STATE-CONSTRAINED SEMILINEAR ELLIPTIC OPTIMIZATION PROBLEMS WITH UNRESTRICTED SPARSE CONTROLS

. In this paper, we consider optimal control problems associated with semilinear elliptic equation equations, where the states are subject to pointwise constraints but there are no explicit constraints on the controls. A term is included in the cost functional promoting the sparsity of the optimal control. We prove existence of optimal controls and derive ﬁrst and second order optimality conditions. In addition, we establish some regularity results for the optimal controls and the associated adjoint states and Lagrange multipliers.


(Communicated by Enrique Zuazua)
Dedicated to Prof. Dr. Fréderic Bonnans on the occasion of his 60th birthday Abstract. In this paper, we consider optimal control problems associated with semilinear elliptic equation equations, where the states are subject to pointwise constraints but there are no explicit constraints on the controls. A term is included in the cost functional promoting the sparsity of the optimal control. We prove existence of optimal controls and derive first and second order optimality conditions. In addition, we establish some regularity results for the optimal controls and the associated adjoint states and Lagrange multipliers.
1. Introduction. In this paper, we analyze the following optimal control problem (P) min u∈U ad J(u) with J(u) = F (u) + κj(u), κ > 0, where y d ∈ L 2 (Ω) and ν > 0 are given. We define with some γ > 0, where y u is the solution of the semilinear elliptic partial differential equation Ay + a(x, y(x)) = u in Ω, y = 0 on Γ. (1.1) Above, Ω ⊂ R n , n = 2 or 3, is a bounded open set with a Lipschitz boundary Γ. For the differential operator A we assume that and ∃Λ > 0 such that Λ|ξ| 2 ≤ n i,j=1 a ij (x)ξ i ξ j ∀ξ ∈ R n and for a.a. x ∈ Ω. (1.3) We also assume that a : Ω × R −→ R is a Carathéodory function of class C 2 with respect to the second variable satisfying a(·, 0) ∈ L 2 (Ω), 0 ≤ ∂a ∂y (x, y) ∀y ∈ R, for almost all x ∈ Ω. There are many papers devoted to the analysis of control problems with pointwise state constraints. As far as we know, control constraints are also included in all of them, except for a few papers dealing with error estimates for the numerical approximation; see [10], [14], [17]. Of course, the analysis is more involved when both type of constraints are present. However, stronger results can be proved if there is no explicit constraint on the controls. In this paper, we want to show these results that are not available under explicit control constraints.
After a second section, where we present some preliminary results, we prove first order optimality conditions for local solutions in §3. A linearized Slater assumption is usually made to derive the first order optimality conditions in a qualified form. However, for our control problem we prove that this condition is automatically fulfilled, it does not have to be assumed. An additional difficulty to get the optimality conditions is the presence of the non-differentiable term j(u) in the cost functional that promotes the sparsity of the optimal control. This difficulty was overcome in [11] by using an abstract result of [3,Theorem 2.1]. In this paper, we provide a new abstract result under less restrictive assumptions; see Theorem 3.2. The presence of j in the cost functional promotes the sparsity of the optimal control; see Corollary 3.5. Actually, the size of the support of the optimal control can be monitored by the sparse parameter κ.
Assuming that the set of points where the state constraint is active has a zero Lebesgue measure, we prove the uniqueness of the Lagrange multiplier associated to the state constraints. This assumption is typically satisfied. Indeed, the set of points x such thatȳ(x) = γ, whereȳ denotes an optimal state, is most of the times reduced to a finite number of points or (most frequently, see Theorem 3.8) it defines a line if n = 2 or a surface if n = 3. As a conclusion, we obtain also the uniqueness of the adjoint stateφ and the multiplierλ corresponding to the non-differentiable term j(u). Finally, under a very weak assumption on the nonlinear term of the state equation, we prove that the adjoint state belongs to L ∞ (Ω) ∩ H 1 0 (Ω). This regularity is transferred to the optimal control. All these results are presented in Section 3.
In §4 we derive second order conditions for local optimality. We also prove that the different notions of local solution are equivalent for our control problem. Furthermore, we obtain that the usual quadratic growth inequality for local solutions is satisfied in an L 2 (Ω)-neighborhood of an optimal control if and only if it is satisfied in an L ∞ (Ω)-neighborhood of the optimal state.
2. Preliminary results. In this section we analyze the state equation and the cost functional J. The results are already known, but we recall them to fix the notation and for convenience of the reader. We start with a well known theorem on existence, uniqueness and regularity of the solution of (1.1).
