Weierstraß-Institut. im Forschungsverbund Berlin e.v. Preprint ISSN

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1 Weierstraß-Institut für Angewandte Analysis und Stocastik im Forscungsverbund Berlin e.v. Preprint ISSN Stability of infinite dimensional control problems wit pointwise state constraints Micael Hinze 1, Cristian Meyer 2 submitted: June 14, Department of Matematics University of Hamburg Bundesstraße Hamburg Germany micael.inze@uni-amburg.de 2 Weierstrass Institute for Applied Analysis and Stocastics Morenstraße Berlin Germany meyer@wias-berlin.de No Berlin Matematics Subject Classification. 49K20, 49N10, 49M20. Key words and prases. Optimal control of semi-linear elliptic equations, pointwise state constraints, finite element approximation.

2 Edited by Weierstraß-Institut für Angewandte Analysis und Stocastik (WIAS) Morenstraße Berlin Germany Fax: preprint@wias-berlin.de World Wide Web:ttp://

3 1 Abstract. A general class of nonlinear infinite dimensional optimization problems is considered tat covers semi-linear elliptic control problems wit distributed control as well as boundary control. Moreover, pointwise inequality constraints on te control and te state are incorporated. Te general optimization problem is perturbed by a certain class of perturbations, and we establis convergence of local solutions of te perturbed problems to a local solution of te unperturbed optimal control problem. Tese class of perturbations include finite element discretization as well as data perturbation suc tat te teory implies convergence of finite element approximation and stability w.r.t. noisy data. 1. Introduction In tis paper, we develop a stability analysis for a general class of optimization problems tat is suitable for te numerical analysis of nonlinear state-constrained optimal control problems. Optimization problems wit pointwise inequality constraints on te state are known to be teoretically and numerically callenging due to te low regularity of te optimal solution (cf. for instance Casas [7] or Alibert and Raymond [2]). Neverteless, in te recent past, tere as been some progress concerning te numerical approximation of linear-quadratic problems wit pointwise state constraints (see Deckelnick and Hinze [14, 15, 16], Deckelnick, Günter and Hinze [17], and Meyer [26]). However, less is known in case of nonlinear problems. To te autors knowledge, te only contribution in tis field as been performed by Casas and Mateos in [8]. Tey considered a full finite element discretization of a semi-linear stateconstrained optimal control problem and proved convergence of global optima of te discrete problems to a global optimum of te infinite dimensional problem. However, it is well known tat, in general, optimization algoritms only compute locally optimal solutions. Terefore, we also address te convergence to local optima in tis paper. To cope wit different types of control problems, we consider a general class of optimization problems, given by minimize f(u) := ϕ(s(u)) + α (PG) 2 u 2 H subject to u C and G(u) K Y. were H is a real Hilbert space, Y a real Banac space, G : H Y a continuous operator, and C H and K Y are closed and convex subsets. For a more precise definition of te quantities in (PG), see Section 2 below. Te second constraint, i.e. G(u) K, may be regarded as state constraint in tis context. In te course of te paper, we will see tat semi-linear elliptic problems wit distributed control in two and tree dimensions and boundary control in two dimensions are covered by (PG). Te discussion of boundary controls in tree dimensions is muc more delicate, but is in parts also included in te general teory (see Section 3.2). Since we aim to analyze different discretization scemes as well as oter perturbations of te problem, we consider a general class of perturbations tat is measured by some parameter wic represents for instance te mes size in case of discretization or te noise level of a certain perturbation of problem data. As already indicated, we focus on te convergence to local optima, more precisely we answer te following question: Consider a fixed unique local optimum of (PG), denoted by u, and a certain perturbation of (PG) wit associated perturbation parameter measuring in some sense te magnitude of te perturbation. Under wic conditions on te perturbation, does a sequence of local optima of te perturbed problems exist wic converges to u if te perturbation parameter tends to zero? We will see tat te required conditions allow for a wide class of perturbations including:

4 2 Semi-discretization in te spirit of [14] Full discretization wit different ansatz functions for te control Perturbation of te problem data Lavrentiev type regularization of te state constraints according to [28]. Hence, te presented analysis implies convergence of finite element scemes as well as stability w.r.t. certain noisy data. Te latter issue was already addressed by Griesse in [20]. However, ere we allow for data perturbations tat in parts differ from te ones considered in [20]. Te paper is organized as follows: After introducing te notation, we present te general class of optimization problems in Section 2. Its discussion is based on a Slater type assumption in combination wit a tecnique developed for te numerical analysis of control-constrained semi-linear problems by Casas and Tröltzsc in [11]. Section 3 is ten concerned wit different specific settings covered by te general teory. It is divided into two parts, te first addressing distributed controls, wile te second part deals wit boundary controls. In Section 4, we present a numerical example tat confirms te teory in te linear-quadratic case Notation. Let us introduce te notation used trougout te paper. If X is a linear normed function space, ten we notate X for a standard norm used in X. Moreover, te dual of X is denoted by X, and for te associated dual pairing, we write.,. X,X. If it is obvious in wic spaces te respective dual pairing is considered, ten te subscript occasionally is neglected. Now, given anoter linear normed space Y, te space of all bounded linear operators from X to Y is denoted by L(X, Y ). For an arbitrary A L(X, Y ), te associated adjoint operator is denoted by A L(Y, X ). If X is continuously embedded in Y, we write X Y and, if te embedding is in addition dense, we write X d Y. Given an optimal control problem, we call a function feasible for tis problem if it satisfies te constraints of te problem, i.e., for instance in case of (PG), u H is feasible if u C and G(u) K old true. Finally, trougout te paper, c denotes a generic constant, wile ε is an arbitrary small number greater zero. 2. A general class of state-constrained problems Let us recall te general class of optimization problems: minimize f(u) := ϕ(s(u)) + α (PG) 2 u 2 H subject to u C and G(u) K. Te constraint G(u) K will be referred to as state constraint in all wat follows, wereas u C is called control-constraint. For te discussion of tis problem, we rely on te following conditions: Assumption A1. Te control space H is a real Hilbert space, and te set C is a closed and convex subset of H. Moreover, te operator G is continuous from H to a real Banac space Y. Te set K is a closed and convex subset of Y, and tere exists a u C suc tat G(u) K. Furtermore, S is a continuous operator from H into a Banac Space W were te latter is continuously embedded in anoter Banac space V. Te functional ϕ : V R is continuous and moreover, if considered wit domain in W, is continuously Frécet-differentiable. In addition to tat, u n u in H implies G(u n ) G(u) in Y, and te same olds for S wen considered wit range in V, i.e. u n u in H ensures S(u n ) S(u) in V. Finally, tere is

