On an Optimal 𝐿¹-Control Problem in Coefficients for Linear Elliptic Variational Inequality

Abstract

We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficients 𝐴(𝑥) in the main part of the elliptic operator as controls in 𝐿¹(Ω;ℝ^{𝑁(𝑁+1)/2}). Since the eigenvalues of such matrices may vanish and be unbounded in Ω, it leads to the “noncoercivity trouble.” Using the concept of convergence in variable spaces and following the direct method in the calculus of variations, we establish the solvability of the optimal control problem in the class of the so-called 𝐻-admissible solutions.

1. Introduction

The aim of this work is to study the existence of optimal controls in the matrix-valued coefficients associated with a linear degenerate elliptic variational inequality with homogeneous Dirichlet boundary conditions. The controls are taken as the matrix 𝐴∈𝐿¹(Ω;ℝ^{𝑁(𝑁+1)/2}) of the coefficients in the main part of the elliptic operator. The most important feature of such controls is the fact that the eigenvalues of the matrix 𝐴 may either vanish on subsets with zero Lebesgue measure or be unbounded. In this case, the precise answer for the question of existence or nonexistence of optimal solutions heavily depends on the class of chosen admissible controls. Using the direct method in the calculus of variations, we discuss the solvability of this optimal control problem in the class of the so-called 𝐻-admissible solutions (see, for instance, [1, 2]). It should be emphasized that in contrast to the paper [3], we do not make use of any relaxations for the original optimal control problem. We note that there are many physical phenomena related to the mathematical theory of cloaking, design of advanced materials, behaviour of the optical meta-materials in the context of cloaking, and others which lead to the appearance of variational inequalities and matrices in the main part of elliptic operators with degenerate spectrum (see, for instance, [4–8]). In particular, the mathematical models of equilibrium of continuous media which are “perfect” insulators or “perfect” conductors (see [9]) need eigenvalues of the matrix 𝐴 either to vanish somewhere or to be unbounded.

Also similar matrices with degenerate eigenvalues arise in an intensively studied nowadays area of complex dynamical networks [10–12]. Namely, such matrices describe an inner configuration of networks; that is, they can be interpreted as connection matrices among nodes [10, 11]. In fact, many properties of complex networks are mainly determined by their configuration matrices, in particular, such as network synchronizability as well as robustness and fragility of a network in synchronization. This fact could be a good motivation to work out techniques for achieving synchronization for dynamical networks which cannot synchronize, taking configuration matrices as controls. Such situation would be similar to the one described in the present paper, when varying a chosen control leads to changing a control object’s structure. Hence, one of the main goals of the paper is to construct a mathematical framework for dealing with problems of this kind.

It is worth noticing that even though numerous articles (see, for instance, [13–18] and references therein) are devoted to variational and nonvariational approaches to problems related to linear and nonlinear elliptic systems, only few deal with optimal control problems for degenerate partial differential equations and variational inequalities (see e.g., [1, 2, 19–23]).

In this paper, we deal with an optimal control problem in coefficients for the following linear degenerate elliptic variational inequality: ^𝑁_{𝑖,𝑗=1Ω}𝑎_𝑖𝑗(𝑥)𝜕𝑦𝜕𝑥_𝑗𝜕𝑣𝜕𝑥_𝑖−𝜕𝑦𝜕𝑥_𝑖𝑑𝑥+_Ω𝑦≥𝑣−𝑦𝑑𝑥_Ω𝑓𝑣−𝑦𝑑𝑥∀𝑣∈𝐾_𝐴. (1) Here,𝑓∈𝐿²(Ω)is a given distribution,𝐾_𝐴is a convex closed subset of the space𝑊₀^1,1(Ω), and𝐴is a measurable nonnegative square symmetric matrix on a bounded open domainΩ⊂ℝ^𝑁(𝑁≥2) such that𝐴∈𝐵𝑉(Ω;ℝ^{𝑁(𝑁+1)/2}).

We furnish this control object by the following cost functional: 𝐼𝐴,𝑦=_Ω𝑦(𝑥)−𝑦_𝑑(𝑥)²+𝑑𝑥_Ω∇𝑦(𝑥),𝐴(𝑥)∇𝑦(𝑥)_ℝ𝑁𝑑𝑥+‖𝐴‖_𝐵𝑉(Ω). (2)

The characteristic feature of this optimal control problem is the fact that the eigenvalues of admissible matrices𝐴may vanish and be unbounded on subdomains of Ω with zero Lebesgue measure. Indeed, we assume that for every admissible matrix𝐴there exist two nonnegative 𝐿¹(Ω)-functions 𝜁 and 𝛽 such that 𝜁⁻¹∈𝐿¹(Ω), 𝜁⁻¹∉𝐿^∞(Ω), and 𝜁(𝑥)𝜉²_ℝ𝑁≤(𝐴(𝑥)𝜉,𝜉)_ℝ^𝑁≤𝛽(𝑥)𝜉²_ℝ𝑁a.e.inΩ∀𝜉∈ℝ^𝑁. (3) Because of this, such matrices are sometimes referred to as matrices with degenerate spectrum. It is clear that conditions (3) lead us to the so-called “noncoercivity trouble.” It means that boundary value problem (1) for some locally integrable matrix-valued functions𝐴may exhibit the Lavrentieff phenomenon, the nonuniqueness of weak solutions, as well as other surprising consequences. So, in general, the mapping𝐴↦𝑦(𝐴)can be multivalued. On the other hand, this problem is ill-posed, in general. It means that there are no reasons to suppose that for every𝑓∈𝐿²(Ω)and𝐴∈𝐿¹(Ω;ℝ^{𝑁(𝑁+1)/2}), the problem (1) admits at least one weak solution in𝐻¹₀(Ω). Thus, it makes it impossible to apply classical theorems (see [24, 25]) to establish solvability of the corresponding variational inequality and, hence, the regularity of the optimal control problem associated with it.

Besides, we show that for every admissible control function𝐴it is plausible to consider weak solutions of problem (1) as elements of the corresponding weighted Sobolev spaces𝑊(Ω,𝐴𝑑𝑥)which are constructed as sets of functions𝑦∈𝑊₀^1,1(Ω)for which the norm 𝑦_𝐴=_Ω𝑦²+∇𝑦,𝐴(𝑥)∇𝑦_ℝ𝑁𝑑𝑥^1/2 (4) is finite. In addition, we note that even if the original elliptic operator is nondegenerate, that is, admissible controls𝐴(𝑥)are such that 𝛼𝜉²_ℝ𝑁≤(𝜉,𝐴(𝑥)𝜉)_ℝ^𝑁≤𝛽𝜉²_ℝ𝑁𝜉∈ℝ^𝑁 (5) with𝛼>0, the majority of optimal control problems in coefficients have no solution in general (see for instance [15, 26–28]).

Since for atypical matrices with degenerate spectrum, namely, such that 𝐴,𝐴⁻¹∉𝐿^∞(Ω;ℝ^{𝑁(𝑁+2)/2}), the space of smooth compactly supported functions 𝐶^∞₀(Ω) is not dense in 𝑊(Ω;𝐴𝑑𝑥) in general, we discuss the solvability of this optimal control problem in the class of the so-called𝐻-admissible solutions. To this end, we consider, as the main solution space, the closure of𝐶^∞₀(Ω)in𝑊(Ω;𝐴𝑑𝑥)-norm and denote this space by 𝐻(Ω;𝐴𝑑𝑥). In order to specify the set𝐾_𝐴in (1), we assume that there exists a closed convex subset 𝐾 of 𝑊₀^1,1(Ω) such that 𝐾∩𝐶^∞₀(Ω)≠∅. In this case, we set 𝐾_𝐴 as the closure of {𝐾∩𝐻(Ω;𝐴𝑑𝑥)} with respect to 𝑊(Ω;𝐴𝑑𝑥)-norm (4).

At the same time, having formulated the optimal control problem in coefficients in terms of the corresponding weighted Sobolev spaces, we are facing another kind of challenge: changing the control matrix changes not only the solution space 𝐻(Ω;𝐴𝑑𝑥) but also the control object itself. In fact, it means that the set of admissible solutions to the previous optimal control problem is a family of pairs, each of which belongs to the correspondent space. Therefore, in this situation there are several possible settings of the variational inequality and optimal control problem associated with it, which depend on the choice of solution space. The main questions are what is the right setting of the optimal control problem with 𝐿¹-controls in coefficients, and what is the right class of admissible solutions to the considered problem? As we show in this paper, the precise answer for the question of existence or nonexistence of optimal solutions heavily depends on the class of chosen admissible controls.

The paper is organized as follows. Section 2 concerns some notation and preliminaries. In Section 3, we describe the main notion of weak and strong convergence in variable𝐿²-spaces. Section 4 gives a collection of auxiliary results that we will use in the sequel. In particular, following Kogut and Leugering [29], we introduce the concept of𝑤-convergence in variable space 𝐿¹(Ω;ℝ^{𝑁(𝑁+1)/2})×𝐻(Ω;𝐴𝑑𝑥) and give sufficient conditions of sequential𝑤-compactness. In Section 5, we introduce the admissible control constrains, the class of 𝐻-optimal admissible solutions, and discuss the regularity of the corresponding optimal control problem. In Section 6, using the direct method in the calculus of variations, we prove the existence of the so-called 𝐻-optimal solutions to the original problem.

2. Notation and Preliminaries

Let Ω be a bounded open subset of ℝ^𝑁(𝑁≥2) with Lipschitz boundary. We assume that the boundary of Ω, denoted by 𝜕Ω, has positive (𝑁−1)-dimensional measure.