By a straightforward application of the chain rule we deduce the differentiability of the functional F . Recall that F is the first summand of the cost functional J. (2.6) Above, A * denotes the adjoint operator of A given by We finish this section by recalling some known properties of the functional j. Obviously, j is convex and Lipschitz. A simple computation shows that λ ∈ ∂j(u) if and only if holds a.e. in Ω. Further, j has directional derivatives given by for u, v ∈ L 1 (Ω), where Ω + u , Ω − u and Ω 0 u represent the sets of points where u is positive, negative or zero, respectively. Finally, the following relation holds We refer to Clarke [ 3. Existence of optimal controls and first order optimality conditions. The goal of this section is to prove the first order optimality conditions satisfied by any local solution of (P) and to deduce some important conclusions from the optimality system. First we observe that the classical approach of taking a minimizing sequence of (P) and to deduce its boundedness in L 2 (Ω), which follows from the structure of the cost functional, produces weak limits that are solutions of (P). It is enough to use Theorem 2.1 in the last step. Hence, we have the following existence theorem.
Theorem 3.1. Under the hypotheses (1.2)-(1.5) and assuming that U ad = ∅, the control problem (P) has at least one solutionū.
In the sequel,ū will denote a local solution of (P). Before proving the first order optimality conditions satisfied byū, we establish two preliminary results. The first result is concerned with the optimality system for an abstract optimization problem that covers (P) as a particular case.
Let U and Y be two topological vector spaces and K ⊂ U and C ⊂ Y two convex sets. Given the mappings G : The next theorem provides the optimality conditions satisfied by any local solution of (Q). The reader is referred to [3] for a related result.
Theorem 3.2. Letū be a local solution of (Q). Assume that f and G are Gâteaux differentiable atū, g is convex and continuous at some point of K, and int C = ∅. Then there exist a real numberμ 0 ≥ 0, a multiplierμ ∈ Y * , andλ ∈ ∂g(ū) such that Moreover, if the linearized Slater condition is satisfied, then (3.3) holds withμ 0 = 1.

Proof. Let us define the sets
and A and B are convex sets and B is open. Moreover, we have From here we infer the existence of a number ρ 0 ∈ (0, 1) such that ∀0 < ρ < ρ 0 and u ρ =ū + ρ(u 0 −ū) Then we have and with the convexity of g The above inequalities imply f (u ρ ) + g(u ρ ) < f (ū) + g(ū) for every 0 < ρ < ρ 0 .
Since we have u ρ ∈ K and G(u ρ ) ∈ C for every 0 < ρ < ρ 0 , this contradicts the local optimality ofū. Therefore, we can separate the sets A and B (see, for instance, [5, pp. 5-7]) by a continuous linear form (μ,μ 0 ) ∈ Y * × R: From the strict inequality, (3.1) follows. Taking y 1 = G(ū) and t 1 > 0 arbitrarily large, and fixing an element (y 2 , t 2 ) ∈ B, (3.5) yields thatμ 0 ≥ 0. Now, by density of B in C × (−∞, 0] and continuity ofμ, we deduce from (3.5) This implies thatū is solution of the optimization problem where I K denotes the indicator function of K. Since g is continuous at some point of K, we can apply the subdifferential calculus to obtain Hence, there exists an elementλ ∈ ∂g(ū) such that (3.3) holds. Finally, we assume the linearized Slater condition and prove thatμ 0 > 0, then we take 1 µ0μ as new Lagrange multiplier. Renaming this multiplier byμ, we conclude the proof. To show thatμ 0 > 0, we argue by contradiction. Assume thatμ 0 = 0. Inserting This implies thatμ = 0, contradicting (3.1).
Before establishing the optimality system (3.1)-(3.3) for our control problem (P), we prove that the linearized Slater condition is satisfied by any local solutionū of (P). We say thatū is a local solution or a local minimizer of (P) if there exists ε > 0 such that The reader is referred to Section 4, Definition 4.1, Theorem 4.2 and Remark 4.3 for additional comments on this definition. Theorem 3.3. Letū be a local solution of (P) with associated stateȳ and assume that (1.2)-(1.5) hold. Then the linearized Slater condition Proof. Let us set v = Aȳ + ∂a ∂y (x,ȳ(x))ȳ. Obviously we have that v ∈ L 2 (Ω). Fix ρ ∈ (0, 1) and take u 0 =ū − ρv. Then we have z u0−ū = −ρȳ. Indeed, it is enough to check that z u0−ū and −ρȳ satisfy the same equation, namely Az + ∂a ∂y (x,ȳ(x))z = −ρv. Therefore, it follows Now, the optimality system satisfied byū follows.