5 3 a constant c > wit and α is a positive real number. c inf f(u), u C G(u) K (2.1) In te following, S is sometimes considered as operator wit range in V, for simplicity also denoted by S. Before we introduce a perturbation of (PG), let us discuss (PG) in more details. It is well known tat Assumption A1 implies te existence of a globally optimal solution. However, due to te nonlinearity of te problem, one cannot expect te uniqueness of te optimal solution. For tis reason, let us define te notion of unique local solutions to (PG): Definition 2.1. A control u C, satisfying G(u ) K, is called local optimum for (PG) if tere exists a real number ε > 0 suc tat f(u) f(u ) u C wit G(u) K, u u H < ε. If te inequality is strict for all u u, ten u is a unique locally optimal solution. Notice tat for instance a quadratic growt condition of te form f(u) = f(u ) + κ u u 2 H, κ > 0, (2.2) ensures uniqueness of a local optimum. Suc a growt condition olds true if second-order sufficient conditions are satisfied by te local optimum (see (SSC) below). In all wat follows, let us consider a fixed unique local optimum u. For te subsequent discussion, we require te following assumption to be satisfied by u : Assumption A2. Let U be a real reflexive Banac space wic is densely embedded in H, and denote te associated dual space wit respect to te inner product in H by U. Tere is a neigborood U(u ) in U, were G can be extended to an operator from U to Y, for simplicity also denoted by G. Moreover, G is twice continuously Frécet-differentiable from U to Y in U(u ). Similarly, also S can be extended to an operator from U to W in U(u ), wic is once continuously Frécet-differentiable. Note tat te dense embedding of U in H immediately implies H U. Hence, Assumption A2 implies tat G and S are continuously Frécet-differentiable from H to Y and W, respectively, at u H if u u H is sufficiently small. Next let us define te projection operator of elements in H on te set C. Assumption A3. Te operator P c : H C, defined by { 1 P c (v) := arg min u C 2 maps U to U. u v 2 H Now, assume tat te interior of K is nonempty. Hence, te supporting yperplane teorem implies tat it can be expressed as int K = η Y {y Y η, y Y,Y < g(η)}, were g : Y R denotes te support functional, i.e. g(η) = sup y K η, y Y,Y (cf. for instance [25, Section 5.13]). Te non-emptiness of K is also part of te following linearized Slater-condition: },

6 4 Assumption A4. A function û U C exists suc tat Tus, tere exists a τ > 0 wit for all η Y. G(u ) + G (u )(û u ) int K. η, G(u ) + G (u )(û u ) Y,Y g(η) τ According to Assumption A2, f is continuously Frécet differentiable from H to R at u. Togeter wit te Slater condition, tis allows to derive first-order necessary optimality conditions by means of te generalized Karus-Kun-Tucker teory (cf. for instance Zowe and Kurcyusz [33]). According to Casas [7, Teorem 5.2], te corresponding optimality system is given by te following Teorem 2.2. Let u be te optimal solution to (PG). Ten, tere exists a Lagrange multiplier µ Y suc tat ( α u + S (u ) ϕ (S(u )) + G (u ) µ, u u ) H 0 u C (2.3) µ, y G(u ) Y,Y 0 y K, (2.4) were S and G are considered as operators wit domain in H. Let us define p = S (u ) ϕ (S u ) + G (u ) µ H. Ten (2.3) implies { 1 } u = arg min u C 2 u ( α 1 p) 2 H = P c ( α 1 p) Tanks to te Frécet-differentiability of S and G at u from U to W and Y, respectively, we ave p U and ence, Assumption A3 yields u U. (2.5) Clearly, Teorem 2.2 is not sufficient for local optimality. It is well known tat te following conditions ensure local optimality of solutions satisfying (2.3) and (2.4): were C(u ) denotes te critical cone defined by f (u ) 2 > 0 C(u ), (SSC) C(u ) := { H = u u, u C, G(u) K}. Observe tat (SSC) requires tat S and ϕ are twice continuously Frécet-differentiable in teir respective spaces wic is not covered by Assumptions A1 and A2. Neverteless tis additional assumption is not necessary for te subsequent discussion. Notice furter tat C(u ) can be srunken by introducing strongly active sets, cf. for instance [10] and te references cited terein. As stated above, conditions (SSC) guarantee a quadratic growt condition of te form (2.2) so tat tey also guarantee unique local optimality. Next we introduce a perturbation of (PG). To tat end, let a positive parameter R be given. Associated to, we consider a subspace U of H (not necessarily a proper subspace) and define te projection operator Π : H U C for an arbitrary u H by Π (u) = arg min u u 2 u U H. (2.6) C

7 5 Given U, te perturbed optimization problem is defined by minimize f (u) := ϕ(s (u)) + α (PG ) 2 u 2 H subject to u U C and G (u) K wit perturbations of S and G, respectively, denoted by S and G. Problem (PG ) sould fulfill te following conditions: Assumption A5. For every > 0, te operators S : H V V and G : H Y Y are continuous in te respective spaces. Furtermore, a function δ : R + R + exists wit δ() 0 as 0 and G(u) G (u) Y δ() u H and S(u) S (u) V δ() u H (2.7) for all u C. Moreover, te space U H is not empty, closed, and convex. Te projection operator, defined by (2.6), satisfies Π (u ) u U c δ() u U, Π (û) û U c δ() û U and Π (u ) u H 0, Π (û) û H 0, as 0, (2.8) were u is te fixed locally optimal solution of (PG) and û is te Slater point from Assumption A4. To ensure te existence of solutions to (PG ), we require te following Assumption A6. Let {u n } H be an arbitrary sequence {u n } H converging weakly in H to u. Ten, for every > 0, ϕ(s (u n )) converges to ϕ(s (u)), i.e. u n u in H ϕ(s (u n )) ϕ(s (u)) in R, as n. (2.9) Furtermore, if in addition G (u n ) K olds for every n N, ten G (u) K follows, i.e., for every > 0, we ave u n u in H and G (u n ) K n N G (u) K. (2.10) Finally, tere exists a constant c p > suc tat olds for every > 0. c p inf f (u), u U C G (u) K (2.11) In te following, we will see tat Assumptions A4 and A5 imply te existence of a feasible point for (PG ), provided is cosen small enoug (cf. Lemma 2.4 below). Consequently, Assumption A6 guarantees te existence of at least one (global) solution to (PG ), cf. Remark 2.5. Analogously to Definition 2.1, we define local optima for (PG ). Definition 2.3. A discrete control u U C, satisfying G (u ) K, is a local optimum for (PG ) if tere exists a real number ε > 0 suc tat f (u ) f (u ) u U C wit G (u ) K, u u H < ε. Now, let us consider te following auxiliary problem: minimize f (u) := ϕ(s (u)) + α (PG,r ) 2 u 2 H subject to u U C B r (u ) and G (u) K,

