Approximation of boundary control problems on curved domains

The boundary control problems associated to a semilinear elliptic equation defined in a curved domain Ω are considered. The Dirichlet and Neumann cases are analyzed. To deal with the numerical analysis of these problems, the approximation of Ω by an appropriate domain Ω<inf>h</inf> (typically polygonal) is required. Here, we do not consider the numerical approximation of the control problems. Instead of it, we formulate the corresponding infinite dimensional control problems in Ω<inf>h</inf> and we study the influence of the replacement of Ω by Ω<inf>h</inf> on the solutions of the control problems. Our goal is to compare the optimal controls defined on Γ = δΩ with those defined on Γ<inf>h</inf> = δΩ<inf>h</inf> and to derive some error estimates. The use of a convenient parametrization of the boundary is needed for such estimates. The results for convex domains are given in [1], the results for nonconvex domains are included in a work in progress.


Introduction.
In this paper we study a Dirichlet control problem (P) defined on a curved domain Ω.To solve numerically this problem, usually it is necessary to approximate Ω by a new domain (typically polygonal) Ω h .Our goal is to analyze the effect of the domain change on the optimal control.More precisely, a new optimal control problem (P h ) in Ω h is defined.The convergence of global or local solutions of problems (P h ) to the corresponding local or global solutions of (P) is investigated for the parameter h tending to zero.We also derive some error estimates.We restrict our study to the case of a convex domain Ω ⊂ R 2 approximated by a polygonal domain Ω h , h being the length of the biggest edge of Ω h .A family of infinite dimensional control problems (P h ) defined in Ω h is considered and the solutions of (P h ) are compared with the solutions of (P).In this way, the influence of small changes in the domain on the solutions of the control problem is analyzed.The case of a Neumann control problem is studied in [6].
In this paper we do not perform the numerical analysis of the optimal control problems.We refer the reader to the related papers, [5] for the numerical discretization of a Dirichlet control problem in the case of a polygonal domain and [7] for the analysis in curved domains.
Let us describe the content of the paper.In §2 control problem (P) is introduced and analyzed.In particular, the second order sufficient optimality conditions are established.In spite of the fact that the cost functional is not of class C 2 in L 2 (Γ), we prove that the standard sufficient optimality conditions imply that the control is a strict local minimum in the L 2 (Γ) norm.This is an improvement of the known results where the optimality is established in the L ∞ (Γ) norm.Approximations Ω h of Ω are defined in §3 along with control problem (P h ).In subsequent §4, the analysis is performed and paper is completed with the full proof of the new error estimates in §5.The second order sufficient optimality conditions are a crucial tool for the derivation of error estimates.

Control Problem (P).
The following control problem is considered in this paper (P) where the state y u associated to the control u is the solution of the Dirichlet problem (2.1) −∆y + a(x, y) = 0 in Ω, y = u on Γ.
The following hypotheses are assumed in the whole paper.(A1) Ω is an open, convex and bounded domain in R 2 , with the boundary Γ of class C 2 .Moreover we assume that N > 0 and −∞ < α < β < +∞.
(A2) L : Ω × R −→ R and a : Ω × R −→ R are Carathéodory functions of class C 2 with respect to the second variable, L(•, 0) ∈ L 1 (Ω), a(•, 0) ∈ L p(Ω), for some 2 ≤ p < +∞.Furthermore, for every M > 0 there exist a constant C L,M > 0 and a function ψ L,M ∈ L p(Ω) such that for almost all x ∈ Ω and all |y|, |y i | ≤ M , i = 1, 2, the following inequalities hold We also assume We say that an element y u ∈ L ∞ (Ω) is a solution of (2.1) if the following integral identity is fulfilled (2.4) where ∂ ν denotes the normal derivative on the boundary Γ.This is the classical definition of a weak solution by transposition.The following result proved by Casas and Raymond [5] is valid for any convex domain Ω.If the domain is not convex, then some smoothness of Γ is required, Γ of class C 1,1 is enough.Theorem 2.1.For every u ∈ L ∞ (Γ) the state equation (2.1) has a unique solution y u ∈ L ∞ (Ω) ∩ H 1/2 (Ω).Moreover, the following Lipschitz properties hold Finally, if u n ⇁ u weakly ⋆ in L ∞ (Γ), then y un → y u strongly in L r (Ω) for all r < +∞.Under the assumptions (A1) and (A2), it can be shown by standard arguments that problem (P) has at least one solution.Since (P) is not convex we cannot expect any uniqueness of solutions.Moreover, (P) may have some local solutions.We formulate the optimality conditions satisfied by such local solutions.To this end, we analyze the differentiability of the cost functional J.