By 𝐶^∞₀(Ω), we denote the set of functions from 𝐶^∞₀(ℝ^𝑁), compactly supported in Ω. We define the Banach space 𝑊₀^1,1(Ω) as the closure of 𝐶^∞₀(Ω) in the classical Sobolev space 𝑊^1,1(Ω). For any subset 𝐸⊂Ω, we denote by |𝐸| its 𝑁-dimensional Lebesgue measure ℒ^𝑁(𝐸).

Symmetric Matrices with Degenerate Eigenvalues. We denote by 𝕊^𝑁∶=ℝ^{𝑁(𝑁+1)/2} the set of all symmetric matrices 𝜉=[𝜉_𝑖𝑗]^𝑁_𝑖,𝑗=1, (𝜉_𝑖𝑗=𝜉_𝑗𝑖). We suppose that 𝕊^𝑁 is endowed with the Euclidian scalar product 𝜉⋅𝜂=tr(𝜉𝜂) and with the corresponding Euclidian norm ‖𝜉‖_𝕊^𝑁=(𝜉⋅𝜉)^1/2. Let 𝐿¹(Ω)^{𝑁(𝑁+1)/2}=𝐿¹(Ω;𝕊^𝑁) be the space of integrable functions whose values are symmetric matrices.

Let 𝛼∈ℝ be a fixed positive value. Let 𝜁_𝑎𝑑∶Ω→ℝ be a given function satisfying the properties 𝜁_𝑎𝑑∈𝐿¹(Ω),𝜁⁻¹_𝑎𝑑∈𝐿¹(Ω),𝜁⁻¹_𝑎𝑑∉𝐿^∞(Ω). (6) Let Ψ_∗ be a nonempty compact subset of 𝐿¹(Ω) such that for any 𝜁_∗∈Ψ_∗ the following conditions hold true: 𝜁_𝑎𝑑(𝑥)<𝜁_∗(𝑥)a.e.in𝜁Ω,_∗∶Ω⟶ℝ₊isasmoothfunctionalongtheboundary𝜁𝜕Ω,_∗=𝛼on𝜕Ω. (7) By𝔐^𝛽_𝛼(Ω), we denote the set of all matrices𝐴(𝑥)=[𝑎_𝑖𝑗(𝑥)]∈𝐿¹(Ω;𝕊^𝑁)such that 𝐴(𝑥)≤𝛽(𝑥)𝐼a.e.inΩ,∃𝜁_∗∈Ψ_∗s.t.𝜁_∗(𝑥)𝐼≤𝐴(𝑥)a.e.inΩ. (8)(9) Here, 𝛽∈𝐿¹(Ω) is a given function such that 𝛽(𝑥)>0 a.e. in Ω and such that 𝛽⋅𝜁⁻¹_𝑎𝑑∈𝐿^∞(Ω),𝐼is the identity matrix in ℝ^𝑁×𝑁, and (8)-(9) should be considered in the sense of quadratic forms. Therefore, (8)-(9) imply the following inequalities: if𝐴∈𝔐^𝛽_𝛼(Ω),then‖𝐴(𝑥)‖_𝐿¹_(Ω;𝕊^𝑁₎≤𝛽_𝐿1(Ω)<+∞,𝜁_∗(𝑥)𝜉²_ℝ𝑁≤(𝐴(𝑥)𝜉,𝜉)_ℝ^𝑁≤𝛽(𝑥)𝜉²_ℝ𝑁a.e.inΩ∀𝜉∈ℝ^𝑁. (10)(11)

Remark 1. Since every measurable matrix-valued function𝐴∶Ω→𝕊^𝑁can be associated with the collection of its eigenvalues{𝜆^𝐴₁,…,𝜆^𝐴_𝑁}, where each𝜆^𝐴_𝑘=𝜆^𝐴_𝑘(𝑥)is counted with its multiplicity, (9), in view of the properties of the classΨ_∗, means that eigenvalues of matrices𝐴∈𝔐^𝛽_𝛼(Ω)may vanish and be unbounded on subdomains ofΩwith zero Lebesgue measure. Because of this, these matrices are sometime referred to as matrices with degenerate spectrum.

Weighted Sobolev Spaces. To each matrix 𝐴∈𝔐^𝛽_𝛼(Ω), we will associate two weighted Sobolev spaces: 𝑊_𝐴=𝑊(Ω;𝐴𝑑𝑥),𝐻_𝐴=𝐻(Ω;𝐴𝑑𝑥), (12) where𝑊_𝐴is the set of functions𝑦∈𝑊₀^1,1(Ω)for which the norm, given by (4), is finite, and𝐻_𝐴is the closure of𝐶^∞₀(Ω)in𝑊_𝐴. Note that due to inequality (11) and estimates _Ω𝑦𝑑𝑥≤_Ω𝑦²𝑑𝑥^1/2|Ω|^1/2≤𝐶𝑦_𝐴,_Ω∇𝑦_ℝ𝑁≤𝑑𝑥_Ω∇𝑦²_ℝ𝑁𝜁_∗𝑑𝑥^1/2_Ω𝜁_∗⁻¹𝑑𝑥^1/2≤𝐶_Ω∇𝑦,𝐴(𝑥)∇𝑦_ℝ𝑁𝑑𝑥^1/2≤𝐶𝑦_𝐴, (13)(14) the space 𝑊_𝐴 is complete with respect to the norm ‖⋅‖_𝐴. It is clear that 𝐻_𝐴⊂𝑊_𝐴, and 𝑊_𝐴, 𝐻_𝐴 are Hilbert spaces. If the eigenvalues {𝜆^𝐴₁,…,𝜆^𝐴_𝑁} of 𝐴∶Ω→𝕊^𝑁 are bounded between two positive constants, then it is easy to verify that 𝑊_𝐴=𝐻_𝐴. However, for a “typical” matrix 𝐴∈𝔐^𝛽_𝛼(Ω) the space of smooth functions 𝐶^∞₀(Ω) is not dense in 𝑊_𝐴. Hence, the identity 𝑊_𝐴=𝐻_𝐴 is not always valid (for the corresponding examples in the case when 𝐴(𝑥)=𝜌(𝑥)𝐼, we refer the reader to [30, 31]).

Weak Compactness Criterion in 𝐿¹(Ω;𝕊^𝑁). Throughout the paper we will often use the concept of weak and strong convergence in𝐿¹(Ω;𝕊^𝑁). Let{𝐴_𝜀}_𝜀>0be a bounded sequence of matrices in𝐿¹(Ω;𝕊^𝑁). We recall that{𝐴_𝜀}_𝜀>0is called equi-integrable onΩ, if for any𝛿>0there is a𝜏=𝜏(𝛿)such that∫_𝑆‖𝐴_𝜀‖_𝕊^𝑁𝑑𝑥<𝛿for every measurable subset𝑆⊂Ωof Lebesgue measure|𝑆|<𝜏. Then the following assertions are equivalent for𝐿¹(Ω;𝕊^𝑁)-bounded sequences:(i)the sequence {𝐴_𝑘}_𝑘∈ℕ is weakly compact in 𝐿¹(Ω;𝕊^𝑁); (ii)the sequence {𝐴_𝑘}_𝑘∈ℕ is equi-integrable.

Lemma 2 (Lebesgue’s Theorem). If a sequence{𝐴_𝑘}_𝑘∈ℕ⊂𝐿¹(Ω;𝕊^𝑁)is equi-integrable and𝐴_𝑘→𝐴almost everywhere in Ω, then 𝐴_𝑘→𝐴 in 𝐿¹(Ω;𝕊^𝑁).

Functions with Bounded Variation. Let𝑓∶Ω→ℝbe a function of𝐿¹(Ω). Define _Ω𝐷𝑓=sup_Ω𝑓div𝜑𝑑𝑥∶𝜑=𝜑₁,…,𝜑_𝑁∈𝐶¹₀Ω;ℝ^𝑁,𝜑(𝑥)≤1for𝑥∈Ω, (15) wherediv𝜑=∑^𝑁_𝑖=1(𝜕𝜑_𝑖/𝜕𝑥_𝑖).

According to the Radon-Nikodym theorem, if ∫_Ω|𝐷𝑓|<+∞, then the distribution 𝐷𝑓 is a measure and there exist a vector-valued function ∇𝑓∈[𝐿¹(Ω)]^𝑁 and a measure 𝐷_𝑠𝑓, singular with respect to the 𝑁-dimensional Lebesgue measure ℒ^𝑁⌊Ω restricted to Ω, such that 𝐷𝑓=∇𝑓ℒ^𝑁⌊Ω+𝐷_𝑠𝑓. (16)

Definition 3. A function 𝑓∈𝐿¹(Ω) is said to have a bounded variation in Ω if ∫_Ω|𝐷𝑓|<+∞. By 𝐵𝑉(Ω), we denote the space of all functions in 𝐿¹(Ω) with bounded variation.

Under the norm‖𝑓‖_𝐵𝑉(Ω)=‖𝑓‖_𝐿1(Ω)+∫_Ω|𝐷𝑓|,𝐵𝑉(Ω)is a Banach space. The following compactness result for𝐵𝑉-functions is well-known.

Proposition 4. Uniformly bounded sets in𝐵𝑉-norm are relatively compact in𝐿¹(Ω).

Definition 5. A sequence{𝑓_𝑘}^∞_𝑘=1⊂𝐵𝑉(Ω)weakly converges to some𝑓∈𝐵𝑉(Ω), and we write𝑓_𝑘⇀𝑓if and only if the following two conditions hold:𝑓_𝑘→𝑓strongly in𝐿¹(Ω), and𝐷𝑓_𝑘⇀𝐷𝑓weakly-∗in the space of Radon measures𝑀(Ω;ℝ^𝑁); that is, lim_𝑘→∞Ω𝜑,𝐷𝑓_𝑘ℝ𝑁=_Ω𝜑,𝐷𝑓_ℝ𝑁∀𝜑∈𝐶₀ℝ^𝑁𝑁. (17)

In the following proposition, we give a compactness result related to this convergence, together with lower semicontinuity (see [32]).

