Convergence and regularization results for optimal control problems with sparsity functional

DOI:10.1051/cocv/2010027 www.esaim-cocv.org

CONVERGENCE AND REGULARIZATION RESULTS

FOR OPTIMAL CONTROL PROBLEMS WITH SPARSITY FUNCTIONAL

Gerd Wachsmuth

and Daniel Wachsmuth

Abstract. Optimization problems with convex but non-smooth cost functional subject to an elliptic partial differential equation are considered. The non-smoothness arises from aL¹-norm in the objective functional. The problem is regularized to permit the use of the semi-smooth Newton method. Error estimates with respect to the regularization parameter are provided. Moreover, finite element approx- imations are studied. A-priori as well asa-posteriori error estimates are developed and confirmed by numerical experiments.

Mathematics Subject Classification.49M05, 65N15, 65N30, 49N45.

Received August 27, 2009. Revised February 9, 2010 and March 18, 2010.

Published online August 6, 2010.

1. Introduction

We investigate optimal control problems with a non-smooth objective functional of the following type:

MinimizeJ(y, u), which is given by J(y, u) =1

2y−y_d²_L2(Ω)+βuL¹(Ω)+α

2u²_L2(Ω) (1.1)

subject to the elliptic equation

Ay =u (1.2)

y|Γ = 0 (1.3)

and to the control constraints

u_a(x)≤u(x)≤u_b(x) a.e.on Ω. (1.4)

Here, Ω ⊂ Rⁿ, n = 2,3, is a bounded domain with boundary Γ. The operator A is assumed to be a lin- ear, elliptic second-order differential operator. The parameters α, β are non-negative parameters. Let us de- note the optimal control problem (1.1)–(1.4) by(P). Such optimal control problems with L¹-functionals arise

Keywords and phrases.Non-smooth optimization, sparsity, regularization error estimates, finite elements, discretization error estimates.

1 Chemnitz University of Technology, Faculty of Mathematics, 09107 Chemnitz, Germany.

2 Johann Radon Institute for Computational and Applied Mathematics (RICAM), Austrian Academy of Sciences, Altenbergerstrae 69, 4040 Linz, Austria. daniel.wachsmuth@ricam.oeaw.ac.at

Article published by EDP Sciences c EDP Sciences, SMAI 2010

if one tries to find the best location of the control actuator, seee.g.Stadler [27]. This is due to the following special property of the solutions: on sets, where the adjoint state is small, the optimal control must be zero.

Consequently, the optimal control has small support, which gives an indication to choose the actuator location.

Analogous observations can be made for optimization problems in sequence spaces involvingl¹-norms, see the comments below.

The optimal control problem under consideration admits a unique optimal control that will be denoted byu_α. For α= 0, the resulting optimization problem is convex but non-smooth, whereas forα > 0 the optimization problem admits a semi-smooth necessary optimality system, in this case, the parameterαacts as regularization and smoothing parameter. We are especially interested in the behavior of solutions for fixedβ≥0 andα→0.

For related estimates in the caseβ = 0 and with additional state constraintsy≤y_c, we refer to the recent work by Lorenz and R¨osch [22].

In this work, we investigate two types of approximations for Problem(P). First, we will study convergence of solutions if the regularization parameterαtends to zero. We prove that the L²-norm of the regularization error of the control obeys

u₀−u_αL² =O(α^1/2),

see below Theorem 3.7. This is a novel result in the context of optimal control problems with inequality constraints. Secondly, we study finite-element approximations for the regularized problem, which yields approx- imationsu_α,h of u_αin a finite-dimensional space. We prove thea-priori estimate

uα,h−u_αL²=O(h),

which coincide with available results for smooth functionals,i.e. forβ= 0, see below Propositions4.5and4.6.

Botha-priori results are combined in Section5 to choose the regularization parameterαin dependence of the mesh sizehto obtain optimal convergence ofu₀−u_α,h. Moreover, localizeda-posteriori error estimators of the type

u_α,h−u_α²_L2 ≤c

T∈Th

η²_T

are considered, where the error indicatorsη_T can be used in an adaptive process to compute approximations of solutionsu_α efficiently.