8 6 were B r (u ) denotes a ball of radius r around u in H. Here, r satisfies r < ε, were ε denotes te parameter tat defines te neigborood were u is locally optimal (see Definition 2.1). Later on, we will see tat tere is at least one (global) solution to (PG,r ) (see Remark 2.5 below). Te overall proof of convergence now proceeds as follows: First, we sow tat, for small enoug, a convex linear combination of te local minimizer u and te Slater point û is feasible for (PG,r ). Afterwards, it is sown tat te weak limit of global solutions of (PG,r ), denoted by ũ, is feasible for te original problem (PG). Togeter wit te uniqueness of te local solution u in te ε-neigborood, tis two-way feasibility gives u = ũ. Finally, we prove tat te global solutions of (PG,r ) are local minimizers of (PG ). Trougout tis section, Assumptions A1 A6 are supposed to be satisfied. Lemma 2.4. Tere exist 0 > 0 and γ > 0 suc tat te sequence {v }, defined by is feasible for (PG,r ) for all 0. v := Π (u ) + γ δ()(π (û) Π (u )), Proof. First, we sow tat v fulfills te control-constraints. To tis end, coose a fixed, but arbitrary γ > 0. Tanks to (2.8), i.e. Π (u ) u H 0 and Π (û) û H 0 as 0, tere is an 1 suc tat, for all 1, v u H r. Moreover, if 1 is cosen sufficiently small, ten γ δ() 1, and consequently v U C by definition of Π and te convexity of U and C. It remains to verify te state-constraint. Recall tat g denotes te support functional of K and define y γ by y γ := G(u ) + γ δ() G (u )(û u ). Ten, te feasibility of u and te linearized Slater condition (cf. Assumption A4) imply for an arbitrary η Y η, y γ Y,Y g(η) γ δ() τ. (2.12) Now, set v := u + γ δ()(û u ) so tat v v in H because of (2.8). Due to te Frécetdifferentiability of G in U(u ) (see Assumption A2), tere exist β, θ, ϑ [0, 1] suc tat G (v ) y γ Y G (v ) G(v ) Y + G(v ) G(v) Y 1 + γ δ() 0 δ() v H + G (u + θ γ δ()(û u )) G (u ) L(U,Y ) dθ û u U 1 + ( γ δ() ) θ 0 G (v + β(v v)) L(U,Y ) dβ v v U G (u + ϑ γ δ()(û u )) L(L(U,Y ),U ) dϑ dθ û u 2 U, were we also used (2.7). Notice tat, tanks to δ() 0 for 0, we ave u + ϑ γ δ()(û u ) U(u ) and v + β(v v) U(u ) for all ϑ, β [0, 1], if is sufficiently small. Tus, te continuous Frécet-differentiability of G from U to Y in tese points is guaranteed by Assumption A2 giving in turn G (v+β(v v)) G (u ) in L(U, Y ) and G (u +ϑ γ δ()(û u )) G (u ) in L(L(U, Y ), U ) as 0. Terefore, 2 > 0 and c > 0 exist suc tat, for all 2, G (v + β(v v)) L(U,Y ) c, G (u + θ 1 θ 2 γ δ()(û u )) L(L(U,Y ),U ) c.

9 7 Moreover, (2.8) implies v v U ( 1 γ δ() ) Π (u ) u U + γ δ() Π (û) û U ( (1 ) ) c δ() γ δ() u U + γ δ() û U c ( δ() + γ δ() 2), were we used u, û U C for te last estimate (cf. (2.5)). Tus, tanks to v C for all 1, we end up wit G (v ) y γ Y c ( δ() + γ δ() 2 + γ 2 δ() 2). (2.13) For te rest of te proof, we argue by contradiction. To tat end, assume tat G (v ) / K for all > 0. Ten, by te minimum norm duality, te distance between G (v ) and K is given by { d(g (v ), K) = max η, G (v ) Y,Y g(η) } η Y =1 { max η, yγ Y,Y + η Y G (v ) y γ Y g(η) } η Y =1 [ δ() γτ c ( 1 + (γ + γ 2 )δ() )], were we used (2.12) and (2.13) for te last estimate. Since γ was arbitrary, we are allowed to take γ 2 c/τ. Ten, due to δ() 0 for 0, tere is an 3 suc tat (γ + γ 2 )δ() < 1, 3, giving in turn tat d(g (v ), K) < 0 for all 3. Consequently, we ave G (v ) K and v U C B r (u ) for all 0 := min( 1, 2, 3 ). Since v is a feasible point for (PG,r ), Assumption A6 implies Remark 2.5. If > 0 is small enoug, ten tere is at least one global solution of (PG,r ), and naturally also for (PG ). In all wat follows, let us consider an arbitrary global optimum of (PG,r ), denoted by u,r. Due to u,r B r(u ), te sequence {u,r }, 0, is uniformly bounded in H. Te reflexivity of H ten gives te existence of subsequence converging weakly in H to a weak limit ũ U C B r (u ). Everyting wat follows is also valid for any oter weakly converging subsequence. Tus, a known argument yields tat w.l.o.g. u,r ũ as 0. Lemma 2.6. Te weak limit ũ of {u,r } is feasible for (PG). Proof. First, since C is convex and closed, and tus weakly closed, we ave ũ C. Furtermore, Assumption A2 ensures tat u,r ũ in H implies G(u,r ) G(ũ) in Y, and consequently Tus, for 0, one finds G(u,r) G(ũ) in Y as 0. G (u,r) G(ũ) Y G(u,r) G(ũ) Y + G (u,r) G(u,r) Y G(u,r) G(ũ) Y + δ() u,r H 0, were we used (2.7) and te uniform boundedness of {u,r } in H because of u,r B r(u ). Because of G (u,r ) K for all > 0, tis implies G(ũ) K, since K is assumed to be closed (cf. Assumption A1). Lemma 2.7. Te sequence {u,r } converges strongly in H to u as 0.

10 8 Proof. Applying again Assumption A2, te same arguments as in te proof of Lemma 2.6 yield S (u,r) S(ũ) V S(u,r) S(ũ) V + S (u,r) S(u,r) V S(u,r) S(ũ) V + δ() u,r H 0, (2.14) Moreover, te optimality of u,r clearly guarantees te existence of a constant c suc tat, togeter wit (2.1) in Assumption A1, c f (u,r ) c. Hence, te continuity of ϕ from V to R (cf. Assumption A1), allows to continue wit lim inf f (u,r) lim ϕ(s (u α 0 0,r)) + lim inf 0 2 u,r 2 H ϕ(s(ũ)) + α (2.15) 2 ũ 2 H = f(ũ) tanks to te weakly lower semicontinuity of. H. Furtermore, te feasibility of v for (PG,r ) by Lemma 2.4 and te global optimality of u,r for (P G,r) give f (u,r) f (v ) 0. Hence, (2.8), i.e. v u in H, implies lim sup 0 f (u,r ) f(u ). On te oter and, since B r (u ) is clearly closed and convex, we ave ũ B r (u ) so tat ũ u H r < ε. Hence, te local optimality of u and te feasibility of ũ by Lemma 2.6 guarantee f(u ) f(ũ) lim inf 0 f (u,r) lim sup f (u,r) f(u ). 0 (2.16) Since u is te unique local optimum, tis implies ũ = u, and tus weak convergence of {u,r } to u by Lemma 2.6. It remains to verify te strong convergence. To tis end, we use f (u,r ) f(u ), wic follows from (2.16). Te definition of f yields u,r 2 H = 2 ( f (u α,r) ϕ(s (u,r)) ). (2.17) Due to (2.14), te rigt and side (2.17) converges to te value at u. Tus, we ave lim u 0,r 2 H = 2 ( f(u ) ϕ(s u ) ) = u 2 α H, and consequently u,r H u H. Togeter wit te weak convergence of {u,r }, tis norm convergence yields strong convergence, i.e. u,r u. Tus, we ave sown tat, for every unique local solution u, a sequence of global solutions of (PG,r ) exists tat converge strongly to u. It remains to verify tat global solutions of (PG,r ) represent local solutions of (PG ), wic is stated by te following teorem tat represents our main result: Teorem 2.8. Let u be a unique locally optimal solution according to Definition 2.1 and suppose tat Assumptions A1 A6 old at u. Ten, tere exists a sequence of local optimal solutions to (PG ), denoted by {u }, tat converges strongly in H to u, i.e. u u as 0. Proof. Take an arbitrary u U C wit G (u ) K and u u,r H Lemma 2.7 yields tat, for sufficiently small, < r/2. Ten, u u H u u,r H + u,r u H < r 2 + r 2 = r,