Under the assumption (A2), (2.6) where y u is the state associated to u and ϕ u ∈ H 2 (Ω) is the unique solution of the problem Furthermore, we have (2.8) where Using (2.6) we obtain the necessary optimality conditions for (P).Theorem 2.2.Let ū be a local minimum of (P).Then ū ∈ W 1−1/ p, p(Γ) and there exist elements ȳ ∈ W 1, p(Ω) and φ ∈ W 2, p(Ω) such that The proof of theorem is given in [5].In order to establish the second order optimality conditions we define the cone of critical directions Now we formulate the second order necessary and sufficient optimality conditions.Theorem 2.3.If ū is a local solution of (P), then J ′′ (ū)v 2 ≥ 0 holds for all v ∈ C ū. Conversely, if ū is an admissible control for problem (P) satisfying the first order optimality conditions given in Theorem 2.2 and the coercivity condition then there exist δ > 0 and ρ > 0 such that for all u such that α ≤ u ≤ β and u − ū L 2 (Γ) ≤ ρ.
Proof.The necessary condition is easy to obtain.The inequality (2.15) is strong when compared with the corresponding inequality of [5].Indeed, here we claim that (2.14) implies that ū is a strict local minimum of (P) in the sense of the L 2 (Γ) topology.In [5] it is shown that condition (2.14) leads to the strict local optimality of ū in the sense of the L ∞ (Γ) topology.A more general result is proved in [2] for a distributed control problem, but in such a case once again only the local optimality in the sense of the L ∞ (Ω) topology is shown.Here we can improve the results because the control appears in a quadratic form within the cost functional.Let us see the precise arguments.
We proceed by contradiction.Let us assume that there is no pair (δ, ρ), with ρ, δ > 0, such that (2.15) holds.Then for every integer k, there exists a feasible control of (P), Let us define (2.17) By taking a subsequence, if necessary, there exists v ∈ L 2 (Γ) such that v k ⇀ v weakly in L 2 (Γ).The proof is divided into three steps: first, we prove that v ∈ C ū, then we deduce that v = 0 and finally we get the contradiction.
Step 2. v = 0. Using again (2.16) we obtain the last inequality being a consequence of (2.12) Once again we denote by y k and ϕ k the state and adjoint state evaluated for ū + . Also we define z k and z v as the elements of H 1/2 (Ω) satisfying . Now, recalling the expression of the second derivative of J given in (2.8) we get Passing to the limit in this expression and using (2.22) we obtain Step 3. Final Contradiction.Using two facts, v k ⇀ v = 0 and v k L 2 (Γ) = 1, we deduce from (2.22) and (2.25) the following contradiction We conclude this section with the following result that provides an equivalent formulation of (2.14), which is more useful for our purposes.
