New regularity results and improved error estimates for optimal control problems with state constraints

DOI:10.1051/cocv/2013084 www.esaim-cocv.org

NEW REGULARITY RESULTS AND IMPROVED ERROR ESTIMATES FOR OPTIMAL CONTROL PROBLEMS WITH STATE CONSTRAINTS

Eduardo Casas

, Mariano Mateos

and Boris Vexler

Abstract. In this paper we are concerned with a distributed optimal control problem governed by an elliptic partial differential equation. State constraints of box type are considered. We show that the Lagrange multiplier associated with the state constraints, which is known to be a measure, is indeed more regular under quite general assumptions. We discretize the problem by continuous piecewise linear finite elements and we are able to prove that, for the case of a linear equation, the order of convergence for the error inL²(Ω) of the control variable ish|logh|in dimensions 2 and 3.

Mathematics Subject Classification. 49K20, 49M05, 49M25, 65N30, 65N15.

Received May 31, 2013. Revised November 25, 2013.

Published online June 5, 2014.

1. Introduction

This paper deals with the following optimal control problem

(P) min

u∈Uad

J(u), where

J(u) = 1 2

Ω

(y_u(x)−y_d(x))²dx+ν 2

Ω

u²(x) dx, U_ad={u∈L²(Ω) :a(x)≤y_u(x)≤b(x) for allx∈Ω}, andy_uis the solution of the Dirichlet problem

Ay=u in Ω,

y= 0 onΓ, (1.1)

Abeing an elliptic operator.

Keywords and phrases.Optimal control, state constraints, elliptic equations, Borel measures, error estimates.

∗ The first two authors were partially supported by the Spanish Ministerio de Econom´ıa y Competitividad under project MTM2011-22711.

1 Departmento de Matem´atica Aplicada y Ciencias de la Computaci´on, E.T.S.I. Industriales y de Telecomunicaci´on, Universidad de Cantabria, 39005 Santander, Spain.[email protected]

2 Departmento de Matem´aticas, E.P.I. Gij´on, Universidad de Oviedo, Campus de Gij´on, 33203 Gij´on, Spain.[email protected]

3 Center for Mathematical Sciences, Technische Universit¨at M¨unchen, Bolzmannstrasse 3, 85748 Garching b. M¨unchen, Germany.

[email protected]

Article published by EDP Sciences c EDP Sciences, SMAI 2014

The inherent difficulty of these control problems is the fact that the Lagrange multiplier associated with the state constraints is a Borel measure ¯μ in Ω ⊂Rⁿ. This leads to an adjoint state ¯ϕ∈ W₀^1,s(Ω) for every 1≤s <_n−1ⁿ . In this paper, under mild assumptions, we prove that the adjoint state ¯ϕbelongs toH¹(Ω)∩L^∞(Ω) and ¯μ∈H⁻¹(Ω). This implies, in particular, that Dirac measures are excluded as Lagrange multipliers forn >1.

As a consequence we also obtain the H¹(Ω)∩L^∞(Ω) regularity for the optimal control ¯u. As far as we know, H¹(Ω) regularity results for ¯uwere proved in [11], where the presence of pointwise control constraints played a crucial role in the proof. However, no additional regularity for the adjoint state ¯ϕand the Lagrange multiplier

¯

μwas obtained there. Our regularity result has been inspired by a recent result by Pieper and Vexler [27] for a sparse control problem with controls in a measure space.

We use this new regularity result to improve the error estimates for the finite element discretization of the control problem. For the error between the optimal control ¯u and its discrete counterpart ¯u_h we prove an estimate of orderO(h|logh|) for both the two and three dimensional case.

Error estimates for optimal control problems with state constraints governed by elliptic equations are derived in several publications. In [9,10,22] error estimates are given for optimal control problems with finitely many state constraints. In Deckelnick and Hinze [15,16] error estimates of orderh^1−εin 2d andh^12−εin 3d are derived for a problem with pointwise state constraints, see also Meyer [26] for a proof of similar results with a different technique. These estimates are obtained for domains with smooth boundaryΓ. For a convex polygonal domain, Meyer [26] obtains an order ofO(h^λ) whereλ∈(1/2,1) depends on the biggest interior angle ofΩ. For the three dimensional case an improvement toh³⁴ is achieved in R¨osch and Steinig [28] and an estimate of orderh|logh|

is shown in [21] for a problem with control and state constraints. For the proof of the later result the presence of control constraints ensuring the uniform boundedness of ¯uand ¯u_h plays a crucial role. Liu, Gong and Yan [23]

and Gong and Yan [19] treat problem (P) by transforming it into a biharmonic obstacle problem. They deal with the linear-quadratic case and the Laplace operator in dimension 2. For the error analysis, they suppose that Ωis polygonal, the obstacle is in H⁴(Ω) and, following [3], some extra assumptions on the active set are needed to have the adjoint state inH₀¹(Ω). Discretization is made with the nonconforming Morley finite element in the first reference and with LagrangeP₁finite elements in the second one. They proveO(h) convergence for the Morley finite element. ForP₁ elements they obtain againO(h) in superconvergent meshes, and O(h^1/2) in quasi-uniform meshes. These results are valid if the optimal state is inH³(Ω).

This means that our result improves the known estimate of almost orderO(h¹²) toO(h|logh|) for a purely state-constrained problem in the three dimensional case and also for plane polygonal convex domains, where in general the optimal state is only inH^2+α(Ω), for someα∈(0,1) depending on the biggest interior angle of the domain (see,e.g.[20], Thm. 5.1.1.4, or [4] for a general regularity result about the biharmonic operator).