Proposition 6. Let{𝑓_𝑘}^∞_𝑘=1be a sequence in𝐵𝑉(Ω)strongly converging to some𝑓in𝐿¹(Ω)and satisfyingsup_𝑘∈ℕ∫_Ω|𝐷𝑓_𝑘|<+∞. Then (i)𝑓∈𝐵𝑉(Ω)  and  ∫_Ω|𝐷𝑓|≤liminf_𝑘→∞∫_Ω|𝐷𝑓_𝑘|;(ii)𝑓_𝑘⇀𝑓  in   𝐵𝑉(Ω).

Elliptic Variational Inequalities. Following Lions [24], let us cite a well-known result concerning solvability and solution uniqueness for variational inequalities which will be useful in the sequel.

Lemma 7 (see [24, Theorem 8.3]). Let𝑉be a Banach space and𝐾⊂𝑉be a closed convex subset. Let𝐵∶𝐾→𝑉^∗be a strictly monotone operator and𝑓∈𝑉^∗be a given element of the dual space. Then the following variational problem: to find an element𝑦∈𝐾such that 𝐵𝑦,𝑣−𝑦_𝑉≥𝑓,𝑣−𝑦_𝑉∀𝑣∈𝐾 (18) admits a unique solution provided operator 𝐵 is corcive in the following sense: there exists an element 𝑣₀∈𝐾 such that 𝐵𝑦,𝑦−𝑣_0𝑉𝑦_𝑉⟶+∞𝑎𝑠𝑦_𝑉⟶∞,𝑦∈𝐾. (19)

3. 𝕊^𝑁-Valued Radon Measures and Weak Convergence in Variable 𝐿²-Spaces

By a nonnegative Radon measure onΩwe mean a nonnegative Borel measure which is finite on every compact subset ofΩ. The space of all nonnegative Radon measures onΩwill be denoted by𝑀₊(Ω). According to the Riesz theory, each Radon measure𝜇∈𝑀₊(Ω)can be interpreted as an element of the dual space to space𝐶₀(Ω)of all continuous functions with compact support. Let𝑀(Ω;𝕊^𝑁)denote the space of all𝕊^𝑁-valued nonnegative Radon measures. Then𝜇=[𝜇_𝑖𝑗]∈𝑀(Ω;𝕊^𝑁)⇔𝜇_𝑖𝑗∈(𝐶₀(Ω))^∗,𝑖,𝑗=1,…,𝑁.

Let 𝜇 and {𝜇_𝑘}_𝑘∈ℕ be matrix-valued nonnegative Radon measures. We say that {𝜇_𝑘}_𝑘∈ℕ weakly-∗converges to 𝜇 in 𝑀(Ω;𝕊^𝑁) if lim_𝑘→∞Ω𝜑⋅𝑑𝜇_𝑘=_Ω𝜑⋅𝑑𝜇∀𝜑∈𝐶₀Ω;𝕊^𝑁. (20) A typical example of such measures is 𝑑𝜇_𝑘=𝐴_𝑘(𝑥)𝑑𝑥,𝑑𝜇=𝐴(𝑥)𝑑𝑥, (21) where 𝐴_𝑘,𝐴∈𝔐^𝛽_𝛼(Ω)∩𝐿¹Ω;𝕊^𝑁,𝐴_𝑘⇀𝐴in𝐿¹Ω;𝕊^𝑁, (22) or 𝐴_𝑘,𝐴∈𝔐^𝛽_𝛼(Ω)∩𝐿^∞Ω;𝕊^𝑁,𝐴⇀_𝑘^∗𝐴in𝐿^∞Ω;𝕊^𝑁. (23) As we will see later (see Lemma 15), the sets𝔐^𝛽_𝛼(Ω)∩𝐿¹(Ω;𝕊^𝑁)are sequentially closed with respect to strong convergence in𝐿¹(Ω;𝕊^𝑁).

In this section, we suppose that the measures 𝜇 and {𝜇_𝑘}_𝑘∈ℕ are defined by (21) and 𝜇⇀_𝑘^∗𝜇 in 𝑀(Ω;𝕊^𝑁). Further, we will use 𝐿²(Ω,𝐴𝑑𝑥)^𝑁 to denote the set of measurable vector-valued functions 𝐟∈ℝ^𝑁 onΩ such that ‖𝐟‖_𝐿²_{(Ω,𝐴𝑑𝑥)}^𝑁=_Ω(𝐟,𝐴(𝑥)𝐟)_ℝ^𝑁𝑑𝑥^1/2<+∞. (24) As follows from estimate (14), any vector-valued function of𝐿²(Ω,𝐴𝑑𝑥)^𝑁is Lebesgue integrable onΩ.

We say that a sequence {𝐯_𝑘∈𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁}_𝑘∈ℕ is bounded if limsup_𝑘→∞Ω𝐯_𝑘,𝐴_𝑘(𝑥)𝐯_𝑘ℝ𝑁𝑑𝑥<+∞. (25)

Definition 8. Let 𝐴 and {𝐴_𝑘}_𝑘∈ℕ be matrices satisfying conditions (22). A bounded sequence {𝐯_𝑘∈𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁}_𝑘∈ℕ is weakly convergent to a function 𝐯∈𝐿²(Ω,𝐴𝑑𝑥)^𝑁 in the variable space 𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁if lim_𝑘→∞Ω𝜑,𝐴_𝑘(𝑥)𝐯_𝑘ℝ𝑁𝑑𝑥=_Ω𝜑,𝐴(𝑥)𝐯_ℝ𝑁∀𝑑𝑥𝜑∈𝐶^∞₀(Ω)^𝑁. (26)

The main property concerning the weak convergence in𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁can be expressed as follows (see for comparison [33]).

Proposition 9 (see [29]). Let 𝐴 and {𝐴_𝑘}_𝑘∈ℕ be matrices satisfying conditions (22). If a sequence {𝐯_𝑘∈𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁}_𝑘∈ℕ is bounded, then it is compact in the sense of weak convergence in 𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁.

The next property of weak convergence in𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁is the lower semicontinuity of the variable𝐿²-norm.

Proposition 10 (see [29]). If the sequence {𝐯_𝑘∈𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁}_𝑘∈ℕ converges weakly to 𝐯∈𝐿²(Ω,𝐴𝑑𝑥)^𝑁, then liminf_𝑘→∞Ω𝐯_𝑘,𝐴_𝑘(𝑥)𝐯_𝑘ℝ𝑁𝑑𝑥≥_Ω(𝐯,𝐴(𝑥)𝐯)_ℝ^𝑁𝑑𝑥. (27)

Definition 11. A sequence {𝐯_𝑘∈𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁}_𝑘∈ℕ is said to be strongly convergent to a function 𝐯∈𝐿²(Ω,𝐴𝑑𝑥)^𝑁 if lim_𝑘→∞Ω𝐛_𝑘,𝐴_𝑘(𝑥)𝐯_𝑘ℝ𝑁𝑑𝑥=_Ω(𝐛,𝐴(𝑥)𝐯)_ℝ^𝑁𝑑𝑥 (28) whenever 𝐛_𝑘⇀𝐛 in 𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁 as 𝑘→∞. We have the following property of strong convergence in the variable 𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁-spaces.

Proposition 12 (see [29]). Weak convergence of a sequence {𝐯_𝑘∈𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁}_𝑘∈ℕ to 𝐯∈𝐿²(Ω,𝐴𝑑𝑥)^𝑁 and lim_𝑘→∞Ω𝐯_𝑘,𝐴_𝑘(𝑥)𝐯_𝑘ℝ𝑁𝑑𝑥=_Ω(𝐯,𝐴(𝑥)𝐯)_ℝ^𝑁𝑑𝑥 (29) is equivalent to strong convergence of {𝐯_𝑘}_𝑘∈ℕ in 𝐿²(Ω,𝐴_𝑘𝑑𝑥)^𝑁 to 𝐯∈𝐿²(Ω,𝐴𝑑𝑥)^𝑁.

4. Auxiliary Results

Following in many aspects Kogut and Leugering [29], we provide some properties of the setΨ_∗⊂𝐿¹(Ω)defined in (7), and give some auxiliary results.

