On unbounded optimal controls in coefficients for ill-posed elliptic dirichlet boundary value problems

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ill-posed elliptic dirichlet boundary value problems

Thierry Horsin, Peter Kogut

To cite this version:

Thierry Horsin, Peter Kogut. On unbounded optimal controls in coefficients for ill-posed elliptic dirichlet boundary value problems. ������ ������������������ ������������. �����: �����������, 2014, 22 (8), pp.3.

�10.15421/141401�. �hal-02946721�

Ïðîáëåìè ìàòåìàòè÷íîãî ìîäåëþâàííÿ òà òåîði¨ äèôåðåíöiàëüíèõ ðiâíÿíü ÓÄÊ 517.95

ON UNBOUNDED OPTIMAL CONTROLS IN

COEFFICIENTS FOR ILL-POSED ELLIPTIC DIRICHLET BOUNDARY VALUE PROBLEMS

T. Horsin^∗, P. I. Kogut^∗∗

∗ Conservatoire National des Arts et Metiers, M2N, IMATH, Case 2D 5000, 292 rue Saint-Martin, 75003 Paris, France ([email protected])

∗∗ Äíiïðîïåòðîâñüêèé íàöiîíàëüíèé óíiâåðñèòåò iì. Îëåñÿ Ãîí÷àðà, ïð. Ãàãàðiíà, 72, 49010, Äíiïðîïåòðîâñüê. E-mail: [email protected]

We consider an optimal control problem associated to Dirichlet boundary value problem for linear elliptic equations on a bounded domain Ω. We take the matrix- valued coecients A(x) of such system as a control in L¹(Ω;R^N×R^N). One of the important features of the admissible controls is the fact that the coecient matrices A(x) are non-symmetric, unbounded on Ω, and eigenvalues of the symmetric part A^sym= (A+A^t)/2may vanish in Ω.

Key words: degenerate elliptic equations, control in coecients, weighted Sobolev spaces, variational convergence.

1. Introduction

The aim of this paper is to study the following optimal control problem (OCP) for a linear elliptic equation with unbounded coecients in the main part of the elliptic operator













MinimizeI(A, y) =∥y−yd∥²_L²_(Ω)+ ˆ

Ω

(∇y, A^sym∇y)_RN dx subject to the constraints

−div(

A(x)∇y)

+a₀(x)y =−divf in Ω, , y= 0 on∂Ω,

(1.1)

where the matrix A = A^sym+A^skew ∈ L¹(Ω;S^Nsym)⊕L^2p(Ω;S^N_skew) is adopted as a control, f ∈ D^′(Ω;R^N) and y_d ∈ L²(Ω)are given distributions, p ≥1, and a₀ ∈ L^∞(Ω) is such that a₀(x) ≥ α > 0 almost everywhere in Ω. We dene a class of admissible controls Aad as a nonempty compact subset of L¹(Ω;S^N_sym)⊕ L^2p(Ω;S^N_skew) such that for every A∈Aad we have

A^∗(x)≼A^skew(x)≼A^∗∗(x) a.e. in Ω, ζad(x)I ≤A^sym(x)≤β(x)I a. e. in Ω,

ˆ

Ω

A^sym(x)dx=M,

c

⃝T. Horsin, P. I. Kogut, 2014

where M ∈ S^Nsym and A^∗, A^∗∗ ∈ L^2p(Ω;S^N_skew) are given nonzero matrices, β ∈ L¹(Ω),β≥ζad, andζ_ad⁻¹∈L^2q(Ω)for q=p/(p−1).

This kind of problems naturally appears in the optimal design theory for linearized elliptic boundary value problems. Their characteristic feature of the problem (1.1) is the fact that the existence, uniqueness, and variational properties of the weak solution to (1.1) are drastically dierent from the corresponding properties of solutions to the elliptic equations with coercive L^∞-matrices in coecients. Typically, in such cases, the boundary value problem (1.1) with unbounded matrices A ∈ Aad may admit many or even innitely many weak solutions [21,22].

Optimal control in coecients for partial dierential equations is a classical subject initiated by Lurie [17], Lions [16], and Pironneau [19]. Since the range of such optimal control problems is very wide, including as well optimal shape design problems, some problems originating in mechanics and others, this topic has been widely studied by many authors. However, most of these results and methods rely on linear PDEs with bounded coecients in the main part of elliptic operators, while only a few articles deal with with unbounded and degenerate coecients, see [1, 3,711,13,14].