The plan of this paper is as follows. In Section 2 we recall the optimality system and we discuss some consequences of it. In particular, the structure of the Lagrange multiplier ¯μis studied. Section 3 is devoted to the proof of our main regularity result. In Section4we consider some extensions of this result to more general control problems. In particular, the case of semilinear state equations is considered. The error estimates for the finite element approximations are proved in Section5and some numerical examples confirming these estimates are given in Section6.

2. Assumptions and preliminary results

Concerning the Dirichlet problem (1.1) we make the following hypotheses.

Assumption 2.1. Ωdenotes an open bounded subset of Rⁿ, n= 2 or 3, with a Lipschitz boundaryΓ. A is the linear operator

Ay=−ⁿ

i,j=1

∂_xj[a_ij(x)∂_xiy] +a₀(x)y,

witha_ij, a₀∈L^∞(Ω),a₀(x)≥0 for almost allx∈Ω. Furthermore, there exists someΛ >0 such that n

i,j=1

a_ij(x)ξ_iξ_j ≥Λ|ξ|²for a.a. x∈Ω and ∀ξ∈Rⁿ.

Some additional regularity will be required for Ω, Γ and the coefficients a_ij in the section devoted to the numerical approximation.

Under this assumption, it is well known that for every u∈H⁻¹(Ω) equation (1.1) has a unique solution in the Sobolev spaceH₀¹(Ω). This solution will be denoted byy_u. Moreover, ifu∈W^−1,p(Ω) for somep > n, then y_u belongs to the H¨older spaceC^θ( ¯Ω) for some 0< θ <1 depending onp; see [18], Theorem 8.29. We also have

the estimate

yu_H¹

0(Ω)≤M₀u_H⁻¹_(Ω),

yu_C^θ_{( ¯Ω)}≤M_pu_W^−1,p_(Ω). (2.1)

Sincen= 2 or 3, we have thatL²(Ω)⊂W^−1,p(Ω), with continuous embedding, for anyp <∞ifn= 2 and any p≤6 if n= 3. Hence, the above estimates remain valid replacing the norm ofuin the corresponding Sobolev space by the L²(Ω)-norm, with the obvious changes of the constants M₀ andM_p.

Assumption 2.2. Along this paper y_d is an element of L²(Ω) and ν > 0. We also assume the following hypotheses on the functionsaandb.

a, b∈C( ¯Ω). (2.2a)

a(x)< b(x)∀x∈Ω.¯ (2.2b)

a(x)<0< b(x)∀x∈Γ. (2.2c)

Aa, Ab∈L^∞(Ω). (2.2d)

Associated with these data we define the set

Yab={y∈C₀(Ω) :a(x)≤y(x)≤b(x) ∀x∈Ω},

whereC₀(Ω) is the space of continuous functions in ¯Ω vanishing onΓ. Then, the admissible control set can be rewritten as follows

U_ad={u∈L²(Ω) :y_u∈ Yab}.

Notice that the assumptions (2.2a)−(2.2d) hold if a and b are constants satisfyinga < 0 < b. This is the case for the typical state constraint |yu(x)| ≤ δ. The assumptions (2.2a)−(2.2c) are natural and similar to assumptions usually required for optimal control problems with state constraints. The additional regularity assumption (2.2d) is crucial for our main regularity result, see Theorem3.1, as well as for the error estimate, see Corollary5.7.

It is obvious that the control problem (P) is strictly convex. Hence, it has at most one solution. The existence of a solution can be proved by standard arguments. Hereafter, ¯u will denote the solution of (P) and ¯y the corresponding state. Before establishing the first-order optimality conditions fulfilled by the optimal control ¯u, we introduce some notation. We denote byM(Ω) the space of real and regular Borel measures inΩ, which is identified with the dual ofC₀(Ω). This is a Banach space for the norm

μ_M(Ω)=|μ|(Ω) = sup

y∈C0(Ω),y∞≤1

Ω

ydμ.

Above|μ| denotes the total variation measure corresponding toμ. We also consider the Jordan decomposition μ=μ⁺−μ⁻. Then, we know that|μ|=μ⁺+μ⁻; see, for instance, Rudin ([29], Chap. 6), for details.

Theorem 2.3. Under the Assumptions 2.1 and (2.2a)–(2.2c), there exist a measureμ¯∈ M(Ω)and an element

¯

ϕ∈W₀^1,s(Ω), for every 1≤s < _n−1ⁿ , such that

A¯y= ¯u inΩ,

¯

y= 0 onΓ, (2.3a)

A^∗ϕ¯= ¯y−y_d+ ¯μ inΩ,

¯

ϕ= 0 onΓ, (2.3b)

Ω

(y−y) d¯¯ μ≤0 ∀y∈ Yab, (2.3c)

¯

ϕ+νu¯= 0, (2.3d)

whereA^∗ is the adjoint operator ofA, given by the expression A^∗ϕ=− ⁿ

i,j=1

∂_xj[a_ji(x)∂_xiϕ] +a₀(x)ϕ.

Moreover, the Lagrange multiplierμ¯ and the adjoint stateϕ¯are unique.

Proof. First, let us prove the uniqueness of ¯ϕ and ¯μ. The uniqueness of ¯ϕ follows from the uniqueness of ¯u and (2.3d). Hence, (2.3b) implies the uniqueness of ¯μas a distribution, which is equivalent to the uniqueness of

¯

μas a measure.