Let us comment on known results ona-priorianda-posteriori analysis of control constrained optimal control problems withα >0, β= 0. Basica-prioriestimates were derived by Falk [9], which yield that theL²-error of the control is controlled by the mesh sizehlikeO(h). Convergence results for the approximation of controls by linear elements can be found in e.g.in the work of Casas and Mateos [4]. The recently introduced variational discretization concept by Hinze [14] gives the error estimateu−u_hL² =O(h²). The same convergence order can be achieved by means of a post-processing step, see Meyer and R¨osch [23].

A-posteriorierror estimators of residual type were studied for instance by Liuet al.[18–20], and Hinterm¨uller et al. [13]. In addition, many papers are devoted to the so-called goal-oriented error estimators, for an outline of the underlying ideas see the survey of Becker and Rannacher [2].

Finally, let us report about existing literature in the context of inverse problems involving minimization problems in l¹. There, e.g., a possibly noisy signal should be reconstructed with as less non-zero coefficients of the solution as possible. That is, the support of the solution should be as small as possible, leading to a minimization with l⁰-functionals. In certain situations, the minimizer of l¹-functionals coincide with the minimizer of the l⁰-problem, see Donoho [7], which justifies the use of l¹-functionals to compute the sparsest solution. Solution methods for the arising non-smooth problems are studied for instance by Daubechieset al.[6], Griesse and Lorenz [11], Jinet al.[16], Ramlau and Teschke [25]. Regularization error estimates under suitable source conditions for (α, β)→(0,0) can be found for instance in Grasmairet al.[10] and Lorenz [21].

Optimization problems in L¹ and l¹ share some major properties: the resulting problems are convex and non-smooth. Furthermore, the optimality conditions imply that their solutions have potentially small support.

The fundamental differences arise from the different underlying functional analytic structure: The space l¹ is

the dual of the Banach spacec₀, which yields that the unit ball l¹ is weak-star compact and that thel¹-norm is weak-star lower semicontinuous. This can be used to prove existence of solutions for optimization problems involving l¹-norms. The same argument does not apply for L¹(Ω): This space is not the dual space of any normed linear space, hence the notation ‘weak-star’ makes no sense. Moreover, bounded sets in L¹(Ω) are in general not relatively weak compact due to the Dunford-Pettis theorem. In particular, the optimization problem (1.1)–(1.3) forα= 0 without additional constraints has no solution inL¹(Ω) in general. Here one has to resort to measures, seee.g. Clason and Kunisch [5]. Hence, the control constraints (1.4) are indispensable to prove existence of solutions of (P).

For inverse problems, the question of convergence of solutions for (α, β) → (0,0) is studied intensively.

There, one is interested to obtain in the limit the solution of an operator equation. Compared to optimal control problems this corresponds to the case that the optimal statey₀ forα= 0 fulfillsy₀=y_d, i.e. that the desired state can be reached. Due to the presence of the inequality constraints and due toβ →0 this cannot be expected in general. One main ingredient in the existing convergence proofs, are the so-called source conditions, where one assumes thatu₀lies in the range of a certain adjoint operatorS^∗. In our case, this would mean that u₀is in the range of the solution operator of an elliptic partial differential equation, which implies the regularity u₀∈H¹(Ω). This is not practical for problems with control constraints, since forα= 0 the optimal controlu₀ is discontinuous in general with jumps along curves, which means thatu₀∈H¹(Ω). This makes the fulfillment of a range condition unlikely. In the proof of our convergence result, we rather used a structural assumption on the active sets, see below Theorem3.7. For a more detailed comparison, we refer to the discussion in Section3.2 below.

Notations and assumptions

Let Ω ⊂ R^d, d = 2,3, be a bounded domain with Lipschitz boundary Γ. The operatorA is a uniformly elliptic differential operator defined by

(Ay)(x) =− N i,j=1

∂

∂x_i

a_ij(x) ∂

∂x_jy(x)

+c₀(x)y(x)

with functions a_ij∈C^0,1( ¯Ω), c₀∈L^∞(Ω), satisfying the conditiona_ij(x) =a_ji(x) and for someδ₀, δ₁>0 δ₀y²_H1(Ω)≤ Ay, yH⁻¹,H¹ ∀y∈H₀¹(Ω),

Ay₁, y_2H⁻¹_,H¹ ≤δ₁y_1H¹_(Ω)y_2H¹_(Ω) ∀y₁, y₂∈H₀¹(Ω).

Let us denote bya(·,·) the bilinear form induced byA

a(u, v) = Au, v_H⁻¹_,H¹.