Lemma 13 (see [29]). Let{𝜁_∗,𝑛}_𝑛∈ℕbe any sequence inΨ_∗. Then, there is an element𝜁_∗∈𝐿¹(Ω)such that, within a subsequence of{𝜁_∗,𝑛}_𝑛∈ℕ, we have 𝜁_∗,𝑛⟶𝜁_∗𝑖𝑛𝐿¹(Ω),𝜁_∗∈Ψ_∗,𝜁⁻¹_∗,𝑛⟶𝜁_∗⁻¹𝑖𝑛𝐿¹(Ω),𝜁⁻¹_∗,𝑛⟶𝜁_∗⁻¹𝑖𝑛𝑣𝑎𝑟𝑖𝑎𝑏𝑙𝑒𝑠𝑝𝑎𝑐𝑒𝐿²Ω,𝜁_∗,𝑛𝑑𝑥. (30)

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

Definition 14. We say that a bounded sequence 𝐴_𝑛,𝑦_𝑛∈𝐿¹Ω;𝕊^𝑁×𝐻Ω;𝐴_𝑛𝑑𝑥_𝑛∈ℕ (31)𝑤-converges to(𝐴,𝑦)∈𝐿¹(Ω;𝕊^𝑁)×𝑊₀^1,1(Ω)as𝑛→∞if 𝐴_𝑛⟶𝐴in𝐿¹Ω;𝕊^𝑁,𝑦_𝑛⇀𝑦in𝐿²(Ω),∇𝑦_𝑛⇀∇𝑦inthevariablespace𝐿²Ω,𝐴_𝑛𝑑𝑥^𝑁; (32)(33)(34) therefore, lim_𝑛→∞Ω𝐴_𝑛⋅𝜂𝑑𝑥=_Ω𝐴⋅𝜂𝑑𝑥∀𝜂∈𝐿^∞Ω;𝕊^𝑁,lim_𝑛→∞Ω𝑦_𝑛𝜆𝑑𝑥=_Ω𝑦𝜆𝑑𝑥∀𝜆∈𝐿²(Ω),lim_𝑛→∞Ω𝜉,𝐴_𝑛∇𝑦_𝑛ℝ𝑁𝑑𝑥=_Ω𝜉,𝐴∇𝑦_ℝ𝑁∀𝑑𝑥𝜉∈𝐶^∞₀(Ω)^𝑁. (35)

In order to motivate this definition, we give the following result.

Lemma 15. Let {(𝐴_𝑛,𝑦_𝑛)∈𝐿¹(Ω;𝕊^𝑁)×𝐻(Ω;𝐴_𝑛𝑑𝑥)}_𝑛∈ℕ be a sequence such that(i)the sequence {𝑦_𝑛∈𝐻(Ω;𝐴_𝑛𝑑𝑥)}_𝑛∈ℕ is bounded; that is, sup_𝑛∈ℕΩ𝑦²_𝑛+∇𝑦_𝑛,𝐴_𝑛∇𝑦_𝑛𝑑𝑥<+∞; (36)(ii){𝐴_𝑛}_𝑛∈ℕ⊂𝔐^𝛽_𝛼(Ω)and there exists a matrix-valued function𝐴(𝑥)∈𝕊^𝑁such that 𝐴_𝑛⟶𝐴,𝐴_𝑛⁻¹⟶𝐴⁻¹𝑖𝑛𝐿¹Ω;𝕊^𝑁𝑎𝑠𝑛⟶∞. (37)Then, 𝐴∈𝔐^𝛽_𝛼(Ω)∩𝐿¹(Ω;𝕊^𝑁); and the original sequence is relatively compact with respect to𝑤-convergence. Moreover, each𝑤-limit pair(𝐴,𝑦)belongs to the space𝐿¹(Ω;𝕊^𝑁)×𝐻(Ω;𝐴𝑑𝑥).

Proof. We note that (36)-(37), (4), and (13)-(14) immediately imply the boundedness of the original sequence in𝐿¹(Ω;𝕊^𝑁)×𝑊^1,1(Ω). Moreover, due to (37), we have 𝑑𝜇_𝑛∶=𝐴_𝑛𝑑𝑥⇀^∗𝐴𝑑𝑥=∶𝑑𝜇inℳΩ;𝕊^𝑁. (38)
Thus, the compactness criterium for weak convergence in variable spaces (see Proposition 9) and (36) imply the existence of a pair (𝑦,𝐯)∈𝐿²(Ω)×𝐿²(Ω,𝐴𝑑𝑥)^𝑁 such that, within a subsequence of{𝑦_𝑛}_𝑛∈ℕ, 𝑦_𝑛⇀𝑦in𝐿²(Ω),∇𝑦_𝑛⇀𝐯invariablespace𝐿²Ω,𝐴_𝑛𝑑𝑥^𝑁. (39)(40) Our aim is to show that𝐴∈𝔐^𝛽_𝛼(Ω),𝐯=∇𝑦, and𝑦∈𝐻(Ω;𝐴𝑑𝑥). It is clear that𝐴(𝑥)∈𝕊^𝑁and this matrix satisfies (8). Since𝐴_𝑛∈𝔐^𝛽_𝛼(Ω)∩𝐿¹(Ω;𝕊^𝑁)for all𝑛∈ℕ, it follows that there is a sequence{𝜁_∗,𝑛}_𝑛∈ℕinΨ_∗such that 𝜁_∗,𝑛(𝑥)𝐼≤𝐴_𝑛(𝑥)𝐼≤𝛽(𝑥)𝐼a.e.inΩ∀𝑘∈{1,…,𝑁}. (41) Then, by𝐿¹-compactness of the setΨ_∗, there exists an element𝜁_∗∈Ψ_∗such that𝜁_∗,𝑛→𝜁_∗in𝐿¹(Ω)as𝑛→∞. Moreover, Lemma 13 implies strong convergence 𝜁⁻¹_∗,𝑛⟶𝜁_∗⁻¹in𝐿¹(Ω) (42) and (7). Hence, passing to the limit in (41) as𝑛→∞, we come to (9). Thus,𝐴∈𝔐^𝛽_𝛼(Ω)and the limit matrix𝐴(𝑥)∈𝕊^𝑁satisfies (10)-(11).
For our further analysis, we fix any test function𝜑∈𝐶^∞₀(Ω)^𝑁 and make use of the following equality: _Ω𝐴_𝑛⁻¹𝜑,𝐴_𝑛𝜓_ℝ𝑁𝑑𝑥=_Ω𝜑,𝜓_ℝ𝑁=𝑑𝑥_Ω𝐴⁻¹𝜑,𝐴𝜓_ℝ𝑁𝑑𝑥, (43) which is obviously true for each𝜓∈𝐶^∞₀(Ω)^𝑁and for all𝑛∈ℕ. Since limsup_𝑛→∞Ω𝐴_𝑛⁻¹𝜑,𝐴_𝑛𝐴_𝑛⁻¹𝜑_ℝ𝑁𝑑𝑥=limsup_𝑛→∞Ω𝜑,𝐴_𝑛⁻¹𝜑_ℝ𝑁𝑑𝑥≤limsup_𝑛→∞Ω𝜁⁻¹_∗,𝑛𝜑²_ℝ𝑁𝑑𝑥_by(42)=_Ω𝜁_∗⁻¹𝜑²_ℝ𝑁≤𝜑𝑑𝑥_{2𝐶(Ω)𝑁}𝜁_∗⁻¹_𝐿1(Ω)<+∞, (44) it follows that the sequence {𝐴_𝑛⁻¹𝜑∈𝐿²(Ω,𝐴_𝑛𝑑𝑥)^𝑁}_𝑛∈ℕ is bounded. Consequently, combining this fact with (43), we conclude that 𝐴_𝑛⁻¹𝜑⇀𝐴⁻¹𝜑 in the variable space 𝐿²(Ω,𝐴_𝑛𝑑𝑥)^𝑁 (see Definition 8). At the same time, strong convergence in (37) implies the relation lim_𝑛→∞Ω𝐴_𝑛⁻¹𝜑,𝐴_𝑛𝐴_𝑛⁻¹𝜑_ℝ𝑁𝑑𝑥=lim_𝑛→∞Ω𝜑,𝐴_𝑛⁻¹𝜑_ℝ𝑁𝑑𝑥=_Ω𝜑,𝐴⁻¹𝜑_ℝ𝑁𝑑𝑥=_Ω𝐴⁻¹𝜑,𝐴𝐴⁻¹𝜑_ℝ𝑁𝑑𝑥. (45) Hence (see Proposition 12), 𝐴_𝑛⁻¹𝜑⟶𝐴⁻¹𝜑stronglyin𝐿²Ω,𝐴_𝑛𝑑𝑥^𝑁∀𝜑∈𝐶^∞₀(Ω)^𝑁. (46) Further, we note that for every measurable subset 𝑆⊂Ω, the estimate _𝑆∇𝑦_𝑛ℝ𝑁≤𝑑𝑥_𝑆∇𝑦_𝑛²_ℝ𝑁𝜁_∗,𝑛𝑑𝑥^1/2_𝑆𝜁⁻¹_∗,𝑛𝑑𝑥^1/2≤_Ω∇𝑦_𝑛,𝐴_𝑛(𝑥)∇𝑦_𝑛ℝ𝑁𝑑𝑥^1/2_𝑆𝜁⁻¹_∗,𝑛𝑑𝑥^1/2≤𝐶_𝑆𝜁⁻¹_∗,𝑛𝑑𝑥^1/2 (47) implies equi-integrability of the family {‖∇𝑦_𝑛‖_ℝ𝑁}_𝑛∈ℕ. Hence, {‖∇𝑦_𝑛‖_ℝ𝑁}_𝑛∈ℕ is weakly compact in 𝐿¹(Ω), which means the weak compactness of the vector-valued sequence {∇𝑦_𝑛}_𝑛∈ℕ in 𝐿¹(Ω)^𝑁. As a result, by the properties of the strong convergence in variable spaces, we obtain _Ω𝜉,∇𝑦_𝑛ℝ𝑁𝑑𝑥=_Ω𝐴_𝑛⁻¹𝜉,𝐴_𝑛∇𝑦_𝑛ℝ𝑁𝑑𝑥_{by(),(),2840and(46)}⟶_Ω𝐴⁻¹𝜉,𝐴𝐯_ℝ𝑁=𝑑𝑥_Ω𝜉,𝐯_ℝ𝑁∀𝑑𝑥𝜉∈𝐶^∞₀(Ω)^𝑁. (48) Thus, in view of the weak compactness property of{∇𝑦_𝑛}_𝑛∈ℕin𝐿¹(Ω)^𝑁, we conclude that ∇𝑦_𝑛⇀𝐯in𝐿¹Ω;ℝ^𝑁as𝑛⟶∞. (49) Since 𝑦_𝑛∈𝑊₀^1,1(Ω) for all 𝑛∈ℕ and the Sobolev space 𝑊₀^1,1(Ω) is complete, (39) and (49) imply ∇𝑦=𝐯, and consequently 𝑦∈𝑊₀^1,1(Ω). It should be also observed that (39)-(40) guarantee the finiteness of the norm ‖𝑦‖_𝐴 (see (4)), hence 𝑦∈𝑊(Ω;𝐴𝑑𝑥) (to elaborate Propositions 9, 10, 12 and Lemma 13, for the sake of reader’s convenience, we closely followed the idea of [29]). However, to show that 𝑦∈𝐻(Ω;𝐴𝑑𝑥), we must be sure that there exists a sequence of smooth functions from 𝐶^∞₀(Ω) strongly converging in 𝑊(Ω;𝐴𝑑𝑥) to this element.
Since for all 𝑛∈ℕ, 𝑦_𝑛∈𝐻(Ω;𝐴_𝑛𝑑𝑥), there exist sequences {𝜑^𝑛_𝑘}_𝑘∈ℕ⊂𝐶^∞₀(Ω), such that 𝜑^𝑛_𝑘→𝑦_𝑛 strongly in 𝐻(Ω;𝐴_𝑛𝑑𝑥) as 𝑘→∞. Let us show that the sequence {𝜑^𝑛_𝑛}_𝑛∈ℕ, chosen with respect to diagonalization procedure, converges weakly to 𝑦 in 𝑊(Ω;𝐴𝑑𝑥). To this end, it is enough to prove that 𝜑^𝑛_𝑛⇀𝑦 in 𝐿²(Ω) and ∇𝜑^𝑛_𝑛⇀∇𝑦 in 𝐿²(Ω;𝐴𝑑𝑥)^𝑁.
Indeed, for an arbitrary 𝜓∈𝐶^∞₀(Ω), we have _Ω𝑦−𝜑^𝑛_𝑛𝜓𝑑𝑥≤_Ω𝑦−𝑦_𝑛+𝜓𝑑𝑥_Ω𝑦_𝑛−𝜑^𝑛_𝑛𝜓𝑑𝑥⟶0,as𝑛⟶∞, (50) since 𝑦_𝑛⇀𝑦 in 𝐿²(Ω), as 𝑛→∞ and for any 𝜀>0 there exists 𝑁₀∈ℕ such that |∫_Ω(𝑦_𝑛−𝜑^𝑛_𝑛)𝜓𝑑𝑥|<𝜀 for all 𝑛>𝑁₀. Similarly, for an arbitrary 𝜓∈𝐶^∞₀(Ω)^𝑁, we have _Ω𝐴∇𝑦−∇𝜑^𝑛_𝑛,𝜓_ℝ𝑁≤𝑑𝑥_Ω𝐴∇𝑦,𝜓_ℝ𝑁𝑑𝑥−_Ω𝐴_𝑛∇𝑦_𝑛,𝜓_ℝ𝑁+𝑑𝑥_Ω𝐴_𝑛∇𝑦_𝑛−∇𝜑^𝑛_𝑛,𝜓_ℝ𝑁+𝑑𝑥_Ω𝐴_𝑛−𝐴∇𝜑^𝑛_𝑛,𝜓_ℝ𝑁𝑑𝑥⟶0,as𝑛⟶∞. (51) It is only the convergence to zero of the third summand that should be explained here: _Ω𝐴_𝑛−𝐴∇𝜑^𝑛_𝑛,𝜓_ℝ𝑁≤𝑑𝑥_Ω𝐴_𝑛−𝐴∇𝜑^𝑛_𝑛,∇𝜑^𝑛_𝑛ℝ𝑁𝑑𝑥^1/2×_Ω𝐴_𝑛−𝐴𝜓,𝜓_ℝ𝑁𝑑𝑥^1/2=𝐽₁⋅𝐽₂. (52) Obviously,𝐽₂→0as𝑛→∞. It is left to prove, that𝐽₁≤𝐶. Indeed, _Ω𝐴_𝑛−𝐴∇𝜑^𝑛_𝑛,∇𝜑^𝑛_𝑛ℝ𝑁≤𝑑𝑥_Ω𝐴_𝑛∇𝜑^𝑛_𝑛,∇𝜑^𝑛_𝑛ℝ𝑁𝑑𝑥+_Ω𝐴∇𝜑^𝑛_𝑛,∇𝜑^𝑛_𝑛ℝ𝑁𝑑𝑥≤_Ω𝐴_𝑛∇𝜑^𝑛_𝑛,∇𝜑^𝑛_𝑛ℝ𝑁+𝑑𝑥_Ω𝛽𝜁⁻¹_𝑎𝑑𝐴_𝑛∇𝜑^𝑛_𝑛,∇𝜑^𝑛_𝑛ℝ𝑁≤𝑑𝑥1+𝛽𝜁⁻¹_{𝑎𝑑𝐿∞(Ω)Ω}𝐴_𝑛∇𝜑^𝑛_𝑛,∇𝜑^𝑛_𝑛ℝ𝑁𝑑𝑥±_Ω𝐴_𝑛∇𝑦_𝑛,∇𝑦_𝑛ℝ𝑁𝑑𝑥≤𝐶, (53) since ‖∇𝜑^𝑛_𝑛‖²_{𝐿2(Ω;𝐴𝑛𝑑𝑥)}−‖∇𝑦_𝑛‖²_{𝐿2(Ω;𝐴𝑛𝑑𝑥)}→0 as 𝑛→∞ and {‖∇𝑦_𝑛‖²_{𝐿2(Ω;𝐴𝑛𝑑𝑥)}}_𝑛∈ℕ is a bounded sequence in ℝ.
Hence, we proved the existence of the sequence from 𝐶^∞₀(Ω) which is weakly convergent in 𝑊(Ω;𝐴𝑑𝑥), to the element 𝑦. However, it is well known that due to Mazur’s lemma in 𝑊(Ω;𝐴𝑑𝑥) there exists a sequence, strongly convergent to 𝑦, which is constructed as a certain linear combinations of elements from {𝜑^𝑛_𝑛}_𝑛∈ℕ. Linear combinations of smooth functions are smooth as well. Hence, as the strong limit of the sequence from 𝐶^∞₀(Ω), the element 𝑦 belongs to the space 𝐻(Ω;𝐴𝑑𝑥).