11 9 giving in turn u B r (u ), i.e. u is feasible for (PG,r ). Since u was cosen arbitrary, te (global) optimality of u,r for (PG,r) ten ensures f (u ) f (u,r) u U C wit G (u ) K, u u,r H < r/2, wic is equivalent to local optimality according to Definition 2.3. Tus, tere is at least one sequence of local solutions of (PG ) tat coincides wit {u,r } for sufficiently small and terefore converges to u. Remark 2.9. Te purely state-constrained case witout furter control-constraints is also covered by te above teory. In tis case, we ave C = H and te uniform boundedness of {u,r } in H, needed for te proofs of Lemma 2.6 and 2.7, ten follows by a standard argument from te optimality of u,r and te Tikonov regularization term α/2 u 2 H witin te objective functional. Remark It is straigt forward to see tat te presented teory also applies to perturbations tat satisfy G(u) G (u) Y δ() and S(u) S (u) V δ() for all u C wit δ() independent of u instead of (2.7). Now suppose tat (PG) admits unique (globally) optimal solution, and te same olds for (PG ) for every. Ten Teorem 2.8 immediately implies: Corollary If (PG) and (PG ) admit a unique global optimum, ten Assumptions A1 A6 ensure te strong convergence in H of te solutions of (PG ) to te solution of (PG) as 0. Clearly, if G and S are linear operators and ϕ is convex, ten, due to α > 0, (PG) is of course strictly convex giving in turn tat it admits a unique (globally) optimal solution u. Taking into account tat G and S arise from discretizations of G and S or possible perturbations of given data (cf. Section 3), it is natural to assume tat G and S are linear as well. Hence, also (PG ) admits a unique optimal solution u and consequently: Corollary Assume tat, in addition to Assumption A1, G, G, S, and S are linear operators and ϕ is a convex functional. Moreover, G and S are continuous as operators from U to Y and V, respectively, and u n u in H implies G u n G u in Y and te same olds for S suc tat Assumption A2 is fulfilled. Furtermore, suppose tat Assumption A5 is satisfied and tat tere is a point û U C wit G û int K (wic implies Assumption A4 in te linear case). Ten, te sequence of unique solutions of (PG ) converges strongly to te solution of (PG) as Specific settings In te subsequent, we present some examples for optimal control problems covered by (PG). Afore let us consider te following general semi-linear PDE y(x) + ϱ(x) y(x) + d(x, y(x)) = f(x) a.e. in Ω n y(x) + b(x, y(x)) = g(x) a.e. on Γ. (3.1) In all wat follows, te dependency on x is frequently neglected suc tat we write ϱ, d(y), and b(y) instead of ϱ(x), d(x, y(x)), and b(x, y(x)). For te discussion of tis equation, we rely on te following conditions on te quantities in (3.1).

12 10 Assumption A7. Te domain Ω R n, n 3, is a bounded Lipscitz domain. Te function ϱ L (Ω) is non-negative and greater tan zero on a set of positive measure. For a fixed y, te functions d(x, y) : Ω R R and b(x, y) : Γ R R are measurable w.r.t. x in Ω and Γ, respectively. Furtermore, tey are supposed to be twice continuously differentiable and monotone increasing w.r.t. y for almost all x in Ω and Γ. Moreover, it olds d(x, 0) + d y (x, 0) + d yy (x, 0) K (3.2) d yy (x, y 2 ) d yy (x, y 1 ) L(M) y 2 y 1 for almost all x Ω and all y 1, y 2 [ M, M], and b fulfills an analogous condition. Te discussion of (3.1) is well known and standard (see for instance Casas et al. [9] or Casas and Mateos [8]). However, for convenience of te reader, we add some details on te underlying analysis in Appendix 5.1 at te end of tis paper. Based on tese results, te control-to-state operator is introduced, wic maps f and g to y and is denoted by G : L q (Ω) L s (Γ) W 1,σ (Ω) wit q > n/2, s > n 1, and σ > n. Notice tat te assumptions on d and b can be weakened, see e.g. [9] for details. Neverteless, to keep te discussion concise, we do not consider te case as general as possible ere Elliptic problems wit distributed control. We start wit te following semi-linear elliptic problem were te control acts in te domain Ω: minimize J(y, u) := ψ(x, y(x)) dx + α u(x) 2 dx 2 (P d ) subject to y + ϱ y + d(y) = u in Ω Ω n y + b(y) = 0 on Γ Ω and y a (x) y(x) y b (x) a.e. in Ω u a u(x) u b a.e. in Ω Assumption A8. In addition to Assumption A7, te function ψ(x, y) : Ω R R is measurable w.r.t. x for every fixed y R. Moreover, it is continuously differentiable for a.a. x Ω and satisfies a condition analogous to (3.2), i.e. ψ(x, 0) + ψ y (x, 0) K, ψ y (x, y 2 ) ψ y (x, y 1 ) L(M) y 2 y 1 for almost all x Ω and all y 1, y 2 [ M, M]. Furtermore, y a and y b are functions in C( Ω) satisfying y a (x) < y b (x) for all x Ω. Te bounds u a and u b are real numbers wit u b u a. Finally, α is a positive real number. To cope wit te teory of Section 2, we set H = L 2 (Ω), U = W 1,σ (Ω), Y = C( Ω), V = W = L (Ω) (3.3) wit σ < n/(n 1). Notice tat W 1,σ (Ω) d L 2 (Ω) since 2 > n/2 for n = 2, 3. Moreover, corresponding to te general framework, te operator G is defined by G(u) = E c G(u, 0), were E c denotes te embedding operator from W 1,σ (Ω) to Y = C( Ω) and G is te solution operator to (3.1) defined above. In addition, we set S(u) = E G(u, 0), were E denotes te embedding operator from W 1,σ (Ω) in L (Ω). Hence, tanks to Teorems 5.3 and 5.5 and Lemma 5.6 (see Appendix 5.1), te conditions for G and S in Assumption A2 are fulfilled. Moreover, due to our assumptions on ψ, a known argument implies tat ϕ(. ) = ψ(x,. ) dx Ω is continuously Frécet-differentiable from L (Ω) to R so tat it satisfies te conditions in

