OPTIMALITY CONDITIONS FOR STATE-CONSTRAINED PDE CONTROL PROBLEMS WITH TIME-DEPENDENT CONTROLS

Transcription

1 OPTIMALITY CONDITIONS FOR STATE-CONSTRAINED PDE CONTROL PROBLEMS WITH TIME-DEPENDENT CONTROLS J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH Abstract. The paper deals with optimal control problems for semilinear elliptic and parabolic PDEs subject to pointwise state constraints. The main issue is that the controls are taken from a restricted control space. In the parabolic case, they are R m -vector-valued functions of the time, while the are vectors of R m in elliptic problems. Under natural assumptions, first- and second-order sufficient optimality conditions are derived. The main result is the extension of second-order sufficient conditions to semilinear parabolic equations in domains of arbitrary dimension. In the elliptic case, the problems can be handled by known results of semi-infinite optimization. Here, different examples are discussed that exhibit different forms of active sets and where second-order sufficient conditions are satisfied at the optimal solution.. Introduction This paper is a further contribution to second-order optimality conditions in the optimal control of partial differential equations. A large number of papers has been devoted to this issue so far and conditions of this type are used as an essential assumption in publications on numerical methods. Sufficient conditions were investigated first for problems with control constraints. Their main focus was on second-order optimality conditions that are close to the associated necessary ones, for instance, by Bonnans [5], Casas, Unger and Tröltzsch [2], Goldberg and Tröltzsch [8]; see also the examples for the control of PDEs in the recent monograph by Bonnans and Shapiro [6]. The situation changes, if pointwise state constraints are given. Here, the theory is essentially more difficult as the Lagrange multipliers associated with the state constraints are Borel measures. Therefore, the associated theory is less complete than that for control constraints. Although considerable progress has been made in this issue, cf. Casas, Tröltzsch and Unger [3], Raymond and Tröltzsch [29], Casas and Mateos [0], Casas, De los Reyes and Tröltzsch [9], there remain open questions, if pointwise state constraints are formulated in the whole domain: In the elliptic case of boundary or distributed control with pointwise state-constraints, the conditions are sufficiently general only for spatial dimension 2 or 3, respectively, [3, 9]. For parabolic problems, only distributed controls in one-dimensional domains can be handled in full generality, [29]. The difficulties mentioned above are intrinsic for problems with pointwise state constraints and it seems that they cannot be entirely avoided. However, a review on the application of optimal control problems for partial differential equations shows that the situation is often easier in practice: In many cases, the control function depends only on finitely many parameters that may also depend on time in parabolic problems. For instance, in all applications the group of the fourth author has been engaged so far, the controls are finite-dimensional in this sense. This finite-dimensionality seems to be characteristic for real applications of control theory. This concerns

2 2 J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH the cooling of steel profiles by water discussed in [6], [3], [32], some problems of flow control, cf. Henning and King [22], the sublimation process for the production of SiC single crystals, cf. [26], and local hyperthermia in tumor therapy, [5]. In all of these problems, the controls are represented by finitely many real values, namely, the intensities of finitely many spray nozzles in cooling steel, amplitude and frequency of controlled suction or blowing in flow control, frequency and power of the electrical current in sublimation crystal growth, and the energy of finitely many microwave antennas in local hyperthermia. In some cases, these finitely many real values depend on time. Moreover, in all the applications mentioned above, pointwise state constraints are very important. Problems of this type are the main issue of our paper. We address the following points: First, we consider semilinear parabolic equations with distributed controls of the type f(x, t) = k i= e i(x)u i (t), where the functions e i are bounded and the control functions u i are taken from a set of admissible controls. Thanks to the boundedness of the fixed functions e i, we are able to extend a result of [9] for one-dimensional parabolic problems to domains of arbitrary dimension. Although our regularity results can be extended to controls f L 2 (0, T ; L ()), we need the restriction to the finite-dimensional ansatz above to deal with second-order sufficient optimality conditions, cf. Remark 2. Moreover, we consider different problems with finite-dimensional controls u R m. If the control is a vector of R m, pointwise state constraints generate a semiinfinite optimization problem. The associated constraint set is infinite by its nature. The Lagrange multipliers for the state-constraints remain to be Borel measures. Therefore, this class of problems is sufficiently interesting for the numerical analysis. This setting belongs to the class of semi-infinite optimization problems. Here, we are able to invoke the known theory of first- and second-order optimality conditions. In the case of partial differential equations, this theory needs special adaptations that are briefly sketched. Moreover, we present different examples of state-constrained control problems, where second-order sufficient conditions are satisfied at the optimal solution. These problems can be used to test numerical methods and show how diverse the active set may look like. 2. Semilinear parabolic problems 2.. Problem statement. We consider the following distributed optimal control problem with time-dependent control functions, min J(u) = L(x, t, y(x, t), u(t)) dxdt + l(x, t, y(x, t)) ds(x)dt u U ad Q Σ + r(x, y(x, T )) dx (P) subject to y t + A y(x, t) + d(x, t, y(x, t)) = m i= e i(x) u i (t) in Q, ν y(x, t) = 0 on Σ, y(x, 0) y 0 (x) = 0 in, g(x, t, y(x, t)) 0 where u = (u,..., u m ) and U ad is defined by for all (x, t) K Q, U ad = u L (0, T ; R m ) : u a (t) u(t) u b (t) a.e. t [0, T ]}. (2.)

3 CONTROL WITH FINITE-DIMENSIONAL CONTROL SPACE 3 In this setting, is a subset of R n (n ) with boundary Γ, we have set Q = (0, T ), Σ = Γ (0, T ), A is a second-order elliptic differential operator, K is a non-empty compact subset of Q, and T > 0 is a fixed real number. Moreover, functions L : Q R R m R, d : Q R R, l : Σ R R, r : R R, g : K R R, an initial state y 0 C( ), fixed functions e i : R, i =,..., m, and fixed bounds u a, u b L (0, T ; R m ) are given such that u a (t) u b (t) holds a.e. on (0, T ) in the componentwise sense. The symbol ν denotes the derivative in the direction of the outward unit normal ν at Γ Main assumptions and well-posedness of the state equation. In the parabolic case, we rely on the following assumptions: (A.) The set R n is an open and bounded Lipschitz domain in the sense of Nečas [28]. The differential operator A is defined by Ay(x) = n xj (a ij (x) xi y(x)) i,j= with coefficients a ij L () satisfying for some λ A > 0. λ A ξ 2 n a ij (x)ξ i ξ j i,j= ξ R n, x (A.2) (Carathèodory type assumption) For each fixed pair (x, t) Q = (0, T ) or Σ = Γ (0, T ), respectively, the functions d = d(x, t, y) and l = l(x, t, y) are twice partially differentiable with respect to y. For all fixed y R, they are Lebesgue measurable with respect to (x, t) Q, or x Σ respectively. Analogously, for each fixed pair (x, t) Q, L = L(x, t, y, u) is twice partially differentiable with respect to (y, u) R m+. For all fixed (y, u) R m+, L is Lebesgue measurable with respect to (x, t) Q. The function g = g(x, t, y) is supposed to be twice continuously differentiable with respect to y on K R, i.e. g, g y, and 2 g are continuous on K R. y2 (A.3) (Monotonicity) For almost all (x, t) Q or (x, t) Σ, respectively, and all y R it holds that d (x, t, y) 0. y (A.4) (Boundedness and Lipschitz properties) The functions e i, i =,..., m, belong to L (). There is a constant C 0 and, for all M > 0, a constant C L (M) such that the estimates d(x, t, 0) + d y (x, t, 0) + 2 d y 2 (x, t, 0) C 0 2 d y 2 (x, t, y ) 2 d y 2 (x, t, y 2) C L (M) y y 2 hold for almost all (x, t) Q and all y i M, i =, 2. The functions l, and g are assumed to satisfy these boundedness and Lipschitz properties on Σ and Q,