The principal feature of OCP (1.1) is that the corresponding boundary value problem (1.1)2(1.1)3 is ill-possed and the class of admissible controls A ∈ A_ad belongs toL¹(Ω;M^N). We note that these assumptions on the class of admissible controls together withL^2p-properties of the skew-symmetric parts are essentially weaker than they usually are in the literature. In Sections 2 and 3, we discuss some auxiliary results that are closely related with the correctness of the notion of weak solutions to the above boundary value problem and describe a mathematical background for convergence formalism in variable Sobolev spaces.

We give the precise denition of the class of admissible controls in Section 4 and, using the direct method in the Calculus of variations, we show that a set of optimal pairs to the above problem is nonempty provided the so-called non- triviality condition on the set of admissible solutions. Since this condition is closely related with the existence of weak solutions to the boundary value problem (1.1)2(1.1)3, we show in Section 5 that this question can be solved due to the approximation approach.

2. Notation and Preliminaries

Let Ω be a bounded open connected subset of R^N (N ≥ 2) with Lipschitz boundary ∂Ω. Let χ_E be the characteristic function of a subset E ⊂ Ω, i.e.

χE(x) = 1 ifx∈E, andχE(x) = 0 if x̸∈E.

LetM^N be the set of allN×N real matrices. We denote byS^N_skewthe set of all skew-symmetric matricesC = [c_ij]^N_i,j=1, i.e.,C is a square matrix whose transpose is also its opposite. Thus, if C ∈ S^N_skew then cij = −cji and, hence, cii = 0. Therefore, the set S^N_skew can be identied with the Euclidean spaceR^N(N2−1). Let

S^Nsym be the set of allN×N symmetric matrices, which are obviously determined byN(N+1)/2scalars. For each matrixB ∈M^N, we have a unique representation

B =Bsym+Bskew, (2.1)

where B_sym := ¹₂(

B+B^t)

∈ S^Nsym and B_skew := ¹₂(

B−B^t)

∈ S^N_skew. In the sequel, we will always identify each matrixB ∈M^N with its decomposition in the form (2.1).

Letp, q∈[1,∞]be given real numbers such that ¹_p+¹_q = 1. LetL^2p(

Ω;S^N_skew) be the normed space of measurable2p-integrable functions whose values are skew- symmetric matrices.

LetA(x)and B(x)be given matrices such thatA, B∈L^2p(Ω;S^N_skew). We say that these matrices are related by the binary relation ≼on the setL^2p(Ω;S^N_skew) (in symbols,A(x)≼B(x) a.e. in Ω), if

L^N(

∪^Ni=1∪^Nj=i+1

{

x∈Ω : aij(x)> bij(x) })

= 0. (2.2)

Here, L^N(E) denotes the N-dimensional Lebesgue measure of E ⊂ R^N dened on the completed borelianσ-algebra.

Letα ∈ R be a xed positive value. Let ζad and β be givenL¹(Ω)-functions satisfying the properties

β > ζad ≥0 a.e. in Ω, ζ_ad⁻¹∈L^2q(Ω) for q =p/(p−1), p≥1, (2.3) β, ζ_ad : Ω→R¹+ are smooth functions along the boundary ∂Ω, (2.4)

ζ_ad=β=α on ∂Ω. (2.5)

ByM^β_ζ

ad(Ω)we denote the set of all matricesA= [ai j(·) ]∈L¹(Ω;S^Nsym)such that

ζadI ≤A(x)≤β(x)I a. e. in Ω (2.6) Here, I is the identity matrix in R^N×N, and (2.6) should be considered in the sense of quadratic forms dened by (Aξ, ξ)_RN for ξ ∈ R^N. Therefore, condition (2.6) implies the following inequalities:

if A∈M^β_ζ

ad(Ω), then ∥A∥L¹(Ω;S^Nsym)≤ ∥β∥L¹(Ω)<+∞, (2.7) ζ_ad(x)∥ξ∥²_RN ≤(A(x)ξ, ξ)_RN a. e. in Ω, ∀ξ ∈R^N (2.8)