13 11 Assumption A1. In addition, it is easy to see tat te ypotesis on ψ in Assumption A8 also yield (2.1). Wit regard to te state-constraint in (P d ), we set K = {y C( Ω) y a (x) y(x) y b (x) x Ω} (3.4) suc tat K is closed, convex, and, due to y a (x) < y b (x) for all x, also non-empty wit non-empty interior. Tus, tere is some ope tat te linearized Slater-condition in Assumption A4 can be fulfilled. Moreover, te set C is given by C = {u L 2 (Ω) u a u(x) u b a.e. in Ω} and tus, closed an convex. Furtermore, te operator P c of Assumption A3 takes te form u a, v(x) < u a P c (v)(x) = v(x), v(x) [u a, u b ] u b, v(x) > u b. To see tat Assumption A3 is fulfilled, we define Ω i := {x Ω v(x) [u a, u b ]}, Ω a := {x Ω v(x) < u a }, and Ω b := {x Ω v(x) > u b }. Ten, we obtain P c (v) W 1,σ (Ω) = ( v σ W 1,σ (Ω i ) + u a σ W 1,σ (Ω + u a) b σ W 1,σ (Ω b ) ( v σ W 1,σ (Ω) + u a σ W 1,σ (Ω) + u ) 1/σ, (3.5) b σ W 1,σ (Ω) suc tat P c indeed maps W 1,σ (Ω) to W 1,σ (Ω). Consequently, Assumptions A1, A2, and A3 are fulfilled and, if in addition te linearized Slater condition in Assumption A4 olds true, ten (P d ) fits into te setting of (PG). However, since te unknown local optimal solution is contained in te Slater condition, it is in general not possible to verify Assumption A4 a priori. Neverteless, as known from first-order teory, it is satisfied in many cases. Analogously, one can verify tat problem (P d ) wit omogeneous Diriclet boundary conditions is also covered by (PG), i.e. (P d ) wit te following state equation y + ϱ y + d(y) = u in Ω y = 0 on Γ. ) 1/σ (3.6) In tis case, an analogon to Lemma 5.1 in Appendix 5.1 follows again from Gröger [21] for n = 2 and from Jerison and Kenig [24] in te tree dimensional case. Moreover, it is straigt forward to see tat te analysis for (3.1), i.e. in particular Teorem 5.3 and Lemma 5.6 (cf. Appendix 5.1), can be transfered to tis case. Te corresponding solution operator of (3.6) is again denoted by G : u y. Now, in view of te omogeneous Diriclet boundary conditions, it is meaningful to require te state constraints on a subset of Ω, i.e. y a (x) y(x) y b (x) a.e. in D Ω. Here, we coose te same setting as above except Y = C( D), G = χ D E c G : L 2 (Ω) C( D), were χ D denotes te restriction operator on D. It is straigt forward to see tat tis modified problem is also of class (PG). Finally we observe tat, due to L 2 (Ω) W 1,σ (Ω) and Remark 2.9, te purely state-constrained case, were u a = and u b =, is also covered by (PG). Now, we turn to possible perturbations of (P d ) fulfilling te conditions in Assumption A5. Semi-discrete approac. First let us consider te case were we do not discretize te control explicitely, i.e. we set U U = L 2 (Ω) so tat (2.8) is trivially satisfies. It remains to verify te conditions on te discretization of G and S in Assumptions A5 and A6. To simplify te argumentation concerning te finite element discretization we pose

14 12 Assumption A9. Te Ω is convex domain wit polygonal (n = 2) or polyedral (n = 3) boundary. Moreover, tere is a family of regular triangulations {T } of Ω wit mes size tat satisfies T T T = Ω. We note tat sligt modifications of te subsequent argumentation also apply to te more general case of domains Ω wit boundary Γ of class C 1,1. States y are discretized by y Y := {y C( Ω) y T P l T T } for some l N, were P k (T ) denotes te set of all polynomials on T of order less or equal k. Ten, te discrete state y Y associated to u L 2 (Ω) solves ( y v + ϱ y v + d(y )v ) dx + b(y )v ds = u v dx v Y. (3.7) Ω Γ Notice tat we do not consider discretizations of te nonlinearities d and b in tis context. Based on te results of Appendix 5, te conditions in Assumption A5 can easily be verified for te case l = 1, i.e. for linear finite elements. Teorem 5.7 and Lemma 5.4 imply G(u) G (u) C( Ω) = c 2 n/2 u L 2 (Ω) S(u) S (u) L (Ω) = c 2 n/2 u L 2 (Ω) Ω (3.8) wic gives in turn (2.7) in view of our settings in (3.3). Here, G(u) = E c G(u, 0), as defined above, and G denotes its FE-discretization, i.e. te solution operator of (3.7) wit range in C( Ω). Moreover, S is defined analogously. Clearly, iger order metods, i.e. l > 1, can be discussed analogously. Consequently, Assumption A5 is fulfilled. Furtermore, Assumption A6 can be verified by Lemma 5.8, wic is demonstrated in te following. To tat end, let a sequence {u n } L 2 (Ω) be given and assume tat u n u in L 2 (Ω). Ten, if S is considered as operator wit range in L 2 (Ω), Lemma 5.8 implies for every > 0 tat S (u n ) S (u) in L 2 (Ω) as n. Moreover, in view of (3.8) and te weak convergence of {u n }, te sequence S (u n ) is uniformly bounded in L (Ω) for all > 0. Terefore, due to te assumptions on ψ in A8 we ave u n u in L 2 (Ω) ψ(s (u n )) ψ(s (u)) L 2 (Ω) as n for every > 0. Hence, condition (2.9) in Assumption A6 is verified. To sow (2.10), also consider G wit range in L 2 (Ω), suc tat G (u n ) G (u) in L 2 (Ω) by Lemma 5.8. Moreover, assume tat G (u n ) K for all n N, were K as defined in (3.4) is seen as a subset of L 2 (Ω). Since K is closed, we ave G (u) K L 2 (Ω), and since G (u) is continuous, also G (u) K according to te original definition of K in (3.4). Consequently also Assumption A6 is fulfilled and tus, Teorem 2.8 implies tat eac local optimum can be approximated by a semi-discrete solution. For te linear-quadratic counterpart of (P d ), tis was already proven by Deckelnick and Hinze in [14, 15], wo also establised an order of convergence of 2 n/2 ε if linear finite elements are used. Full finite element (FE) discretization. In contrast to te semi-discrete approac, te control is now also discretized by or u U (0) := {u L 2 (Ω) u T = const. T T } u U (k) := {u C( Ω) u T P k T T } for some k = 1, 2,...

15 13 Clearly, te finite element error analysis, presented in above section, is also applicable ere so tat we only ave to verify condition (2.8), i.e. te convergence properties of te convex projection operator Π as defined in (2.6). Let us first consider te case k = 0, were te control is discretized by piecewise constant functions. It is easy to see tat Π (u) T = 1 T T u dx T T, (3.9) satisfies Π (u)(x) [u a, u b ] if u(x) [u a, u b ] a.e. in Ω. Moreover, based on results of Stampaccia [30], it is proven in [26] tat, for eac ε > 0, u Π (u) L 2 (Ω) c 2 n/2 ε u W 1,σ (Ω) olds. Tis implies (2.8) since U = W 1,σ (Ω) by construction. Remark 3.1. Note tat one can also allow for varying bounds u a, u b L (Ω) W 1,σ (Ω). In tis case, te assumptions in (2.8) can be verified using a tecnique introduced by Falk in [19, Lemma 5]. However, since te arguments a rater tecnical, tis is not considered ere. Let us furter note tat (2.8) may be substituted by allowing for a convex, closed set C in (P G ) wic depends on te parameter and approximates te set C of (P G) sufficiently well for tending to zero, compare [16]. Now, we turn to te case k = 1 and introduce te Clément interpolation operator Ĩ (u)(x) := N (Π i u)(x i )φ i (x), i=1 were x i denotes a node of te triangulation, N is te number of nodes, and φ i denotes te usual linear finite element ansatz function, i.e. φ i U wit φ i (x j ) = δ ij. Furtermore, Π i denotes te L 2 -projection on supp{φ i }. Using results of Clément [12] and standard embedding teorems, we find u Ĩ(u) L 2 (Ω) c 2 n/2 ε u W 1,σ (Ω). (3.10) However, Ĩ(u) need not satisfy Ĩ(u)(x) [u a, u b ] a.e. in Ω even if u itself is feasible w.r.t. control constraints. Hence, we define u = I P c (Ĩ(u)), were I is te standard Lagrange interpolation operator and P c again denotes te pointwise projection on [u a, u b ]. We continue wit u u L 2 (Ω) u P c (Ĩ(u)) L 2 (Ω) + P c (Ĩ(u)) u L 2 (Ω). (3.11) Te latter norm is estimated by standard interpolation error estimates: P c (Ĩ(u)) u L 2 (Ω) c 2 n/2 ε P c (Ĩ(u)) W 1,σ (Ω) c ( ) (3.12) 2 n/2 ε u a W 1,σ (Ω) + u b W 1,σ (Ω) + u W 1,σ (Ω), were we used (3.5) and Ĩ(u) W 1,σ (Ω) c u W 1,σ (Ω) (cf. for instance Steinbac [31]). For te estimation of te first addend in (3.11), let us define Ω b := {x Ω Ĩ(u)(x) > u b },

