A. UNGER Remark 2.1. Notice that the estimates in (i){(iii) imply boundedness and Lipschitz properties of b; f; g; b 0 ; f 0 ; g 0 in several L{spaces. We omit them, because they follow from the mean value theorem. (A3) (i) Let us dene the norm kyk 2 = kyk C(A) + kyk Lr () + kyk Ls () for y 2 C(), where A is a certain measurable compact subset. Here A stands for a subset, where we know y 2 C(A) for Neumann boundary data given in L 2 (). In the case n = 2 we may take A =, while A is needed for n > 2. For A = ; we put kyk C(A) = 0. We assume for a xed reference state y 2 C() that jfi(y)yj 0 C F kyk 2 8y 2 C() jfi 00 (y)[y 1 ; y 2 ]j C F ky 1 k 2 ky 2 k 2 8y 1 ; y 2 2 C() holds with some C F > 0. Moreover, we require with a C M > 0 jfi(y 0 1 )y Fi(y 0 2 )yj C M ky 1 y 2 k 2 kyk 2 j(fi 00(y 1) Fi 00(y 2))[y; v]j C M (ky 1 y 2 k C() )kyk 2 kvk 2 for all y j with ky j k C() M, j = 1; 2, all y, v from C(), and all i = 1; : : :; m: (ii) Analogous assumptions are imposed on E : C()!, where k k is to be substituted for j j. For instance, ke 0 (y)yk C E kyk 2 8y 2 C() is supposed. We shall explain the main constructions of our paper by the following canonical example (P) that ts in the general setting: Example (P) Minimize subject to and 1 (y y d ) 2 dx u 2 ds 4y + y = 0 y = u y 3 on juj 1 y(0) y 0 in the open unit ball R 3 around zero, where > 0, y 0 2 R, and y d 2 L 1 () are given. Here, we have = R, K = R, A = f0g, E(y) = y(0)y 0, and we need y 2 C() to make E well dened.

5 SUFFICIENT SECOND ORDER OPTIMALITY CONDITIONS 5 3. The state equation, rst order necessary optimality conditions. It can be shown that the equation (2.2) admits for each u 2 U ad a unique weak solution y = y(u) 2 Y, where U ad = fu 2 L 1 () j u a (x) u(x) u b (x) a. e. on g. Moreover, there is a constant M such that (3.1) ky(u)k Y M 8u 2 U ad : In particular, it holds kyk C() M. Casas and Troltzsch [8] have proved that the mapping u 7! y(u) from L 1 () into Y is of class C 2. Furthermore, there is a constant C 2 such that the Lipschitz property ky(u 1 ) y(u 2 )k 2 C 2 ku 1 u 2 k L 2 () holds for all u 1 ; u 2 2 U ad (k k 2 as dened in (A3)). For xed u 2 U ad we have b(; y; u) 2 L p (), hence the weak solution y 2 Y of (2.2) belongs to the space Y q;p = fy 2 H 1 () j y + y 2 L q y 2 L p ()g; which is known to be continuously embedded into Y = C()\H 1 () for each q > n=2 and each p > n 1. In all what follows we assume that a reference pair (y; u) 2 Y U ad is given, satisfying together with an associated adjoint state ' 2 W 1; (), 8 < n=(n 1), and Lagrange multipliers = ( 1 ; : : :; m ) T 2 R m ; z 2 the associated standard rst order necessary optimality conditions, which we shall recall below for convenience. They can be proved following Casas [7], Bonnans and Casas [6], or owe and Kurcyusz [23]. We will just assume them. The rst order optimality system to be satised by (y; u) consists of the state equations (2.2), the constraint u 2 U ad, the adjoint equation (3.2) (3.3) ' + ' = f y (; y) + mx ' = b y (; y; u)' + g y (; y; u) + i F 0 i (y)j + (E 0 y) z j for the adjoint state ', the complementary slackness condition mx i=1 in i F 0 i(y)j + (E 0 y) z j on (3.4) hz ; E(y)i K; and the variational inequality (3.5) (g u (x; y(x); u(x)) + '(x)b u (x; y(x); u(x)))(u(x) u(x)) ds(x) 0 for all u 2 U ad. We have F 0 i (y) 2 C(), i = 1; : : :; m, and E 0 (y) z 2 C(), hence these quantities can be identied with real Borel measures on. Let a nonnegative function 2 L 1 () and real Borel measures and concentrated on and, respectively, be given. Then the problem (3.6) ' + ' = ' + ' = on

6 6 E. CASAS AND F. TR OLTSCH AND A. UNGER admits a unique solution ' 2 W 1; () for all < n=(n 1) (see Casas [7]). In view of this, we may write ' = ' 0 + mx i=1 i ' i + ' E ; where ' 0, ' i, and ' E solve (3.6) for = f y, Fi 0(y)j, E 0 (y) z j and = g y, Fi 0(y)j, E 0 (y) z j, respectively. We have at least ' 0, ' i and ' E in W 1; (). Moreover, ' satises the formula of integration by parts (3.7) (y + y)' dx + (@ y + y)' ds(x) = y d + y d for all y 2 Y q;p, where q > n=2, p > n 1. Therefore, it is easy to verify that the optimality conditions can be expressed by the Lagrange function (3.8) L(y; u; '; ; z ) = F 0 (y; u) + mx j=1 (y + y) ' dx j F j (y) + hz ; E(y)i; (@ y b(; y; u))' ds L : Y q;p U W 1; () R m! R. The regularity of y and ' ts together, as ' 2 W 1; () for all < n=(n 1) ensures ' 2 L s () for all s < n=(n 2) ( Necas [17], Thm. 3.4, p. 69) and 'j 2 L r () holds for all r < 1 + 1=(n 2) ([17], Thm. 4.2, p.84). Therefore, this denition makes sense. In (3.8), h; i denotes the duality pairing between and its dual space. The Lagrange function L is of class C 2 with respect to (y; u) for xed ';, and z. Thanks to (3.7), the optimality system can be rewritten in terms of L. Then it is expressed by (2.6), the constraints on the state (2.3), (2.4), the constraints on the control u 2 U ad, and (3.9) (3.10) (3.11) L y (y; u; '; ; z )y = 0 L u (y; u; '; ; z )(u u) 0 8y 2 Y 8u 2 U ad hz ; E(y)i K: This form is more convenient for our later evaluations. Example: In (P), adjoint equation and variational inequality are given by 4' + ' = y y d + z ' + 3 y 2 ' = 0 (u + ')(u u)ds 0 8juj 1; where (0) is the Dirac measure. To shorten our notation, derivatives taken at (y; u; '; ; z ) will be indicated by a bar. For instance, L y y, L u (u u) will stand for the derivatives in (3.9) and (3.10), respectively. L yy [y 1 ; y 2 ] denotes the second order derivative of L in the directions y 1 ; y 2 taken at (y; u; '; ; z ). Moreover, L ww [w 1 ; w 2 ] is the second order derivative