The elliptic equation is solved in the weak sense,i.e. the weak solutiony satisfies

a(y, v) = (u, v) ∀v∈H₀¹(Ω). (1.5)

The corresponding solution mapping is denoted by S, which is a continuous linear injective operator from H⁻¹(Ω) toH₀¹(Ω). Thanks to the assumptions on the differential operator Aabove, the operatorS as well as its adjoint operatorS is continuous fromL²(Ω) toL^∞(Ω), seee.g.[28].

Furthermore, functionsy_d∈L²(Ω), u_a, u_b∈L^∞(Ω)∩H¹(Ω),u_a(x)≤0≤u_b(x)a.e.on Ω, are given. Please note, that the assumptionu_a ≤0≤u_b is not a restriction. If one has,e.g.,u_a>0 on a subset Ω₁⊂Ω, we can decompose theL¹-norm asu_L¹_(Ω)=u_L¹_(Ω\Ω1)+

Ω1u. Hence, on Ω₁ theL¹-norm inU_adis in fact a linear functional, and thus the problem can be handled in an analogous way.

2. Existence of solutions and optimality conditions

In this section we prove existence and uniqueness of solutions. Moreover, we derive optimality conditions.

In [27] this is done already for the caseα >0, but we will also handle the caseα= 0.

Lemma 2.1. The problem (P)has a unique solution even in the cases α= 0 orβ= 0.

Proof. Since the solution mappingS is injective, it is easy to see that the reduced objective ˆJ(u) :=J(Su, u) is strictly convex and continuous. Furthermore, the set U_ad is convex and weakly compact inL²(Ω). Therefore, the existence and uniqueness of the optimal control follows from standard arguments [29].

Let us remark that it is also possible to prove the existence and uniqueness of the solution for α = 0 in a pure L¹ setting. That is, if we assume only u_a, u_b ∈ L¹(Ω) we have to state the problem in L¹(Ω) since U_ad⊂L²(Ω). Therefore, we need higher regularity assumptions of the domain Ω to solve the elliptic equation with a right-hand side in L¹(Ω). Caused by the fact that L¹(Ω) is not reflexive, we can not prove the weak compactness of U_ad by its boundedness. However, weak compactness can be proven directly, which gives the existence and uniqueness of a solution in L¹(Ω), see [30], p. 8.

Since the objective function is not smooth but convex with respect tou, we can use the calculus of subdif- ferentials, see e.g.[15], Chapter 0.3.2. The subdifferential of the L¹(Ω)-norm is given by

v∈∂u_L¹⇔v(x)

⎧⎪

⎨

⎪⎩

= 1 u(x)>0

∈[−1,1] u(x) = 0

=−1 u(x)<0

for almost allx∈Ω, v∈L^∞(Ω). (2.1)

Now we can characterize the solution of (P) by a variational inequality, which is necessary and sufficient for the optimality ofu_α.

Lemma 2.2. The functions u_α ∈ U_ad and y_α =Suα are the optimal solution of (P) if and only if u_α, the adjoint state p_α=S(y_d−y_α)and a subgradientλ_α∈β∂uα_L¹ satisfy the variational inequality

(−pα+αu_α+λ_α, u−u_α)≥0 for allu∈U_ad. (2.2) Proof. Following [15] we can compute the necessary and sufficient optimality condition for the convex problem min_u∈UadJ(u) as follows:ˆ u_α is a solution if there existsλ_α∈∂Jˆ(u_α) such that for everyu∈U_ad

(λ_α, u−u_α)≥0 (2.3)

holds. We derive the subdifferential as

∂Jˆ(u_α) =−p_α+αu_α+β∂u_αL¹,

and so the variational inequality directly follows.

As in [29], p. 57, and [27], p. 4, one can discuss the variational inequality pointwise and get a pointwise relation of u_α and p_α as displayed in Figure 1. We see that |p_α| < β implies u_α = 0, which promotes the sparsity property ofu_α. See [27] for a more detailed discussion.

3. Estimates of the regularization error

As already mentioned, one can compute solutions of (P) with a semi-smooth Newton method in the case α >0, where the method converge locally super-linearly, see [27], Theorem 4.3. This however does not hold for α= 0. Hence, it is natural to approximate the solutionu₀ forα= 0 with the solutionsu_αforα >0.

Figure 1. Relationship between u_αandp_α.

3.1. Estimation by regularity of the active sets

At first, we derive an inequality that will be the starting point to obtain error estimates for the states and adjoints.