5. Setting of the Optimal Control Problem

Let𝑀∈𝕊^𝑁be a given constant matrix satisfying the condition (𝑀𝜉,𝜉)_ℝ^𝑁≥𝑚𝜉²_ℝ𝑁forsome𝑚>0,∀𝜉∈ℝ^𝑁. (54)

Let𝐾be a convex closed subset of the space𝑊₀^1,1(Ω) such that𝐾∩𝐶^∞₀(Ω)≠∅, and let𝑓∈𝐿²(Ω)be a given distribution. Having fixed𝐴(𝑥)∈𝐿¹(Ω;𝕊^𝑁), we denote by𝐾_𝐴the closure of{𝐾∩𝐻(Ω;𝐴𝑑𝑥)}with respect to the norm‖⋅‖_𝐴, and consider the following elliptic variational inequality: to find an element𝑦∈𝐾_𝐴such that −div𝐴(𝑥)∇𝑦+𝑦,𝑣−𝑦_𝐻𝐴≥𝑓,𝑣−𝑦_𝐻𝐴,∀𝑣∈𝐾_𝐴. (55)

Remark 16. The distinctive feature of this variational problem is the following fact: choosing a matrix of coefficients𝐴(𝑥)∈𝐿¹(Ω;𝕊^𝑁), we choose a space in which the problem is considered; or, speaking in terms of optimization problem in coefficients for given inequality, changing a control matrix, we change not only the state or even the space of states but also the control object itself.

To introduce the class of admissible controls in coefficients, we adopt the following concept.

Definition 17. We say that a matrix-valued function𝐴=𝐴(𝑥)∈𝐿¹(Ω;𝕊^𝑁)is an admissible control for variational inequality (55) (it is written as𝐴∈𝔄_𝑎𝑑) if 𝐴∈𝐵𝑉Ω;𝕊^𝑁,𝐴∈𝔐^𝛽_𝛼(Ω),_Ω𝐴(𝑥)𝑑𝑥=𝑀. (56) Hereinafter, we assume that the set𝔄_𝑎𝑑is nonempty.

The optimal control problem we consider here is to minimize the discrepancy (tracking error) between a given distribution𝑦_𝑑∈𝐿²(Ω) and the solution of elliptic variational inequality (55)-(56) by choosing an appropriate coefficients matrix𝐴∈𝔄_𝑎𝑑. More precisely, we are concerned with the following optimal control problem: Minimize𝐼𝐴,𝑦=_Ω𝑦(𝑥)−𝑦_𝑑(𝑥)²𝑑𝑥+_Ω∇𝑦(𝑥),𝐴(𝑥)∇𝑦(𝑥)_ℝ𝑁+𝑑𝑥_{𝑁𝑖,𝑗=1Ω}𝐷𝑎_𝑖𝑗(𝑥) (57) subject to the constraints (55)-(56).

Remark 18. The second term in (57) plays a special role in this problem. Its appearance in the cost function (57) is motivated by the fact that there are no appropriate a priori estimates in the𝐻_𝐴-norm for solutions 𝑦=𝑦(𝐴,𝑓) of problem (55)-(56). Hence, the term ∫_Ω(∇𝑦(𝑥),𝐴(𝑥)∇𝑦(𝑥))_ℝ𝑁𝑑𝑥 together with the first one in (57) ensures the coercivity of the cost function.