∥A^−1/2(x)ξ∥²_R^N ≤ζ_ad⁻¹(x)∥ξ∥²_R^N a. e. in Ω, ∀ξ∈R^N, (2.9) and, therefore,

A^−1/2∈L^4q(Ω;S^N_sym) and ∥A^−1/2∥_L^4q_(Ω;S^N_sym)≤√

∥ζ_ad⁻¹∥L^2q(Ω). (2.10)

To each matrix A ∈ M^β_ζ

ad(Ω) ⊂ L¹(Ω;S^Nsym) we will associate two weighted Sobolev spaces: W_A(Ω) =W(Ω;A dx) and H_A(Ω) =H(Ω;A dx),whereW_A(Ω) is the set of functions y∈W₀^1,1(Ω)for which the norm

∥y∥A= (ˆ

Ω

(y²+ (∇y, A(x)∇y)_RN

)dx )1/2

(2.11) is nite, and H_A(Ω) is the closure of C₀^∞(Ω) in W_A(Ω)-norm. It is well-known that due to the inequality (2.8) the spaceWA(Ω)is complete with respect to the norm ∥ · ∥A (see [11]). It is clear that HA(Ω)⊂WA(Ω), andWA(Ω),HA(Ω)are Hilbert spaces.

For our further analysis, we make use of the following observation.

Remark 2.1. If N ≥ 2 and there exists a value ν ∈ (_N

2,+∞)

such that u^−ν ∈ L¹(Ω), then the expressions (for more details see [4, pp.46]):

||y||1,Hu = [ˆ

Ω

u|∇y|²dx ]1/2

and (2.12)

∥y∥2,Hu = (ˆ

Ω

(y²+u|∇y|²) dx

)1/2

(2.13) can be considered as equivalent norms on H_u := cl_∥·∥2,HuC₀^∞(Ω). Moreover, in this case the embedding Hu ,→L²(Ω)is compact. Taking this fact and denition of the classM^β_ζ

ad(Ω)into account, we deduce that the norm∥·∥A, given by (2.11), is equivalent to the following one

∥y∥1,A= (ˆ

Ω

(∇y, A(x)∇y)_RN dx )_1/2

(2.14) on HA(Ω)provided A∈M^β_ζ

ad(Ω),ζ_ad⁻¹ ∈L^2q(Ω)withq=p/(p−1), where p∈[1,∞) if N ∈ {2,3}, and 1≤p≤ N

N −4 if N ≥4. (2.15) Indeed, since the conditions (2.15) implies the fullment of inequality q=p/(p− 1)> N/4, it follows that ν := 2q∈(_N

2,+∞)

andζ_ad^−ν ∈L¹(Ω). Let A = A_sym+A_skew ∈ L¹(

Ω;M^N)

be a given matrix matrix such that Askew ∈ L^2p(

Ω;S^N_skew)

. In what follows, we associate with A the bilinear skew- symmetric form

Φ(y, v)_A= ˆ

Ω

(∇v, A_skew(x)∇y)

R^Ndx, ∀y, v∈W_Asym(Ω), and introduce the matrix C(x)∈S^N_skew following the rule

C(x) =A⁻_sym^1/2(x)Askew(x)A⁻_sym^1/2(x) a.e. in Ω. (2.16)

It is easy to see that C ∈ L²(

Ω;S^N_skew)

. Indeed, by the Cauchy-Bunyakowsky inequality and estimate (2.10), we have

∥C∥²_L2(Ω;S^N_skew)≤ ˆ

Ω

∥A_skew∥²_L(R^N_,R^N₎∥A⁻

1

sym2 ∥⁴_L(R^N_,R^N₎dx

≤ (ˆ

Ω

∥A_skew∥^2p_L(RN,R^N)dx

)1/p(ˆ

Ω

∥A⁻

1

sym2 ∥^4q_L(RN,R^N)dx )1/q

=∥A_skew∥²_L2p(Ω;S^Nskew)∥A⁻

1

sym2 ∥⁴_L4q(Ω;S^Nsym)

≤ ∥ζ_ad⁻¹∥²_L^2q_(Ω)∥A_skew∥²_L2p(Ω;S^N_skew)<+∞. (2.17) Hence, the form Φ(y, v)_Ais unbounded onW_Asym(Ω), in general.