4 4 J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH respectively. The function r is assumed to obey these assumptions with x substituted for (x, t) Q. Analogously, L(x, t, 0, 0) + L (x, t, 0, 0) + L (x, t, 0, 0) C 0 L (x, t, y, u ) L (x, t, y 2, u 2 ) C L (M) ( y y 2 + u u 2 ) hold for almost all (x, t) Q and all y i M, u i M, i =, 2. Here, L and L denote the gradient and the Hessian matrix of L with respect to (y, u) R m Well-posedness of the state equation State equation and existence of optimal controls. In this subsection, we show that the control-to-state mapping G : u y is well defined for all admissible controls u. Moreover, we show certain continuity properties of G. Later, we will also discuss the differentiability of G. In all what follows, we denote the state function y associated with u by y u, i.e. y u = G(u). Theorem. Suppose that the assumptions (A.) (A.4) are satisfied. Then, for every u L q (0, T ; R m ) with q > n/2 +, the state equation (2.) has a unique solution y u C( Q) W (0, T ). If u k u weakly in L q (0, T ; R m ) with q = maxq, 2}, then y uk y u strongly in C( Q). Proof. The functions e i are bounded, hence the right-hand side m v(x, t) = e i (x) u i (t) i= of equation (2.) belongs to L q (Q) with q > n 2 +. Therefore, existence, uniqueness and continuity of the solution y W (0, T ) of (2.) follow from Casas [8]. It remains to show that u k u weakly in L q (0, T, R m ) implies y uk y u strongly in C( Q). To this aim, we first re-write the parabolic equation as dy k dt + A y k = v k := m i= e i(x) u k,i (t) d(x, t, y k ) in Q, ν y k = 0 on Σ, y k (x, 0) = y 0 (x) in, (2.2) where we introduced y k := y uk. The sequence of right-hand sides (v k ) is bounded in L q (Q). Therefore, the boundedness of the sequence y k in C( Q) is a standard conclusion, [8], so that (d(x, t, y k )) is bounded in L (Q). In view of this, the sequence (v k ) is bounded in L 2 (0, T ; L p ()), where p is taken sufficiently large to meet the assumptions of Theorem 3. Therefore, (v k ) contains a sub-sequence (v kl ) with v kl v weakly in L 2 (0, T ; L p ()), l, W.l.o.g we can assume y 0 = 0, since the solution ŷ of ŷ t + Aŷ = 0, ν ŷ = 0 and ŷ(x, 0) = y 0 (x) can be subtracted from the sequence y k. This fixed function does not influence the convergence properties. Thanks to Theorem 3, the mapping v k y k defined by (2.2) is linear and continuous from L 2 (0, T ; L p ()) to the Hölder space C α (Q) for some α > 0. By the compactness of the injection C α (Q) C( Q), (y kl ) converges uniformly to some y C( Q). This, implies (d(x, t, y kl )) (d(x, t, y)) in L (Q) L 2 (0, T ; L p ()) so that the right-hand side converges weakly to v = m i= e i u i d(, y). Therefore, y is associated with this function v and solves the semilinear equation with control u. By uniqueness, it must hold y = y u, hence the same result is obtained for any subsequence of (v k ), and a standard argument yields y k y in C( Q) as k. It is obvious that only those parts of the assumption (A.4) are needed in the theorem that are related to d and d y.

5 CONTROL WITH FINITE-DIMENSIONAL CONTROL SPACE 5 Theorem 2. Assume that the assumptions (A.) (A.4) are fulfilled, the function L = L(x, t, y, u) is convex with respect to u R m and the set of feasible controls is nonempty. Then the control problem (P) has at least one solution. This theorem is a standard consequence of Theorem and the lower semicontinuity of J that needs the convexity of L with respect to u Hölder regularity for linear parabolic equations. The results of this subsection are needed for the extension of second-order sufficient conditions to the case of n- dimensional domains, if the controls depend only on time. To derive second-order sufficient conditions, linearized equations are considered for L 2 -controls, and the regularity of associated states is derived in Theorem 3, which has already been used in the proof of Theorem. The theorem is proved by recent results on maximal parabolic regularity. Throughout this section, is allowed to be a Lipschitz domain in the sense of Grisvard [2]. This is more general than domains with Lipschitz boundary in the sense of Nečas [28]. We consider the linear parabolic problem dz dt + A z + c 0z = v in Q, ν z = 0 in Σ, z(x, 0) = 0 in, (2.3) where c 0 L (Q) is given fixed. Later, c 0 stands for the partial derivative d/ y taken at the optimal state. To deal with equation (2.3), we provide some basic facts on maximal parabolic regularity. We denote by A L (; R n n ) the matrix function A(x) = (a ij (x)), with a ij, i, j,..., n} defined in (A.). Associated with our differential operator A, the linear and continuous operator A : H () H () is defined by < A v, w >:= A v w dx v, w H (). (2.4) The restriction of this operator to the spaces L p () with p 2 is denoted by A p. Its domain is given by D(A p ) = y H () : A y L p ()} equipped with the associated graph norm. It is known that A p incorporates a (homogeneous) Neumann boundary condition, see [7] Ch. II.2 or [4] Ch..2. Let us next recall the concept of maximal regularity: Let X be a Banach space and A denote a closed operator with dense domain D X equipped with the graph norm. Moreover, let S = (T 0, T ) R be a bounded interval. Suppose r (, ). Then A is said to satisfy maximal parabolic L r (S; X)-regularity iff for any f L r (S; X) there is a unique function w W,r (S; X) L r (S; D) that satisfies w t + Aw = f, w(t 0) = 0. (2.5) By W,r (S; X), we denote the set of those elements from L r (S; X) whose distributional derivative also belongs to L r (S; X). If X, Y are Banach spaces which form an interpolation couple, then we denote by [X, Y ] θ the corresponding complex interpolation space and by (X, Y ) θ,r the real interpolation space, see [30]. Below we well employ the following fact: There is a continuous injection E : W,r (S; X) L r (S; D) C( S; (X, D) r,r ), (2.6) see [4] Ch. III Thm , cf. also [30] Ch..8. Moreover, for any η (0, r ) there is a continuous embedding (X, D) r,r [X, D] η and, consequently, a