The existence of ¯μand ¯ϕsatisfying (2.3b)−(2.3d) is well known under the assumption of the Slater condition, see, for instance, [6]. Let us check that the Slater condition is fulfilled:

∃u₀∈L²(Ω) such thata(x)< y_u0(x)< b(x)∀x∈Ω.¯ Define

ρ₁= min

x∈Ω¯(b(x)−a(x)),

Due to (2.2a) and (2.2b), we have thatρ₁>0. Moreover, using (2.2c) we can findε >0 andρ₂>0 such that a(x)<−ρ2< ρ₂< b(x) ∀x∈Ω¯ such that d(x, Γ)< ε. (2.4) Setρ= min{ρ1, ρ₂}. Now, we use Uryshon’s lemma to obtain a functionφ∈C₀(Ω) such that

0≤φ≤1 and φ(x) =

0 ifd(x, Γ)≤ ε 2, 1 ifd(x, Γ)≥ε.

Setting a_φ =φaandb_φ=φb, we have thata_φ, b_φ∈C₀(Ω) and

a(x)≤a_φ(x)≤b_φ(x)≤b(x) ∀x∈Ω.¯

We know that the spaceC₀^∞(Ω) of smooth functions having a compact support inΩis dense inC₀(Ω). Therefore, we can select one of these functions, denoted byy, such that

y−1

2(a_φ+b_φ) L^∞(Ω)

< ρ 4·

It is obvious thatAy∈W^−1,p(Ω) for everyp≥1. Fix somep∈(n,+∞) and take u₀∈L^p(Ω) such that u0−Ay_W^−1,p_(Ω)< ρ

4M_p,

whereM_p is given by (2.1). Let us prove thatu₀ satisfies the Slater assumption. First, we observe y_u0−1

2(a_φ+b_φ) L^∞(Ω)

≤ yu0−y_L^∞_(Ω)+ y−1

2(a_φ+b_φ) L^∞(Ω)

< M_pu₀−AyW^−1,p(Ω)+ρ 4 <ρ

2· (2.5)

To prove thata(x)< y_u0(x)< b(x) in ¯Ωwe distinguish three cases.

Case I. d(x, Γ)≤ ^ε₂. In this case,a_φ(x) =b_φ(x) = 0, therefore (2.4) and (2.5) lead to a(x)<−ρ₂≤ −ρ < y_u0(x)< ρ≤ρ₂< b(x).

Case II. d(x, Γ)≥ε. For this xwe have thata_φ(x) =a(x) andb_φ(x) =b(x). From the definition ofρ₁ andρ we get

−ρ≥ −ρ1≥a(x)−b(x) and ρ≤ρ₁≤b(x)−a(x).

Thus, we obtain with (2.5) a(x)≤1

2(a(x) +b(x))−ρ₁ 2 ≤1

2(a_φ(x) +b_φ(x))−ρ

2 < y_u0(x)

<1

2(a_φ(x) +b_φ(x)) +ρ 2 ≤1

2(a(x) +b(x)) +ρ₁

2 ≤b(x).

Case III. ^ε₂ < d(x, Γ)< ε. Using again (2.4) and (2.5), it follows a(x)≤ 1

2a_φ(x) +1

2a(x)≤1

2(a_φ(x) +b_φ(x))−ρ₂ 2

≤ 1

2(a_φ(x) +b_φ(x))−ρ

2 < y_u0(x)< 1

2(a_φ(x) +b_φ(x)) +ρ 2

≤ 1

2(a_φ(x) +b_φ(x)) +ρ₂ 2 ≤1

2b_φ(x) +1

2b(x)≤b(x).

Remark 2.4. Let us notice that the assumption (2.2d) was not necessary for the proof of the Slater condition.

We only used the assumptions (2.2a)−(2.2c). Moreover, the proof can be simplified if we assume that the coefficientsa_ij of the operatorA are Lipschitz continuous in ¯Ω. Indeed, under this assumption, if we takeφof class C² in Ω and satisfying the conditions of the proof, thenu₀=A[¹₂(a_φ+b_φ)]∈L²(Ω) satisfies the Slater condition. Furthermore, ifaandb are constants, witha <0< b, thenu₀= 0 satisfies the Slater condition.

Regarding the adjoint state equation (2.3b), some explanation is necessary. Following Stampacchia [31], given a measureμ∈ M(Ω), we say that an elementϕ∈L¹(Ω) is a solution of the Dirichlet problem

A^∗ϕ=μ in Ω,

ϕ= 0 onΓ, (2.6)

if

Ω

ϕAzdx=

Ω

zdμ ∀z∈ Z, with

Z ={z∈H₀¹(Ω) :Az∈C₀(Ω)}.

Using again ([18], Thm. 8.29), we deduce thatZ ⊂C₀(Ω), hence the above integrals are well defined. With this definition, we know that there exists a unique solution that additionally belongs toW₀^1,s(Ω) for everys <_n−1ⁿ .

Moreover, if E denotes the support ofμ, thenϕ∈H_loc¹ (Ω\E)∩C( ¯Ω\E), and for any compact K⊂Ω\E the following estimate holds

ϕC(K)≤C_Kμ_M(Ω), (2.7)

see ([31], Thm. 9.3). Of course, all these properties are enjoyed by ¯ϕ.

The following complementarity result is well known, but we have not found any proof in the literature for the case of non-constant bounds aandb. For the sake of completeness we give a proof, see alsoe.g.[6] for the proof in the case of constant bounds.