However, if we temporary assume that C ∈ L^∞(Ω;S^N_skew), then the bilinear form Φ(·,·)_Ais obviously bounded on W_Asym(Ω). In this case we have

ˆ

Ω

(∇φ, A_skew∇y)

R^Ndx² = ˆ

Ω

(A

1

sym2 ∇φ, [

A⁻

1

sym2 A_skewA⁻

1

sym2

] A

1

sym2 ∇y)

R^Ndx ²

≤ ∥C∥L^∞(Ω;S^N_skew)

ˆ

Ω

|A

1

sym2 ∇φ|²_R^Ndx ˆ

Ω

|A

1

sym2 ∇y|²_R^Ndx

≤ ∥C∥_L∞(Ω;S^N_skew)∥φ∥Asym∥y∥Asym.

In order to deal with the case C ̸∈ L^∞(Ω;S^N_skew), we notice that the value Φ(y, v)A is always nite providedy∈WAsym(Ω)and φ∈C₀^∞(Ω). Indeed,

|Φ(y, v)_A|² :=

ˆ

Ω

(∇φ, A_skew∇y)

R^Ndx

² ≤ ∥φ∥²_C1(Ω)

(ˆ

Ω

|A_skew∇y|_R^Ndx )₂

≤ ∥φ∥²_C1(Ω)

ˆ

Ω

A_skewA⁻

1

sym2 ²

L(R^N,R^N)dx ˆ

Ω

A

1

sym2 ∇y²

R^Ndx

≤ ∥φ∥²_C1(Ω)

ˆ

Ω

A_skew²

L(R^N,R^N)ζ_ad⁻¹dx ˆ

Ω

(∇y, A_sym∇y)

R^Ndx

≤ ∥φ∥²_C1(Ω)∥y∥²A

(ˆ

Ω

∥A_skew∥^2p_L(RN,R^N)dx

)1/p(ˆ

Ω

ζ_ad^−qdx )1/q

≤ ∥φ∥²_C1(Ω)∥y∥²A|Ω|^1/2q∥ζ_ad⁻¹∥L^2q(Ω)∥A_skew∥_L²2p(Ω;S^N_skew) <+∞. Hence, if C ∈ L²(

Ω;S^N_skew)

then the integral ˆ

Ω

(∇φ, A_skew(x)∇y)

R^Ndx is well dened for every y ∈ W_Asym(Ω) and φ ∈ C₀^∞(Ω). Taking this fact into account, we set

[y, φ]_A= ˆ

Ω

(∇φ, A_skew(x)∇y)

R^Ndx= ˆ

Ω

(A^1/2_sym∇φ, C(x)A^1/2_sym∇y)

R^Ndx,

∀y∈W_Asym(Ω), ∀φ∈C₀^∞(Ω),

where the matrixC is dened by (2.16), and introduce of the following notion.

Let V_Asym(Ω) be some intermediate space with H_Asym(Ω) ⊆ V_Asym(Ω) ⊆ W_Asym(Ω).

Denition 2.1. Let A = A_sym+A_skew ∈ L¹(

Ω;M^N)

be a given matrix such thatA_skew ∈L^2p(

Ω;S^N_skew)

. We say that an elementy∈V_Asym(Ω)belongs to the setD(VAsym) if

ˆ

Ω

(∇φ, A_skew∇y)

R^Ndx

≤c(y, A) (ˆ

Ω

φ²dx+ ˆ

Ω

(∇φ, A_sym∇φ)_RNdx )1/2

=c(y, A)∥φ∥Asym, ∀φ∈C₀^∞(Ω) (2.18) with some constantc depending on y andA.

As a result, if y∈D(VAsym) then the mappingφ7→[y, φ]Acan be dened for all φ∈HAsym(Ω)using (2.18) and the standard rule

[y, φ]A= lim

ε→0[y, φε]A, (2.19) where{φε}_ε>0⊂C₀^∞(Ω)and φε →φstrongly inHAsym(Ω)(it is the case where we essentially use the fact that C₀^∞(Ω) is dense in H_Asym(Ω)). In particular, if y ∈ D(H_Asym), then we can dene the value [y, y]_A and this one is nite for every y ∈ D(HAsym), although the "integrand" (

∇y, Askew∇y)

R^N needs not be integrable onΩ, in general.

