SECOND-ORDER AND STABILITY ANALYSIS FOR STATE-CONSTRAINED ELLIPTIC OPTIMAL CONTROL PROBLEMS WITH SPARSE CONTROLS

An optimal control problem for a semilinear elliptic partial differential equation is discussed subject to pointwise control constraints on the control and the state. The main novelty of the paper is the presence of the L1−norm of the control as part of the objective functional that eventually leads to sparsity of the optimal control functions. Second-order sufficient optimality conditions are analyzed. They are applied to show the convergence of optimal solutions for vanishing L2-regularization parameter for the control. The associated convergence rate is estimated. AMS subject classifications. 90C48, 49K20, 35J61

Thanks to the presence of the L 1 -norm in the objective functional, a convex but not differentiable functional is to be minimized.This term accounts for sparsity of optimal controls: With increasing parameter κ, the support of the optimal controls shrinks to finally have the measure zero for sufficiently large κ.

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Recently, such effects of sparsity attracted increasing interest.The first paper addressing this issue in the context of PDE control was [22], where the case of a linear elliptic equation was considered with the focus on numerical aspects.The associated analysis was extended in [25].Later, the control of semilinear elliptic equations was investigated in [7] and [8].For parabolic equations, sparsity can occur in different ways with respect to time and space variables.In particular, striped sparsity patterns (directional sparsity) can be obtained by suitable formulations of the cost functional.This type of sparsity was introduced in [19] in the context of linear elliptic and parabolic equations.Sparsity in time and space turned out to be important for (nonlinear) reaction-diffusion equations of Schlögl and FitzHugh-Nagumo type; cf.[12].By admitting measures as controls instead of integrable control functions, the support of the optimal control can even have a zero Lebesgue measure; see [5], [6], and [14].
The main novelty of our contribution is the study of sparsity under the presence of pointwise state constraints, in particular the discussion of second-order sufficient optimality conditions and the convergence analysis with respect to a vanishing regularization parameter ν.Second-order conditions were investigated for problems without sparsity in many papers; see the references in [24, p. 352].For sparse control problems, the nondifferentiability of the objective functional causes new difficulties.This was addressed in [7] for the case without state constraints.Complementing the control constraints by state constraints in our paper further increases the level of difficulty.Moreover, our results even extend the whole theory of second-order conditions for state constrained problems.In particular, we allow for a vanishing Tikhonov parameter and compare various formulations of second-order conditions known in the literature.
We also discuss the stability of optimal solutions of the problems (P ν ) for ν → 0. We show that, under a second-order sufficient optimality condition for (P 0 ), the optimal solutions of (P ν ) converge strongly, and we estimate the order of convergence.
The plan of our paper is as follows: After stating the main assumptions in section 2.1, we shall discuss the well-posedness of the optimal control problem and investigate the differentiability properties of the control-to-state mapping.In section 2.2, we derive first-order necessary optimality conditions in a fairly standard way, while section 3 is devoted to second-order sufficient optimality conditions.Here, we distinguish between the cases ν > 0 and ν = 0. Finally, we perform a convergence analysis for the optimal control problem when ν tends to zero.The second-order and convergence analysis of sections 3 and 4 constitute the main novelty of our paper, although the first-order necessary optimality conditions are also new.

Notation and main assumptions.
Let us fix right here the following sets: Moreover, to shorten the notation, we introduce for ν ≥ 0 the family of functionals Notice that j is convex and continuous but not differentiable.
On the state equation (1.2), we impose the following assumptions.Assumption 1.A is the linear operator with a ij ∈ L ∞ (Ω), and there exists some Λ > 0 such that n i,j=1 Assumption 2. a : Ω × R −→ R is a Carathéodory function of class C 2 with respect to the second variable, with a(•, 0) ∈ L p(Ω) for some p > n 2 , and satisfying Moreover, we assume some uniform continuity of ∂ 2 a ∂y 2 : For every M > 0 and ε > 0, there exists ρ ε,M > 0 such that We finish this section by recalling some known properties of the functional j.Since j is convex and Lipschitz, the subdifferential in the sense of convex analysis and the generalized gradients introduced by Clarke coincide.Moreover, a simple computation shows that λ ∈ ∂j(u) if and only if λ is measurable and satisfies (2.1) holds a.e. in Ω.Further, j has directional derivatives given by (2.2) , 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: Remark 2.1.By (P ν ) , we discuss a simplified version of the control problem for better readability.Our theory can be extended to more general formulations by obvious modifications.This includes the case of state constraints of the type γ 1 (x) ≤ y(x) ≤ γ 2 (x), where one of the equalities might be missing.Instead of assuming α < 0, sparsity can be also obtained for α = 0; see [12].Notice that in many applications only nonnegative controls are meaningful.Finally, the more general cost functional can be treated in an analogous way; see [7].