Proposition 2.5. Let a, b be two functions satisfying (2.2a)–(2.2c) and let y¯ ∈ C₀(Ω) and μ¯ ∈ M(Ω) sat- isfy (2.3c). Then, the following embeddings hold

supp ¯μ⁺⊂ {x∈Ω: ¯y(x) =b(x)},

supp ¯μ⁻⊂ {x∈Ω: ¯y(x) =a(x)}, (2.8)

whereμ¯= ¯μ⁺−μ¯⁻ is the Jordan decomposition ofμ.¯ Proof. Let us denote

K_a ={x∈Ω: ¯y(x) =a(x)} and K_b={x∈Ω: ¯y(x) =b(x)}.

For every integerk≥1, we set Ω_k =

x∈Ω:a(x) + 1

k <y(x)¯ < b(x)−1 k

.

From the assumptions (2.2a)−(2.2c) we infer that Ω_k is a nonempty open set for every k sufficiently large.

Let us take an arbitrary element y ∈C₀(Ω_k) such that y∞≤1. We extendy by zero toΩ and denote this extension again byy. Then,y∈C₀(Ω) andy_k= ¯y+¹_ky∈ Yab, hence (2.3c) implies

Ωk

yd¯μ=k

Ω

(y_k−y) d¯¯ μ≤0.

This implies that|μ¯|(Ω_k) = 0 for everyk, therefore

|μ¯|(Ω\(K_a∪K_b)) = lim

k→∞|μ¯|(Ω_k) = 0.

This means that the support of ¯μis contained in K_a∪K_b. It is enough to prove that ¯μis nonnegative onK_b and nonpositive onK_a to conclude (2.8). We show that ¯μis nonpositive onK_a; the proof of the nonnegativity of ¯μonK_b is analogous. Take a numberρsatisfying

0< ρ < 1 2 inf

x∈Ω¯(b(x)−a(x)).

This choice is possible thanks to (2.2a)−(2.2c). Now, we define the sets

Ω_a={x∈Ω: ¯y(x)< a(x) +ρ} and Ω_b ={x∈Ω: ¯y(x)> b(x)−ρ}.

Since ¯y∈C₀(Ω), we get with (2.2a)−(2.2c) thatΩ_a andΩ_b are open sets and K_a ⊂Ω_a, K_b⊂Ω_b and ¯Ω_a∩Ω¯_b=∅.

Lety∈C(K_a)\{0}be a nonnegative function. Using Tietze’s extension theorem, we can extendytoΩ, denoted yagain, in such a way that suppy⊂Ω_a. Moreover, taking max{y(x),0}, we can assume thaty≥0 inΩ. Then, we have

y_ρ = ¯y+ ρ

y∞y∈ Yab.

Indeed, ifx∈Ω_a, theny_ρ= ¯y anda(x)≤y_ρ(x)≤b(x). Ifx∈Ω_a, the definition ofρandΩ_a implies a(x)≤y(x)¯ ≤y_ρ(x)≤y(x) +¯ ρ < a(x) + 2ρ < b(x).

Finally, from suppy ⊂Ω_a, supp ¯μ⊂K_a∪K_b and (2.3c) it follows

Ka

yd¯μ=

Ω

yd¯μ= y∞

ρ

Ω

(y_ρ−y) d¯¯ μ≤0.

Since this inequality holds for every nonnegative functiony∈C(K_a), we conclude that ¯μis nonpositive onK_a,

as desired.

The classical regularity result for ¯uis deduced from the equality (2.3d): ¯u∈W₀^1,s(Ω) for every 1≤s <_n−1ⁿ . In the next section, we will show that ¯u∈H₀¹(Ω).

3. A regularity result for the adjoint state ϕ ¯ and the lagrange multiplier μ ¯

The goal of this section is to prove the following theorem.

Theorem 3.1. Under the Assumptions 2.1 and 2.2, the following regularity result holds:

¯

ϕ,u¯∈H₀¹(Ω)∩L^∞(Ω) and μ¯∈ M(Ω)∩H⁻¹(Ω).

We state two auxiliary lemmas before proving the theorem.

Lemma 3.2. Let μ ∈ M(Ω) be a positive measure with a compact support in Ω and ϕ ∈ W₀^1,s(Ω) be the solution of (2.6). Define

ϕ^∗(x) :=

Ω

g_A(x, ξ) dμ(ξ) ∀x∈Ω,

whereg_Ais Green’s function corresponding to the Dirichlet problem (2.6). Then, we have that ϕ∈L^∞(Ω)⇔ sup

x∈suppμ

ϕ^∗(x)<∞.

Proof. For the case A = −Δ this lemma is proven in [27]. For the general case, let us consider the solution z∈W₀^1,s(Ω) of the Dirichlet problem −Δz=μ inΩ,

z= 0 onΓ.

Observe that the positivity ofμimplies thatϕandzare nonnegative almost everywhere in Ω. Moreover, since A^∗ϕ=Δz= 0 in the open setΩ\suppμandϕ=z= 0 onΓ, we deduce thatϕ, z ∈C( ¯Ω\suppμ). Therefore, givenε >0 we can choose a compact setK_εsuch that suppμ⊂K_ε⊂Ω and

ϕ(x) +z(x)< ε for a.a.x∈Ω\K_ε. (3.1) We know from [31] that there exists a positive numberC_ε such that

1

C_εg(x, ξ)≤g_A(x, ξ)≤C_εg(x, ξ) ∀x, ξ∈K_ε,

wheregdenotes the Green’s function associated with the Dirichlet problem corresponding to−Δ. Analogously toϕ^∗ we define

z^∗(x) =

Ω

g(x, ξ) dμ(ξ).