Let f : Ω→R be a function ofL¹(Ω). We dene T V(f) :=

ˆ

Ω

|Df|= sup {ˆ

Ω

f(∇, φ)_RNdx :

φ= (φ1, . . . , φN)∈C₀¹(Ω;R^N), |φ(x)| ≤1 forx∈Ω }

, where(∇, φ)_RN =∑_N

i=1

∂φi

∂xi.

According to the Radon-Nikodym theorem, ifT V(f)<+∞then the distribu- tionDf is a measure and there exist a vector-valued function∇f ∈[L¹(Ω)]^N and a measure Dsf, singular with respect to the N-dimensional Lebesgue measure L^N⌊Ωrestricted toΩ, such that Df =∇fL^N⌊Ω +D_sf.

Denition 2.2. A functionf ∈L¹(Ω)is said to have a bounded variation in Ω if T V(f) <+∞. By BV(Ω) we denote the space of all functions in L¹(Ω)with bounded variation, i.e.BV(Ω) ={

f ∈L¹(Ω) : T V(f)<+∞} .

Under the norm∥f∥BV(Ω)=∥f∥L¹(Ω)+T V(f), BV(Ω)is a Banach space. For our further analysis, we need the following properties ofBV-functions (see [5]):

Proposition 2.1. Let {f_k}^∞_k=1 be a sequence in BV(Ω) strongly converging to some f inL¹(Ω)and satisfying condition sup_k∈NT V(fk)<+∞. Then

f ∈BV(Ω) and T V(f)≤lim inf

k→∞ T V(f_k)

and for every bounded sequence {f_k}^∞_k=1 ⊂ BV(Ω) there exists a subsequence, still denoted by fk, and a functionf ∈BV(Ω)such thatfk→f inL¹(Ω).

Let I_k :Uk×Yk → R be a cost functional, Yk be a space of states, and Uk

be a space of controls. Let min{I_k(u, y) : (u, y)∈Ξ_k} be a parameterized OCP, where

Ξ_k⊂ {(u_k, y_k)∈Uk×Yk : u_k ∈U_k, I_k(u_k, y_k)<+∞}

is a set of all admissible pairs linked by some state equation. Hereinafter we always associate to such OCP the corresponding constrained minimization problem:

(CMPk) :

⟨

(u,y)inf∈Ξ_kIk(u, y)

⟩

. (2.20)

Since the sequence of constrained minimization problems (2.20) lives in variable spaces Uk×Yk, we assume that there exists a Banach space U×Ywith respect to which a convergence in the scale of spaces {Uk×Yk}_k∈N is well dened (for the details, we refer to [12, 20]). In the sequel, we use the following notation for this convergence (uk, yk)−→^µ (u, y) inUk×Yk. Moreover, we assume that every bounded sequence in variable space Uk×Yk is sequentially compact with respect to the µ-convergence.

In order to study the asymptotic behavior of a family of(CMPk), the passage to the limit in (2.20) as the parameter k tends to +∞ has to be realized. The expression passing to the limitmeans that we have to nd a kind of limit cost functional I and limit set of constraints Ξwith a clearly dened structure such that the limit object ⟨

inf_(u,y)∈ΞI(u, y)⟩

may be interpreted as some OCP.

Following the scheme of the direct variational convergence [12], we adopt the following denition for the convergence of minimization problems in variable spaces.

Denition 2.3. A problem ⟨

inf_(u,y)∈ΞI(u, y)⟩

is the variational µ-limit of the sequence (2.20) as k→ ∞, if and only if the following conditions are satised:

(d) If sequences {kn}_n∈N and {(un, yn)}_n∈N are such that kn → 0 as n → ∞, (un, yn)∈Ξkn ∀n∈N, and (un, yn)−→^µ (u, y) inUkn×Ykn, then

(u, y)∈Ξ; I(u, y)≤lim inf

n→∞ I_kn(u_n, y_n); (2.21) (dd) For every (u, y) ∈Ξ ⊂U×Y, there are an integer k⁰ > 0 and a sequence

{(u_k, y_k)}_k∈N (called aΓ-realizing sequence) such that

(u_k, y_k)∈Ξ_k, ∀k≥k⁰, (u_k, y_k) −→^µ (u,b y)b in Uk×Yk, (2.22) I(u, y)≥lim sup

k→∞ I_k(u_k, y_k). (2.23) Then the following result takes place [12].