Lemma 3.1. The inequality

yα−y_α²_L2+αuα−u_α²_L2≤(α−α) (u_α, u_α−u_α) holds for all α >0,α ≥0.

Proof. The solutionsu_α,u_αfulfill the variational inequalities

(−p_α+αu_α+λ_α, v₁−u_α)≥0 (−p_α +α u_α +λ_α, v₂−u_α)≥0

each for all admissiblev₁, v₂∈U_ad. Testing withv₁=u_α andv₂=u_α, and adding the inequalities leads to (p_α−p_α−αu_α+αu_α−λ_α+λ_α, u_α−u_α)≥0.

Sinceλ_α andλ_α are subgradients of · _L¹, we obtain

(−λ_α+λ_α, u_α−u_α)≤0 by using the monotonicity of the subdifferential. This gives

(p_α −p_α−αu_α+αu_α, u_α−u_α)≥0 which directly leads to

0≤(S(y_α−y_α), u_α−u_α)−α(u_α−u_α, u_α−u_α) + (α−α) (u_α, u_α−u_α)

=−yα−y_α²_L2−αuα−u_α²_L2+ (α−α) (u_α, u_α−u_α).

This entails our claim.

Usingα= 0 in the previous lemma we obtain the estimate

y0−y_α²_L2+αu0−u_α²_L2 ≤α(u₀, u₀−u_α). (3.1)

Table 1. Partition of Ω, used in Proof of Lemma3.3.

p₀<−β |p0|< β p₀> β p_α≤ −β+αu_a u₀=u_α=u_a I₁ I₁

p_α∈(−β+αu_a,−β] I₂ I₁ I₁

|p_α|< β I₁ u₀=u_α= 0 I₁

p_α∈[β, β+αu_b) I₁ I₁ I₃

p_α≥β+αu_b I₁ I₁ u₀=u_α=u_b

Since the admissible set is bounded due to the control constraints, we can conclude a first convergence result for the states and adjoints.

Corollary 3.2. The estimate

y₀−y_αL²≤C α^1/2, p₀−p_αL^∞ ≤C α^1/2 holds for all α >0.

Proof. The operatorS maps right-hand sides inL²(Ω) to adjoints inL^∞(Ω), which yields with (3.1)

p₀−p_α²_L∞ ≤αS²_2→∞(u₀, u₀−u_α). (3.2) Since the scalar product (u₀, u₀−u_α) is bounded due to the control constraints, we find directly the stated

convergence rates.

We will now show convergence rates for the error in the adjoint states imply convergence rates for the error in the controls. To this end, we have to make an assumption on the boundary of the set{|p₀|=β}. Analogous assumptions on the boundary of active sets can be found in connection with finite element error estimates for elliptic optimal control problems, see [4,23].

Lemma 3.3. Let us assume that there exists a constantC_p>0 such that for every ε≥0 the estimate for the Lebesgue measure μof |p₀| −β≤ε

is bounded as:

μ|p₀| −β≤ε

≤C_pε. (3.3)

Then we have for all d∈(0,1]

p₀−p_αL^∞≤C α^d ⇒ u₀−u_αL² ≤C α^d/2 and p₀−p_αL^∞ ≤Cα^d+24 .

Proof. Let us divide Ω in disjoint sets depending on the values ofp₀ andp_α, see also Table 1, I₁:={x∈Ω :β or −β lies betweenp₀ andp_α}

I₂:={x∈Ω :p₀, p_α≤ −β andp_α≥ −β+αu_a} I₃:={x∈Ω :p₀, p_α≥+β andp_α≤β+αu_b}.

Note that we can ignore the set {|p0| =β}, since it has measure zero by assumption (3.3). Let us define the unionU =I₁∪I₂∪I₃. On Ω\U we haveu₀=u_α, while we can bound the measures of the setsI₁,I₂ andI₃. OnI₁the assumption ensures |p0| −β≤p₀−p_α| ≤C α^d. OnI₂ we have||p0| −β|=|p0+β| ≤αu_a+C α^d and onI₃ we have analogously||p₀| −β| ≤αu_b+C α^d. So on the unionU we have||p₀| −β| ≤C_bα+C α^dwith the constant C_b = max(uaL^∞,ubL^∞) depending on the bounds u_a, u_b. Using d≤1 and α≤1 we have

||p₀| −β| ≤(C+C_b)α^d. Now we can bound the measureM =μ(U) of this set and getM ≤C_p(C+C_b)α^d. Due to the L^∞control constraints andu₀=u_αon Ω\U, we have by (3.1)

u₀−u_α²_L2 ≤Mu₀L^∞u₀−u_αL^∞ ≤2C_p(C+C_b)C_b²α^d. By (3.2) we obtain

p₀−p_αL^∞ ≤C_S(C_b)^1/2(2C_p(C+C_b)C_b²)^1/4α^d+24 ,

with the constants C_b = max_u∈UaduL² = max(u_b,−u_a)L² and C_S = S2→∞. Let us define C :=