2.2.
Well-posedness of the optimal control problem and first-order optimality conditions.We start with known properties of the control-to-state mapping associated with the state equation (1.2).
Theorem 2.2.Under Assumptions 1 and 2, to each u ∈ L 2 (Ω) there exists a unique solution where the functions z vi are defined by (2.4) above.
The proof of the existence, uniqueness, and regularity of y u is well known; see [24, section 4.2] and the references therein.Let us show the differentiability of G.We set with q = min{2, p}.Endowed with the graph norm, V is a Banach space.Moreover, we deduce from [18,Theorem 8.30] that V is embedded in C( Ω).Now, we consider Due to Assumption 2, F is well defined.Moreover, it is easy to check that F is of class C 2 , F (y u , u) = (0, 0) for every u ∈ L 2 (Ω), and where . Indeed, by using a classical approach, see [23, Theorems 4.1 and 4.2], along with the monotonicity of a(x, y) with respect to y, we can get a uniform bound for {y u k } k in L ∞ (Ω).Then, [18,Theorem 8.29] shows the boundedness of {y u k } k in a space of Hölder functions C θ ( Ω) for some 0 < θ < 1.The compactness of the embedding , which allows us to conclude the strong convergence y u k → y u in H 1 0 (Ω).This property implies that u ∈ U ad if {u k } k ⊂ U ad .Hence, by the continuity and convexity of the integrals in (1.1) involving the control, we get the following existence result in a standard way.
Theorem 2.4.Let U ad be nonempty.Then, for every ν ≥ 0, problem (P ν ) has at least one optimal solution denoted by u ν .
Notice that (P ν ) is a nonconvex problem, and hence more than one optimal solution might exist for fixed ν.The assumption U ad = ∅ is satisfied in particular if a(x, 0) = 0 holds for a.a.x ∈ Ω; then u = 0 belongs to U ad .Throughout the paper, we use the notation y ν := y uν .
To establish first-order necessary optimality conditions of Karush-Kuhn-Tucker type, we assume the following linearized Slater condition.
Assumption 3 (linearized Slater condition).For given ν, there exists We shall prove later that this assumption is satisfied for all sufficiently small ν > 0 if it holds for ν = 0; see Theorem 4.3.
Let us now recall the first-order necessary optimality conditions for a given optimal solution u ν which are a straightforward consequence of an abstract result proved by Bonnans and Casas in [1, Theorem 2.1].
The next relations that we formulate for the cases ν > 0 and ν = 0 are immediate conclusions of (2.10) and (2.11).
where P [s,t] : R → R is the projection function on the interval [s, t].We refer to Casas, Herzog, and Wachsmuth [7].From (2.12) and (2.14) we deduce the following regularity results for u ν and λ ν .In what follows, we denote by M (Ω) the Banach space of all real and regular Borel measures on Ω.
In what follows, we shall use for short the notation Next, we prove the boundedness of the adjoint states, uniform with respect to ν and an extra regularity property of μ ν .The next theorem is inspired by a recent regularity result by Pieper and Vexler [21]; see also [9] for a posterior extension to the semilinear case.They considered the Poisson equation with measures as controls and observed that the optimal control enjoys H −1 (Ω) regularity.It became clear to us that we could extend this approach to analyze the regularity of the Lagrange multiplier μ ν and the corresponding adjoint state.The reader is also referred to [10], where the authors have obtained recently a similar result for more general pointwise state constraints and a linear state equation.Theorem 2.8.Let (u ν , y ν , ϕ ν , μ ν , λ ν ) satisfy the optimality system (2.7)-(2.11)with u ν ∈ U ad .Assume that α ≤ a(x, 0) ≤ β, a(x, +γ) > α, and a(x, −γ) < β hold a.e. in Ω.Then, there exists M ν > 0 such that Downloaded 07/09/14 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php We prepare the proof of this theorem by some auxiliary results.The next result permits one to invoke certain regularity properties for the Laplace operator in investigating the Dirichlet problem for A.
Lemma 2.9.Let μ ∈ M (Ω) be a positive measure with a compact support in Ω and g A be the Green's function corresponding to the Dirichlet problem associated with A+a 0 I, a 0 ≥ 0 belonging to L ∞ (Ω).Define y μ and z μ as the solutions to the problems Then, for every ε > 0 there exists a compact set K ε and a constant In particular, it holds that Then it holds that y * μ (x) = y μ (x) a.e. in Ω and Proof.First, let us observe that the function y * μ defined in the statement of the lemma is a particular function in the L 1 (Ω)-equivalence class of the solution y μ ; see, for instance, [23,Theorem 9.4] or [20].The same holds for z μ and the function z * μ defined by where g is the Green's function in Ω associated with the operator −Δ.Therefore, y * μ and z * μ are univocally defined at every point x ∈ Ω, possibly being infinite at some points.However, y μ and z μ are only defined almost everywhere.
Observe that the positivity of μ implies that y μ and z μ are nonnegative almost everywhere in Ω.Moreover, since Ay μ + a 0 y μ = Δz μ = 0 in the open set Ω \ supp(μ) and y μ = z μ = 0 on Γ, we deduce that y μ , z μ ∈ C( Ω \ supp(μ)).Therefore, given ε > 0 we can choose a compact set K ε such that supp(μ) ⊂ K ε ⊂ Ω and the second inequality of (2.22) holds.Let us prove the first inequality of (2.22).We know from [23, p. 252] that there exists a positive number C ε such that Integration with respect to μ and taking into account that μ ≥ 0 and supp(μ These inequalities imply the first inequality of (2.22).Downloaded 07/09/14 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpBy the first part of our proof, we have where the last equivalence is due to a result by Pieper and Vexler [21].