Integration with respect to μ in the above inequalities and taking into account that μ≥ 0 and suppμ ⊂K_ε yields for allx∈K_ε

1

C_εz^∗(x) = 1 C_ε

Ω

g(x, ξ) dμ(ξ) = 1 C_ε

Kε

g(x, ξ)dμ(ξ)≤

Kε

g_A(x, ξ) dμ(ξ)

=

Ω

g_A(x, ξ) dμ(ξ) =ϕ^∗(x)≤C_ε

Kε

g(x, ξ) dμ(ξ) =C_εz^∗(x).

Sinceϕ=ϕ^∗almost everywhere in Ω, these inequalities imply that ϕ∈L^∞(Ω)⇔ϕ^∗∈L^∞(Ω)⇔z^∗∈L^∞(Ω)⇔ sup

x∈suppμ

z^∗(x)<∞, where the last equivalence is due to a result by Pieper and Vexler [27].

Finally, the inequalities 1

C_εz^∗(x)≤ϕ^∗(x)≤C_εz^∗(x) ∀x∈suppμ⊂K_ε and the above equivalences imply that

x∈suppsup μ

ϕ^∗(x)<∞ ⇔ sup

x∈suppμ

z^∗(x)<∞ ⇔ϕ∈L^∞(Ω).

For any functionϕand α≤β, we denote Proj_[α,β](ϕ)(x) = min{max{α, ϕ(x)}, β}.

Lemma 3.3. Let μ ∈ M(Ω) and let ϕ ∈ W₀^1,s(Ω) for all s < n/(n−1) be the solution of (2.6). Then, Proj_[−M,M](ϕ)∈H₀¹(Ω)for everyM >0.

Proof. This result can be deduced from ([14], Thm. 10.1 and Eq. (2.22)) or ([17], Eq. (7)). For convenience, we provide the reader with a simple proof.

Let us take a sequence of functions{μk}k⊂L²(Ω), such that μ_k μ^∗ inM(Ω) and μkL¹(Ω)≤ μ_M(Ω). Letϕ_k∈H₀¹(Ω) be the unique solution of

A^∗ϕ_k=μ_k inΩ, ϕ_k= 0 onΓ.

Due to the compact embedding of M(Ω) into W^−1,s(Ω) for all s < _n−1ⁿ , we have thatϕ_k → ϕstrongly in W^1,s(Ω). Take M >0 fixed and define ϕ^M_k = Proj_[−M,M](ϕ_k). By the continuity of Proj_[−M,M]: W^1,s(Ω)→ W^1,s(Ω) we also have

limk ϕ^M_k = Proj_[−M,M](ϕ) strongly in W₀^1,s(Ω)∀1≤s < n

n−1· (3.2)

On the other hand, using the uniform ellipticity of the operatorA^∗, we have Λ∇ϕ^M_k ²_L2(Ω)≤

n i,j=1

Ω

a_ij(x)∂_xjϕ^M_k ∂_xiϕ^M_k dx+

Ω

a₀(x)ϕ^M_k ϕ^M_k dx

≤ n i,j=1

Ω

a_ij(x)∂_xjϕ_k∂_xiϕ^M_k dx+

Ω

a₀(x)ϕ_kϕ^M_k dx

=

Ω

μ_kϕ^M_k dx≤ ϕ^M_{k L}^∞_(Ω)μk_L¹_(Ω)≤Mμ_M(Ω).

Therefore the sequence{ϕ^M_k }kis bounded inH₀¹(Ω) and there existϕ_M ∈H₀¹(Ω) and a subsequence of{ϕ^M_k }k, denoted in the same way, such thatϕ^M_k ϕ_M weakly inH₀¹(Ω). Using this fact and (3.2) we readily have that

Proj_[−M,M](ϕ) =ϕ_M ∈H₀¹(Ω).

Proof of Theorem 3.1. Let us consider the Jordan decomposition of ¯μ: ¯μ= ¯μ⁺−μ¯⁻. We also decompose

¯

ϕ=ϕ₀+ϕ₊−ϕ₋

whereϕ₀,ϕ₊ andϕ₋ are the solutions of (2.6) with right hand side equal to ¯y−y_d, ¯μ⁺ and ¯μ⁻, respectively.

Since ¯y−y_d ∈L²(Ω), then we know thatϕ₀∈H₀¹(Ω)∩C( ¯Ω). We will prove that ϕ₊∈L^∞(Ω), the proof of the boundedness ofϕ₋ being analogous, which implies the boundedness of ¯ϕ. Then, theH₀¹(Ω)-regularity of ¯ϕ follows from Lemma3.3. The proof of the theorem is concluded by (2.3b) and (2.3d): ¯μ=A^∗ϕ−¯ y+y¯ _d∈H⁻¹(Ω) and ¯u=−_ν¹ϕ¯∈H₀¹(Ω)∩L^∞(Ω).

Let us prove the boundedness of ϕ₊. First, we observe that ϕ₊ and ϕ₋ are nonnegative functions and ϕ₋∈C( ¯Ω\supp ¯μ⁻). By Lemma3.2, we have thatϕ₊∈L^∞(Ω) if and only ifϕ^∗₊ is upper bounded in supp ¯μ⁺, whereϕ^∗₊ is defined as in Lemma3.2by

ϕ^∗₊(x) :=

Ω

g_A(x, ξ) dμ⁺(ξ) ∀x∈Ω.