16 14 suc tat we ave u(x) u b = P c (Ĩ(u))(x) < Ĩ(u)(x) a.e. in Ω b. Tis immediately implies P c (Ĩ(u))(x) u(x) < Ĩ(u)(x) u(x) a.e. in Ω b and consequently P c (Ĩ(u)) u L 2 (Ω b ) Ĩ(u) u L 2 (Ω b ). (3.13) Togeter wit an analogous argument for te lower bound, it follows P c (Ĩ(u)) u L 2 (Ω) Ĩ(u) u L 2 (Ω) c 2 n/2 ε u W 1,σ (Ω), were we used (3.10) for te last estimate. In view of (3.11) and (3.12), tis gives (2.8) for k = 1. Notice tat, strictly speaking, te bounds u a and u b enter te first inequality in (2.8) via (3.12). Neverteless, it is easy to see tat tis does not influence te teory. Tanks to U (k) U (k+1), te above arguments also guarantee (2.8) if te control is discretized wit iger order ansatz functions. Terefore, also te full discretization of (P d ) is covered by te presented analysis giving in turn tat local optima of (P d ) can be approximated by a full finite element discretization. Perturbation of te data. Te setting in Assumption A5 also includes perturbations of te problem data wic is demonstrated in te following. To tis end, we consider te following optimal control problem wit box constraints on te state and a tracking type objective functional wit desired state y d L 2 (Ω); minimize J(y, u) := 1 2 y y d 2 L 2 (Ω) + α 2 u 2 L 2 (Ω) subject to y + ϱ y + d(y) = u in Ω (P ex ) n y + b(y) = 0 on Γ and y(x) y b (x) a.e. in Ω. For te remaining quantities, we suppose te same conditions as for (P d ). Now, ϕ is defined by ϕ(x, y(x)) = 1/2 Ω (y(x) y d(x)) 2 dx and ence, te conditions in Assumption A8 are fulfilled. Moreover, ϕ is clearly also continuous from L 2 (Ω) to R. Tus, we set V = L 2 (Ω) wereas we coose te same spaces as in (3.3) for H, U, Y, and W. It is straigt forward to see tat Assumptions A1 and A2 are still fulfilled in tis setting. Now we add some noise on te problem data, i.e. te bounds and te desired state, for instance y d + ε and y b + δ wit ε L 2 (Ω) and δ C( Ω). By setting S (u)(x) = S(u)(x) ε (x), G (u)(x) = G(u)(x) δ (x), (3.14) suc a perturbation is covered by te general teory. To fulfill te conditions in Assumption A5 (cf. Remark 2.10), we require ε L 2 (Ω) 0 and δ (x) 0, if 0. Notice tat Assumption A6 is automatically fulfilled due to te properties of S and G tat follow from Lemma 5.6. Tus, Teorem 2.8 yields tat (P ex ) is stable w.r.t. perturbation of tis form in te sense tat, for every unique local solution u, tere is a sequence of local solutions of te perturbed problems converging strongly in L 2 (Ω) to u if te noise level tends to zero. A possible coice for δ is for instance δ (x) = cos( 4 π x 1 ) cos( 4 π x 2 ) as in te numerical example in Section 4 below. Notice tat perturbations of te desired state in te context of state constraints were already discussed by Griesse in [20]. Lavrentiev type regularization. Next, we replace te pointwise state constraints in (P d ) by mixed constraints of te form y a (x) u(x) + y(x) y b (x) a.e. in Ω wit some R, > 0. Tis regularization tecnique was proposed in [28] to tackle (P d ) in te purely state constrained case. It is one advantage of tis regularization tecnique tat te related

17 15 problems (P d ) admit multipliers wit iger regularity tan tose associated to problem (P d ). Concerning linear-quadratic problems, convergence of te solutions of te regularized problems to te solution of te original problem is sown in [28] and [27]. In te semi-linear case, convergence of global solutions is addressed by Hintermüller et al. in [22]. However, by setting G (u)(x) = u(x) + G(u)(x), te general teory of Section 2 allows to analyze te convergence to local solutions. To tis end, let us coose Y = L (Ω). It is easy to see tat tat te above teory can also be carried out wit L (Ω) instead of C( Ω) since K = {y L (Ω) y a (x) y(x) y b (x) a.e. in Ω} admits a non-empty interior. Wit tis setting at and, te conditions in Assumptions A5 and A6 can easily be verified. Since tere is no perturbation of S, we only ave verify te assumptions on G. We start wit (2.10); if a sequence {u n } converges weakly in L 2 (Ω) to u, ten G (u n ) G (u) in L 2 (Ω) follows immediately. Consequently, since te set K, considered as a subset of L 2 (Ω), is convex and closed, ence weakly closed, we ave G (u) K so tat (2.10) and tus Assumption A6 is satisfied. Notice tat, in case of unilateral state constraints, one as to require C L (Ω) to ensure G (u) L (Ω). Concerning Assumption A5, we find G(u) G (u) L (Ω) u L (Ω). Hence if C L (Ω), ten Teorem 2.8 and Remark 2.10, respectively, imply te existence of a sequence of local solutions of te regularized problems tat converges strongly in L 2 (Ω) to a unique local solution of te original problem as 0. Notice owever tat additional control constraints are necessary to ensure te boundedness of te controls in L (Ω) Elliptic problems wit boundary control. Next, let us consider a problem wit boundary control in two dimensions: minimize J(y, u) := ψ(y(x)) dx + α u(x) 2 ds 2 (P b ) subject to y + ϱ y + d(y) = 0 in Ω R 2 Ω n y + b(y) = u on Γ Γ and y a (x) y(x) y b (x) a.e. in Γ u a u(x) u b a.e. in Γ In all wat follows, te quantities in (P b ) are assumed to fulfill te conditions in Assumption A7 and A8. In tis case, we coose H = L 2 (Γ), U = W 1 1/σ,σ (Ω), Y = C( Ω), V = W = L (Ω), were σ is as above given by σ < n/(n 1). According to Teorem 5.3 below, tere is a unique solution of te state equation in W 1,σ (Ω) C( Ω) if u L s (Γ), s > n 1 = 1. Tus, similarly to Section 3.1, we define G(u) = E c G(0, u) and S(u) = E G(0, u). Moreover, since te trace operator is continuous from W 1,σ (Ω) to W 1 1/σ,σ (Γ), one can associate to every u W 1 1/σ,σ (Γ) an element of W 1,σ (Ω), also denoted by u. Consequently, Teorem 5.5