From this theorem, we get the following properties.
Corollary 3.5. Under the assumptions of Theorem 3.4, the following properties hold (1)λ andφ are related by the formulā Let y 0 be the state associated to the null control. We assume that y 0 belongs to the open ball B γ (0) ⊂ C 0 (Ω). Then, there exists κ 0 > 0 such thatū ≡ 0 is the unique solution of (P) for every κ ≥ κ 0 .
Next we analyze the uniqueness of the Lagrange multiplierμ. If y 0 ∈ B γ (0), as assumed in Corollary 3.5, (5) and (6), andū ≡ 0, thenμ = 0 is obviously the only Lagrange multiplier associated withū. In the next theorem, we consider the case of a non zero locally optimal control. First, we introduce some notation. We assume thatȳ is the state associated withū and K γ = {x ∈ Ω : |ȳ(x)| = γ} denotes the set of points where the state constraint is active. Because of the continuity ofȳ and the boundedness of Ω, the set K γ is compact.
Theorem 3.7. Let the assumptions (1.2)-(1.5) be fulfilled and suppose that a ij ∈ C 0,1 (Ω) for 1 ≤ i, j ≤ n,ū is a nonzero local minimizer of (P), and K γ has a zero Lebesgue measure. Then the Lagrange multiplierμ satisfying the optimality conditions (3.9)-(3.11) is unique. As a consequence,φ andλ are unique as well.
Proof. We introduce the set whereφ is the adjoint state corresponding toū. We observe that equation (3.9) implies thatφ is continuous in the open set Ω \ K γ ; see [20,Theorem 9.3]. Hence, Ωū is also an open set. The identity (3.12) implies thatλ ∈ C(Ω \ K γ ) as well. Finally, we get from (3.11) thatū is also continuous in Ω \ K γ . Moreover, according to Corollary 3.5-(3), we have thatū(x) = 0 holds for all x ∈ Ωū. Now, we define the linear operator where the elements v ∈ L 2 (Ωū) are extended by 0 to Ω and z v = G (ū)v is the solution of (2.2) for u =ū. The remaining proof is split into two parts.

4.
Second order optimality conditions. The goal of this section is to set up sufficient second order optimality conditions for a local solution of (P). First, let us define the notion of local solution depending of the selected topology.
Definition 4.1. We say thatū is an L p (Ω)-weak local solution of (P), p ∈ [1, +∞], if there exists some ε > 0 such that We say thatū is a strong local solution if there exists some ε > 0 such that We say thatū is a strict (weak or strong) local solution if the above inequalities are strict for u =ū.
As far as we know, the concept of strong local solution was introduced for the first time in the framework of control of partial differential equations in [1]; see also [2]. We argue by contradiction. Ifū is not an L 1 (Ω)-weak local solution, then there exists a sequence {u k } ∞ k=1 ⊂ U ad such that From the first inequality we infer This estimate and the second inequality of (4.4) imply the existence of a subsequence, denoted in the same way, such that u k ū in L 2 (Ω). We get with Theorem 2.1 and the weak lower semicontinuity of the last two terms of J This implies the convergence J(u k ) → J(ū). Since y k = y u k →ȳ in L ∞ (Ω), we deduce From (4.5) and (4.6) it follows hence we have the convergence of the norms u k L 2 (Ω) → ū L 2 (Ω) . This convergence and the weak convergence u k ū in L 2 (Ω) are equivalent to the strong convergence u k →ū in L 2 (Ω). Therefore, there exists k ε such that that u k −ū L 2 (Ω) ≤ ε for all k ≥ k ε . Hence, (4.3) implies that J(ū) ≤ J(u k ) for all k ≥ k ε , wich contradicts (4.4).
Then, there exists a constant C 2 independent of u such that see [20, §4]. Next we select ε = ε/C 2 and deduce with (4.2) Henceū is a local solution of (P). Then, 1 and 2 imply thatū is an L p (Ω)-weak local solution of (P) for all p ∈ [1, ∞].