7 SUFFICIENT SECOND ORDER OPTIMALITY CONDITIONS 7 of L in the directions w 1 = (y 1 ; u 1 ); w 2 = (y 2 ; u 2 ). If w 1 = w 2 = w, we will write for short L ww [w; w] = L ww [w] 2. Next we provide some useful results on linearized versions of the state equation. Regard rst the linear system y + y = f (3.12) y + y = g on ; where 2 L 1 () is nonnegative. For each pair (f; g) 2 L 1 () L 1 (), this system admits a unique solution y 2 W 1; (), where < n=(n 1), see Casas [7]. (Notice that a function of L 1 can be considered as a Borel measure.) On the other hand, the solution y of (3.12) belongs to H 1 () \ C(), if (f; g) 2 L q () L p (). This regularity result is well known for domains with C 1 {boundary. Moreover, it remains true for domains with Lipschitz boundary in the sense of Necas [17] (see Stampacchia [19] and Murthy and Stampacchia [16]). On account of this, the mapping D : (f; g) 7! (y; yj ) is continuous from L 1 () L 1 () into L s () L t () for s < n=(n 2) and t < (n1)=(n2). D is continuous also from L q ()L p () into L 1 ()L 1 (). We obtain these spaces by embedding results for W 1; () (Necas [17]). In both cases, this mapping is linear and continuous. Interpolation theory applies to show the following results for D considered as a mapping dened on L 2 () L 2 (): (3.13) y 2 8 >< >: C(); n = 2; L r () 8r < 1; n = 3; L r () 8r < 2n yj n 3 ; n 2 4; 8 >< >: C(); n = 2; L s () 8s < 1; n = 3; L s () 8s < 2(n 1) n 3 ; n 4: 4. Regularity condition and linearization theorem. Let us recall that we consider throughout the paper a xed reference pair (y; u) satisfying together with ('; ; z ) the rst order necessary conditions (3.9) { (3.11). The linearized cone of U ad at u is the set C(u) = fv 2 L 1 () j v = %(u u); % 0; u 2 U ad g. Let F = F (y) denote the mapping y 7! (F 1 (y); : : : ; F m (y)) T from Y to R m. For convenience, we introduce the set of all feasible pairs M = fw = (y; u) 2 Y U ad j y = G(u) and y satises the state{constraintsg (notice that G is the nonlinear control-state-mapping). Following Maurer and owe [15], the linearized cone L(M; w) at w = (y; u) is dened by where (4.1) (4.2) (4.3) L(M; w) = fw j w = (y; u); u 2 C(u) and (y; u) satises (4.1) {(4.3)g; y + y = = F 0 (y)y = 0 E 0 (y)y 2 K(E(y)): 0 in b y (; y; u)y + b u (; y; u)u on Here, K(E(y)) = fz 2 j z = %( E(y)); % 0; 2 Kg is the conical hull of K E(y). Remark 4.1. The choice = R k, E(y) = (E 1 (y); : : : ; E k (y)) T, K = (R k ) for E : Y! is of particular interest. Then (4.3) reduces to E 0 i(y)y 0

8 8 E. CASAS AND F. TR OLTSCH AND A. UNGER for all active i 2 f1; : : :; kg, that is for all i, where E i (y) = 0 holds. Example: The linearized cone for (P) is the set of all pairs (y; u), such that u 2 C(u) and (4.4) (4.5) 4y + y = y + 3y 2 y = u; y(0) 0; if y(0) = y 0 (active state constraint). If the state constraint is not active, then (4.5) disappears. The following regularity assumption (R) is basic for our further analysis: To formulate (R) we combine the two state constraints to one general constraint. We therefore take = R m, K = f0g K, dene T : Y! by T (y) = (F (y); E(y)) and put K(T (y)) = f0gk(e(y)). The regularity condition was introduced by owe and Kurcyusz [23] and requires (R) T 0 (y)g 0 (u)c(u) K(T (y)) =. This condition is sucient for the existence of a (non{degenerate) Lagrange multiplier associated to the state{constraint E(y) 2 K, see [23]. We should underline that (R) does not rely on the condition int K 6= ;. In Appendix 7.1 we shall present some sucient conditions for (R) which, however, require int K 6= ;. Moreover, (R) is discussed for the canonical example (P) there. For = R k, K = (R k ), the condition (R) is equivalent to the well{known Mangasarian{Fromowitz condition. Theorem 4.2. Suppose that (R) is satised. Then for all pairs (^y; ^u) 2 M there is a pair (y; u) 2 L(M; w) such that the dierence r = (r y ; r u ) = (^y; ^u)(y; u)(y; u) can be estimated by (4.6) (4.7) krk Y L 1 () C L;p k^u uk L 1 () k^u uk Lp () 8p > n 1 krk C L;2 k^u uk L 1 () k^u uk L 2 () ; where krk = kr y k 2 + kr u k L 2 (). In the particular case b(x; y; u) = b 1 (x; y) + b 2 (x)u we have (4.8) krk Y L 1 () C L;p k^u uk 2 L p () 8p > n 1: This theorem is proved in Appendix 7.2. Let us conclude this section by considering some useful estimates for L 00 and for certain remainder terms. First, we evaluate the expression for L 00 [(y 1 ; u 1 ); (y 2 ; u 2 )] = L 00 (y; u; '; ; z )[(y 1 ; u 1 ); (y 2 ; u 2 )]; where L 00 denotes the second order derivative of L with respect to (y; u). We have (4.9) L 00 [(y 1 ; u 1 ); (y 2 ; u 2 )] = + + f yy (; y)y 1 y 2 dx + mx i=1 ' (y 1 ; u 1 )b 00 (; y; u)(y 2 ; u 2 ) T ds (y 1 ; u 1 )g 00 (; y; u)(y 2 ; u 2 ) T ds i F 00 i (y)[y 1 ; y 2 ] + hz ; E 00 (y)[y 1 ; y 2 ]i:

9 SUFFICIENT SECOND ORDER OPTIMALITY CONDITIONS 9 Example: In the case of (P), L 00 admits the form L 00 [(y 1 ; u 1 ); (y 2 ; u 2 )] = y 1 y 2 dx + (6' y y 1 y 2 + u 1 u 2 )ds: (4.10) The term connected with ' causes troubles, more precisely, I = ' (b yy (; y; u)y 1 y 2 + b yu (; y; u)(y 1 u 2 + y 2 u 1 ) + b uu (; y; u)u 1 u 2 ) ds: An estimate of I is needed with respect to the norm kyk 2 + kuk L 2 () (cf. (4.19)). We therefore have to require at least ' 2 L 2 () in the second item and ' 2 L 1 () in the third one. On the other hand, only ' 2 L r () follows from ' 2 W 1; () for r < (n 1)=(n 2), see Necas [17], p. 84. For n = 2 we obtain ' 2 L r () for all r < 1, while n = 3 yields the regularity ' 2 L r () for all r < 2. On account of this, the following additional assumption is crucial for our analysis: (A4) Let one of the following statements be true: (i) ' 2 L 1 (). (ii) b uu (x; y; u) = 0 on R 2 and, if n 3, then ' 2 L r () for some r > n 1. (iii) b uu (x; y; u) = b yu (x; y; u) = 0 on R 2 and, if n 4, then ' 2 L r () for some r > (n 1)=2. (iv) b 00 (; y; u) = 0. Let us briey comment on the consequences of these assumptions: (i) is true, if f y 2 L q (), g y 2 L p (), and if the restrictions of Fi 0, i = 1; : : :; m, and E 0 (y) z to and, respectively, belong to L q (), L p (), as well. Moreover, (i) holds for functionals Fi 0, i = 1; : : :; m, and E0 (y) z of C(), where the associated real Borel measures are concentrated on the set A. In addition to some assumptions on the regularity of ' for n 3; 4, (ii) requires linearity of b with respect to u, that is b(x; y; u) = b 0 (x; y) + b 1 (x; y)u; (iii) means that b(x; y; u) = b 1 (x; y) + b 2 (x)u, while (iv) is only true for an ane{linear boundary condition (but yet for a nonlinear functional F 0 ). (A4) is obviously satised in the example (P). As a consequence of (A3) and (A4), pointwise state{constraints on the whole set can only be handled by the standard part of our theory, if u appears linearly in the boundary condition and n = 2. In the considerations below, we denote by ri T the remainder terms associated with the i{th order Taylor expansion of a mapping T. For instance, the following rst and second order expansions of b(x; y; u) will be used at triplets (x; y; u) and (x; y; u) 2 R n+2 : (4.11) b(x; y; u) b(x; y; u) = b 0 (x; y; u)(y y; u u) + r b 1; where (4.12) r b 1 = (b # y b y )(y y) + (b # u b u )(u u);

10 10 E. CASAS AND F. TR OLTSCH AND A. UNGER and b # y; b # u; b y ; b u denote b y ; b u taken at (x; y + #(y y); u + #(u u)) and (x; y; u), respectively, with some # 2 (0; 1). Expanding the same expression up to the order two, we have (4.13) b(x; y; u) b(x; y; u) = b 0 (x; y; u)(y y; u u) (y y; u u)b00 (x; y; u) y y + r u u 2; b with the second order remainder term (4.14) r b 2 = 1 2 (y y; u u)[b00;# b 00 ](y y; u u) T : Here, b 00;# ; b 00 denote the Hessian matrix of b with respect to (y; u) taken at the same triplets as above. Due to our assumptions on b 0 and b 00, the estimates (4.15) (4.16) jr b 1j C M (jy yj 2 + ju uj 2 ) jr b 2j C M (jy yj + ju uj)(jy yj 2 + ju uj 2 ) are valid for all jyj; jyj; juj; juj M. We continue with the discussion of the remainders r L and 1 rl 2. A Taylor expansion of L gives L(y; u; '; ; z ) L(y; u; '; ; z ) = L y (y y) + L u (u u) + r L 1 = L y (y y) + L u (u u) Lyy [y y] 2 + 2L yu [y y; u u] + L uu [u u] 2 + r L 2 ; where L indicates that L and its derivatives are taken at (y; u; '; ; z ). We have r L 1 = (L # y L y )(y y) + (L # u L u )(u u) r L 2 = 1 2 (L # yy L yy )[y y] 2 + 2(L # yu L yu )[y y; u u] + (L # uu L uu )[u u] 2 : L # indicates that (y + #(y y); u + #(u u); '; ; z ) is substituted for (y; u; '; ; z ) in L 0 and L 00 with some # 2 (0; 1). On account of the assumptions (A1){(A4), we are able to verify (4.17) jr L 1 j C L (ky yk ku uk 2 L 2 ) () (4.18) jr L 2 j C L (ky yk C() + ku uk L 1 () )(ky yk ku uk 2 L 2 ) () and (4.19) jl 00 [(y 1 ; u 1 ); (y 2 ; u 2 )]j C L (ky 1 k 2 + ku 1 k L 2 () )(ky 2 k 2 + ku 2 k L 2 () ) with some C L > 0, depending in particular on '. For the denition of we refer to the assumption (A2). The analysis of (4.17){(4.19) is performed in Appendix Standard second order sucient optimality condition. Our main aim is to establish sucient optimality conditions close to the necessary ones derived in Casas and Troltzsch [8]. Therefore, we consider also certain rst order sucient optimality conditions. We shall combine an approach going back to owe and Maurer [15] with a splitting technique introduced by Dontchev, Hager, Poore, and Yang [10]. The