Theorem 2.1. Assume that the constrained minimization problem

⟨ inf

(u,y)∈Ξ0

I₀(u, y)

⟩ (2.24)

is the variationalµ-limit of sequence (2.20) in the sense of Denition 2.3 and this problem has a nonempty set of solutions

Ξ^opt₀ :=

{

(u⁰, y⁰)∈Ξ0 : I0(u⁰, y⁰) = inf

(u,y)∈Ξ0

I0(u, y) }

.

For every k ∈ N, let (u⁰_k, y_k⁰) ∈ Ξ_k be a minimizer of I_k on the corresponding set Ξk. If the sequence {(u⁰_k, y⁰_k)}k∈N is relatively compact with respect to the µ- convergence in variable spaces Uk×Yk, then there exists a pair (u⁰, y⁰) ∈ Ξ^opt₀ such that

(u⁰_k, y⁰_k) −→^µ (u⁰, y⁰) in Uk×Yk, (2.25) inf

(u,y)∈Ξ0

I₀(u, y) =I₀( u⁰, y⁰)

= lim

k→∞I_k(u⁰_k, y⁰_k) = lim

k→∞ inf

(uk,yk)∈Ξk

I_k(u_k, y_k).

(2.26) 3. Weak Convergence in Variable L²-Spaces Associated with S^Nsym-Valued Radon Measures

By a nonnegative Radon measure on Ωwe mean a nonnegative Borel measure which is nite on every compact subset ofΩ. The space of all nonnegative Radon measures on Ω will be denoted by M+(Ω). According to the Riesz theory, each Radon measure µ ∈M+(Ω) can be interpreted as an element of the dual of the space C₀(Ω) of all continuous functions with compact support. Let M(Ω;S^Nsym) denote the space of allS^Nsym-valued Borel measures. Thenµ= [µ_ij]∈M(Ω;S^Nsym)

⇔ µij ∈C₀^′(Ω),i, j= 1, . . . , N.

Let µ and the sequence {µ_k}_k∈N be matrix-valued Radon measures. We say that{µ_k}_k∈Nweakly-∗ converges to µinM(Ω;S^Nsym)if

klim→∞

ˆ

Ω

φ·dµ_k= ˆ

Ω

φ·dµ ∀φ∈C₀(Ω;S^N_sym).

A typical example of such measures is

dµ_k=A_k(x)dx, dµ=A(x)dx, (3.1) where A_k, A∈L¹(Ω;S^N_sym) and A_k→A in L¹(Ω;S^N_sym). (3.2) Hereinafter we suppose that the measuresµand{µ_k}_k∈Nare dened by (3.1) (3.2). Thenµ_k ⇀ µ^∗ inM(Ω;S^Nsym). Further, we will use L²(Ω, A dx)^N to denote the Hilbert space of measurable vector-valued functionsf ∈R^N on Ωsuch that

∥f∥L²(Ω,A dx)^N = (ˆ

Ω

(f, A(x)f)_RNdx )1/2

<+∞. We say that a sequence {

vk ∈L²(Ω, Akdx)^N}

k∈Nin is bounded if lim sup

k→∞

ˆ

Ω

(vk, Ak(x)vk)_RN dx <+∞.

Denition 3.1. A bounded sequence {

v_k∈L²(Ω, A_kdx)^N}

k∈N is weakly con- vergent to a function v∈L²(Ω, A dx)^N in variable spaceL²(Ω, A_kdx)^N if

klim→∞

ˆ

Ω

(φ, A_k(x)v_k)_RN dx= ˆ

Ω

(φ, A(x)v)_RN dx ∀φ∈C₀^∞(Ω)^N. (3.3) Denition 3.2. A sequence {

v_k∈L²(Ω, A_kdx)^N}

k∈N is said to be strongly convergent to a function v∈L²(Ω, A dx)^N if

klim→∞

ˆ

Ω

(b_k, A_k(x)v_k)_RN dx= ˆ

Ω

(b, A(x)v)_RN dx (3.4) whenever b_k⇀ binL²(Ω, A_kdx)^N ask→ ∞.