C_b(2C_p(C+C_b))^1/2. Then we obtain the desired implication

p0−p_αL^∞ ≤C α^d ⇒ u0−u_αL² ≤Cα^d/2

and p0−p_αL^∞ ≤C_S(C_bC)^1/2α^d+24

for alld∈(0,1].

With this lemma, we can prove a first convergence result.

Corollary 3.4. Under the assumptions of Lemma 3.3, there is for each d <1/3 a constant C_d>0 such that u0−u_αL² ≤C_dα^d

holds.

Proof. By Lemma3.3, p₀−p_αL^∞ ≤C α^d implies u₀−u_αL² ≤Cα^d/2 and p₀−p_αL^∞ ≤Cα^(d+2)/4. By Corollary 3.2, we know already p0−p_αL^∞ ≤ C α^1/2. Now, let us consider the sequence d₀ = 1/2, d_k+1 = (d_k + 2)/4, which corresponds to the convergence rates of the adjoint states. It is monotonically increasing and has the limit 2/3. So we get for all d < 2/3 a constant C_d with p0−p_αL^∞ ≤ C_dα^d and

u₀−u_αL² ≤C_dα^d/2. This proves the claim.

This corollary provides us with convergence rates for the controls up to order 1/3−ε. However, we observed higher convergence rates in our numerical experiments. We will now prove higher rates using sensitivity infor- mation with respect to the parameterα. Similar techniques are used for path-following algorithms, seee.g.[26]

for an application to an optimal control problem with state constraints.

At first let us state the following differentiability result, which is proven in the appendix.

Lemma 3.5. The mappings α→u_α, α→y_α and α→p_α are Gˆateaux-differentiable intoL²(Ω),L²(Ω) and L^∞(Ω), respectively, at almost all α >0. With help of the sets

I₁={pα∈(β, β+αu_b)}, I₂={pα∈(−β+αu_a,−β)}

the derivatives are given as solutions of the system

˙ u_α= 1

α²[(β−p_α)χ_I1−(β+p_α)χ_I2] + 1

αp˙_αχ_I1∪I2 (3.4a)

˙

y_α=Su˙_α (3.4b)

˙

p_α=−Sy˙_α. (3.4c)

Please note, that the claim of the lemma is only valid for almost all α > 0. To obtain differentiability for allα >0, one has to resort to directional (Bouligand) derivatives. Then the resulting system for ( ˙u_α,y˙_α,p˙_α) is not longer linear. However, the quantities ( ˙u_α,y˙_α,p˙_α) can be interpreted as the solution of a suitably chosen inequality-constrained optimization problem. See [12] for further discussions and references.

Using the sensitivity information, we can estimate the distance fromy_α to by an integral over the solutions of the sensitivity system.

Lemma 3.6. For all α >0 we have the following estimate y₀−y_αL² ≤

_α

0 y˙_α˜L²d ˜α.

Proof. Let us consider the functionf :α→ y₀−y_αL². This function is Lipschitz continuous on each interval [ε, M],ε >0 by Lemma3.1, and thus also absolutely continuous and almost everywhere differentiable. Moreover, it holds

f(α₁)−f(α₂) = _α2

α1

f(t) dt forα_i∈[ε, M]. Now, let us estimate the difference quotient

y₀−y_α+δL²− y₀−y_αL²

δ

≤ 1

δ(y_α+δ−y_α) L²

, and taking the limit δ→0 yields

|f(α)| ≤ y˙_αL² for almost allα >0. Therefore we have

f(α)−f(ε)≤ _α

ε

y˙_α˜L²d ˜α≤ _α

0 y˙_α˜L²d ˜α for allε >0, and passing to the limitε→0 yields

y₀−y_α ≤ _α

0 y˙_α˜L²d ˜α

sincef is continuous in 0 withf(0) = 0.