Theorem 2.4.Let ū be a feasible control of problem (P) satisfying the first order optimality conditions (2.10)-(2.12).Then the condition (2.14) holds if and only if where Proof.Since C ū ⊂ C ϑ ū for any ϑ > 0, it is obvious that (2.27) implies (2.14).Let us prove the reciprocal implication.We proceed again by contradiction.We assume that (2.14) holds, but there is no pair of positive numbers (µ, ϑ) such that (2.27) is fulfilled.Then for every integer k there exists and element Dividing v k by its norm and denoting the quotient by v k again, and taking a subsequence if necessary, we have that (2.28) Arguing as in the proof of Theorem 2.3, we obtain that v satisfies (2.13).On the other hand, from the fact that v k ∈ C 1/k ū and denoting by Γ k the subset of Γ formed by those points This inequality and the fact that v satisfies (2.13) imply that v vanishes whenever But from (2.28) we deduce that Consequently we have that v ≡ 0. However, if we argue as in the proof of Theorem 2.3, we have that 0 < N ≤ lim inf k→∞ J ′′ (ū)v 2 k ≤ 0, which is a contradiction.

Control Problem (P h
).Now we define Ω h .We follow the notation introduced in [6, Section 4].Given a set of points {x j } N (h) j=1 ⊂ Γ, we put where x N (h)+1 = x 1 .Γ h is the polygonal line defined by the nodes {x j } j=1 and Ω h is the polygon delimited by Γ h .Since Ω is convex, then Ω h ⊂ Ω.Now, for every 1 ≤ j ≤ N (h), we denote by x j x j+1 the arc of Γ delimited by the points x j and x j+1 .Let us define ψ j : [0, h j ] −→ x j x j+1 ⊂ Γ by where ν j represents the unit outward normal vector to Ω h on the boundary edge (x j , x j+1 ) and Now, we define we define the one-to-one mapping g h : Γ h −→ Γ in the following way For every point x ∈ Γ, ν(x) denotes the unit outward normal vector to Γ at the point x.By τ (x) is denoted the unit tangent vector to Γ at the point x such that {τ (x), ν(x)} is a direct reference system in R 2 .For each point x ∈ Γ h the corresponding reference system is denoted by {τ h (x), ν h (x)}.If x ∈ (x j , x j+1 ) then ν h (x) = ν j and τ h (x) = τ j .The following relations are proved in [6] (3.1) max{|τ (g and In the domain Ω h we define the problem (P h ) as follows where y h,u is the solution of the problem Theorem 2.1 can be applied to (3.5) to get the existence and uniqueness of a solution Moreover, inequalities (2.5) hold.(P h ) has at least one global solution and possibly there are some other local solutions of (P h ).For each local solution we have the first order optimality conditions analogous to the conditions in Theorem 2.2.Theorem 3.1.Let ūh be a local minimum of (P h ).Then ūh ∈ H 1/2 (Γ h ) and there exist elements ȳh ∈ H 1 (Ω h ) and φh ∈ H 2 (Ω h ) such that We observe that ūh is less regular than ū.The same is true for ȳh and φh with respect to ȳ and φ.The reason of the lost of regularity is the lack of regularity of Γ h .Γ is of class C 2 and consequently we can deduce the W 2, p(Ω) regularity of φ (see, for instance, Grisvard [8]), which leads to the W 1−1/ p(Γ) regularity of ū and consequently to the W 1, p(Ω) regularity of ȳ.Using the results for polygonal domains of [8], we can establish W 2,p (Ω) regularity of φh for some 2 < p ≤ p (assuming p > 2), with p depending on the angles of Ω h .The point is that p → 2 if the maximal angle of Ω h tends to π.This is exactly the case for h → 0, therefore we cannot deduce the boundedness of { φh W 2,p (Ω h ) } h>0 for any p > 2.
4. Convergence Analysis.In this section we prove the convergence of the local or global solutions of (P h ) to the solutions of (P) with h → 0. To prove the convergence, first we establish the convergence of the solutions of the state and adjoint state equations.