We argue by contradiction and we assume that ϕ^∗₊ is not bounded in supp ¯μ⁺. Take 0 < ρ < 1 such that a(x)< ρb(x) for everyx∈Ω. The existence of¯ ρfollows from (2.2a) and (2.2c). We define the compact set

K={x∈Ω: ¯y(x)≥ρb(x)}. By (2.7) and (2.8) we have

ϕ_−C(K)≤C_Kμ¯⁻_M(Ω). Let us set

M =ϕ0_C(K)+C_K¯μ⁻_M(Ω)+ν

a0_L^∞_(Ω)b−a_L^∞_(Ω)+Ab_L^∞_(Ω) .

Since ϕ^∗₊ is not bounded in supp ¯μ⁺, there exists an elementx⁰ ∈supp ¯μ⁺ such thatϕ^∗₊(x⁰)> M. From the positivity of Green’s functiong_Awe deduce, with Fatou’s Lemma, thatϕ^∗₊is lower semicontinuous. Hence, the set {x∈ Ω : ϕ^∗₊(x) > M} is open. Let us takeε > 0 such that ϕ^∗₊(x) > M ∀x∈ B_ε(x⁰) and B_ε(x⁰) ⊂K.

Therefore, we have for almost everyx∈B_ε(x⁰)

¯

ϕ(x) =ϕ₀(x) +ϕ^∗₊(x)−ϕ^∗₋(x)

> M− ϕ0C(K)− ϕ^∗₋C(K)

≥ν

a₀L^∞(Ω)b−aL^∞(Ω)+AbL^∞(Ω) .

We consider the difference ˜y = ¯y−b. There holds ˜y ≤0 and due to the fact that x₀ ∈ supp ¯μ⁺ we have by Proposition2.5that ˜y(x₀) = ¯y(x₀)−b(x₀) = 0 takes its maximum atx=x₀. Now, (2.2a), (2.2d) and the above inequality imply

− n i,j=1

∂_xj[a_ij(x)∂_xi(˜y(x))] = ¯u(x)−a₀(x)(¯y(x)−b(x))−Ab

=−1

νϕ(x)¯ −a₀(x)(¯y(x)−b(x))−Ab

<−a0L^∞(Ω)b−aL^∞(Ω)−a₀(x)(¯y(x)−b(x))≤0

for almost allx∈B_ε(x⁰). Hence, the maximum principle implies that ˜y is constant in the ballB_ε(x⁰), which

contradicts the above inequality.

Remark 3.4. The assumptions (2.2a)−(2.2c) are quite natural for the study of pointwise state constraints.

However, the assumption (2.2d) is unusual, but it is necessary to prove the regularity results of Theorem3.1.

Indeed, Theorem3.1can fail if it does not hold. There are several examples in the literature where the multipliers

¯

μ are Dirac measures; see e.g. ([24], Sect. 8.1), ([25], Sect. 6.2) or [12]. In these examples, Ab ∈ L²(Ω). We provide an example of a linear quadratic control problem such thatAb∈L^p(Ω) for all p <+∞, and however the Lagrange multiplier is a Dirac measure.

ConsiderΩ=B₁(0)⊂R²the unit ball,y_d≡1,ν= 1 andA=−Δ. Let ˜ube the solution of the unconstrained linear quadratic control problem

( ˜P) min

u∈L²(Ω)J(u) =1

2yu−y_dL²(Ω)+1

2u²_L2(Ω). Let ˜y=y_u˜. From the optimality system

−Δ˜y=−ϕ,˜ −Δϕ˜= ˜y−1 inΩ, y˜= ˜ϕ= 0 onΓ, u˜=−ϕ˜in Ω,

we deduce that ˜y ≡ 0. Now, consider the point x₀ = (0,0) and take someb < y(x˜ ₀). We define the problem with a state constraint only at the pointx₀

(P₀) min

u∈Uad

J(u), withU_ad={u∈L²(Ω), y_u(x₀)≤b}.

This problem has a unique solution ¯uwith related state ¯y. Moreover, the state constraint at x₀ is attained:

¯

y(x₀) =b. Indeed, if ¯y(x₀)< b, then ¯μ= 0 and ¯uwould satisfy the optimality system for problem ( ˜P). Therefore,

¯

uwould be a solution of the unconstrained problem ( ˜P), and by uniqueness ¯u= ˜u. This would imply ¯y(x₀)> b, which is a contradiction. The optimality system for (P₀) reads like

−Δ¯y=−ϕ,¯ −Δϕ¯= ¯y−1 + ¯λδ_x0 in Ω, y¯= ¯ϕ= 0 onΓ,u¯=−ϕ¯in Ω,

and ¯λ∈ R, ¯λ >0. Again, if ¯λ = 0 we would have ¯u= ˜uleading to a contradiction. Recall that n= 2; then we have that ¯ϕ∈W^1,s(Ω) for all s <2 and hence ¯ϕ∈L^p(Ω) for all p < ∞, but ¯ϕ∈L^∞(Ω). Therefore, we conclude that−Δ¯y∈L^p(Ω) for allp <∞, but−Δy¯∈L^∞(Ω).

Considerb(x) = ¯y(x) +|x|², where|x|denotes the Euclidean norm ofxinR² and the problem (P) min

u∈Uad

J(u), withU_ad={u∈L²(Ω), y_u(x)≤b(x)∀x∈Ω}.