6 6 J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH continuous embedding C( S; (X, D) r,r ) C( S; [X, D] η ). (2.7) Next, we need the following Lemma. Assume τ (0, r ). Then there is an index κ > 0 such that W,r (S; X) L r (S; D) continuously embeds into C κ (S; [X, D] τ ). Proof. First of all, the estimate t ( t ) ( t w(t) w(t 0 ) X = w (s) ds X w (s) r r X ds t 0 t 0 w W,r (S;X) t t 0 r r t 0 ds ) r r gives us a (continuous) embedding from W,r (S; X) into C δ (S; X), where δ = r r. Let η be a number from (τ, r ). Then, putting λ = τ η we obtain by the reiteration theorem for complex interpolation, see [30] Ch..9.3, w(t) w(s) [X,D]τ t s δ( λ) c w(t) w(s) [X,[X,D] η] λ t s δ( λ) c w(t) w(s) λ X t s δ( λ) w(t) w(s) λ [X,D] η c ( w(t) w(s) X t s δ ) λ( 2 w C( S;[X,D]η)) λ c2 w W,r (S;X) L r (S;D). Corollary. Assume that A satisfies maximal parabolic L r (S; X)-regularity and let τ (0, r ) be given. Then the mapping that assigns to every f Lr (S; X) the solution of (2.5) is continuous from L r (S; X) into C κ (S; [X, D] τ ) for a certain κ > 0. Proof. The mapping t + A : W,r (S; X) L r (S; D) w w(t 0 ) = 0} L r (S; X) is continuous and bijective. Hence, the inverse is also continuous by the open mapping theorem. Combining this with the preceding lemma gives the assertion. Now we are able to show our main result on parabolic regularity: Hölder regularity can be obtained for functions which exhibit only to L 2 -regularity in time, if their spatial regularity is L p and p is sufficiently large. Theorem 3. If f belongs to L r (S; L p ()) with r > and sufficiently large p, then the solution w of w t + A pw = f, w(0) = 0 (2.8) is from a space C κ (S; C β ()). Moreover, the mapping f w is continuous from L r (S; L p ()) to C κ (S; C β ()). Proof. We apply the general results on maximal parabolic regularity to our operator A p. It is known that A p enjoys maximal parabolic L r (S; L p ())-regularity for every p (, ) and r >. We refer to [9], Thm Moreover, the following interpolation result is known: If θ (0, ) and β := θα ( θ) n > 0, (2.9) p

7 CONTROL WITH FINITE-DIMENSIONAL CONTROL SPACE 7 then [L p (), C α ()] θ C β (). (2.0) The result is shown in [9], see the proof of Thm. 7.. In general, the domain of A p is difficult to determine. However, the following embedding result is helpful: For p > n 2, there exists α > 0 such that the continuous embedding dom(a p ) C α () (2.) holds true. This result is proved in [20]. Keeping α > 0 fixed we can increase p so that also (2.9) is satisfied. Clearly, (2.0) and (2.) remain true. Taking now into account Corollary for X = L p (), then the assertion follows from (2.0) and (2.). Corollary 2. If r > and p is sufficiently large, then for all v L r (S; L p ()), the weak solution of (2.3) belongs to C α ( Q) with some α > 0. The mapping v z is continuous from L r (0, T ; L p ()) to C α ( Q). Proof. The operator t + A p is a topological isomorphism between L r (S; D(A p )) W,r (S; L p ()) z : z(t 0 ) = 0} and L r (S; L p ()), and is therefore a Fredholm operator of index zero. Obviously, the multiplication operator induced by c 0 is bounded on L 2 (S; L p ()); hence, the domain of t + A p + c 0 equals the domain of t +A p which, due to Theorem 3, compactly embeds into L r (S; L p ()). Thus, the multiplication operator induced by c 0 is relatively compact with respect to t +A p. By a well known perturbation theorem, t + A p + c 0 then also must be a Fredholm operator and is also of index zero (see Kato [23], Ch. IV.5.3). Let us show that it is injective: Let z be a solution of z t + A pz + c 0 z = 0 or, equivalently, z t + A pz = c 0 z. As a solution of this parabolic equation with right hand side c 0 z, z has then the representation z(t) = t c 0 e(s t)ap 0 (s, )z(s) ds. But the semigroup e tap is contractive on L p () (see [9] Thm. 4.) and c 0 is essentially bounded; thus Gronwall s lemma yields z 0. Therefore, t + A p + c 0 is injective. Because it is Fredholm and, additionally, of index zero, it is also surjective. Consequently, the inverse operator maps L r (S; L p ()) continuously into the domain of t + A p, and this is continuously embedded into a space C α ( Q). Notice that in particular L 2 (0, T ; L ()) is mapped continuously into a space C α ( Q). This is what we need for the discussion of second-order sufficient conditions. The theorem refers to the PDE (2.3) with homogeneous initial condition. It is obvious that the result extends to inhomogeneous Hölder continuous initial data. Remark. The Hölder regularity might also be deduced from Ladyzhenskaya et al. [25] under the assumption that Γ is a Lipschitz boundary in the sense of Nečas [28], cf. also [8]. Under this assumption, the continuity of the state function for our restricted class of controls was also shown in [29]. The use of maximal parabolic regularity permits to extend these results to Lipschitz domains in the sense of Grisvard [2]. Moreover, this approach is not restricted to Neumann conditions. It also allows for mixed boundary conditions and, of course, for Dirichlet conditions. Last, but not least, our approach essentially shortens and unifies the associated proofs Necessary optimality conditions. The control-to-state mapping G(u) = y u, G : L (0, T ; R m ) C( Q) W (0, T ), and the reduced objective functional J are of class C 2 from L (0, T ; R m ) to their image spaces, provided that the assumptions (A.)-(A.4) are satisfied. This follows by the arguments of [8].

8 8 J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH We define, for v L (0, T ; R m ), the function z v as the unique solution to dz v dt + Az v + d y (x, t, y u)z v = m i= e i(x)v i (t) in Q, ν z v = 0 in Σ, (2.2) y(x, 0) = 0 in. Then G (u), G : L (0, T ; R m ) C( Q) W (0, T ) is given by G (u)v = z v. Moreover, for v, v 2 L (0, T ; R m ), we introduce z vi = G (u)v i, i =, 2, and obtain G (u)v v 2 = z vv 2, where z vv 2 is the solution to dz vv 2 dt + Az vv 2 + d y (x, t, y u)z vv d y 2 (x, t, y u)z v z v2 = 0 in Q, ν z vv 2 = 0 in Σ, z vv 2 (x, 0) = 0 in. (2.3) The adjoint state ϕ 0u W (0, T ) associated with u and J is introduced as the unique solution to dϕ dt + A ϕ + d y (x, t, y u)ϕ = L y (x, t, y u, u) in Q, ν ϕ = l y (x, t, y u) in Σ, (2.4) ϕ(x, T ) = r y (x, y u(x, T )) in, where A is the formally adjoint operator to A. Standard computations show T m ( ) } L J (u)v = (x, t, y u, u) + ϕ 0u e i (x) dx v i (t) dt, (2.5) 0 i= u i [ J 2 L (u)v v 2 = Q y 2 (x, t, y u, u)z v z v2 + 2 L y u (x, t, y u, u) (z v v 2 + z v2 v ) +v 2 L u 2 (x, t, y 2 ] d u, u)v 2 ϕ 0u y 2 (x, t, y u)z v z v2 dxdt, 2 l + Σ y 2 (x, t, y u)z v z v2 ds dt 2 r + y 2 (x, y u(x, T ))z v (x, T )z v2 (x, T ) dx. (2.6) As in (2.6), we will use a dot to denote the inner product of R m. Notice that the z vi and ϕ 0u depend on (x, t), while the v i depend on t only. We require the following linearized Slater condition: There exists a function u 0 L (0, T ; R m ) with u a (t) u 0 (t) u b (t) for a.e. t (0, T ) such that g(x, t, ȳ(x, t)) + g y (x, t, ȳ(x, t))z u 0 ū(x, t) < 0 (x, t) K. (2.7) Inserting t = 0, this imposes a condition on the initial state y 0. Therefore, to satisfy (2.7), we have to assume that g(x, 0, y 0 (x)) < 0 x with (x, 0) K. (2.8) Defining the Hamiltonian H by [ m ] H(x, t, y, u, ϕ) = L(x, t, y, u) + ϕ e i (x) u i d(x, t, y), the first-order necessary conditions admit the following form: i=