Now we can proof the main result of this section. The idea is to estimate the norm of the derivatives ˙y_α, and then apply the previous lemma to obtain an estimate of the error.

Theorem 3.7. Let the regularity assumption

μ|pα| −β≤ε

≤C_pε (3.5)

be satisfied forα= 0. Then for each d <1 there is a constantC_d, such that

u0−u_αL² ≤C_dα^d/2, (3.6a)

y₀−y_αL² ≤C_dα^d, (3.6b)

p0−p_αL^∞ ≤C_dα^d, (3.6c)

holds asα→0.

If, in addition, the regularity condition (3.5)holds uniformly for almost allα≥0, then there is a constantC₁ such that (3.6)is true for d= 1.

The first part of the theorem uses the same regularity condition as Lemma3.3. Using the sensitivity infor- mation, we can however show a stronger result. If the regularity condition holds uniformly forα≥0, then we can prove the full convergence rate. In this case the regularity condition implies that the set{|pα| −β= 0}

has zero measure for α≥0.

Proof. In this proofC denotes a generic constant that is independent of α. Testing (3.4b) with ˙y_α and (3.4c) with ˙u_αyields

y˙_α²_L2=−

Ωu˙_αp˙_αdx.

Combining this result with equation (3.4a) we obtain with the notationsI₁andI₂ from Lemma3.5 y˙_α²_L2+ 1

αp˙_αχ_I²_L2= 1 α²

I1

(p_α−β) ˙p_αdx+

I2

(p_α+β) ˙p_αdx

. (3.7)

Then by definition of these sets, we have p_α−β ∈(0, α u_b) onI₁ andp_α+β ∈(α u_a,0) onI₂. Let us define I:=I₁∪I₂.

Together withu_a, u_b∈L^∞(Ω) we can estimate the right hand side of (3.7) and obtain y˙_α²_L2+ 1

αp˙_αχ_I²_L2≤ C

αp˙_αχ_IL²χI_L², which implies

y˙_αL² ≤ C

√αχI_L². (3.8)

Now, it remains to boundχI_L² = μ(I).

Case 1. Inequality (3.5)holds forα= 0 and for almost allα >0:

The regularity assumption (3.5) yields

μ(I)≤C α, and with (3.8) we get

y˙_αL²≤C.

Now integration (Lem. 3.6) yields

y₀−y_αL² ≤C α and we conclude with the smoothing property ofS and Lemma 3.3

p0−p_αL^∞ ≤C α, u0−u_αL² ≤C α^1/2. Case 2. Inequality (3.5)holds for α= 0:

In this case we use a bootstrapping argument similar to Corollary3.4. Let us defineC_b := max(u_a∞,u_b∞).

Then we have

I⊂|pα| −β≤C_bα .

Now, suppose that p₀−p_α∞≤C α^d withd <1 holds. Using|p₀| −β≤|p_α| −β+|p_α−p₀|we have for α <1

I⊂|p_α| −β≤C_bα

⊂|p₀| −β≤C α^d .

Utilizing assumption (3.5), we conclude μ(I) ≤ C α^d. Following the same steps as in Case 1, we obtain y˙_αL² ≤C α^(d−1)/2,y0−y_αL² ≤C α^(d+1)/2andp0−p_α∞≤C α^(d+1)/2. Thus, we proved the implication

p0−p_α∞≤C α^d ⇒ p0−p_α∞≤C α^(d+1)/2.

Starting with d = 1/2 (Cor. 3.2) and observing that the sequence a₀ = 1/2, a_n+1 = (a_n+ 1)/2 converges

towards 1 ends the proof.

3.2. Discussion of the regularity assumption and source conditions

μ|p0| −β≤ε

≤C_pε.

This assumption implies, that the set|p₀| −β= 0

has zero measure. Consequently, the optimal controlu₀ satisfiesu₀(x)∈ {u_a(x),0, u_b(x)} almost everywhere. This is typically observed for solutions of optimal control problems forα= 0. There the optimal control has discontinuities on the set|p₀| −β= 0

, which means that in generalu₀∈H¹(Ω) holds.

Let us comment on available source conditions for our optimal control problem. The considerations will be based on the inequality (3.1),i.e.

y₀−y_α²_L2+αu₀−u_α²_L2 ≤α(u₀, u₀−u_α). (3.9) Following Lorenz and R¨osch [22], let us assume that the following source condition is fulfilled: There exists w∈L²(Ω) such that

u₀=P_Uad(S^∗w).