The function bsatisfies the assumptions (2.2a)−(2.2c), but not (2.2d) because−Δb=−Δ¯y−4. We have that

¯

y(x) ≤ b in ¯Ω since b−y¯ = |x|² and ¯u, ¯y, ¯ϕand ¯μ = ¯λδ_x0 satisfy the optimality system for (P). Since the problem is convex, necessary conditions are also sufficient and hence we have found that the solution does not satisfy the claims of Theorem3.1.

We finish this section establishing a corollary of Theorem3.1that will be useful to prove the error estimates in Section5.

Corollary 3.5. Let us suppose that the Assumptions 2.1 and 2.2 hold, and a_ij ∈ C^0,1(Ω) for 1 ≤ i, j ≤n.

Then, for every open setΩ⊂Ω¯⊂Ω and any1≤p <∞, there exists a constantC >0 independent ofu¯and psuch that

y¯W^2,p(Ω)≤Cpu¯L^∞(Ω). (3.3)

The fact that ¯y ∈ W^2,p(Ω) follows by elliptic regularity. The exact dependence of the constant on pcan be traced for example from Theorem 9.9 in [18].

4. Some extensions

More general formulations for state-constrained optimal control problems of elliptic equations can be found in the literature, see for instance [7] or [11]. The formulation of the problem (P) was given in a simple way for an easier reading of the paper and the technique used in the Proof of Theorem3.1. In this section, we point out under which assumptions Theorem3.1can be extended to more general formulations.

4.1. A general cost functional and control constraints

Instead of the quadratic cost functional considered in the formulation of (P), a more general functional can be treated

J(u) =

Ω

L(x, y_u(x), u(x)) dx,

with L: Ω×R² →Ra Carath´edory function. Some differentiability hypotheses onL must be assumed to get the corresponding optimality system (2.3a)−(2.3d). We make the following assumption: L is of classC¹ with respect to the second and third variables,L(·,0,0)∈L¹(Ω), and for allM >0 there is a functionψ_M ∈L²(Ω)

such that

∂L

∂u(x, y, u) +

∂L

∂y(x, y, u)

≤ψ_M(x), for a.a.x∈Ωand|y|,|u| ≤M.

Additionally, we can consider control constraintsu∈ Uαβ, with

Uαβ={u∈L^∞(Ω) :−∞< α≤u(x)≤β <+∞ for a.a.x∈Ω}

with α, β ∈ R and α < β. In this case, the control problem is in general not convex and we can have local and global solutions. To prove the existence of a solution of the control problem, besides the Assumptions 2.1 and 2.2, we need the convexity ofL with respect to u. Assuming that the Slater condition is fulfilled, if ¯uis a local solution then the optimality system (2.3a)−(2.3d) holds with the following changes. Instead of (2.3b) and (2.3d), we have

⎧⎨

⎩

A^∗ϕ¯= ∂L

∂y(x,y,¯ u) + ¯¯ μ inΩ,

¯

ϕ= 0 onΓ,

(4.1)

Ω

¯ ϕ+∂L

∂u(x,y,¯ u)¯

(u−u) dx¯ ≥0 ∀u∈ Uαβ. (4.2) Relations (2.3b) and (2.3d) played an important role in the Proof of Theorem3.1. Relations (4.1) and (4.2) can be used in a similar way to get the same regularity results. If L is independent of u, then we get from from (4.2) that ¯u(x) =αfor a.a.x∈B_ε(x⁰). IfLdepends onu, then we additionally assume thatLis of class C² with respect touand

∃κ >0 such that ∂²L

∂u²(x, y, u)≥κ for a.a. x∈Ω and ∀y∈R.

Then, (4.2) leads to the formula

¯

u(x) = Proj_[α,β](¯s(x)), where ¯s(x) is (uniquely) defined through the equation

¯

ϕ(x) + ∂L

∂u(x,y(x),¯ ¯s(x)) = 0 for a.a.x∈Ω;

see [1] for more details. The above equality and the assumption ^∂_∂u^2L₂(x, y, u)≥ κimplies that ¯s(x) <0 is as small as needed assuming that ¯ϕ(x)> M forM sufficiently large. To argue as in the Proof of Theorem3.1, the assumption (2.2d) is not enough. We need new one involvingα,β,aandb:α < Abandβ > Aain Ω. Ifaand b are constants satisfyinga <0< b, then the precedent conditions are simplified:α <0< β.

Remark 4.1. The conditions relatingαwithbandβ withacannot be removed. We provide a model example where b is constant and α≡ 0 and the multiplier is not an element in H⁻¹(Ω). Consider problem ( ˜P) as in Remark3.4, but withy_d(x) =|x|⁴−4|x|²+67. The unique solution of this problem is given by ˜u(x) = 16(1−|x|²), whose associate state is ˜y(x) =y_d(x)−64. Defineα(x)≡0, fix some 0< <1, define b=˜y(0) and consider the problem

(P⁰) minJ(u) = 1

2yu−y_d²_L2(Ω)+1

2u²_L2(Ω), u(x)≥0 for allx∈Ω, y(x¯ ₀)≤b

Sinceu=¹₂˜uis admissible for (P⁰), it is clear that this problem has a unique solution ¯u. The optimality system reads like:

−Δ¯y= ¯uinΩ, y= 0 onΓ,−Δϕ¯= ¯y−y_d+ ¯λδx₀ inΩ, ϕ¯= 0 onΓ, u(x) = max(−¯ ϕ(x),¯ 0) in ¯Ω Notice that the state constraint must be attained and the Lagrange multiplier ¯λ∈ R associated to the state constraint must be strictly positive (in other case, the solution of the optimality system would be ˜u, which is impossible). Notice also that due to the radial symmetry of the problem and the fact that−Δy¯= ¯u≥0 in Ω, we gather that ¯y attains its absolute maximum atx₀, with valueb. Therefore ¯uis the solution of problem

(P) minJ(u) = 1

2y_u−y_d²_L2(Ω)+1

2u²_L2(Ω), u(x)≥0 for allx∈Ω, y(x)¯ ≤b for allx∈Ω¯

and the associated multiplier to the state constraint is ¯μ= ¯λδ_x0, which does not satisfy the claims of Theorem3.1.