9 CONTROL WITH FINITE-DIMENSIONAL CONTROL SPACE 9 Theorem 4. Let ū be a local solution of (P). Suppose that the assumptions (A.) (A.4) hold and assume the Slater condition (2.7) with some u 0 L (0, T ; R m ), u a (t) u 0 (t) u b (t) for a.e. t (0, T ). Then there exists a measure µ M(K) and a function ϕ L s (0, T ; W,σ ()) for all s, σ [, 2) with (2/s)+(n/σ) > n+ such that d ϕ dt + A ϕ + d L g (x, t, ȳ) ϕ = (x, t, ȳ, ū) + y y y (x, t, ȳ) µ Q, ν ϕ(x, t) = l y (x, t, y u(x, t)) + g y (x, t, ȳ(x, t)) µ Σ, ϕ(x, T ) = r g (x, ȳ(x, T )) + y y (x, T, ȳ(x, T )) µ T } (2.9) for a.a. x, t (0, T ), where µ Q, µ Σ, and µ T } denote the restrictions of µ to Q, Σ, and T }, respectively, (z(x, t) g(x, t, ȳ(x, t))d µ(x, t) 0 z C(K) with z(x, t) 0 (x, t) K, K (2.20) and, for almost all t (0, T ), H u (x, t, ȳ(x, t), ū(t), ϕ(x, t)) dx (u ū(t)) 0 u [u a(t), u b (t)]. (2.2) This theorem follows from Casas [8] using the admissible set Ũ ad = v v = m e i ( )u i ( ), u a,i (t) u i (t) u b,i (t) a.e. on (0, T )} i= and writing the associated variational inequality in terms of u. The inequality (2.20) implies the well-known complementary slackness condition g(x, t, ȳ(x, t))d µ(x, t) = 0. (2.22) K Notice that H/ u is a m vector function. Since we need the integrated form of H and its derivatives at the optimal point, we introduce the vector function H H u (t) = (x, t, ȳ(x, t), ū(t), ϕ(x, t)) dx (2.23) u and the (m, m)-matrix valued function 2 H H uu (t) = (x, t, ȳ(x, t), ū(t), ϕ(x, t)) dx (2.24) u2 with entries H uiu j (t) := ( H uu ) ij (t), i, j,..., m}. The (reduced) Lagrange function is defined in a standard way by L(u, µ) = L(x, t, y u (x, t), u(t)) dxdt + l(x, t, y u (x, t)) dsdt Q Σ + r(x, y u (x, T )) dx + g(x, t, y u (x, t)) dµ(x, t). For later use, we establish the second-order derivative of L with increments v i L (0, T ; R m ), i =, 2. The expressions below contain functions z vi (x, t) and v i (t), K

10 0 J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH but we suppress their arguments for short. 2 [ L 2 u 2 (u, µ)v L v 2 = y 2 (x, t, y u, u)z v z v2 Q + 2 L y u (x, t, y u, u) (z v v 2 + z v2 v ) + v 2 ] d ϕ u y 2 (x, t, y u)z v z v2 dxdt + T 2 l + 0 y 2 (x, t, y u)z v z v2 dt + K 2 L u 2 (x, t, y u, u)v 2 2 r y 2 (x, y u(x, T ))z v (x, T )z v2 (x, T ) dx 2 g y 2 (x, t, y u)z v z v2 d µ(x, t), (2.25) where ϕ u is the solution of (2.9) with u taken for ū, y u for ȳ, and µ for µ, respectively Second-order sufficient optimality conditions. The regularity results of the preceding section at hand, we are able to discuss second-order sufficient conditions for arbitrary dimensions of. To establish them and to show their sufficiency, we follow the recent paper [9] by Casas et al.. The presentation here is similar to the one in [9], but there occur some differences due to the appearance of vectorvalued control functions. We do not assume that the reader is familiar with the results of [9]. Therefore, we sketch the main steps of the analysis. For an easier comparison with the arguments presented in [9], we adopt also part of the notation used there. First, we define the cone of critical directions associated with ū by Cū = h L 2 (0, T ; R m ) : h satisfies (2.26), (2.27) and (2.28) below}, 0 if ū i (t) = u a,i (t), i,..., m}, h i (t) = 0 if ū i (t) = u b,i (t), = 0 if H u,i (t) 0, (2.26) g y (x, t, ȳ(x, t))z h(x, t) 0 if g(x, t, ȳ(x, t)) = 0, (2.27) g y (x, t, ȳ(x, t))z h(x, t) d µ(x, t) = 0. (2.28) K The vector function H u was defined in (2.23). Moreover, we define, for fixed τ > 0 and all i,..., m}, the sets of sufficiently active control constraints E τ i = t [0, T ] : H u,i (t) τ}. (2.29) In the next theorem, we write diag(χ E τ i (t)) for the matrix diag(χ E τ (t),..., χ E τ m (t)). The second-order sufficient optimality conditions for ū are stated in the following result: Theorem 5. Let ū be a feasible control of problem (P) that satisfies, together with the associated state ȳ and ( ϕ, µ) L s (0, T ; W,σ ()) M(K) for all s, σ [, 2) with (2/s) + (n/σ) > n +, the first-order conditions (2.9)-(2.2). Assume in addition that there exist constants ω > 0, α 0 > 0, and τ > 0 such that for all α > α 0 d ( Huu(t) + α diag(χeτ i (t)) )d ω d 2 a.e. t [0, T ], d R m, (2.30) 2 L u 2 (ū, µ)h2 > 0 h Cū \ 0}. (2.3)