Due to the properties of the projection, we have

(u₀− S^∗w, u_α−u₀)≥0.

Then (3.9) becomes

y0−y_α²_L2+αu0−u_α²_L2≤α(u₀, u₀−u_α)≤α(S^∗w, u₀−u_α) =α(w,S(u0−u_α)) =α(w, y₀−y_α), which gives

y0−y_αL² ≤αw_L², u0−u_αL²≤√

αw_L².

However, the assumption on u₀ implies the regularity u₀ ∈ H¹(Ω) for bounds u_a, u_b ∈ H¹(Ω), and is thus not compatible with the observation thatu₀ is typically not in H¹(Ω). Other source conditions as developed in [10,21] are not directly applicable, since they are tailored to the case (α, β)→(0,0), while we are considering the convergence forα→0 and fixedβ.

4. A

finite element error analysis, α > 0

As indicated, the optimal control problem with α > 0 is better suited for numerical computations. After studying the regularization error, we will now turn to the finite element analysis of the regularized problems.

Let us fix the assumptions on the discretization of Problem(P)by finite elements. First let us specify the notation of regular meshes. Each meshT consists of closed cellsT (for example triangles, tetrahedra,etc.) such that ¯Ω =

T∈T T holds, which implies in particular that cells with edges/faces lying on the boundary are curved for smooth, non-polygonal Ω. We assume that the mesh is regular in the following sense: For different cells T_i, T_j ∈ T,i=j, the intersectionT_i∩T_j is either empty or a node, an edge, or a face of both cells,i.e. hanging nodes are not allowed. Let us denote the size of each cell by h_T = diamT and defineh(T) = max_T∈T h_T. For eachT ∈ T, we defineR_T to be the diameter of the largest ball contained inT.

We will work with a family of regular meshes F = {Th}h>0, where the meshes are indexed by their mesh size,i.e.h(Th) =h. We assume in addition that there exist two positive constantsρandRsuch that

h_T

R_T ≤R, h h_T ≤ρ

hold for all cells T ∈ Th and all h > 0. With each mesh Th ∈ F we associate a finite-dimensional subspace V_h⊂V. For a given right-hand sideu, we definey_h∈V_h as the solution of the discrete weak formulation

a(y_h, v_h) = (u, v_h) ∀vh∈V_h, (4.1)

and we denote the corresponding solution operator by Sh, i.e. y_h = Shu. In the following, we rely on an assumption on the spaces V_h, which is met by standard finite element choices.

Assumption 4.1. Letu∈L²(Ω)be given. Letyandy_hbe the solutions of (1.5)and (4.1), respectively. There exists a constant c_A>0 independent ofh, usuch that

y−y_hL²+hy−y_hH¹ ≤c_Ah²uL². This assumption implies in particularSh− S_L²_→H¹ ≤c_Ah.

Now, let us introduce the control discretization. We will discretize the control utilizing positive basis func- tions. Here, we follow an approach introduced by Meyeret al. in [24]. Alternatively, one can follow the so-called variational approach of [14], in which one setsU_h:=U, see the corresponding arguments in Section 4.3.

Assumption 4.2. To each mesh we associate a finite-dimensional space U_h ⊂ U. There is a basis Φ_h = {φ¹_h, . . . , φ^N_h^h}of U_h,e.g. U_h= span Φ_h, where the basis functions φⁱ_h∈L^∞(Ω)have the following properties:

φⁱ_h≥0, φⁱ_h∞= 1 ∀i= 1. . . N_h,

Nh

i=1

φⁱ_h(x) = 1 for a.a.x∈Ω. (4.2) Furthermore, there are numbers M, N such that following conditions are fulfilled for allhand all i= 1. . . N_h: each support ω_hi := suppφ_hi is connected, and it is contained in the union of at most M adjacent cellsT ∈ Th

sharing at least one vertex. Each cell T ∈ Th is subset of at mostN supportsωⁱ_h.