4.2. A semilinear elliptic equation

We can extend our results to semilinear equations replacing (1.1) Ay+a₀(x, y) =u in Ω,

y= 0 onΓ, with

Ay=− n i,j=1

∂_xj[a_ij(x)∂_xiy].

Here, a₀: Ω×R −→ R is a Carath´eodory function of class C¹ with respect to the second variable, with a(·,0)∈L^p(Ω) for somep > ⁿ₂, and satisfying

⎧⎪

⎪⎨

⎪⎪

⎩

∂a₀

∂y (x, y)≥0 for a.a. x∈Ω and ∀y∈R,

∀M >0 ∃CM >0 s.t. ∂a₀

∂y (x, y)≤C_M for a.a.x∈Ω and∀|y| ≤M.

In this situation, the regularity results of Theorem 3.1 still hold under the assumption (2.2d) and assuming that the functionsa₀(x, a(x)) anda₀(x, b(x)) belong toL^∞(Ω). The proof follows the same steps of the one of Theorem3.1. It is enough to take into account the monotonicity ofa₀ with respect to the second variable and to take

M =ϕ0_C(K)+C_Kμ¯⁻_M(Ω)+ν

Ab_L^∞_(Ω)+a0(x, a(x))_L^∞_(Ω) . Then, we have with ˜y= ¯y−b

Ay˜= ¯u(x)−a₀(x,y(x))¯ −Ab

≤ −1

νϕ(x)¯ −a₀(x, a(x))−Ab <0 inB_ε x⁰ , and we can use again the maximum principle to get the contradiction.

5. Numerical approximation

In this section, we make some additional assumptions on Ω as well as on the coefficients a_ij and on the boundsa, b, which avoid some technicalities and which are necessary to assure a higher regularity of the optimal state and adjoint state. This extra regularity is needed to improve the error estimates.

Assumption 5.1. Hereafter we will suppose thatΩis convex anda_ij ∈C^1,θ( ¯Ω), for some 0< θ≤1.

Under Assumptions 2.1 and 5.1, the solutiony_u of (1.1) belongs toH²(Ω)∩H₀¹(Ω) ifu∈L²(Ω), and there exists a constantC such that

y_uH²(Ω)≤CuL²(Ω) ∀u∈L²(Ω); (5.1) see in Chapter 3 of [20].

Let {Th}h be a quasi-uniform family of triangulations of ¯Ω and letΩ_h be the interior of ∪{T : T ∈ Th}. As usual, we assume that |Ω\Ω_h| ≤ch². This holds if Γ is of classC^1,1 or if Ωis a polygonal or polyhedral domain. For the discretization of the control, the state and the adjoint state we use the space of linear finite elementsY_h0⊂H₀¹(Ω)

Y_h0={y∈C( ¯Ω) :y_h∈P¹(T)∀T ∈ Th, y_h≡0 in ¯Ω\Ω_h}.

For the discrete Lagrange multiplier we use the space Mh ⊂ M(Ω) which is spanned by Dirac measures corresponding to the interior nodes {x_j}ⁿ_j=1^h of the finite element mesh. For every u ∈ L²(Ω), y_h(u) is the unique element inY_h0 such that

a_A(y_h(u), z_h) =

Ωh

uz_hdx ∀zh∈Y_h0,

where a_A: H₀¹(Ω)×H₀¹(Ω) → R denotes the bilinear form associated to the elliptic operator A. Problem (P_h) reads like

(P_h)

⎧⎪

⎨

⎪⎩

minJ_h(u) =1

2y_h(u)−y_d²_L2(Ωh)+ν

2u²_L2(Ωh)

u∈Y_h0, y_h(u)∈ Yab,h, where

Yab,h={y_h∈Y_h0: a(x_j)≤y_h(x_j)≤b(x_j) for allj = 1, . . . , n_h}.

Proposition 5.2. Under Assumptions 2.1, 2.2 and 5.1, problem (P_h)has a unique solution u¯_h ∈ Y_h0, with related state y¯_h =y_h(¯u_h). Moreover, for every h≤h₀, for some h₀ >0, there exist ϕ¯_h ∈Y_h0 and μ¯_h ∈ Mh

such that the following optimality system is satisfied.

a_A(¯y_h, z_h) = (¯u_h, z_h)∀z_h∈Y_h0, (5.2a) a_A(z_h,ϕ¯_h) = (¯y_h−y_d, z_h) +

Ωh

z_hd¯μ_h ∀z_h∈Y_h0, (5.2b)

Ωh

(y_h−y¯_h) d¯μ_h≤0∀yh∈ Yab,h, (5.2c)

ϕ_h+νu¯_h= 0, (5.2d)

where(·,·)denotes the inner product in L²(Ω).

Before proving this proposition, we establish a technical lemma that we will use several times in the sequel.