11 SUFFICIENT SECOND ORDER OPTIMALITY CONDITIONS 11 method of [10] was focussed on the optimal control of ordinary dierential equations and has been extended later by the authors in [9] to the case of elliptic equations without state{constraints. In [15], Maurer and owe introduced rst order sucient optimality conditions for dierentiable optimization problems subject to a general constraint g(w) 0. For our problem, the application of their approach in its full generality turned out to be rather technical. Therefore, we introduce in an initial step the rst order sucient optimality condition only for the constraints on the control. Later, we shall deal in the same way with additional state{constraints. Dene for xed > 0 (arbitrarily small) the set = fx 2 j jg u (x; y(x); u(x)) + '(x)b u (x; y(x); u(x))j g: is a subset of "strongly active" control constraints (cf. (3.5)). Let us mention at this point the relation (5.1) hz ; E 0 (y)yi 0 for all (y; u) 2 L(M; w), which follows from hz ; E 0 (y)yi = %hz ; E(y)i 0 in view of (3.4). Let P : L 1 ()! L 1 () denote the projection operator u 7! n u = P u. In other words, (P u)(x) = u(x) holds on n, while (P u)(x) = 0 holds on. We begin with our rst and at the same time simplest second order sucient optimality condition. (SSC) There exist positive numbers and such that (5.2) L 00 (y; u; '; ; z )[w 2 ; w 2 ] ku 2 k 2 L 2 () holds for all pairs w 2 = (y 2 ; u 2 ) constructed in the following way: For every w = (y; u) 2 L(M; w) we split up the control part u in u 1 = (u P u) and u 2 = P u. The solutions of the linearized state equation (5.3) yi + y i = 0 y i = b y (; y; u)y i + b u (; y; u)u i on : associated to u i are denoted by y i, i = 1; 2. By this construction, we get the representation w = w 1 + w 2 = (y 1 ; u 1 ) + (y 2 ; u 2 ). Remark 5.1. By (SSC), the coercitivity condition (5.2) is required on the whole set L(M; w), if is empty. This quite strong second order condition is obtained by the formal setting = 1. Let the feasible pair w = (y; u) satisfy the regularity condition Theorem 5.2. (R), the rst order necessary optimality conditions (3.9){(3.11), and the second order sucient optimality condition (SSC). Suppose further that the general assumptions (A1){(A4) are satised. Then there are constants % > 0 and 0 > 0 such that (5.4) F 0 (^y; ^u) F 0 (y; u) + 0 k^u uk 2 L 2 () holds for all feasible pairs ^w = (^y; ^u) such that (5.5) k^u uk L 1 () < %:

12 12 E. CASAS AND F. TR OLTSCH AND A. UNGER Proof. We denote by l = ('; ; z ) the triplet of Lagrange multipliers appearing in the rst order necessary optimality conditions. Let an arbitrary feasible pair ^w = (^y; ^u) be given. Then (5.6) F 0 ( ^w) F 0 (w) = L( ^w; l) L(w; l) hz ; E(^y) E(y)i follows from F ( ^w) = F (w) = 0. The complementary slackness condition implies hz ; E(^y) E(y))i 0; hence we can avoid this term, and a second order Taylor expansion yields F 0 ( ^w) F 0 (w) L( ^w; l) L(w; l) l u (^u u) ds L00 (w; l)[ ^w w] 2 + r L 2 (w; ^w w); where l u (x) = g u (x; y(x); u(x)) + '(x)b u (x; y(x); u(x)). Using the variational inequality, we nd (5.7) F 0 ( ^w) F (w) j^u uj ds L00 (w; l)[ ^w w] 2 + r L 2 (w; ^w w): Let us introduce for convenience the bilinear form B = L 00 (w; l). Next we approximate ^w w by w = (y; u) 2 L(M; w), according to Theorem 4.2. In this way we get a remainder r = (r y ; r u ) satisfying ^w w = w + r together with the estimate (5.8) krk C L k^u uk L 1 () k^u uk L 2 () : It follows that B[ ^w w] 2 = B[w] 2 + 2B[r; w] + B[r] 2. We have w 2 L(M; w), hence (SSC) applies to B[w] 2. Now we substitute in B[w] 2 the representation w = w 1 + w 2 described in (SSC) and deduce B[w] 2 = B[w 2 ] 2 + 2B[w 1 ; w 2 ] + B[w 1 ] 2 ku 2 k 2 L 2 () 2C L (ky 1 k 2 + ku 1 k L 2 () )(ky 2 k 2 + ku 2 k L 2 () ) C L (ky 1 k 2 + ku 1 k L 2 () ) 2 from (SSC) and (4.19). In the following, c will denote a generic constant. Suppose that % < 1 is given and assume k^u uk L 1 () < %. Then ky i k 2 cku i k L 2 () and Young's inequality together yield B[w] 2 ku 2 k 2 L 2 () 2 ku 2k 2 L 2 () cku 1 k 2 L 2 () 2 2 c n n u 2 ds c u 2 ds j^u uj 2 ds c n j^u uj jr u j ds c j^u uj jr u j ds c jr u j 2 ds: j^u uj 2 ds

13 SUFFICIENT SECOND ORDER OPTIMALITY CONDITIONS 13 The expression under the third integral is estimated by k^u uk L 1 () j^u uj. In the other integrals (except the rst) we insert (5.8) and derive (5.9) B[w] 2 j^u uj 2 ds c% j^u uj ds c%k^u uk 2 L 2 2() : n The treatment of B[r; w] and B[r] 2 is simpler, we nd jb[r; w]j ckrkkuk L 2 () = ckrkk^u u + r u k L 2 () c%k^u uk 2 L 2 () : The same type of estimate applies to B[r] 2. Altogether, (5.10) B[ ^w w] 2 j^u uj 2 ds c% j^u uj ds c%k^u uk 2 L 2 2() n is obtained. By substituting (5.10) in (5.7), we get F 0 ( ^w) F 0 (w) ( c%) j^u uj ds + j^u uj 2 ds c%k^u uk 2 L 2 2() n jr L 2 (w; ^w w)j j^u uj ds + j^u uj 2 ds c%k^u uk 2 L 2 2 2() n jr L 2 (w; ^w w)j: Since k^uuk L 1 () 1 was assumed, we have j^uuj j^uuj 2 almost everywhere. Using this in the rst integral, setting 0 = minf=2; =2g, and substituting the estimate (4.18) for r L 2, we complete our estimation by F 0 ( ^w) F 0 (w) k^u uk 2 L 2 () (0 c% (ck^u uk L 1 () )) 0 2 k^u uk2 L 2 () for suciently small % > 0. Our condition (SSC) does not have the form expected from a comparison with second order conditions in nite dimensional spaces. In particular, the pair (y 2 ; u 2 ) constructed in (SSC) does not in general belong to L(M; w). To overcome this diculty, we introduce another regularity condition (R) being stronger than (R). This new constraint qualication is similar to that one used in Casas and Troltzsch [8] to derive second order necessary conditions. Let C (u) denote the set of controls u 2 C(u) having the property u(x) = 0 if x 2. We strengthen (R) to (R) T 0 (y)g 0 (u)c (u) K(T (y)) =. On using (R), we are able to show that the following second order sucient optimality condition implies (5.4) as well:

14 14 E. CASAS AND F. TR OLTSCH AND A. UNGER (SSC) There exist positive numbers and such that (5.11) L 00 (y; u; '; ; z )[w; w] kuk 2 L 2 () holds for all pairs w = (y; u) of L(M; w) with the property u(x) = 0 for almost every x 2. Theorem 5.3. Let the assumptions of Theorem 5.2 be fullled with (R) and (SSC) replaced by (R) and (SSC). Then the assertion of Theorem 5.2 remains true. Proof. The proof is almost identical to that of Theorem 5.2. The only dierence consists in a more detailed splitting. In the rst part of the proof we repeat the steps up to the splitting w = w 1 + w 2 after (5.8). Dene = T G. Then we have as w 1 + w 2 2 L(M; w). Therefore, 0 (u)(u 1 + u 2 ) 2 K((u)); 0 (u)u 2 2 K((u)) 0 (u)u 1 holds so that w 2 does not in general belong to the linearized cone. Thanks to the regularity condition (R), the linear version of the Robinson{Ursescu theorem (see Robinson [18]) implies the existence of u H in C (u) such that holds and 0 (u)u H 2 K((u)) ku 2 u H k L 2 () cku 1 k L 2 (): is satised (see the proof of Theorem 4.2 in the appendix). In other words, we nd a pair w H = (y H ; u H ) in L(M; w) with u H = 0 on. Hence, (SSC) applies to B[w H ] 2. Moreover, the control u H is suciently close to u 2. Now we dene ~u 2 = u H and ~u 1 = u 1 + (u 2 u H ). Further, let ~y i = G 0 (u)~u i denote the corresponding solution of the linearized state equation. In what follows, ~w i = (~y i ; ~u i ) is substituted for w i = (y i ; u i ), i = 1; 2. Then we perform all remaining steps of the proof of Theorem 5.2 with the new quantities. Example: Let us briey comment on (SSC) in the case of (P) for an active state constraint y(0) = y 0. Then L(M; w) is expressed through (4.4), (4.5), and a quite strong second order condition is formulated by (5.12) L 00 (y; u; '; z )[w; w] kuk 2 L 2 () for all w = (y; u) 2 L(M; w). In this way, we would not take advantage of strongly active control constraints. This type of constraints is considered by = fx 2 j ju(x) + '(x)j jg. We split (y; u) = (y 1 ; u 1 ) + (y 2 ; u 2 ), where u 2 = 0 on and u 1 = 0 on n. (SSC) requires the coercitivity condition (5.12) only for (y 2 ; u 2 ), while (y 1 ; u 1 ) is handled by rst order sucient optimality conditions. Notice that y 2 might violate the state{constraint y(x 0 ) 0. This behaviour is avoided by (SSC) : It requires the coercitivity condition for all u 2 C(u) vanishing on and satisfying together with the associated solution y of the linearized partial dierential equation the state{constraint y(x 0 ) 0.

15 SUFFICIENT SECOND ORDER OPTIMALITY CONDITIONS 15 A study of the paper [15] reveals that rst order sucient optimality conditions can be extended also to state{constraints. However, this leads to a quite technical construction and more restrictive assumptions. We have to suppose that the function b is linear with respect to the control u and n = 2. The corresponding theorem is stated below. We dene for xed > 0 and > 0 the following subset of L(M; w): L ; (M; w) = fw j w = (y; u) 2 L(M; w) and w satises (5.13) belowg: The decisive inequality characterising L ; is (5.13) hz ; E 0 (y)yi n ju(x)j ds(x): L ; (M; w) is the subset of L(M; w), where rst order sucient optimality conditions are not very supported by the term hz ; E(y)i. It is only this set, where we have to require second order conditions, namely (SSC') There exist positive numbers,, and such that (5.14) L 00 (y; u; '; ; z )[w 2 ; w 2 ] ku 2 k 2 L 2 () holds for all w 2 = (y 2 ; u 2 ) obtained in the same way introduced in (SSC) by elements w taken from the smaller set L ; (M; w). Using this condition, we formulate the Let the feasible pair w = (y; u) satisfy the regularity condition Theorem 5.4. (R), the rst order necessary optimality conditions (3.9){(3.11) and the second order sucient optimality condition (SSC'). Suppose further that the general assumptions (A1){(A4) are satised. Moreover, assume that n = 2 and b(x; y; u) = b 1 (x; y) + b 2 (x)u. Then there are constants % > 0 and 0 > 0 such that (5.15) F 0 (^y; ^u) F 0 (y; u) + 0 k^u uk 2 L 2 () holds for all feasible pairs ^w = (^y; ^u) satisfying (5.16) k^u uk L 1 () < %: Proof. We begin exactly in the same way we have shown Theorem 5.2 and derive in a rst step (5.17) F 0 ( ^w) F 0 (w) = L( ^w; l) L(w; l) hz ; E(^y) E(y)i: Once again, the representation ^w w = w + r is obtained. Now we distinct between two cases. Case I: w = (y; u) 2 L(M; w) n L ; (M; w) This is the case, where we deduce (5.15) from rst order suciency. Here, the inequality (5.18) hz ; E 0 (y)yi > ju(x)j ds(x); n