Second-order sufficient optimality conditions.
In order to perform the second-order analysis of the control problem (P ν ), we introduce the Lagrangian function From Theorems 2.2 and 2.3, we obtain that L is of class C 2 and According to (3.2), the variational inequality (2.11) can be written in the form Moreover, from (2.2) we also have In this section, u ν will denote an element of U ad satisfying with (y ν , ϕ ν , λ ν , μ ν ) the optimality system (2.7)-(2.11).Associated with u ν , we introduce the following cone of critical directions for every τ ≥ 0: In the case τ = 0, we simply write C uν instead of C 0 uν .As proved in [7, Lemma 3.5], if v ∈ L 2 (Ω) satisfies (3.8), then As a consequence of this, for τ = 0 we have In the second-order analysis, we will distinguish between two cases depending on whether the parameter ν is strictly positive or zero.Downloaded 07/09/14 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php3.1.Case ν > 0. The role played by the Tikhonov regularization term in the cost functional is crucial for the second-order analysis; see [11] for κ = 0. Indeed, surprisingly some formulations of the second-order sufficient optimality conditions are equivalent, which is not true for general optimization problems in infinite dimension.In particular, this is not true for ν = 0.The next theorem states these equivalent formulations.
Theorem 3.1.The following statements are equivalent: where , it is obvious that (3.17) implies (3.18) with the same τ and replacing σ by σ/C 2 in (3.18).The implication (3.18)⇒(3.16) is obvious.To prove (3.16)⇒(3.17),we proceed by contradiction.We assume that (3.16) holds but (3.17) is false.Then, for every integer k ≥ 1, there exists an element We divide v k by its L 2 (Ω)-norm and, selecting a subsequence if necessary, we obtain an element v ∈ L 2 (Ω) such that Let us prove that v ∈ C uν .First we observe that v satisfies (3.13) because every v k does it.Now, since the functional L 2 (Ω) v → j (u ν ; v) ∈ R is convex and continuous, and v k satisfies (3.7) with τ = 1/k, we can pass to the limit below, see (3.2), and deduce This inequality and (3.11) imply that (3.12) holds for v.Moreover, the weak convergence . Therefore, it is easy to pass to the limit in (3.9) and (3.10) for v k and τ = 1/k and to obtain (3.14) and (3.15).This completes the proof of v ∈ C uν .On the other hand, we can pass to the limit in (3.19), see (3.3), and get According to the assumption (3.16), this is only possible if v = 0.This implies that z v k → 0 strongly in C 0 (Ω).Hence, using again the expressions (3.3) and (3.19), it follows that Since ν > 0, we have a contradiction.Downloaded 07/09/14 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpTheorem 3.2.Assume that u ν satisfies (3.16).Then there exist δ > 0 and ε > 0 such that where B ε (u ν ) denotes the L 2 (Ω)-ball centered at u ν with radius ε.
Proof.Let us fix τ > 0 and σ > 0 such that (3.17) holds.We prove this theorem by contradiction and assume that there exists a sequence We shall show the existence of k τ > 0 such that there holds i.e., u k − u ν belongs to the critical cone for all sufficiently large k.For this purpose, we have to confirm the conditions (3.7)-(3.10).To verify (3.7), we observe first that (2.9) implies By a Taylor expansion, it follows from (3.21) and (2.3) that with some ϑ k ∈ (0, 1).This implies Next, let us verify condition (3.9).We have the equations Downloaded 07/09/14 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php We move the last two terms to the other side of (3.24) and use (3.23) to deduce This shows (3.10), and altogether we have shown u k − u ν ∈ C τ uν for k large enough.Next, we derive the contradiction to our initial hypotheses.We proved above that With (3.6) we obtain 1 2 We rewrite the left-hand side of this inequality and apply (3.17) to deduce 1 2 where y ϑ k and ϕ ϑ k denote the state and adjoint state associated with u ν +ϑ k (u k −u ν ), and To proceed with our estimation, we consider the following equations: Subtracting the two equations, we obtain with ŷϑ k being intermediate functions between y ϑ k and y ν .Therefore, it follows that This estimate yields for k > c Next, we estimate the term I defined in (3.26).To this aim, we first consider the term where we have used (3.27) and (3.28).For the other part of the integrand, we find where To further estimate the terms (3.30)-(3.32),we subtract the equations and get with some intermediate function ŷϑ Therefore, we can estimate ϕ ϑ k − ϕ ν by For deriving the last line, we have considered the equations Downloaded 07/09/14 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.php The estimates (3.28) and (3.33) imply Now we consider the terms II and III: From Assumption 2 and y k , y ν ∈ Y γ , we know that for every ε > 0 there exists ρ ε,γ > 0 such that Therefore, with (3.28) it holds that where the L ∞ -norm above tends to zero as k → ∞.
The estimate of III is an immediate consequence of (3.29).With all obtained estimates, we found where ε k → 0. This is only possible if u k = u ν holds for all sufficiently large k, which contradicts (3.21).