11 CONTROL WITH FINITE-DIMENSIONAL CONTROL SPACE Then there exist ε > 0 and δ > 0 such that, for every admissible control u of problem (P), the following inequality holds J(ū) + δ 2 u ū 2 L 2 (0,T ;R m ) J(u) if u ū L (0,T ;R m ) < ε. (2.32) Proof. The proof is by contradiction. It follows the one presented in Casas et al. [9] for an elliptic control problem. Nevertheless, we sketch the main steps, since there are some essential changes due to the different nature of our problem. Suppose that ū does not satisfy the quadratic growth condition (2.32). Then there exists a sequence u k } k= L (0, T ; R m ) of feasible controls for (P) such that u k ū in L (0, T ; R m ) and Define ρ k = u k ū L 2 (0,T ;R m ) and J(ū) + k u k ū 2 L 2 (0,T ;R m ) > J(u k) k. (2.33) h k = ρ k u k ū L 2 (0,T ;R m ). Since h k L 2 (0,T ;R m ) =, a weakly converging subsequence can be extracted. W.l.o.g. we can assume that h k h weakly in L 2 (0, T ; R m ), k. Now, the proof is split into several steps. Step : It is shown that L (ū, µ)h = 0. (2.34) u The arguments are the same as in [9]. Moreover, they are analogous to the classical proof for finite-dimensional problems. Therefore, we omit them. Step 2: h Cū. We have to confirm (2.26) (2.28). It is easy to verify that h satisfies the sign conditions of (2.26). To see that h i (t) vanishes, where H u,i (t) 0, we notice that (2.2) implies after a standard discussion that, for all i,..., m} Therefore, T 0 H u,i (t)h i (t) = H u,i (t)h i (t) 0 for a.a. t [0, T ]. (2.35) m H u,i (t) h i (t) dt = i= T 0 H u (t) h(t) dt = L (ū, µ)h = 0 u must hold in view of (2.34). This implies h i (t) = 0 if H u,i (t) 0, hence (2.26) is verified. The proof of (2.27) is fairly standard and follows from z h = G yū+ρk h (ū)h = lim k ȳ k ρ k in C( Q), and from g(x, t, yū+ρk h k (x, t)) = g(x, t, y k (x, t)) 0 for every (x, t) K, since u k is feasible. Notice that g(x, t, ȳ(x, t)) = 0 holds for all (x, t) considered in (2.27). It remains to show (2.28). We get from the complementary slackness condition (2.22) that g K y (, ȳ)z [ h d µ = lim g(, yū+ρk h k ρ k ) g(, ȳ) ] d µ = k K = lim g(, y uk ) d µ 0, (2.36) k ρ k since µ 0 and g(x, t, y uk (x, t)) 0 on K. On the other hand, (2.33) yields J J(ū + ρ k h k ) J(ū) J(u k ) J(ū) ρ k (ū)h = lim = lim lim k ρ k k ρ k k k K = 0. (2.37)

12 2 J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH Now, (2.34), (2.36), (2.37) and the simple identity L u (ū, µ)h = J g (ū)h + y (x, t, ȳ(x, t))z h(x, t) d µ(x, t) imply that J (ū)h = K K g y (x, t, ȳ(x, t))z h(x, t) d µ(x, t) = 0, hence (2.28) holds and we have shown h Cū. Step 3: h = 0. In view of (2.3), it suffices to show h 2 L (ū, µ)h 0. (2.38) u2 For this purpose, we perform a second-order Taylor expansion of the Lagrangian, L L(u k, µ) = L(ū, µ) + ρ k u (ū, µ)h k + ρ2 k 2 L 2 u 2 (w k, µ)h 2 k, (2.39) where w k is an intermediate point between ū and u k. Re-writing this equation, we get L ρ k u (ū, µ)h k + ρ2 k 2 L 2 u 2 (ū, µ)h2 k [ = L(u k, µ) L(ū, µ) + ρ2 k 2 ] L 2 u 2 (ū, µ) 2 L u 2 (w k, µ) h 2 k. (2.40) Moreover, we mention that (2.33) can be written in terms of L as L(u k, µ) L(ū, µ) ρ2 k k. (2.4) Taking into account the expression (2.25) of the second derivative of the Lagrangian, the assumptions (A.)-(A.4), Theorem, (2.2) and the fact that u k ū in L (0, T ; R m ) and h k L 2 (0,T ;R m ) =, we get that [ 2 ] L u 2 (ū, µ) 2 L u 2 (w k, µ) h 2 k 2 L u 2 (ū, µ) 2 L u 2 (w k, µ) h k 2 L 2 (0,T ;R m ) B(L 2 (0,T ;R m )) = 2 L u 2 (ū, µ) 2 L u 2 (w k, µ) 0 when k, (2.42) B(L 2 (0,T ;R m )) where B(L 2 (0, T ; R m )) is the space of quadratic forms in L 2 (0, T ; R m ). From (2.35) and the definition of h k we know that H u,i (t) h k,i (t) 0 for a.a. t [0, T ] and all i,..., m}, therefore L u (ū, µ)h k = T 0 m τ i= H u (t) h k (t) dt For any ε > 0, there exists k ε such that E τ i m i= E τ i H u,i (t) h k,i (t) dt h k,i (t) dt. (2.43) ρ k h k L (0,T ;R m ) = ū u k L (0,T ;R m ) < ε k k ε, therefore ρ 2 k h2 k,i (t) ρ k h k,i (t) k k ε, ε a.e. t [0, T ]. From this inequality and (2.43) it follows that L m ρ k u (ū, µ)h k ρ k τ h k,i (t) dt ρ2 k τ m ε h 2 k,i(t) dt. (2.44) i= E τ i i= E τ i

13 CONTROL WITH FINITE-DIMENSIONAL CONTROL SPACE 3 Collecting (2.40)-(2.42) and (2.44) and dividing by ρ 2 k /2 we obtain for any k k ε 2τ m h 2 ε k,i(t) dt + 2 L i= Ei τ u 2 (ū, µ)h2 k 2 k + 2 L u 2 (ū, µ) 2 L u 2 (w k, µ). B(L 2 (0,T ;R m )) (2.45) Let us consider the left-hand side of this inequality. First of all we notice that from (2.25) and 2 L H uu (t) = (x, t, ȳ(x, t), ū(t)) dx (2.46) u2 it follows that 2 L T [ ] u 2 (ū, µ)h2 k = h k (t) H uu (t)h k (t) + 2 L uy (x, t)z hk (x, t) dx h k (t) dt 0 + L yy (x, t)zh 2 k (x, t) dxdt + l yy (x, t)zh 2 k (x, t) dsdt + Q Σ r yy (x)zh 2 2 g k (x, T ) dx + K y 2 (x, t, ȳ(x, t))z2 h k (x, t) d µ(x, t), where H uu (t) was defined in (2.24) and L uy, L yy, l yy, and r yy are defined by L uy (x, t) = 2 L (x, t, ȳ(x, t), ū(t)), Lyy u y l yy (x, t) = 2 l (x, t, ȳ(x, t)), y2 r (x, t) = 2 L (x, t, ȳ(x, t), ū(t)), y2 yy(x) = 2 r (x, ȳ(x, T )). y2 The first integral of 2 L u (ū, µ)h 2 2 k needs special care. We notice that 2τ m T h 2 ε k,i(t) dt = h k (t) ( 2τ ) ε diag(χ Ei τ (t)) h k (t)dt. (2.47) i= E τ i Thanks to assumption (2.30), it holds that 0 d ( H uu (t) + 2τ ε diag(χ E τ i (t)) )d ω d 2 0 < ε < ε 0, (2.48) if ε 0 is taken sufficiently small. Therefore, the matrix function H uu (t)+ 2τ ε diag(χ Ei τ (t)) is uniformly positive definite, and hence we infer that T lim inf h k (t) (H uu (t) + 2τ ) k 0 ε diag(χ E τi )(t)) h k (t) dt T h (t) (H uu (t) + 2τ ) ε diagχ E τi (t)) h(t) dt. (2.49) 0 Finally, taking into account that, by Corollary 2 with c 0 = d y (, ȳ), z h k z h strongly in C( Q), we deduce from (2.45)-(2.49) and (2.42) T h (t) (H uu (t) + 2τ ) 0 ε diagχ E τi (t)) h(t) dt + 2 L uy (x, t)z h (x, t) h(t) dxdt Q + L yy (x, t)zh(x, 2 t) dxdt + l yy (x, t)zh(x, 2 t) dsdt Q Σ + r yy (x)zh(x, 2 T ) dx + K This expression can be written as 2τ ε m i= E τ i 2 g y 2 (x, t, ȳ(x, t))z2 h(x, t) d µ(x, t) 0. h 2 i (t) dt + 2 L u 2 (ū, µ)h2 0, which along with (2.3) and the fact that h Cū implies that h = 0.