Remark 3.1. Note that in the caseAk ≡A, Denitions 3.13.2 leads to the usual notion of convergence in weighted Hilbert space L²(Ω, A dx)^N.

The main properties of the weak and strong convergences in L^p(Ω, dµε) can be expressed as follows (see [11] for the details):

Proposition 3.1. If a sequence {

v_k ∈L²(Ω, A_kdx)^N}

k∈N is bounded and the condition (3.2) holds true, then it contains a weakly convergent subsequence in L²(Ω, Akdx)^N.

Proposition 3.2. If the sequence{

v_k∈L²(Ω, A_kdx)^N}

k∈Nconverges weakly to v∈L²(Ω, A dx)^N and the condition (3.2) holds true, then

lim inf

k→∞

ˆ

Ω

(v_k, A_k(x)v_k)_RN dx≥ ˆ

Ω

(v, A(x)v)_RN dx. (3.5) Proposition 3.3. Assume the condition (3.2) holds true. Then the weak conver- gence of a sequence {

vk∈L²(Ω, Akdx)^N}

k∈N to v∈L²(Ω, A dx)^N and

klim→∞

ˆ

Ω

(v_k, A_k(x)v_k)_RN dx= ˆ

Ω

(v, A(x)v)_RN dx (3.6) are equivalent to the strong convergence of {v_k}_k∈N in L²(Ω, A_kdx)^N to v ∈ L²(Ω, A dx)^N.

In what follows, we make use of the following result.

Lemma 3.1. Let A^skew ∈L^2p(Ω;S^N_skew) be a given matrix, and let{ A^sym_k }

k∈N⊂ M^β_ζ

ad(Ω) be a sequence such that

A^sym_k →A^sym₀ in L¹(Ω;S^Nsym). (3.7) Let φ∈C₀^∞(Ω) be an arbitrary test function. Then

v_k :=(

A^sym_k )₋₁

∇φ→(A^sym₀ )⁻¹∇φ=:v0 and w_k:=(

A^sym_k )₋₁

A^skew∇φ→(A^sym₀ )⁻¹A^skew∇φ=:w0

strongly in variable space L²(Ω, A^sym_k dx)^N ask→ ∞.

Proof. Letφ∈C₀^∞(Ω)be an arbitrary test function. Since

∥v_k∥²_L²_(Ω,A^sym

k dx)^N = ˆ

Ω

(v_k, A^sym_k v_k)

R^N dx= ˆ

Ω

(∇φ,(

A^sym_k )₋1

∇φ )

R^N dx

≤ ˆ

Ω

|∇φ|²_R^Nζ_ad⁻¹dx≤ ∥φ∥²_C¹_(Ω)|Ω|^p+12p ∥ζ_ad⁻¹∥L^2q(Ω)<+∞,

∥w_k∥²_L2(Ω,A^sym_k dx)^N = ˆ

Ω

(w_k, A^sym_k w_k)

R^N dx

= ˆ

Ω

(

A^skew∇φ,(

A^sym_k )₋1

A^skew∇φ )

R^N dx

≤ ˆ

Ω

ζ_ad⁻¹|A^skew∇φ|²dx≤ ∥φ∥²_C¹_(Ω)∥ζ_ad⁻¹∥L^q(Ω)∥A^skew∥²_L2p(Ω;S^Nskew)<+∞

and, for each ψ∈C₀^∞(Ω), ˆ

Ω

(∇ψ, A^sym_k v_k)

R^N dx= ˆ

Ω

(∇ψ, A^sym_k (

A^sym_k )₋₁

∇φ )

R^N dx

= ˆ

Ω

(∇ψ,∇φ)_RN dx= ˆ

Ω

(∇ψ, A^sym₀ v₀)_RN dx, ˆ

Ω

(∇ψ, A^sym_k w_k)

R^N dx= ˆ

Ω

(∇ψ, A^sym_k (

A^sym_k )₋1

A^skew∇φ )

R^N dx

= ˆ

Ω

(∇ψ, A^skew∇φ )

R^N dx= ˆ

Ω

(∇ψ, A^sym₀ w₀)_RN dx, (3.8) it follows that the sequences {v_k}_k∈N and {w_k}_k∈N are bounded and weakly convergent in variable space L²(Ω, A^sym_k dx)^N to vector-valued functions v0 ∈ L²(Ω, A^sym₀ dx)^N and w₀ ∈L²(Ω, A^sym₀ dx)^N, respectively.

In order to show that the sequence {v_k}_k∈N is strongly convergent to v₀ :=

(A^sym₀ )⁻¹∇φ, we make use of Proposition 3.3. Following this assertion, it is enough to prove the equality

klim→∞

ˆ

Ω

(vk, A^sym_k vk

)

R^N dx= lim

k→∞

ˆ

Ω

(∇φ,(

A^sym_k )₋₁

∇φ)

R^N dx

= ˆ

Ω

(∇φ,(A^sym₀ )⁻¹∇φ )

R^N dx= ˆ

Ω

(v0, A^sym₀ v0)_RN dx. (3.9) In view of estimate(

∇φ,(

A^sym_k )₋₁

∇φ )

R^N ≤ ∥φ∥²_C¹_(Ω)ζ_ad⁻¹ <+∞, ∀k∈N,the sequence {(

∇φ,(

A^sym_k )₋1

∇φ )

R^N

}

k∈N ie equi-integrable. On the other hand, property (3.7) implies that, within a subsequence, we have the pointwise conver- gence(

A^sym_k )₋1

→(A^sym₀ )⁻¹almost everywhere inΩ. Hence, up to a subsequence, (∇φ,(

A^sym_k )₋1

∇φ )

R^N →(

∇φ,(A^sym₀ )⁻¹∇φ )

R^N a.e. in Ω.

Thus, the equality (3.9) is a direct consequence of Lebesgue Dominated Theorem, and hence,

(A^sym_k )₋1

∇φ→(A^sym₀ )⁻¹∇φ strongly in L²(Ω, A^sym_k dx)^N ∀φ∈C₀^∞(Ω).

Following the same arguments, it can be shown that

klim→∞

ˆ

Ω

(w_k, A^sym_k w_k)

R^N dx=− lim

k→∞

ˆ

Ω

(∇φ, A^skew(

A^sym_k )₋1

A^skew∇φ )

R^N dx

=− ˆ

Ω

(∇φ, A^skew(A^sym₀ )⁻¹A^skew∇φ )

R^N dx= ˆ

Ω

(w₀, A^sym₀ w₀)_RN dx.

Combining this fact with relation (3.8), by Proposition 3.3 we have:

(A^sym_k )₋1

A^skew∇φ→(A^sym₀ )⁻¹A^skew∇φ strongly in L²(Ω, A^sym_k dx)^N

∀φ∈C₀^∞(Ω).The proof is complete.

4. Setting of the Optimal Control Problem

Letp≥1be a given exponent and letf : Ω→R^N be a vector-valued function such that f ∈ L^4p/(p+1)(Ω;R^N). Let M ∈ S^N_sym be a constant matrix satisfying the condition

(M ξ, ξ)_RN ≥m∥ξ∥²_R^N for some m >0.

The optimal control problem we consider in this paper is to minimize the discre- pancy (tracking error) between a given distribution y_d ∈L²(Ω)and a solution y of the Dirichlet boundary value problem for the linear elliptic equation

−div(

A(x)∇y)

+a₀(x)y =−divf in Ω, (4.1)

y= 0 on ∂Ω. (4.2)

by choosing an appropriate matrix-valued control A(x) = Asym(x) +Askew(x).

Here,a0 ∈L^∞(Ω)is a given function such thata0(x)≥α >0almost everywhere inΩ.

More precisely, we are concerned with the following OCP MinimizeI(A, y) =∥y−yd∥²_L²_(Ω)+

ˆ

Ω

(∇y, Asym∇y)_RN dx (4.3) subject to the constraints (4.1)(4.2) withA∈Aad ⊂L¹(Ω;M^N). (4.4) In order to dene the class of admissible controls A_ad, we begin with some preliminaries. Let A^∗, A^∗∗ ∈ L^2p(Ω;S^N_skew) be given nonzero matrices such that A^∗ ≼A^∗∗ a.e. in Ω, let c be a given positive constant, and let Qbe a nonempty