This assumption covers several commonly-used control discretizations, such as piecewise constant or linear functions, see [24]. Following the approach of [3,24], let us introduce a quasi-interpolation operator Π_h:L¹(Ω)→ U_h. The operator Π_h is given by

Π_h(u) :=

Nh

i=1

πⁱ_h(u)φⁱ_h withπⁱ_h(u) :=

Ωuφⁱ_h

Ωφⁱ_h ·

Please note, that Π_h is not a projection with respect to the L²-scalar product. Nevertheless, the following orthogonality relation holds foru∈L²(Ω)

Ω

(u−π_hⁱ(u))φⁱ_h= 0. (4.3)

Based on the assumptions on the mesh and on the control discretization, we have the following interpolation estimates. For the proofs, we refer to [3,24].

Lemma 4.3. There is a constant c_I independent ofhsuch that

hu−Π_huL²(Ω)+u−Π_hu_H⁻¹_(Ω)≤c_Ih²∇uL²(Ω)

is fulfilled for all u∈H¹(Ω).

It remains to describe the discrete admissible setU_ad,h. We use the quasi-interpolation operator Π_hto define new bounds by

u_a,h= Π_hu_a =

i

uⁱ_a,hφⁱ_h=

i

πⁱ_h(u_a)φⁱ_h, u_b,h= Π_hu_b=

i

uⁱ_b,hφⁱ_h.

Let us set

U_ad,h:={u∈U_h: u_a,h≤u≤u_b,h a.e.on Ω}.

Here it may happen, thatu_a,horu_b,hare no longer admissible,i.e.u_a,h∈U_adoru_b,h∈U_ad, which gives in the end a not admissible discretizationU_ad,h ⊂U_ad. For the special case of constant upper and lower boundsu_a andu_b, it holdsU_ad,h⊂U_ad. Nevertheless, the admissible setU_ad,h can be written equivalently in the following way.

Lemma 4.4. Let u_a,h, u_b,h, U_ad,h be defined as above. Then it holds

U_ad,h=

u=

i

uⁱφⁱ_h, uⁱ_a,h≤uⁱ≤uⁱ_b,h

. (4.4)

Proof. The first part ‘⊂’ of (4.4) follows directly from Assumption4.2, which givesuⁱ_a,h≤uⁱ_h≤uⁱ_b,h. Summation ofuⁱ_a,hφⁱ_h(x)≤uⁱφⁱ_h(x)≤uⁱ_b,hφⁱ_h(x) yields also the second inclusion ‘⊃’.

Thanks to this lemma, the constraint u_h ∈ U_ad,h can be transformed in simple box constraints of the coefficients ofu_h, which enables to use efficient solution techniques for the resulting optimization problem.

Let us now define the discrete optimal control problem as: Minimize J(y_h, u_h) subject tou_h∈U_ad,h and a(y_h, v_h) = (u, v_h) ∀vh∈V_h.

This represents an optimization problem, which is uniquely solvable. Let us denote its solution by (y_α,h, u_α,h) with associated adjoint state p_α,h and subgradient λ_α,h ∈ ∂u_hL¹. Analogously to the continuous problem, one obtains the variational inequality

(αu_α,h−p_α,h+λ_α,h, u_h−u_α,h)≥0 ∀uh∈U_ad,h (4.5) as necessary and sufficient optimality condition, see Lemma2.2.

We will now derive error estimates in terms of the mesh size h. At first, we will derive upper bounds of uα−u_α,hL² andyα−y_α,hL². For different choices ofU_h, we have to proceed differently, which amounts in a number of analogous error estimates. Now, let us start to derive the basic error bound with the help of the variational inequalities (2.2) and (4.5).

Here, it would be nice if we could use u_α as a test function in the variational inequality (4.5), which char- acterizes u_α,h. However, in general the function u_α does not belong to U_ad,h and cannot be utilized as test function. To overcome this difficulty, let us introduce an approximation ˜u_h ≈ u_α with ˜u_h ∈ U_ad,h, which is suitable as test function in (4.5).

The same arguments apply tou_α,h. Here one cannot expectu_α,h∈U_adif for instance the control bounds are not constants. Thus, let us take a function ˜u≈u_α,h that is feasible for the continuous problem,i.e. u˜∈U_ad, and can be used as test function in the variational inequality (2.2).

Now let us use the test function ˜uin (2.2) and the test function ˜u_hin (4.5). Adding the resulting inequalities we obtain

αu_α−u_α,h²_L2 ≤(αu_α,h−p_α,h,u˜_h−u_α) + (αu_α−p_α,u˜−u_α,h)−(p_α,h−p_α, u_α−u_α,h)

+β(˜u_L¹− uα,h_L¹+˜u_hL¹− uα_L¹). (4.6)