16 16 E. CASAS AND F. TR OLTSCH AND A. UNGER is fullled. We transform (5.17) as follows, F 0 ( ^w) F 0 (w) = L 0 (w; l)( ^w w) + r L(w; ^w w) 1 hz ; E(^y) E(y)i = L y (w; l)(^y y) + L u (w; l)(^u u) hz ; E 0 (y)(^y y)i +r L 1 (w; ^w w) hz ; r E 1 (y; ^y y)i = 0 + l u (x)(^u(x) u(x)) ds(x) hz ; E 0 (y)yi (5.19) +r L 1 (w; ^w w) hz ; E 0 (y)r y + r E 1 (y; ^y y)i; where l u (x) stands for g u (x; y(x); u(x)) + '(x)b u (x; y(x); u(x)). Owing to n = 2 and b(x; y; u) = b 1 (x; y) + b 2 (x)u, we are able to apply the strong estimate (4.8) with p = 2. This yields (5.20) krk Y L 1 () C L;2 k^u uk 2 L 2 () : By Theorem 4.2, (5.20), (4.17), and (A3), (ii) we have maxfkr y k 2 ; jr L 1 j; kr E 1 k g c(k^y yk k^u uk2 L 2 () ): Now the Lipschitz property of the mapping u 7! y(u) = G(u) from L 2 () into C() (note that n = 2) permits to estimate the last three items of (5.19) by c k^u uk 2 L 2 (). To the second one, we apply (5.18), while the rst one is treated by : We know that l u (x)(^u(x) u(x)) 0 a. e. on ; hence l u (^u u) ds l u (^u u) ds = jl u j j^u uj ds j^u uj ds: Inserting this in (5.19) we continue by F 0 ( ^w) F 0 (w) j^u uj ds + j^u uj ds + n juj ds ck^u uk 2 L 2() n j^u uj ds ck^u uk 2 L 2() in view of kr u k L 1 () ck^u uk 2 L. Proceeding with the estimation, we deduce 2 () F 0 ( ^w) F 0 (w) minf; gk^u uk L 1 () c%k^u uk L 1 () 0 k^u uk L 1 () with some 0 > 0, provided that k^u uk L 1 () % % 1 holds with suciently small % 1. Assume additionally that % 1 1. Then j^uuj 2 j^uuj holds almost everywhere, hence (5.21) F 0 ( ^w) F 0 (w) 0 k^u uk 2 L 2 ()

17 SUFFICIENT SECOND ORDER OPTIMALITY CONDITIONS 17 follows for k^u uk L 1 () % 1. Case II: w 2 L ; (M; w) (Partial use of rst order sucient optimality conditions) Here, we avoid the term hz ; E(^y) E(y)i and proceed word for word as in the proof of Theorem 5.2, using L ; instead of L. Remark 5.5. In applications, it will be dicult to describe in an explicit way, which (y; u) 2 L(M; w) belong to the dierent classes, where case I or case II applies. Therefore, this type of rst order sucient condition seems to be only of limited value. Example: To illustrate (SSC') for (P) in comparison with (SSC), let us assume for simplicity u 2 intu ad, hence = ;. Then (SSC) requires the coercitivity condition (5.12) on the whole set L(M; w). If y(0) = y 0 and z > 0 (strong complementarity), then (SSC') is weaker than (SSC): (5.12) is not needed for all (y; u) 2 L(M; w) satisfying (5.22) z y(0) ju(x)jds: Assume that y can be represented by a positive Green's function G = G(x; ), y(0) = G(0; )u()ds(); such that G(0; ) > 0 on. Then (5.22) is fullled with = z for all u 0. Moreover, all u 0, u 6= 0 do not contribute to L(M; w). Therefore, the coercitivity condition (5.12) is only needed for all u having "suciently large negative parts". However, this information does not essentially improve (SSC). Remark 5.6. Theorem 5.2 follows from Theorem 5.4 by setting = 0, where we can avoid the restrictions n = 2 and b(x; y; u) = b 1 (x; y) + b 2 (x)u. The cone C(u) is dened by C(u) = f(u u) j u 2 U ad ; 0g. Its closure in L 2 () is cl C(u) = fv 2 L 2 () j v(x) 0; if u(x) = u b (x); v(x) 0; if u(x) = u a (x)g: Let us re{dene L(M; w) by substituting cl C(u) for C(u), and require (SSC) in this form. Then Theorem 5.2 holds as well, since cl C(u) C(u). It can be proved by (R) and the generalized open mapping theorem that (SSC) based on cl C(u) is in fact equivalent to (SSC) established with C(u). 6. Extended second order conditions. A study of the preceding sections reveals that (SSC) is sucient for local optimality in any dimension of without restrictions on the form of the nonlinear function b, whenever we know ' 2 L 1 (). ' is bounded and measurable, if pointwise state-constraints are given only in compact subsets of with the other quantities being suciently smooth. We can allow for pointwise state-constraints up to the boundary in two{dimensional domains, if b(x; y; u) is linear with respect to u. An extension to ' 2 L r () requires stronger assumptions on b. However, we shall briey sketch in this section that some extended form of (SSC) may partially improve the results for n 3.