We can indicate the set of admissible solutions to the original optimal control problem as follows: Ξ=𝐴,𝑦∣𝐴∈𝔄_𝑎𝑑,𝑦∈𝐻_𝐴,𝐴,𝑦arerelatedby(55). (58)

Remark 19. Note that due to (13)-(14), we have the following obvious inclusion for the set of admissible solutions: Ξ⊂𝐿¹(Ω;𝕊^𝑁)×𝑊₀^1,1(Ω).

To discuss solvability of variational inequality (55) and uniqueness of its solution, we make use of the following results.

Lemma 20. For every fixed admissible control𝐴∈𝔄_𝑎𝑑, the operator𝐵_𝐴∶𝐾_𝐴→𝐻(Ω;𝐴𝑑𝑥), defined as 𝐵_𝐴𝑦,𝑣_𝐻𝐴=^𝑁_{𝑖,𝑗=1Ω}𝑎_𝑖𝑗(𝑥)𝜕𝑦𝜕𝑥_𝑗𝜕𝑣𝜕𝑥_𝑖+𝑑𝑥_Ω𝑦𝑣𝑑𝑥,∀𝑣∈𝐻(Ω;𝐴𝑑𝑥), (59) is strictly monotone and coercive.

Proof. Let𝐴∈𝔄_𝑎𝑑be a fixed matrix. Then the relation 𝐵_𝐴𝑦,𝑦_𝐻𝐴=_Ω∇𝑦,𝐴∇𝑦_ℝ𝑁𝑑𝑥+_Ω𝑦²𝑑𝑥=𝑦²_𝐴 (60) implies the strict monotonicity and coercivity property of the operator𝐵_𝐴. Indeed, by setting in (60)𝑦=𝑢−𝑣,𝑢,𝑣∈𝐻_𝐴, we immediately obtain 𝐵_𝐴(𝑢)−𝐵_𝐴(𝑣),𝑢−𝑣_𝐻𝐴=‖𝑢−𝑣‖²_𝐴>0∀𝑢≠𝑣. (61)

Proposition 21. For every control𝐴∈𝔄_𝑎𝑑and every 𝑓∈𝐿²(Ω), there exists a unique solution to degenerate elliptic variational inequality (55)-(56).

Proof. Taking into account Lemmas 7 and 20, it is enough to show that there exists an element𝑣₀∈𝐾_𝐴such that −div𝐴(𝑥)∇𝑦+𝑦,𝑦−𝑣_0𝐻𝐴𝑦_𝐴⟶+∞as𝑦_𝐴⟶∞,𝑦∈𝐾_𝐴. (62) In order to do this, we fix an arbitrary element𝑣₀∈𝐾_𝐴and a matrix𝐴∈𝔄_𝑎𝑑. Then for all𝑦∈𝐾_𝐴, in view of estimate (60), we have −div𝐴(𝑥)∇𝑦+𝑦,𝑦−𝑣_0𝐻𝐴=𝑦²_𝐴−−div𝐴(𝑥)∇𝑦+𝑦,𝑣_0𝐻𝐴≥𝑦²_𝐴−−div𝐴(𝑥)∇𝑦+𝑦,𝑣_0𝐻𝐴, (63) where −div𝐴(𝑥)∇𝑦+𝑦,𝑣_0𝐻𝐴=_Ω𝐴(𝑥)∇𝑦,∇𝑣_0ℝ𝑁𝑑𝑥+_Ω𝑦𝑣₀≤𝑑𝑥_Ω𝐴^1/2∇𝑦²_ℝ𝑁𝑑𝑥^1/2_Ω𝐴^1/2∇𝑣₀²_ℝ𝑁𝑑𝑥^1/2+𝑦_𝐿2(Ω)𝑣_0𝐿2(Ω)=_Ω∇𝑦,𝐴∇𝑦_ℝ𝑁𝑑𝑥^1/2_Ω∇𝑣₀,𝐴∇𝑣_0ℝ𝑁𝑑𝑥^1/2+𝑦_𝐿2(Ω)𝑣_0𝐿2(Ω)=∇𝑦_{𝐿2(Ω;𝐴𝑑𝑥)}^𝑁𝑣_{0𝐿2(Ω;𝐴𝑑𝑥)}^𝑁+𝑦_𝐿2(Ω)𝑣_0𝐿2(Ω)≤𝑦_𝐴𝑣_0𝐴. (64) Combining (63) and (64) we come to the inequality −div𝐴(𝑥)∇𝑦+𝑦,𝑦−𝑣_0𝐻𝐴𝑦_𝐴≥𝑦_𝐴−𝑣_0𝐴⟶+∞,as𝑦_𝐴⟶∞. (65) Hence, problem (55)-(56) satisfies all conditions of Lemma 7. Therefore, for every control 𝐴(𝑥)∈𝔄_𝑎𝑑 and every 𝑓∈𝐿²(Ω), the considered problem has a unique solution.

Thus, as follows from Proposition 21, the set of admissible solutions Ξ is nonempty. Hence, the minimization problem inf_{(𝐴,𝑦)∈Ξ}𝐼(𝐴,𝑦) is regular. We say that a pair (𝐴⁰,𝑦⁰)∈𝐿¹(Ω;𝕊^𝑁)×𝐻_𝐴 is 𝐻-optimal for problem (55)-(56), (57) if 𝐴⁰,𝑦⁰∈Ξ,𝐼𝐴⁰,𝑦⁰=inf_{𝐴,𝑦∈Ξ}𝐼𝐴,𝑦. (66)

Referring to Lions [24], we make use of the following assumption, which allows to establish a certain smoothness property for solutions of variational inequality (55).

Hypothesis 1. For a given set 𝐾⊂𝑊₀^1,1(Ω), we have the following: for all 𝐴∈𝔄_𝑎𝑑, for all 𝑦∈𝐾_𝐴, and for all 𝜀=𝜀(𝐴)>0there exists an element 𝑦_𝜀∈𝐾_𝐴 such that 𝐵_𝐴𝑦_𝜀=−div𝐴∇𝑦_𝜀+𝑦_𝜀∈𝐿²(Ω),𝑦_𝜀+𝜀𝐵_𝐴𝑦_𝜀=𝑦. (67)
Then, as a direct consequence of Theorem  8.7 in [24] and Proposition 21, we come to the following result.

Theorem 22. Assume Hypothesis 1 holds true. Let𝐴∈𝔄_𝑎𝑑be any admissible control and let𝑦=𝑦(𝐴)be the corresponding solution of variational inequality (55). Then𝐵_𝐴(𝑦)∈𝐿²(Ω)provided𝑓∈𝐿²(Ω).

As an example of the set𝐾∈𝑊₀^1,1(Ω)satisfying the Hypothesis 1, we propose the following one.

Example 23. Let𝜓∈𝐶^∞₀(Ω)be a given function such that −div𝐴(𝑥)∇𝜓+𝜓≤0∀𝐴∈𝔄_𝑎𝑑. (68) Let us define the closed convex set𝐾as follows. 𝐾=𝑣∣𝑣∈𝑊₀^1,1(Ω),𝑣≥𝜓a.e.inΩ. (69) Then, for an arbitrary matrix𝐴∈𝔄_𝑎𝑑, we have 𝐾_𝐴=𝑐𝑙{𝐾∩𝐻(Ω;𝐴𝑑𝑥)}_𝐻𝐴=𝑣∣𝑣∈𝐻(Ω;𝐴𝑑𝑥),𝑣≥𝜓a.e.inΩ. (70) Let us show that for all 𝑦∈𝐾_𝐴, for all 𝜀>0 there exists an element 𝑦_𝜀∈𝐾_𝐴 satisfying the following conditions: −div𝐴∇𝑦_𝜀+𝑦_𝜀∈𝐿²(Ω),𝑦_𝜀+𝜀𝐵_𝐴𝑦_𝜀=𝑦. (71) As a result, we would have 𝐵_𝐴𝑦_𝜀−𝑦−𝑦_𝜀𝜀=0, (72) and having set 𝑦_𝜀−𝑦=𝑤_𝜀∈𝐿²(Ω), we come to the equation 𝐵_𝐴𝑤_𝜀+𝑤_𝜀𝜀=−𝐵_𝐴𝑦. (73) Obviously, the mapping𝐻(Ω;𝐴𝑑𝑥)∋𝑤_𝜀→𝐵_𝐴(𝑤_𝜀)+𝑤_𝜀/𝜀is strictly monotone and coercive (see Proposition 21). Hence, there exists a unique solution𝑤_𝜀to (73). Moreover, in this case𝐵_𝐴(𝑦_𝜀)=(𝑦−𝑦_𝜀)/𝜀∈𝐿²(Ω).
Let us show that𝑦_𝜀∈𝐾_𝐴, namely,𝑦_𝜀≥𝜓almost everywhere inΩ. In order to do this, we set𝑧_𝜀=sup{𝜓−𝑦_𝜀,0}. Then it is enough to prove that𝑧_𝜀=0. From (72) it follows that 𝐵_𝐴𝜓−𝑦_𝜀−𝐵_𝐴𝜓=𝑦_𝜀−𝑦𝜀. (74) Having noticed⟨𝐵_𝐴(𝜓−𝑦_𝜀),𝑧_𝜀⟩_𝐻𝐴=⟨𝐵_𝐴(𝑧_𝜀),𝑧_𝜀⟩_𝐻𝐴≥0, we conclude that −𝐵_𝐴𝜓,𝑧_𝜀𝐻𝐴≤𝑦_𝜀−𝑦𝜀,𝑧_𝜀𝐻𝐴. (75) Since𝑧_𝜀≥0and𝐵_𝐴(𝜓)≤0by the initial assumptions, we have ⟨(𝑦_𝜀−𝑦)/𝜀,𝑧_𝜀⟩_𝐻𝐴≥0; that is, _𝐸𝑦_𝜀−𝑦𝜀𝑧_𝜀(𝑥)𝑑𝑥≥0,𝐸=𝑥∣𝑥∈Ω,𝜓≥𝑦_𝜀(𝑥). (76) At the same time, since𝑦∈𝐾_𝐴, then𝑦≥𝜓and, hence,𝑦≥𝑦_𝜀on𝐸. As a result, we have𝑦_𝜀−𝑦≤0on𝐸. Therefore, _𝐸𝑦_𝜀−𝑦𝜀𝑧_𝜀𝑑𝑥≤0⟹_𝐸𝑦_𝜀−𝑦𝜀𝑧_𝜀𝑑𝑥=0. (77) Thus, we arrive at the following conclusion: either𝑦_𝜀=𝑦or𝑧_𝜀=0a.e. on𝐸. Hence,𝑧_𝜀=0and this concludes the proof.