18 16 yields te continuous Frécet differentiability of G and S in a neigborood in W 1 1/σ,σ (Γ) at u L 2 (Γ). Togeter wit Lemma 5.6, tis ensures Assumption A2. Moreover, by setting K = {y C( Ω) y a (x) y(x) y b (x) x Γ} C = {u L 2 (Γ) u a u(x) u b a.e. on Γ}, we see tat also Assumption A1 is satisfied. Furtermore, by similar arguments as in (3.5), it can be seen tat Assumption A3 olds. Terefore, if in addition te Slater condition in Assumption A4 is fulfilled, also (P b ) is covered by te general setting of (PG). As in case of (P d ), it is a priori not possible to ensure te Slater condition. Notice tat Teorem 5.3 does not yield continuous solutions for u L 2 (Γ) in case of n = 3. Moreover, W 1 1/σ,σ (Γ) is not embedded in L 2 (Γ) for n = 3. Tis already indicates tat boundary control in tree dimensions causes some trouble and is not covered by te above teory. We will add some comments on tis complex of questions in a section below. Discretization of problem (P b ). As for (P d ), we suppose Assumption A9, i.e. in particular Ω is convex and possesses a polygonal boundary. Ten, Lemma 5.4 and Teorem 5.7 yield G(u) G (u) C( Ω) = c 1/2 u L 2 (Γ), S(u) S (u) L (Ω) = c 1/2 u L 2 (Γ), were, as above, G and S denote te finite element approximations of G and S. Tus, (2.7) is guaranteed. Moreover, based on Lemma 5.8, te conditions in Assumption A6 can be verified analogously to te case wit distributed control in Section 3.1. Hence, Teorem 2.8 already implies convergence in te semi-discrete case. As above, concerning te full discretization, (2.8) as to be verified. To tis end, let us define E T := T Γ T T and E := {E T T T }. According to Assumption A9, te triangulations of Ω exactly fit te boundary suc tat E = Γ. Now, assume first tat te control is discretized by constant ansatz functions, i.e. Analogously to (3.9) we define u U := {u L 2 (Γ) u E = const. E E }. Π (u) E = 1 E E u ds E E. (3.15) Moreover, in view of σ < n/(n 1) = 2, embedding teorems yield W 1 1/σ,σ (Γ) H t (Γ) wit t < (3 n)/2 = 1/2. Tus, from (3.15) and [12, Lemma 1], it follows tat u Π (u) L 2 (E) c diam(e) t u H t (E). Notice in tis context tat Ω is polygonally bounded by Assumption 5.4 so tat standard interpolation error estimates are applicable. Consequently, tanks to Γ = E and diam(e), u Π (u) 2 L 2 (Γ) c 2t u 2 H t (Ω) suc tat (2.8) in Assumption A5 is fulfilled. If linear and continuous ansatz functions are used for te discretization of te control, similar arguments as in case of distributed control can be applied (in particular (3.11) and (3.13)). In tis case, te Clément interpolation operator is defined by Ĩ (u)(x) := x i Γ(Π i u)(x i )φ i (x), x Γ,

19 17 were Π i is te L 2 -projection on supp(φ i ) Γ. Standard interpolation error estimates ten imply u u L 2 (Γ) c t u H t (Γ) c 1/2 ε u W 1 1/s,/s (Γ) (cf. for instance [6]). Hence (2.8) is also fulfilled if iger order ansatz functions are used. In summary, we see tat, in te case n = 2, a standard discretization of (P b ) also fits into te setting of te above teory and terefore, Teorem 2.8 implies tat local optima can be approximated by common discrete scemes. Neverteless, te situation canges in tree dimensions, as we will see in te following. Some remarks on boundary control in tree dimensions. Let us restrict on te linear-quadratic case, i.e. (P b ) wit b = d 0, ϱ 1, and ψ(x,. ) = 1/2. y d (x) 2 wit a given function y d L 2 (Ω). As already mentioned above, a control in L 2 (Γ) is not sufficient to guarantee continuity of te solution to te state equation, even in te linear case (cf. Lemma 5.1). However, if additional control constraints guarantee u L (Γ), ten continuous solutions are obtained. Neverteless, te analysis of Section 2 is not applicable and as to be modified by introducing a new control space L s (Γ) wit sufficiently large s > n 1 = 2, tat is embedded in W 1 1/σ,σ (Γ). It is straigt forward, but rater tecnical to see tat te arguments of Section 2 can be adapted to tis case by using L s (Γ) instead of L 2 (Γ). To be more precise, te teory in Section 5.1 yields tat Assumptions A1 and A2 old wit L s (Γ) instead of H = L 2 (Γ). Notice tat s = is not allowed, since te proof of Lemma 2.6 exploits tat L s (Γ) is reflexive. Wile Assumption A6 can be verified by arguments analogously to tose of Section 3.1, te crucial part is now Assumption A5 wic in general is ard to guarantee. As we will see in te following one as to require strong conditions on te setting to ensure tis assumption. Instead of (2.7) and (2.8), we now ave to require G(u) G (u) C( Ω) δ() u L s (Γ), S(u) S (u) L 2 (Ω) δ() u L s (Γ) (3.16) Π (u) u W 1 1/σ,σ (Γ) c δ() u L s (Γ), Π (u) u L s (Γ) 0 as 0 (3.17) for all u C. Wit tese conditions at and, te proofs of Lemma can easily be modified suc tat Teorem 2.8 also olds in tis case. Note in tis context tat G is twice continuously Frécet-differentiable from W 1 1/σ,σ (Γ) to C( Ω) around u L s (Γ) as already demonstrated for n = 2. In case of perturbations of te data of te form (3.14), conditions (3.16) and (3.17) are clearly satisfied. However, if discretizations of te control problem are considered te situation becomes more difficult. Subsequently, we state te assumptions tat are needed to treat te semi-discrete as well as te fully discrete approac. As above, let us first turn to semi-discretization, were just (3.16) as to be verified. To tis end, assume tat te boundary of Ω is smoot. Ten, for every finite p, te state equation admits a unique solution in W 1,p (Ω) for every rigt and side f in W 1,p (Ω), p = p/(p 1), and tere olds y W 1,p (Ω) c f W 1,p (Ω) (cf. for instance [1, Teorem 15.3 ]). By te trace teorem, v W 1,p (Ω) implies τ v L r (Γ) wit r = (n 1)p/((n 1)p n) = 2p/(2p 3) suc tat r tends to 1 if p. Terefore, u L s (Γ) is an element of W 1,p (Ω) if s r = (n 1)p/n = 2p/3. Consequently, since u C implies its boundedness in L (Γ), we ave y W 1,p (Ω) for all p <. Moreover, assume tat te triangulation of Ω is curvilinear and exactly fits te boundary. Ten, as sown by Deckelnick and Hinze in [14], tere is an extension of a result of Scatz [29, Teorerm 2.1] wic states y y L (Ω) c log y I y L (Ω),