18 18 E. CASAS AND F. TR OLTSCH AND A. UNGER Let us assume ' =2 L 1 (). Then it seems to be natural to introduce in L 1 () another norm kuk ' = (1 + j'(x)j)u 2 (x) ds(x) 1 A 1=2 : This denition is justied, as u 2 L 1 () and y 2 C() holds in all parts of our paper. For ' 2 L 1 (), the new norm is equivalent to kuk L 2 (). To get rid of the restrictions imposed on b in (A4) we redene the set of strongly active control constraints by (6.1) ;' = fx 2 j jg u (x; y(x); u(x)) + '(x)b u (x; y(x); u(x))j (1 + j'(x)j)g: Moreover, we substitute the condition (6.2) L 00 (y; u; '; ; z )[w 2 ; w 2 ] ku 2 k 2 ' for (5.2). R If ' =2 L 1 (), then (6.2) is stronger than (5.2). On the other hand, the term 'b uu u 2dS contributes to 2 L00. (SSC) implies (at least) the non{negativity of 'b uu, hence 'b uu u 2 2dS = j'j jb uu ju 2 2dS j'ju 2 2dS holds, provided that jb uu j. In view of this, (6.2) appears quite natural. Now Theorem 5.2 remains true for n 3 without assumption (A4). This statement is easy to verify. Apart from the estimates (4.17){(4.19), our theory is not inuenced by introducing kuk '. Therefore, the discussion of (4.17){(4.19) is the decisive point. We are able to replace kk L 2 () by kk ' there, as the basic inequalities (7.14){(7.16) (Appendix 7.3.) can be slightly reformulated: (7.14) is nothing more than (6.3) j'ju 2 ds kuk 2 '; while (7.16) remains unchanged (n = 2; 3). Only (7.15) has to be substituted by (6.4) j'j jyj juj ds = j'j 1=2 jyj j'j 1=2 juj ds kuk ' k'k 1=2 L s=(s2) () kyk L s ()kuk ' : j'jy 2 ds Here we have invoked (7.15) for suciently large s (n = 2; 3). Now a careful study of the proof of Theorem 5.2 shows that (A4) can be removed on using (6.3) and (6.4). Assuming (6.2), we arrive at the estimate (5.4) with k^uuk 2 ' instead of k^uuk 2 L. 2 () Then (5.4) follows from kuk ' kuk L 2 (). The same arguments apply to the rst order sucient conditions in Theorem 5.4 for n = 2, if we redene L ; (M; w) by substituting for (5.13) the inequality (6.5) hz ; E 0 (y)yi (1 + j'j)juj ds: 1 A 1=2 n

19 SUFFICIENT SECOND ORDER OPTIMALITY CONDITIONS Appendix On the regularity condition. Let ^Y H 1 () denote the set of all solutions of the state equation (4.1) linearized at (y; u), associated to u 2 L 1 (). In other words, we have ^Y = G 0 (u)l 1 (). (R) is satised in the following particular cases: a) K = (no inequality constraints) Then (R) means F 0 (y)g 0 (u)c(u) = R m. This condition is satised, if in addition to the surjectivity property F 0 (y) ^Y = R m there is a ~u 2 int L 1 () U ad with F 0 (y)~y = 0. Here, ~y denotes the solution of the linearized state equation (4.1) associated to ~u u, that is ~y = G 0 (u)(~u u). The proof follows from [22], Lemma b) F = 0 (no equality constraints) In this case, (R) reads E 0 (y)g 0 (u)c(u) K(E(u)) =. Once again, two separate conditions may be checked: (R) follows from E 0 (y) ^Y K(E(y)) = and from the existence of an ~u 2 int L 1 () U ad having the property that E 0 (y)~y 2 K(E(y)) holds for ~y = G 0 (u)(~u u) ([22], Lemma 1.2.2). It should be mentioned that case a) follows from b). Example: (P) is worth to be discussed in this context. If the state constraint y(0) y 0 is not active at y, then (R) is obviously satised. Therefore, we assume y(0) = y 0 and get K(E(y)) = fz 2 Rj z 0g = R. Then E 0 (y) ^Y K(E(y)) = reduces to the requirement that for every z 2 R there exists a function u 2 L 1 () such that the equation y(0) = z is valid for the corresponding solution y of the linearized equation (4.4). This property is fullled, since we may nd at least one u 2 L 1 () such that y(0) 6= 0. Hence, (R) is implied by the following conditions: There are ~u 2 L 1 () and " > 0 such that j~uj 1 " holds and that the solution ~y of (4.4) corresponding to ~u u satises ~y(0) 0. c) General case Let us assume int K 6= ; and int L 1 () U ad 6= ;. Moreover we require (7.1) F 0 (y) ^Y = R m ; together with the existence of an ~u 2 int L 1 () U ad such that (7.2) (7.3) E(y) + E 0 (y)~y 2 int K; F 0 (y)~y = 0 holds for ~y = G 0 (u)(~u u). Then (R) is fullled. To show this, we rst mention the simple fact that ~z 2 int K implies ~z + z=% 2 K for arbitrary z 2, if % is suciently large. We have to verify that the system (7.4) (7.5) F 0 (y)y = z 1 E 0 (y)y %(k E(y)) = z 2 is solvable for all z 1 2 R m, z 2 2 by some y 2 G 0 (u)c(u), k 2 K, and % 0: From (7.1) we nd u 1 2 L 1 () such that y 1 = G 0 (u)u 1 solves the equation Now we add to y 1 a multiple of ~y. Then F 0 (y)y 1 = z 1 : F 0 (y)(y 1 + %~y) = F 0 (y)y 1 = z 1

20 20 E. CASAS AND F. TR OLTSCH AND A. UNGER is obtained from (7.3). Consequently, (7.4) holds for y = y 1 + %~y. Moreover, we deduce from (7.2) for suciently large % that This relation is equivalent to E(y) + E 0 (y)~y 1 % (z 2 E 0 (y)y 1 ) = k 2 K: E 0 (y)(y 1 + %~y) %(k E(y)) = z 2 : Therefore, (7.5) is satised by y = y 1 + %~y. Furthermore, u 1 + %(~u u) = %(~u + (1=%)u 1 u) 2 C(u) holds for suciently large %, as ~u + (1=%)u 1 2 U ad for % large enough (notice that ~u 2 int L 1 () U ad ). Thus we have also shown y 2 G 0 (u)c(u) Proof of the linearization theorem. To prove Theorem 4.2 we need the following auxiliary result: Lemma 7.1. Let u; ^u 2 U ad be given with associated states y; ^y dened by (2.2). Introduce y 2 Y as the solution of the linearized state equation y + y = 0 (7.6) Then the estimates (7.7) (7.8) y = b y (; y; u)y + b u (; y; u)(^u u) on : k^y y yk Y C p k^u uk L 1 () k^u uk Lp () 8p > n 1 k^y y yk 2 C 2 k^u uk L 1 () k^u uk L 2 () are satised with certain constants C p, C 2. If b u (x; y; u) does not depend on y and u, then we have (7.9) k^y y yk Y C p k^u uk 2 L p () 8p > n 1: Proof. We use the rst order expansion of b at (x; y; u) and obtain from (2.2), (7.6), and (4.11) the system where (^y y y) + (^y y y) = 0 (^y y y) b y (; y; u)(^y y y) = r b 1 on ; jr b 1(x)j C M (j^y(x) y(x)j 2 + j^u(x) u(x)j 2 ) and M depends on U ad (notice that the boundedness of U ad implies a uniform bound on all admissible states). Therefore, the discussion of (3.12) yields for p > n 1 k^y y yk Y ckr b 1k Lp () c 00 j^y yj 2p ds 1 A 1 p + j^u uj 2p ds 1 A 1 p 1 C A : The mapping u 7! y = G(u) is Lipschitz from L p () to C() for p > n 1. If p = 2, then the Lipschitz property holds in the norm kyk 2 for y. For p > n 1, we continue by k^y y yk Y c k^u uk 2 L p () + k^u uk L 1 ()k^u uk L p () ;