6. Existence of Optimal Solutions

Since our prime interest is the solvability of optimal control problem (55)-(56), (57), we begin with the study of the topological properties of the set of admissible solutionsΞ. To do so, we give some auxiliary results.

Definition 24. We say that a sequence{(𝐴_𝑛,𝑦_𝑛)∈Ξ}_𝑛∈ℕis bounded if sup_𝑛∈ℕ𝐴_{𝑛𝐵𝑉(Ω;𝕊𝑁)}+𝑦_𝑛𝐴𝑛<+∞. (78)

Lemma 25. Let{(𝐴_𝑛,𝑦_𝑛)∈Ξ}_𝑛∈ℕbe a bounded sequence in the sense of Definition 24. Then there exists a pair(𝐴,𝑦)∈𝐿¹(Ω;𝕊^𝑁)×𝑊₀^1,1(Ω)such that, up to a subsequence, 𝐴_𝑛,𝑦⟶_𝑛^𝑤𝐴,𝑦,𝐴∈𝔄_𝑎𝑑,𝑦∈𝐻(Ω;𝐴𝑑𝑥). (79)

Proof. By the compactness of 𝐵𝑉-functions (see Proposition 6), there exist a subsequence of {𝐴_𝑛}_𝑛∈ℕ, still denoted by the same indices, and a matrix 𝐴∈𝐵𝑉(Ω;𝕊^𝑁) such that 𝐴_𝑛→𝐴 in 𝐿¹(Ω;𝕊^𝑁). Thus, 𝐴∈𝐵𝑉Ω;𝕊^𝑁,_Ω𝐴(𝑥)𝑑𝑥=lim_𝑛→∞Ω𝐴_𝑛(𝑥)𝑑𝑥=𝑀, (80) and the condition (32) of Definition 14 holds true. In order to check the remaining conditions (33)-(34) of this definition and to show that𝐴∈𝔄_𝑎𝑑, we make use of the following observation.
We have (𝐴_𝑛,𝑦_𝑛)∈Ξ for all 𝑛∈ℕ. Hence, there is a sequence {𝜁_∗,𝑛}_𝑛∈ℕ in Ψ_∗ such that (see Lemma 13 for the details) 𝜁_∗,𝑛→𝜁_∗ and 𝜁⁻¹_∗,𝑛→𝜁_∗⁻¹ in 𝐿¹(Ω) as 𝑛→∞. Moreover, by properties of the set Ψ_∗, the 𝐿¹-limit element 𝜁_∗ satisfies (7). Then, in view of 𝐿¹-convergence 𝐴_𝑛→𝐴, we may assume that 𝐴_𝑛⁻¹→𝐴⁻¹ almost everywhere in Ω. Since 𝐴_𝑛(𝑥)≥𝜁_∗,𝑛𝐼 a.e. in Ω, it follows that _𝑆𝜉,𝐴_𝑛⁻¹𝜉_ℝ𝑁𝑑𝑥≤_𝑆𝜁⁻¹_∗,𝑛𝑑𝑥𝜉²_ℝ𝑁∀𝑛∈ℕ (81) for any subset𝑆⊂Ω. Hence, due to the strong𝐿¹-convergence𝜁⁻¹_∗,𝑛→𝜁_∗⁻¹, the sequence{𝐴_𝑛⁻¹}_𝑛∈ℕis equi-integrable. Then, by Lebesgue’s Theorem (see Lemma 2) we obtain𝐴_𝑛⁻¹→𝐴⁻¹in𝐿¹(Ω;𝕊^𝑁)as𝑛→∞. As a result, 𝐴∈𝔐^𝛽_𝛼(Ω)∩𝐿¹Ω;𝕊^𝑁 (82) by Lemma 15. Combining this fact with properties (80), we conclude𝐴∈𝔄_𝑎𝑑.
It remains to notice that conditions (33)-(34) of Definition 14 and 𝑦∈𝐻_𝐴 for the 𝑤-limiting component (𝐴,𝑦) of the sequence {(𝐴_𝑛,𝑦_𝑛)}_𝑛∈ℕ are ensured by Lemma 15. This concludes the proof.

Our next step deals with the study of topological properties of the set of admissible solutionsΞto problem (55)-(56), (57). The following theorem is crucial for our next analysis.

Theorem 26. For any admissible initial data𝑓∈𝐿²(Ω), the set of admissible solutionsΞis sequentially closed with respect to𝑤-convergence.