3.2.
Case ν = 0.In this section, the functions L 0 , J 0 , and F 0 are simply denoted by L, J, and F , respectively.Let ū ∈ U ad be a control that satisfies the first-order necessary optimality system together with (ȳ, φ, λ, μ).For ν = 0, the second-order conditions (3.16)- (3.18) are not equivalent.The issue is to find out if any of these three conditions is sufficient for local optimality of ū.The assumption (3.16) is too weak; see [17] for an example.The condition (3.17) is too strong and it is never fulfilled when ν = 0; see [4].The correct assumption is (3.18), as stated in the next theorem.
Theorem 3.3.If the second-order condition (3.18) is satisfied, then there exist ε > 0 and δ > 0 such that The proof of this theorem follows the lines of the proof of Theorem 3.2 just changing u k − ū by z u k −ū when necessary.For instance, (3.21) must be substituted by The inequality (3.25) has to be replaced by 1 2 The final contradiction admits the form σ . Downloaded 07/09/14 to 193.144.185.28.Redistribution subject to SIAM license or copyright; see http://www.siam.org/journals/ojsa.phpwhere Bε (ū) denotes the L 2 (Ω)-closed ball with center at ū and radius ε.Analogously, for every ν > 0 we consider the problems For every ν > 0 there exists at least one solution u ν of (P ν,ε ).We can apply Theorem 4.1 and deduce that, for a subsequence if necessary, {u ν } converges strongly in L 2 (Ω) to a solution of (P ε ).But the only solution of (P ε ) is ū, and hence the whole sequence {u ν } converges strongly to ū in L 2 (Ω).For ν sufficiently small, u ν − ū L 2 (Ω) < ε, and consequently u ν is a local solution of (P ν ) .Assumption 4 (linearized slater condition).There exists u s ∈ U α,β such that where In what follows, we write for short z ν,s := z ν,us−uν .Theorem 4.3.Under Assumption 4, there exists some ν s > 0 such that Proof.The sequence {y ν } ν converges to ȳ, uniformly in Ω.Therefore, it suffices to prove that z ν,s − z us−uν C0(Ω) → 0 for ν → 0.
Proof.The relations (4.4)-(4.7)constitute the necessary optimality conditions for the problem (P ν ), which follow from the linearized Slater condition.Therefore, it remains to show the convergence properties. (i) ] is obvious.The identity (2.14) shows the boundedness of {λ ν } in L ∞ (Ω).If we prove the boundedness of {μ ν }, then the boundedness of {ϕ ν } follows from (2.8).Let us study the sequence {μ ν }.Let ν s > 0 be as defined in Theorem 4.3.