14 4 J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH Step 4: h k 0 strongly in L 2 (0, T ; R m ). We have already proved that h k 0 weakly in L 2 (0, T ; R m ), therefore z hk 0 strongly in C( Q) by Corollary 2. In view of (2.45) and h k L 2 (0,T ;R m ) = we get 0 < ω = ω lim sup K k T T 0 h k (t) 2 dt lim sup h k (t) (H uu (t) + 2τ ) k 0 ε diag(χ E τi (t)) h k (t) dt 2 lim sup k k + 2 L u 2 (ū, µ) 2 L u 2 (w k, µ) B(L 2 (0,T ;R m )) [ 2 L uy (x, t)z h (x, t) h(t) L ] yy (x, t)zh(x, 2 t) dxdt Q l yy (x, t)zh(x, 2 t) dsdt r yy (x)zh(x, 2 T ) dx Σ 2 g y 2 (x, t, ȳ(x, t))z2 h(x, t) d µ(x, t) = 0. Thus we have got a contradiction so that our hypotheses (2.33) cannot be true. Remark 2. A study of the proof reveals, why the special form of time-dependent controls is important: The sequence (h k ) converges in L 2 (0, T ; R m ). This space is associated with the domain (0, T ), where the controls can vary (notice that the functions e i L () are fixed). Controls of L 2 (0, T ; R m ) are mapped continuously to continuous state functions. This fact is essential for the theory to work. For more general control functions f L 2 (0, T ; L ()), the associated domain of variability is (0, T ). Here, the space L 2 ( (0, T )) would underly the proof. However, state functions associated with controls of L 2 ( (0, T )) are not in general continuous so that the proof does not go through. Let us briefly comment on the consequences of the assumption (2.30) for the matrix function H uu. To this aim, we consider a certain index set I,..., m} and assume that E = i I E τ i (2.50) is non-empty. To avoid a re-numbering, let us assume that I =,..., k}. We write H uu + α diag(χ E (t)) as a block matrix in the form [ ] A(t) + α Id B(t) H uu (t) + α diag(χ E (t)) = B. (t) C(t) Then (2.30) is satisfied on the set E, if C(t) is uniformly positive definite there: There must exist β > 0 such that ξ C(t)ξ β ξ 2 ξ R m k, but no condition on A(t) is needed. This is seen as follows: We split d R m in d R k, containing the first k components and d 2 R m k with the remaining m k components. Then d ( ) H uu (t) + α diag(χ E (t)) d = d A(t)d + α d 2 + 2d B(t)d 2 + d 2 C(t)d 2 α d 2 c A d 2 c B d 2 ε d β d 2 2 α 2 d 2 + β 2 d 2 2 ω d 2 by the Young inequality, if α is sufficiently large, ε > 0 is taken sufficiently small, and ω = minα, β}/2.

15 CONTROL WITH FINITE-DIMENSIONAL CONTROL SPACE 5 In the particular case k = m, i.e. on E = m i= Eτ i, we do not need any condition on H uu, while H uu must be uniformly positive definite on [0, T ] \ m i= Eτ i, where no strong activity helps. Certainly, there are many possible index sets I, and the discussion of all different cases is mainly a matter of combinatorics. We do not dwell upon this point. Instead, let us discuss the case m = 2, where H uu + α diag(χ E τ, χ E τ 2 ) is given as follows: In E τ E2 τ, we have [ ] H (t) + α H H uu + α diag(χ E τ, χ E τ 2 ) = 2 (t). H 2 (t) H 22 (t) + α This matrix is uniformly positive definite on E τ E τ 2 for all sufficiently large α. Therefore, the matrix H uu satisfies (2.30) on this subset without any further assumption on positive definiteness of H uu. In E τ \ (E τ E2 τ ), it holds [ ] H (t) + α H H uu + α diag(χ E τ, χ E τ 2 ) = 2 (t) H 2 (t) H 22 (t) so that we need H 22 (t) β > 0 to make the matrix positive definite on E τ \ (E τ E τ 2 ) for sufficiently large α. An analogous condition must be imposed on H (t) on E τ 2 \ (E τ E τ 2 ). In [0, T ] \ (E τ E τ 2 ), the matrix H uu (t) must be uniformly positive definite to satisfy (2.30), since diag(χ E τ, χ E τ 2 ) vanishes here. Finally, we mention the case of a diagonal matrix H uu (t) = diag(h uiu i (t)). Here, (2.30) is satisfied, if and only if i,..., m} : H uiu i (t) ω t [0, T ] \ E τ i. This happens in the standard setting where u appears only in a Tikhonov regularization term, i.e. L = L(x, t, y) + λ u Semilinear elliptic case 3.. Problem statement. In this section, the following elliptic problem with pointwise state constraints is considered: min J(u) = L(x, y(x), u) dx + l(x, y(x)) ds(x) u U ad (P2) subject to y(x) + y(x) + d(x, y(x), u) = 0 in ν y(x) + b(x, y(x)) = 0 on Γ, g(x, y(x)) 0 for all x, Γ (3.) where U ad := u R m : u a u u b }. In this setting, R n (n 2) is a bounded domain with Lipschitz boundary Γ. Moreover, sufficiently smooth functions L, d : R R m R, g : R R, and b, l : Γ R R are given; u a and u b with u a u b are vectors of R m. We adopt the notation from the preceding sections. Similar to [0], [2], the following assumptions are imposed on L, l, d, b:

16 6 J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH (A.5) (Carathèodory type assumption) For each fixed x or Γ respectively, the functions L = L(x, y, u), l = l(x, y), d = d(x, y, u), b = b(x, y), are of class C 2 with respect to (y, u). For all fixed (y, u) or fixed y, respectively, they are Lebesgue measurable with respect to the variable x, or x Γ respectively. The function g = g(x, y) is supposed to be twice continuously differentiable with respect to y on R. (A.6) (Monotonicity) For almost all x, or x Γ, respectively, and any fixed u U ad when applies, it holds d b (x, y, u) 0, y (x, y) 0. y (A.7) (Boundedness and Lipschitz properties) There is a constant C 0 and, for all M > 0, an C L (M) such that the estimates d(x, 0, 0) + d (x, 0, 0) + d (x, 0, 0) C 0 d (x, y, u ) d (x, y 2, u 2 ) C L (M)( y y 2 + u u 2 ) hold for almost all x, all u, u i U ad, and all y i M, i =, 2. The functions L, ϕ, b, and g are assumed to satisfy (A.7) as well First-order necessary conditions. In this section, we consider optimality conditions for problem (P2). Since the controls are in R m and infinitely many pointwise state constraints are given, this problem belongs to the class of semiinfinite mathematical programming problems. Therefore, the first- and second-order optimality conditions might be deduced from the theory of semi-infinite programming. Nevertheless, the transfer of these results to the control of PDEs needs the handling of the associated partial differential equations and to discuss the differentiability properties of the underlying control-to-state mappings so that these facts are worth mentioning. First, we derive some basic results for the control-to-state mapping that are standard for problems with control functions appearing linearly on the right hand side of the PDE. Since our setting includes controls appearing nonlinearly, we have to slightly update the arguments, and we state them for convenience of the reader. Theorem 6. Let R n be a bounded and Lipschitz domain. Suppose that the conditions (A.5) and (A.6) are satisfied. Then, for all u U ad, the partial differential equation y(x) + y(x) + d(x, y(x), u) = 0 in ν y(x) + b(x, y(x)) = 0 on Γ, has a unique solution y u H () C( ) and the estimate (3.2) y u H () + y u C( ) c M (3.3) holds with a constant c M that does not depend on u, if u belongs to U ad. Proof. The result follows by setting d(x, y) := d(x, y, u), and applying regularity results for semilinear elliptic equations obtained in [3] or [7]. Notice that d is a monotone function with respect to y. Remark 3. Due to (3.3), the control-to-state mapping G : R m H () C( ) that assigns to each u U ad the solution y u of equation (3.2), satisfies the boundedness condition G(u) H () C( ) c M for all u U ad.

17 CONTROL WITH FINITE-DIMENSIONAL CONTROL SPACE 7 Remark 4. Due to our assumptions, the weak solution of (3.2) lies in the space Y q,p = y H () : y + y L q (), ν y L p (Γ)} for all p, q. Y q,p is known to be continuously embedded in H () C( ) for each q > n/2 and each p > n, see [3] or [3]. Remark 5. The operator G : U ad H () C( ) obeys the Lipschitz property y y 2 H () + y y 2 C( ) C L (M) u u 2 (3.4) for all y i such that y i = G(u i ), with u i R m and u i M for i =, 2. Remark 6. Under Assumption (A.7), the control-to-state mapping G : R m H () C( ) is twice continuously Fréchet-differentiable. For arbitrary elements ū and u of U ad, and for ȳ = G(ū), the function y = G (ū)v is the unique solution of the problem z v + z v + d y (x, ȳ, ū)z v = d (x, ȳ, ū)v u in ν z v + b (3.5) y (x, ȳ)z v = 0 on Γ. Moreover, y satisfies the inequality z v H () + z v C( ) c v (3.6) for some constant c independent of u. The function z vv 2, defined by z vv 2 = G (u)[v, v 2 ], is the unique solution of z vv 2 + z vv 2 + d y (x, y u, u)z vv 2 = (y u, u ) d (x, y u, u) (y u2, u 2 ) in ν z vv 2 + b y (x, y u)z vv 2 = 2 b (3.7) y 2 (x, y u)y u y u2 on Γ. The proofs of these results are fairly standard. For control functions appearing linearly in the right hand side of (3.2), it is given in []. It can also be found in [33]. The adaptation to the vector case needed here is more or less straightforward. We omit the associated arguments. We first note that the set feasible set of this problem is closed and bounded in R m due to the continuity of g. Therefore, the reduced functional f defined above is continuous and compactness guarantees existence of an optimal control ū provided that the feasible set is non-empty. Notice that this existence result is not in general true for control functions appearing nonlinearly. In the sequel, ū stands for an optimal control with state ȳ = G(ū). Later, in the context of second-order conditions, it is again a candidate for local optimality. Henceforth we assume the following linearized Slater condition: (A.8) (Regularity Condition) There exists u 0 U ad such that g(x, ȳ(x)) + g y (x, ȳ(x))z u0 ū(x) < 0 for all x, (3.8) where ȳ satisfies (3.2). It is well known, see [24], that this condition guarantees the existence of a nonnegative Lagrange multiplier µ in the space M( ) = C( ) of regular Borel measures on, and an associated adjoint state ϕ W,σ () for all σ < n/(n ), defined by the adjoint equation ϕ + ϕ + d (, ȳ, ū) ϕ y = ν ϕ + b (, ȳ) ϕ y = L y (, ȳ, ū) + µ g (, ȳ) in y l (3.9) y (, ȳ) + µ g Γ y (, ȳ) on Γ,

18 8 J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH where µ and µ Γ denote the restrictions of µ M( ) to and Γ, respectively. Following [7] or [3], we know that equation (3.9) admits a unique solution ϕ W,σ (), for all σ < n/(n ). Moreover the variational inequality H T u(u ū) 0, u U ad, (3.0) holds, where the vector H u R m is defined by H u = (H u,i ) i=,...,m, with L H u,i := (x, ȳ(x), ū) ϕ(x) d (x, ȳ(x), ū) dx, u i u i and the complementarity condition g(x, ȳ(x))d µ(x) = 0 (3.) is satisfied. For the ease of later computations, we introduce a different form of the Lagrange function L : Y q,p R m W,σ () M( ) R, L(y, u, ϕ, µ) =J(y, u) ( y + y + d(x, y, u))ϕ dx ( ν y + b(x, y))ϕ ds + µ, g(, y), (3.2) Γ which accounts also for the state-equation. Clearly, it holds that L = L, if y satisfies the state equation. Moreover, it is known that it holds for the partial derivatives of L with respect to y and u L y (ȳ, ū, ϕ, µ)y = 0 y H () (3.3) L u (ȳ, ū, ϕ, µ)(u ū) = H u T (u ū) 0 u U ad. (3.4) Therefore, the Lagrange function is an appropriate tool to express optimality conditions in a convenient way, in particular it is useful to verify second-order sufficient conditions by checking L (ȳ, ū, ϕ, µ)[(z h, h)] 2 > 0 for all z h satisfying the linearized equation (3.5). This second derivative of L with respect to (y, u) is expressed by L (ȳ, ū,ϕ, µ)[(y, u ), (y 2, u 2 )] = 2 L y 2 y y L u y y 2 u + 2 L y u u T 2y + u 2 L u 2 u 2 dx 2 l + Γ y 2 y ( 2 d y 2 ds y 2 y y d u y y 2 u + 2 d y u u T 2y + u 2 d u 2 u ) 2 b 2 ϕ dx Γ y 2 y y 2 ϕ ds + µ, 2 g y 2 (, ȳ)y y 2, (3.5) where the derivatives of L and d are taken at x, ȳ, and ū, respectively Second-Order Sufficient Optimality Conditions. In this section, we discuss second-order sufficient conditions for problem (P2). Since our controls belong to R m, the low regularity of the adjoint state does not cause troubles in the estimations of L. Therefore, second-order sufficient conditions can be obtained for arbitrary dimension of the domain. The proof of these conditions can either be performed analogous to Section 2.4. or derived by transferring the known second-order conditions from the theory of semi-infinite programming problems to our control problem. Therefore, we state the second-order sufficient optimality conditions without proof. Let d = (H u ) denote the first-order derivative of the Lagrangian function