Proof. Let {(𝐴_𝑛,𝑦_𝑛)∈Ξ}_𝑛∈ℕ be a bounded 𝑤-convergent sequence of admissible solutions to optimal control problem (55)-(56), (57). Let (𝐴,𝑦) be its 𝑤-limit. Our aim is to prove that (𝐴,𝑦)∈Ξ. By Lemma 25, we have 𝐴∈𝔄_𝑎𝑑 and 𝑦∈𝐻(Ω;𝐴𝑑𝑥).
Further, let us show that 𝑦∈𝐾_𝐴. Indeed, the condition 𝑦_𝑛∈𝐾_𝐴𝑛 implies 𝑦_𝑛∈𝐾 for all 𝑛∈ℕ, whereas relations (33)-(34) imply the weak convergence of {𝑦_𝑛}_𝑛∈ℕ to 𝑦 in 𝑊₀^1,1(Ω) within a subsequence (see (13)-(14)). Since the set 𝐾 is closed in 𝑊₀^1,1(Ω) and convex, then, in view of Mazur’s lemma, 𝐾 is weakly closed in 𝑊₀^1,1(Ω). Therefore, 𝑦∈𝐾 and 𝑦∈𝐻(Ω;𝐴𝑑𝑥). Combining these facts, we conclude that 𝑦∈𝐾_𝐴.
It remains to show that the pair (𝐴,𝑦) satisfies relation (55) for all 𝑣∈𝐾_𝐴. In order to do this, we divide the rest of the proof into three steps.
Step 1. Let us show that for every𝑣∈𝐾_𝐴, there exists a sequence{𝑣_𝑛∈𝐾_𝐴𝑛}_𝑛∈ℕ, such that 𝑣_𝑛⟶𝑣in𝐿²(Ω),∇𝑣_𝑛⇀∇𝑣invariablespace𝐿²Ω;𝐴_𝑛𝑑𝑥^𝑁. (83) Indeed, since𝑣∈𝐻(Ω;𝐴𝑑𝑥), it follows that there exists a sequence{𝜑_𝑛}_𝑛∈ℕin𝐶^∞₀(Ω)such that 𝜑_𝑛⟶𝑣in𝐿²(Ω),∇𝜑_𝑛⟶∇𝑣in𝐿²(Ω;𝐴𝑑𝑥)^𝑁as𝑛⟶∞. (84) Moreover, by the initial assumptions, 𝐾_𝐴 contains only smooth functions and their limits with respect to the strong convergence in 𝐻(Ω;𝐴𝑑𝑥). Hence, 𝜑_𝑛∈𝐾_𝐴 for all indices large enough. Let us show that {∇𝜑_𝑛}_𝑛∈ℕ converges to ∇𝑣 weakly in 𝐿²(Ω;𝐴_𝑛𝑑𝑥)^𝑁. To begin with, we show the boundedness of the sequence {∇𝜑_𝑛}_𝑛∈ℕ in 𝐿(Ω;𝐴_𝑛𝑑𝑥)^𝑁. Indeed, limsup_𝑛→∞Ω𝐴_𝑛∇𝜑_𝑛,∇𝜑_𝑛ℝ𝑁𝑑𝑥≤limsup_𝑛→∞Ω𝛽(𝑥)∇𝜑_𝑛(𝑥)²_ℝ𝑁𝑑𝑥≤limsup_𝑛→∞Ω𝛽(𝑥)𝜁⁻¹_∗,𝑛(𝑥)𝐴∇𝜑_𝑛,∇𝜑_𝑛ℝ𝑁𝑑𝑥<limsup_𝑛→∞Ω𝛽(𝑥)𝜁⁻¹_𝑎𝑑(𝑥)𝐴∇𝜑_𝑛,∇𝜑_𝑛ℝ𝑁≤𝑑𝑥𝛽𝜁⁻¹_{𝑎𝑑𝐿∞(Ω)}limsup_𝑛→∞Ω𝐴∇𝜑_𝑛,∇𝜑_𝑛ℝ𝑁𝑑𝑥<∞. (85) In view of Proposition 9, the sequence {∇𝜑_𝑛}_𝑛∈ℕ, within a subsequence, is weakly convergent in the variable space 𝐿(Ω;𝐴_𝑛𝑑𝑥)^𝑁. Let us show that ∇𝑣 is its weak limit. Let 𝜓∈𝐶^∞₀(Ω)^𝑁 be an arbitrary smooth function. Then _Ω𝐴_𝑛∇𝜑_𝑛,𝜓_ℝ𝑁𝑑𝑥−_Ω𝐴∇𝑣,𝜓_ℝ𝑁≤𝑑𝑥_Ω𝐴_𝑛−𝐴∇𝜑_𝑛,𝜓_ℝ𝑁+𝑑𝑥_Ω𝐴∇𝜑_𝑛−∇𝑣,𝜓_ℝ𝑁𝑑𝑥=𝐼₁+𝐼₂. (86) Obviously,𝐼₂→0as𝑛→∞provided by the strong convergence𝜑_𝑛→𝑣in𝐻(Ω;𝐴𝑑𝑥). Further, using the Cauchy-Bunjakowsky inequality, we have _Ω𝐴_𝑛−𝐴∇𝜑_𝑛,𝜓_ℝ𝑁≤𝑑𝑥_Ω𝐴_𝑛−𝐴∇𝜑_𝑛,∇𝜑_𝑛ℝ𝑁𝑑𝑥^1/2⋅_Ω𝐴_𝑛−𝐴∇𝜓,∇𝜓_ℝ𝑁𝑑𝑥^1/2⟶0,as𝑛⟶∞ (87) as the product of a bounded sequence and a sequence converging to zero. It follows that 𝜑_𝑛⟶𝑣stronglyin𝐿²(Ω),∇𝜑_𝑛⇀∇𝑣weaklyin𝐿²Ω;𝐴_𝑛𝑑𝑥as𝑛⟶∞. (88)
Step 2. Here we have to substantiate the correctness of the limit passage in the following variational inequality: −div𝐴_𝑛(𝑥)∇𝑦_𝑛+𝑦_𝑛,𝑣_𝑛−𝑦_{𝑛𝐻𝐴𝑛}≥𝑓,𝑣_𝑛−𝑦_{𝑛𝐻𝐴𝑛}∀𝑣_𝑛∈𝐾_𝐴𝑛, (89) where𝑦_𝑛∈𝐾_𝐴𝑛is its solution. To do so, it is enough to show that div𝐴_𝑛(𝑥)∇𝑦_𝑛⇀div𝐴(𝑥)∇𝑦in𝐿²(Ω)as𝑛⟶∞. (90) Since for every 𝑛∈ℕ relation (89) holds true for every𝑣_𝑛∈𝐾_𝐴𝑛, we may consider only those elements{𝑣_𝑛∈𝐾_𝐴𝑛}_𝑛∈ℕwhich form convergent sequences to elements𝑣∈𝐾_𝐴in a way that was established in Step 1.
Taking into account Hypothesis 1 and Theorem 22, we obtain div𝐴_𝑛(𝑥)∇𝑦_𝑛∈𝐿²(Ω)∀𝑛∈ℕ, (91) and, moreover, for an arbitrary 𝜓∈𝐶^∞₀(Ω), we have div𝐴_𝑛(𝑥)∇𝑦_𝑛,𝜓_𝐿2(Ω)=_Ω𝐴_𝑛∇𝑦_𝑛,∇𝜓_ℝ𝑁⟶𝑑𝑥_Ω𝐴∇𝑦,∇𝜓_ℝ𝑁𝑑𝑥=div𝐴(𝑥)∇𝑦,𝜓_𝐿2(Ω)as𝑛⟶∞. (92) Hence, the assertion (90) is valid.
Step 3. In view of the initial assumptions, we have(1)𝐴_𝑛→𝐴strongly in𝐿¹(Ω;𝕊^𝑁);(2)𝑦_𝑛⇀𝑦in𝐿²(Ω);(3)∇𝑦_𝑛⇀∇𝑦in𝐿²(Ω;𝐴_𝑛𝑑𝑥);(4)div(𝐴_𝑛(𝑥)∇𝑦_𝑛)⇀div(𝐴(𝑥)∇𝑦)in𝐿²(Ω);(5)forall𝑣∈𝐾_𝐴 there exists a sequence {𝑣_𝑛∈𝐾_𝐴𝑛}_𝑛∈ℕ, such that 𝑣_𝑛→𝑣 strongly in 𝐿²(Ω) and ∇𝑣_𝑛⇀∇𝑣 in 𝐿²(Ω;𝐴_𝑛𝑑𝑥).Hence, using this, Proposition 10 and the lower semicontinuity property of𝐿²-norm with respect to the weak convergence, we obtain −div𝐴(𝑥)∇𝑦+𝑦,𝑣−𝑦_𝐻𝐴=−div𝐴(𝑥)∇𝑦+𝑦,𝑣_𝐿2(Ω)−_Ω𝐴∇𝑦,∇𝑦_ℝ𝑁+𝑦²𝑑𝑥≥limsup_𝑛→∞−div𝐴_𝑛(𝑥)∇𝑦_𝑛+𝑦_𝑛,𝑣_{𝑛𝐿2(Ω)}−liminf_𝑛→∞Ω𝐴_𝑛∇𝑦_𝑛,∇𝑦_𝑛ℝ𝑁+𝑦²_𝑛𝑑𝑥≥limsup_𝑛→∞−div𝐴_𝑛(𝑥)∇𝑦_𝑛+𝑦_𝑛,𝑣_{𝑛𝐻𝐴𝑛}−_Ω𝐴_𝑛∇𝑦_𝑛,∇𝑦_𝑛ℝ𝑁+𝑦²_𝑛𝑑𝑥≥lim_𝑛→∞𝑓,𝑣_𝑛−𝑦_{𝑛𝐻𝐴𝑛}=lim_𝑛→∞𝑓,𝑣_𝑛−𝑦_{𝑛𝐿2(Ω)}=𝑓,𝑣−𝑦_𝐻𝐴∀𝑣∈𝐾_𝐴. (93) As a result, we come to the desired relation: −div𝐴(𝑥)∇𝑦+𝑦,𝑣−𝑦_𝐻𝐴≥𝑓,𝑣−𝑦_𝐻𝐴∀𝑣∈𝐾_𝐴, (94) which means that the limit pair (𝐴,𝑦) belongs to the set of admissible solutions of problem (55)-(56), (57). Hence, set Ξ is sequentially closed with respect to 𝑤-convergence in 𝐿¹(Ω;𝕊^𝑁)×𝐻(Ω;𝐴_𝑛𝑑𝑥), and this concludes the proof.

We are now in a position to state the existence of𝐻-optimal solution to problem (55)-(56), (57).

Theorem 27. Let 𝑓∈𝐿²(Ω) and 𝑦_𝑑∈𝐿²(Ω) be given functions. Assume that the Hypothesis 1 is valid. Then optimal control problem (55)-(56), (57) admits at least one solution (𝐴⁰,𝑦⁰)∈𝐿¹(Ω;𝕊^𝑁)×𝐻(Ω;𝐴₀𝑑𝑥).

Proof. Since the cost functional 𝐼=𝐼(𝐴,𝑦) is bounded below and Ξ≠∅, it provides the existence of a minimizing sequence {(𝐴_𝑛,𝑦_𝑛)∈Ξ}_𝑛∈ℕ to the problem (66). Then, inf_{𝐴,𝑦∈Ξ}𝐼𝐴,𝑦=lim_𝑛→∞𝐼𝐴_𝑛,𝑦_𝑛=lim_𝑛→∞Ω𝑦_𝑛(𝑥)−𝑦_𝑑(𝑥)²+𝑑𝑥_Ω∇𝑦_𝑛(𝑥),𝐴(𝑥)∇𝑦_𝑛(𝑥)_ℝ𝑁+𝑑𝑥_{𝑁𝑖,𝑗=1Ω⧵𝑄}𝐷𝑎^𝑛_𝑖𝑗(𝑥)<+∞ (95) implies the existence of a constant 𝐶>0 such that sup_𝑛∈ℕ∇𝑦_{𝑛𝐿2(Ω,𝐴𝑛𝑑𝑥)𝑁}≤𝐶,sup_𝑛∈ℕ𝑦_{𝑛𝐿2(Ω)}≤𝐶,sup_𝑛∈ℕ𝐴_{𝑛𝐵𝑉(Ω⧵𝑄;𝕊𝑁)}≤𝐶. (96) Hence, the minimizing sequence {(𝐴_𝑛,𝑦_𝑛)∈Ξ}_𝑛∈ℕ is bounded in the sense of Definition 24. Then, by Lemma 25, there exist functions 𝐴⁰∈𝐿¹(Ω;𝕊^𝑁) and 𝑦⁰∈𝐻(Ω;𝐴₀𝑑𝑥) such that, up to a subsequence, (𝐴_𝑛,𝑦_𝑛)−→^𝑤(𝐴⁰,𝑦⁰). Since the set Ξ is sequentially closed with respect to the𝑤-convergence (see Theorem 26), it follows that the 𝑤-limit pair (𝐴⁰,𝑦⁰) is an admissible solution to (55)-(56), (57) (i.e., (𝐴⁰,𝑦⁰)∈Ξ). To conclude the proof, it is enough to observe that the cost functional 𝐼 is sequentially lower 𝑤-semicontinuous. Hence, 𝐼𝐴⁰,𝑦⁰≤liminf_𝑛→∞𝐼𝐴_𝑛,𝑦_𝑛=inf_{𝐴,𝑦∈Ξ}𝐼𝐴,𝑦; (97) that is, (𝐴⁰,𝑦⁰) is an optimal solution. The proof is complete.