Proof of 4.-Sinceū is an L 2 (Ω)-weak local solution, (4.3) holds for some ε > 0. We argue again by contradiction. Ifū is not a strong local solution, then there exists a sequence {u k } ∞ k=1 ⊂ U ad such that where y k is the state associated with u k . Now we can argue as in the proof of (1) and to deduce thatū k →ū strongly in L 2 (Ω). Hence, (4.3) implies that J(ū) ≤ J(u k ) for all sufficiently large k, which contradicts (4.8) Remark 4.3. Let us discuss a few consequences of this result, first for the problem (P) posed here, where control constraints are not given and the feasible set is not bounded in general. Observe that we assumed ν > 0 from the very beginning.
Since J(u) < ∞ if and only if u ∈ L 2 (Ω), this is the natural space to study the control problem (P). In L 2 (Ω), · L p (Ω) is a norm if and only if p ∈ [1,2]. Hence, the natural concepts for a local solution of (P) are the L p (Ω)-weak local solutions with p ∈ [1, 2] and strong local solutions. However -for p ∈ [1, 2] -all these concepts are equivalent as we deduce from Theorem 4.2.
Let us complete this observation by discussing the case of control problems with pointwise control constraints and L ∞ -bounds, where the set of admissible controls is bounded in L ∞ (Ω). Here, all the norms · L p (Ω) with p ∈ [1, ∞] are reasonable. Hence, it makes sense to consider L p (Ω)-weak local solutions and strong local solutions for arbitrary p ∈ [1, ∞]. We can distinct between two cases: Case ν = 0. In this case, to assure the existence of a solution, we have to assume some constraints on the controls. If the set of feasible controls is bounded in L ∞ (Ω), then -a priori -there are three different type of local solutions: L ∞ (Ω)-weak local solutions, L p (Ω)-weak local solutions with 1 ≤ p < ∞, and strong local solutions. Since all the topologies induced in U ad for the L p (Ω) norms with 1 ≤ p < ∞ are equivalent in an L ∞ -bounded set, the concepts of local solutions in the L p (Ω) sense are equivalent for p ∈ [1, ∞). Once again arguing as in the proof of Theorem 4.2, we are able to deduce that any strong local solution is an L p (Ω)-weak local solution for every p ∈ [1, ∞], and every L p (Ω)-weak local solution is an L ∞ (Ω)-weak local solution, but the converses are maybe false: We cannot deduce that an L p (Ω)-weak local solution is a strong local solution or that an L ∞ (Ω)-weak local solution is an L p (Ω)-weak local solution for p < ∞.
In order to establish a sufficient second order condition for local optimality, we introduce the cone of critical directions. Thus, givenū ∈ U ad satisfying the first order optimality conditions (3.9)-(3.11), we define where z v = G (ū)v is the solution of (2.2) for u =ū. |μ| =μ + +μ − is the total variation measure associated withμ, and L : L 2 (Ω)×M (Ω) −→ R is the Lagrangian function defined by From Theorems 2.3 and 2.4, we deduce that L is of class C 2 and ∂L ∂u where ϕ u ∈ W 1,p 0 (Ω), for all p < n n−1 , is the solution of    A * ϕ + ∂a ∂y (x, y u )ϕ = y u − y d + µ in Ω, ϕ = 0 on Γ. Moreover, from (2.9) we also infer ∂L ∂u The next theorem provides sufficient second order conditions for local optimality.
Proof. This proof is inspired in [7] and [11]. We argue by contradiction. Suppose thatū does not satisfy the quadratic growth condition (4.19). Then there exists a sequence {u k } ∞ k=1 ⊂ U ad such that Let us take Since v k L 2 (Ω) = 1, we can extract a subsequence, denoted in the same way, such that v k v weakly in L 2 (Ω). Now we split the proof into several steps.
2. There exists ε p > 0 such that J(ū) + δ 2 u −ū 2 L 2 (Ω) ≤ J(u) ∀u ∈ U ad : u −ū L p (Ω) ≤ ε p . (4.30) 3. There exists ε ∞ > 0 such that Proof. Let us consider the optimization problem Notice that J δ (u) is obtained from J by subtracting δ 2 ( u −ū 2 L 2 (Ω) − ū 2 L 2 (Ω) ). The statements of the corollary are equivalent to the claims thatū is an L 2 (Ω)weak local solution of (P δ ), an L p (Ω)-weak local solution of (P δ ), and a strong local solution of (P δ ), respectively. But, from Theorem 4.2, we know that these three notions are equivalent for p ∈ [1, 2). Indeed, the last term of J δ is linear and continuous with respect to u and we have that ν − δ > 0. Consequently, the proof of Theorem 4.2 is exactly the same if we replace the cost functional J by J δ .