) be the corresponding solutions of (2.1) and (3.5), respectively.Then there exists a constant C M > 0 independent of h such that Proof.Let us take From (2.5) and (3.2) we get Let us estimate φ h = y u − y h .By substraction of the equations satisfied by y u and y h and using the mean value theorem, we get (4.5) where w h = y h + θ h (y h,u h − y h ) and 0 < θ h < 1.Now we have Finally, by using the inequality (see Bramble and King [1, Lemma 1]) we conclude This inequality along with (4.4) proves (4.2).Now we proceed with the analysis of the adjoint state equation.Let ϕ u ∈ H 2 (Ω) and ϕ h,u h ∈ H 2 (Ω h ) be given as the solutions of the equations Then we have the following estimate.Theorem 4.2.Let (u, y u ) and (u h , y h,u h ) be as in Theorem 4.1.Let ϕ u ∈ H 2 (Ω) and ϕ h,u h ∈ H 2 (Ω h ) be the corresponding solutions of (4.7) and (4.8), respectively.Then there exists a constant C M > 0, independent of h, such that the following estimate holds (4.9) Proof.Let us define φ h = ϕ u − ϕ h,u h ∈ H 2 (Ω h ).From (4.7) and (4.8) we get (4.10) From assumption (A2), taking into account that y u and y h,u h are bounded and using (4.2), we get (see Kenig [10]) Let us estimate ϕ u in H 1 (Γ h ).The norm in H 1 (Γ h ) is given by , where ∂ τ h ϕ u (x) = ∇ϕ u (x) • τ h (x), τ h (x) being the unit tangent vector to Γ h at the point x; see §3.The estimate of the first term of the norm follows easily from (4.6) and the fact that Now the L 2 (Γ) norm of the tangential derivative is estimated.To this end we observe that ϕ u = 0 on Γ, therefore ∂ τ ϕ u = 0 on Γ as well.Thus, we also have This along with (4.6) and (3.1) leads to Finally, (4.9) follows from (4.11), (4.12) and (4.13).Corollary 4.3.Under the assumptions of Theorem 4.2, the following inequality holds for some see [?] and [10].From this inequality, Assumption (A2), estimates (4.2), (4.9), (4.10) and (4.11) we get We complete this section by proving that the family of problems (P h ) realizes a correct approximation of (P).More precisely we prove that the solutions of problems (P h ) converge to the solutions of (P).Reciprocally, we also prove that any strict local solution of (P) can be approximated by a sequence of local solutions of problems (P h ).
Theorem 4.4.Let ūh be a solution of problem (P h ).Then {ū h • g −1 h } h>0 is a bounded family in H 1/2 (Γ).If ū is a weak limit for a subsequence, still denoted in the same way, ūh where ȳ and ȳh denote the solutions of (2.1) and (3.5) corresponding to ū and ūh , respectively.
Proof.First of all we recall definition of norm Let us estimate each of two integrals.In Remark 3.2, we establish the boundedness of { ūh By the change of variables in the second integral of (4.15), in view of (3.4), we get Therefore, Now, we assume that x ∈ [x j , x j+1 ] and On the other hand, Analogously, we can prove that Finally using (4.18) we obtain Using this inequality in (4.17) we conclude that From (4.15), (4.16) and (4.19) it follows Therefore, there exists a subsequence and an element ū denote by ȳh the states associated to ūh and by ȳ the state associated to ū, we deduce from (4.2) that lim Hence, it is easy to prove that J h (ū h ) → J(ū).It remains to prove that ū is a solution of (P).Let us take any feasible control u for (P), then u • g h is also feasible for (P h ).Therefore, since ūh is a solution of (P h ), we obtain which completes the proof.Theorem 4.5.Let ū be a strict local minimum of (P), then there exists a family {ū h } such that each control ūh is a local minimum of (P h ) and ūh Proof.Let ε > 0 be such that ū is the unique global solution of problem Now, for every h we consider the problems It is obvious that ū • g h is a feasible control for each problem (P hε ), therefore there exists at least one solution u hε of (P hε ).Let us show that u hε • g −1 h ⇀ ū weakly in H 1/2 (Γ) with h → 0. Since , we can extract a subsequence, still denoted by the same symbol, and an element ũ the state associated to u hε and consider an extension of y hε to Ω, still denoted by y hε , such that . Therefore, by taking a subsequence, we can assume that We are going to prove that ỹ is the state associated to ũ.According to the definition given in §2, we have to prove that the following identity holds (4.20) For a given w ∈ H 2 (Ω) ∩ H 1 0 (Ω) we take As in the proof of Theorem 4.2 we have Hence Since y hε is the state associated to u hε we have In view of (4.21), this identity can be rewritten as follows (4.24) Now we want to pass to the limit with h → 0 in (4.24).Using the compactness of the imbedding H 1/2 (Ω) ⊂ L 2 (Ω) it is easy to pass to the limit in the first two integrals, which are also the first two integrals of (4.20).Let us consider the right-hand side term of (4.24).Applying (4.23) we get (4.25) Now from Lemma 4.6 below we deduce (4.26) Finally, combining (4.25) and (4.26) we get Thus, we show that (4.20) follows from (4.24) by the limit passage.Now, using that u hε • g −1 h ⇀ ũ weakly in L 2 (Γ), y hε → ỹ strongly in L 2 (Ω), {y hε } h>0 is bounded in L ∞ (Ω) and the fact that u hε is a solution of (P hε ) and ū • g −1 h is feasible for problems (P hε ) we obtain Since ū is the unique solution of (P ε ), the above inequality leads to ũ = ū and J h (u hε ) → J(ū), which implies This identity and the weak convergence imply the strong convergence u hε • g −1 h → ū in L 2 (Γ).First consequence of this strong convergence is that the constraint u • g −1 h − ū L 2 (Γ) ≤ ε is not active at the controls u hε for h small enough.Therefore, u hε is a local minimum of problem (P h ) for every h small enough.Since { u hε L 2 (Γ h ) } is bounded, then we can argue as in the proof of Theorem 4.4 and conclude that Lemma 4.6.Let w ∈ H 2 (Ω) and v ∈ L 2 (Γ), then there exists a constant C > 0 independent of w and v such that Proof.First, we observe that (3.3) implies that On the other hand, From this identity we get, in view of (3.1), (3.2) and (4.6), Now, (4.28) and (4.29) imply (4.27).

Error Estimates.
In this section we assume that ūh is a local minimum of (P h ) such that ūh • g −1 h converges weakly in H 1/2 (Γ) to a local minimum ū of (P) with h → 0; see Theorems 4.4 and 4.5.The goal of this section is to derive an estimate for ū − ūh • g −1 h L 2 (Γ) , which is established in the following theorem.Theorem 5.1.Let ū and ūh be as above and let us denote by ȳ, ȳh and φ, φh the states and adjoint states associated to ū and ūh respectively.Let us assume that the second order sufficient optimality condition (2.14) is fulfilled for ū.Then there exists a constant C, independent of h such that the following estimates hold Before proving this theorem we provide a preliminary result.The proof of Lemma 5.2 is inspired by [5, Lemma 7.2], however there are some important differences.
Lemma 5.2.Let µ > 0 be taken from Theorem 2.4.Then there exists h 0 > 0 such that Proof.By applying the mean value theorem there is an intermediate element Let us take Taking a subsequence, if necessary, we can assume that v h ⇀ v weakly in L 2 (Γ).We show that v belongs to the critical cone C ū defined in §2.First of all, observe that v satisfies the sign condition (2.13) since every element v h satisfies the same condition.Let us prove that v(x) = 0 if N ū(x)−∂ ν φ(x) = 0. To this end it is enough to establish the limit passage Indeed, from (5.4) we deduce, in view of (3.8), that which proves the required property.Let us show (5.4).By the strong convergence ūh • g −1 h → ū in L 2 (Γ) combined with (4.14) and (3.2), we have On the other hand, from Lemma 4.6 we get (5.6) Finally, from (3.3) we obtain (5.7) Thus, (5.4) follows from (5.5), (5.6) Taking into account that v L 2 (Γ) ≤ 1, the above inequality leads to lim h→0 J ′′ (û h )v 2 h ≥ min{µ, N } > 0, which proves the existence of h 0 > 0 such that From this inequality, by the definition of v h and (5.3), we deduce (5.2), which completes the proof.