19 CONTROL WITH FINITE-DIMENSIONAL CONTROL SPACE 9 with respect to u introduced in (3.0). For convenience, we introduce the following sets: A + := i : d i > 0}, A := i : d i < 0}, and A := A + A. Let τ := min d i : i A} > 0. We also define the critical cone associated with ū Cū = h R m : h i = 0 i A, and satisfies (3.6)-(3.8)} 0 if ūi = u h i = a,i (3.6) 0 if ū i = u b,i where z h stands for G (ū)h. g y (x, ȳ(x))z h(x) 0, if g(x, ȳ(x)) = 0 (3.7) g y (x, ȳ(x))z h(x)dµ = 0, (3.8) Remark 7. It is not difficult to see that this definition of the critical cone Cū is equivalent with the one used for semi-infinite programing in Bonnans and Shapiro [6]. We only include the restrictions on the control in the restriction function of the semi-infinite problem. Theorem 7. Let ū be a feasible control for problem (P 2) with associated state ȳ satisfying the first-order necessary conditions formulated in Section 3.2. Assume that h 2 L u 2 (ū, µ)h > 0, h C ū\0}. (3.9) Then there exist ε > 0 and δ > 0 such that for every feasible control of (P 2) J(ū) + δ 2 u ū 2 J(u) holds for all feasible controls with u ū < ε. Remark 8. As mentioned above, this theorem might be proven as for our parabolic problem. The proof is even simpler, since our controls belong to the space R m, where the unit sphere is compact. Moreover, it follows from regularity results Casas [7] or Alibert and Raymond [3] that the control-to-state mapping u y is of class C 2 from R m to H () C( ). However, the second-order conditions follow also from the theory of semi-infinite programming, cf. Bonnans and Shapiro [6], if the associated results are re-written in terms of partial differential equations. In the same way, parabolic problems for the type min J(y, u) = L(x, t, y(x, t)) dxdt + l(x, t, y(x, t), u) ds(x)dt u U ad subject to Q y t y(x, t) + d(x, t, y(x, t)) = 0 in Q, ν y(x, t) + b(x, t, y(x, t), u) = 0 on Σ, y(x, 0) y 0 (x) = 0 on, g(x, t, y(x, t)) 0 for all (x, t) Q Σ (3.20) can be dealt with, where U ad is defined as before. Notice that here the control appears in the boundary condition, where second-order sufficient conditions have not yet been proven for control functions under the presence of pointwise state constraints. Here, the control-to-state mapping is from R m to C( Q), cf. Casas [8]. Again, the second-order conditions can also be derived from the conditions for semi-infinite programming in [6].

20 20 J.C. DE LOS REYES P. MERINO J. REHBERG F. TRÖLTZSCH 3.4. Examples. Let us finally illustrate the situation by some examples, which exhibit different types of active sets and some kinds of nonlinearities. Example. Here we study the optimal control problem min J(y, u) = u R 4 2 y y d 2 L 2 () + 2 u u d 2 where (E) b(x) = subject to y(x) + y(x) + 7 y(x)3 = 4 i= u ie i (x) in ν y(x) = 0 on Γ, y(x) b(x) for all x = [0, ] [0, ], 2x +, x < 2,, x i, 2 x 2, and e i = 0 otherwise, i =, 2, 3, 4, and = (0, 2 ] (0, 2 ], 2 = (0, 2 ] ( 2, ), 3 = ( 2, ) [ 2, ), and 4 = ( 2, ) (0, 2 ). We shall see that the optimal state is ȳ = 2, hence the active set for this problem is 3 4. The first-order conditions for this problem are as follows: For the optimal solution (ȳ, ū) in H 2 () R 4 of the optimal control problem (E), there exist a Lagrange multiplier µ M( ) and an adjoint state ϕ W,σ () such that ȳ + ȳ + 7ȳ3 = 4 i= ūie i in ν ȳ = 0 on Γ, ϕ + ϕ ϕ = ȳ y d µ in ν ϕ = 0 on Γ, [ (ū u d ) + ϕe dx,..., ϕe 4 dx] = 0, (ȳ b)dµ = 0, ȳ(x) b(x) x and µ 0. The following quantities satisfy the optimality system: ȳ(x) = 2, ū = 22 x 2 7 [,,, + ], ϕ(x, x 2 ) = 2, x < 2, 4, x 2, (3.2) together with the Lagrange multiplier µ = δ x ( 2 ), where δ x i (z) denotes the Dirac measure with respect to the variable x i, concentrated at x i = z. In fact, it is easy to see that ȳ and ū fulfill the state equation in (3.2). Since ϕ does not depend on x 2, we find that ϕ = x 2 ϕ = δ x ( 2 ) ψ(x ), where 2, x < 2 ψ(x ) =, 0, x 2. Therefore, ϕ + ϕ ϕ = δ x ( 2 ) ( )( 2 x2 ), x < 2, 4 ( ), x 2,

21 CONTROL WITH FINITE-DIMENSIONAL CONTROL SPACE 2 and a simple computation shows that this is equal to ȳ y d µ, with (x 2 2 y d = )( ) x < 2, [ ( ) x 2, u d = ū + 48, 5 48, 6, ]. 6 To check the second order conditions notice that ϕ 2, hence we have L (ȳ, ū, ϕ, µ)[(z h, h)] 2 = z h 2 L 2 () + 2 h 2 7 ϕz2 h dx h 2 + zh 2 dx h 2 7 for all h R 4. Example 2. This example includes a semilinear elliptic equation with controls appearing nonlinearly. The active set has measure zero. The problem is the following (E2) where min J(y, u) = u R 2 2 y y d 2 L 2 () + 2 u u d 2 subject to y(x) + y(x) + 5 y(x)3 = 5 (u e (x) + u 2 e 2 (x)) 2 in ν y(x) = 0 on Γ, y(x) b(x) for all x = [0, ] [0, ], 2 b(x) = 4 (x + x 2 ), x + x 2, 4 (x + x 2 ), x + x 2 <, 8 e =, x + x 2, 4 e 2 =, x + x 2 <, 0 otherwise, 0 otherwise. (3.22) The optimality system is as in the previous example, with the gradient equation [ (ū u d ) + 2 ] (ū e (x) + ū 2 e 2 (x)) ϕe dx 5 (ū e (x) + ū 2 e 2 (x)) ϕe 2 dx [ = (ū u d ) + 2 ūe ] (x) 2 ϕ dx 5 ū2e 2 (x) 2 = 0. (3.23) ϕ dx To satisfy the optimality system, we define the quantities ȳ(x) = [ 4, ū = 9, 9 (x 2 2x + 2), x + x 2,, ϕ(x) = 2] (x ), x + x 2 <, note that ϕ satisfies ν ϕ = 0. Moreover, we define the Lagrange multiplier µ by the positive measure µ = 2( x )δ x2 ( x ) + 2x 2 δ x ( x 2 ). where δ x2 ( x ) and δ x ( x 2 ) are the Dirac measures with respect to x 2 and x concentrated at x 2 = x and x = x 2, respectively. These quantities satisfy the state equation in (3.22) and the identity (3.23). The active set is the diagonal of the unit square: x 2 = x. Let us discuss the adjoint equation. It holds 2 ϕ x 2 (x, x 2 ) = 2x 2 δ x ( x 2 ) + 2, x + x 2, 0, x + x 2 <,,