21 while p = 2 yields only SUFFICIENT SECOND ORDER OPTIMALITY CONDITIONS 21 k^y y yk 2 ck^u uk L 1 () k^u uk L 2 () : We have shown (7.7) and (7.8). If b u does not depend on (y; u), then b u (; y + #(^y y); u + #(^u u)) = b u (; y; u), hence This yields jr b 1j = j(b # y b y )(^y y)j cj^y yj 2 : kr b 1k Lp () c k^y yk C() k^y yk Lp () ck^u uk 2 Lp () ; that is (7.9). Proof of Theorem 4.2. Dene v = ^uu and let ~y denote the solution of the linear system (4.1) associated to u := v. We have ~y = G 0 (u)v, where G : L 1 ()! Y is the control{state mapping u 7! y = G(u) for the nonlinear system (2.2). By Lemma 7.1, (7.10) k^y y ~yk Y e(v); where e(v) denotes the right{hand side of the estimates (7.7) and (7.9), respectively, depending on the assumptions on b. Let us introduce the mapping (u) = T (G(u)). Thus 0 (u)v = T 0 (y)g 0 (u)v, and the regularity condition (R) can be rewritten as 0 (u)c(u) K((u)) = : We know that (^u) 2 K, hence a Taylor expansion yields (7.11) where the norm of r 1 (7.12) (^u) = (u) + 0 (u)(^u u) + r 1 ; can be estimated by kr 1 k c e(v): Since (^u) and k = (u) belong to K, (7.11) implies 0 (u)(^u u) + k + r1 also 2 K, thus (7.13) 0 (u)(^u u) 2 r 1 + K((u)): In other words, we have ^uu 2 C(u) and 0 (u)(^uu) K((u)) r 1, where z K((u)) 0 is dened by z 2 K((u)). Owing to (R), this inequality is regular in the sense of Robinson [18]. Therefore, we are able to apply the linear version of the Robinson{ Ursescu theorem (see [18]): It implies the existence of a constant C R > 0 and a u 2 C(u) satisfying ku (^u u)k L 1 () C R kr 1 k together with 0 (u)u 2 K((u)): Consequently, for y = G 0 (u)u, we have (y; u) 2 L(M; w) and ku (^u u)k L 1 () ~ce(v): The estimates stated in (4.6) and (4.8) follow immediately. (4.7) is proved completely analogous. Here, e(v) is dened by (7.8), k k Y is to be replaced by k k 2, and k k L 2 () is to be substituted for k k L 1 (). We rely on the continuity of 0 (y) in the L 2 {norm.

22 22 E. CASAS AND F. TR OLTSCH AND A. UNGER 7.3. Estimates of the Lagrange function. In this subsection we derive the estimates (4.17){(4.19) for r L 1, r L 2, and L 00. They depend mainly on the estimation of I dened in (4.10), which is performed by the discussion of the following integrals: (7.14) j'j u 2 ds ckuk 2 L 2 () provided that assumption (A4), (i) is fullled, and (7.15) j'j jyj juj ds ck'yk L 2 ()kuk L 2 () ck' 2 k 1=2 L (s=2) 0 () ky 2 k 1=2 L s=2 () kuk L 2 () ck'k L 2s=(s2) () kyk Ls ()kuk L 2 () : These estimates are justied by (A4), (ii): For n = 2 we know y 2 C() and ' 2 L r () 8r < 1. If n 3, then y 2 L s () holds for all s < 2(n 1)=(n 3) (including s < 1 for n = 3). The function 2s=(s 2) = 2=(1 1=s) is monotone decreasing. Therefore, s " 2(n1)=(n3) implies 2s=(s2) # n1, so that ' 2 L r () for some r > n 1 justies (7.15) with a suciently large s. Finally, (7.16) j'jy 2 ds k'k L (s=2)0 () ky 2 k L s=2 () = k'k L s=(s2) () kyk 2 L s () is estimated by (A4), (iii): In the case n = 2 we can take s = 1, as y 2 C() and ' 2 L 1 () is true without any additional assumption. For n = 3 we know y 2 L s () for all s < 1. If s " 1, then s=(s 2) # 1 < n=(n 1). Since ' 2 L r () holds for all r < n=(n 1), (7.16) is true for suciently large s. In the case n 4 the same analysis as in the case n 3 above leads to the additional assumption ' 2 L r () for some r > n1 : Now it is easy to derive the estimates (4.17){(4.19) for 2 L 00, r L 1, and r L 2 : For instance, I in (4.10) is handled by (7.14){(7.16) and jij j'j(jb yy j jy 1 y 2 j + jb yu j(jy 1 u 2 j + jy 2 u 1 j) + jb uu j ju 1 u 2 j ds c(ky 1 k 2 + ku 1 k L 2 ())(ky 2 k 2 + ku 2 k L 2 ()); as b yy, b yu, and b uu belong to L 1 (). The other parts of L 00 are discussed by means of (A1){(A3). This yields (4.19) after easy evaluations. In the same way, the remainder terms are investigated. Here, the quantities in I are the most dicult ones again. For instance, (7.14){(7.16) applies to discuss jr I 2j = j'j jb # yy b yy j jy yj 2 + 2jb # yu b yu j jy yj ju uj +jb # uu b uu j ju uj 2 ds c(ky yk C() + ku uk L 1 () )(ky yk ku uk 2 L 2 () ); which contributes to r L 2. The other terms of r L 2 are handled by the estimates for second order derivatives in (A1){(A3) in a direct way. Simple evaluations of this type verify (4.17){(4.18). We leave the details to the reader.

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