20 18 were y denotes te approximation of y wit piecewise linear continuous finite elements and I denotes te associated Lagrange interpolation operator. Now, interpolation error estimates for curved domains (cf. Bernardi [5]) give y y L (Ω) c 1 n/p ε y W 1,p (Ω) c 1 n/p ε u L s (Γ) wit s (n 1)p/n = 2p/3 according to te above considerations. Hence, if we coose s large enoug, i.e. s > n 1 = 2, it follows tat 1 n/p ε = 1 3/p ε > 0 giving in turn (3.16) wit δ() = 1 3/p ε. Tus, Teorem 2.8 remains valid in te semi-discrete case. Notice owever tat, wit regard to te proof of Lemma 2.7, only strong convergence in L 2 (Γ), and not in L s (Γ), is obtained in tis way. If full discretization is applied, ten, in addition, (3.17) as to be verified. Concerning te first condition in (3.17), we benefit from te fact tat uniform convergence of te projection operator is only needed w.r.t. te W 1 1/σ,σ (Γ) -norm. Let us restrict to piecewise constant ansatz function for te control, were u C immediately implies Π (u) C wit Π defined in (3.15). Using ortogonality properties of te projection operator, we obtain Π (u) u W 1 1/σ,σ (Γ) = sup (Π Γ (u) u)v ds v 0 v W 1 1/σ,σ (Γ) = sup (Π Γ (u) u)(π (v) v) ds v 0 v W 1 1/σ,σ (Γ) Π (u) u L σ (Γ) sup v 0 Π (v) v L σ (Γ). v W 1 1/σ,σ (Γ) Now, interpolation error estimates on curved domains (cf. again [5]) yield (3.18) Π (v) v L σ (Γ) c 1 1/σ v W 1 1/σ,σ (Γ). (3.19) Moreover, Douglas et al. sowed in [18] tat te projection operator is stable w.r.t. L p -norms, 1 p, suc tat Π (u) u L σ (Γ) c u L σ (Γ) c u L s (Γ) provided tat s > σ = σ/(σ 1) > n = 3. Togeter wit (3.18) and (3.19), tis implies te first condition in (3.17). A verification of te second condition in (3.17) is still an open question. Neiter te additional regularity of an optimal solution nor a density argument yields te desired convergence. 4. Numerical example For te numerical verification of te above teory, we consider problem (P ex ) wit Ω = (0, 1) 2, ϱ 1, d = b 0, and y b 1, i.e. minimize J(y, u) := 1 2 y y d 2 L 2 (Ω) + α 2 u 2 L 2 (Ω) subject to y + y = u in Ω (P 1 ) n y = 0 on Γ and y(x) 1 a.e. in Ω. Moreover, y d is a given function in L 2 (Ω) tat satisfies y d (x) > 1 + 1/α a.e. in Ω. According to Casas [7] and Alibert and Raymond [2], te necessary and sufficient conditions for (P 1 )

21 19 read y + y = u n y = 0 in Ω on Γ p + p = y y d + µ Ω n p = µ Γ α u (x) + p (x) = 0 a.e. in Ω (y (x) 1) dµ(x) = 0 Ω y(x) dµ(x) 0 y C + ( Ω), y (x) 1 x Ω. Ω wit C( Ω) + := {y C( Ω) y(x) 0 x Ω}. Moreover, µ is an element of C( Ω), i.e. a regular Borel measure, and µ Ω and µ Γ denote its restrictions to Ω and Γ, respectively. It is easy to verify tat tis optimality system is satisfied by u = y 1, p 1 α, µ = y d 1 α 1. in Ω on Γ (4.1) Note tat µ is a proper function ere and tat µ > 0 since y d > 1 + 1/α. Moreover, we observe tat te state constraint is active everywere in Ω. Now, let us consider te controlto-state operator wit range in L 2 (Ω) and denote tis operator by S. Note tat S is clearly linear in case of (P 1 ). Te inequality constraint ten implies S u = 1 as equation for u, wic is clearly an ill-posed equation due to te compactness of S : L 2 (Ω) L 2 (Ω) and terefore unstable w.r.t. a certain class of perturbations. For instance, a similar equation wit a perturbation of small amplitude and ig frequency on te rigt-and side, i.e. admits te solution S u = 1 + δ λ wit δ λ (x) := λ cos(λ 4 π x 1 ) cos(λ 4 π x 2 ), λ > 0, (4.2) u λ (x) = ( ) 2 π 2 λ 7 + λ cos(λ 4 π x 1 ) cos(λ 4 π x 2 ), suc tat u λ u L 2 (Ω) = u λ 1 L 2 (Ω) for λ 0. On te oter and, we impose te same perturbation on te state constraint in (P 1 ), i.e. y(x) 1 + δ λ (x) a.e. in Ω. (4.3) In view of Remark 2.10 and te considerations in Section 3.1, (P 1 ) is naturally covered by te general teory. Tus, since (P 1 ) is in addition a linear-quadratic problem, Corollary 2.12 implies tat te unique solutions of te perturbed problems converge strongly in L 2 (Ω) to te unique solution of (P 1 ) if λ 0. Hence, in contrast to (4.2), (P 1 ) is stable w.r.t. te afore mentioned perturbation. To numerically confirm tis assertion, we solve (P 1 ) wit (4.3) as inequality constraint as well as (4.2) using a full discretization wit linear ansatz functions for te state and te control. Hence, te discretization fits into te setting of Section 3.1. Te discrete version of state equation ten is equivalent to (3.7) wit ϱ 1 and d = b 0 and can be written in te form (K + M) y = M u, were K and M denote te stiffness and te mass matrix associated to linear finite elements, wile y and u are te vectors associated to te discrete versions of state and control. Notice tat, strictly speaking, we ave two perturbations in tis context: te first one arising from te discretization wit mes size, and a second perturbation induced by te function δ λ wit associated parameter λ. However, as demonstrated in Section 3.1, bot perturbations fulfill te conditions in Assumption A5. In order to illustrate te effects of te ill-posedness

22 20 on te numerical treatment of (4.2), we compute te solution of te discrete version of (4.2) given by f = M 1 (K + M ) (1 + δλ, ) (4.4) were δλ, denotes te vector of values of δλ at te nodes of te triangulation. Te numerical results for different mes sizes and λ = 10 5 are sown in Figure 4.1 and 4.2. Altoug te discretization clearly as a smooting property, we observe tat te numerical solutions are indeed fairly irregular and tat tis beavior is even worsened by a decrease of. f x x 1 Figure 4.1. Solution f of (4.4) for = 0.01 and λ = Figure 4.2. Solution f of (4.4) for = and λ = u x x1 Figure 4.3. Optimal control for = 0.01, λ = 10 5, and α = Figure 4.4. Optimal control for = 0.005, λ = 10 5, and α = On te oter and, we investigate te perturbed optimal control problem, i.e. problem (P1 ) wit an inequality constraint of te form (4.3) instead of y(x) 1 a.e. in Ω. Te associated discrete optimal control problem were solved by a primal-dual active set strategy (see for instance Bergounioux, Ito, and Kunisc [3] or Bergounioux and Kunisc [4]). Figures 4.3 and 4.4 sow te discrete optimal solutions. In accordance wit te teory (cf. Corollary 2.12), te optimal solutions appear stable wit respect to perturbations of te form (4.3), wic is also demonstrated by te fact tat te solution does not become irregular if te mes size is decreased. Tis is also confirmed by Table 4.1 sowing te L2 -errors of control and state for different values of α. Note tat te results wit respect to te control are improved by increasing te Tikonov parameter α. In summary, we obtain tat state constrained optimal control problems in general beave stable (and well) wit respect to a wide class of perturbations.