# On an OptimalL1-Control Problem in Coefficients for Linear Elliptic Variational Inequality

Authors: Olha P. Kupenko; Rosanna Manzo
Journal: Abstract and Applied Analysis
 (2013)

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## Abstract

We consider optimal control problems for linear degenerate elliptic variational inequalities with homogeneous Dirichlet boundary conditions. We take the matrix-valued coefficientsA(x) in the main part of the elliptic operator as controls in L1(Ω;ℝN(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.


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## Body

## 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 matrixA∈L1(Ω;ℝN(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:(1)∑i,j=1N‍∫Ω‍(aij(x)∂y∂xj)(∂v∂xi-∂y∂xi)dx+∫Ω‍y(v-y)dx≥∫Ω‍f(v-y)dx∀v∈KA.
Here,f∈L2(Ω)is a given distribution,KAis a convex closed subset of the spaceW01,1(Ω), andAis a measurable nonnegative square symmetric matrix on a bounded open domainΩ⊂ℝN(N≥2) such thatA∈BV(Ω;ℝN(N+1)/2).We furnish this control object by the following cost functional:(2)I(A,y)=∫Ω‍|y(x)-yd(x)|2dx+∫Ω‍(∇y(x),A(x)∇y(x))ℝNdx+∥A∥BV(Ω).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
(3)ζ(x)∥ξ∥ℝN2≤(A(x)ξ,ξ)ℝN≤β(x)∥ξ∥ℝN2a.e.inΩ∀ξ∈ℝN.
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;ℝN(N+1)/2), the problem (1) admits at least one weak solution inH01(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∈W01,1(Ω)for which the norm
(4)∥y∥A=(∫Ω‍(y2+(∇y,A(x)∇y)ℝN)dx)1/2
is finite. In addition, we note that even if the original elliptic operator is nondegenerate, that is, admissible controlsA(x)are such that
(5)α∥ξ∥ℝN2≤(ξ,A(x)ξ)ℝN≤β∥ξ∥ℝN2ξ∈ℝN
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 thatA,A-1∉L∞(Ω;ℝN(N+2)/2), the space of smooth compactly supported functions C0∞(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 ofC0∞(Ω)inW(Ω;Adx)-norm and denote this space by H(Ω;Adx). In order to specify the setKAin (1), we assume that there exists a closed convex subset K of W01,1(Ω) such that K∩C0∞(Ω)≠∅. In this case, we set KA 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 spaceH(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. Section2 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;ℝN(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

LetΩ be a bounded open subset of ℝN(N≥2) with Lipschitz boundary. We assume that the boundary of Ω, denoted by ∂Ω, has positive (N-1)-dimensional measure.ByC0∞(Ω), we denote the set of functions from C0∞(ℝN), compactly supported in Ω. We define the Banach space W01,1(Ω) as the closure of C0∞(Ω) in the classical Sobolev space W1,1(Ω). For any subset E⊂Ω, we denote by |E| its N-dimensional Lebesgue measure ℒN(E).Symmetric Matrices with Degenerate Eigenvalues. We denote by 𝕊N:=ℝN(N+1)/2 the set of all symmetric matrices ξ→=[ξij]i,j=1N, (ξij=ξji). We suppose that 𝕊N is endowed with the Euclidian scalar product ξ→·η→=tr(ξ→η→) and with the corresponding Euclidian norm ∥ξ→∥𝕊N=(ξ→·ξ→)1/2. Let L1(Ω)N(N+1)/2=L1(Ω;𝕊N) be the space of integrable functions whose values are symmetric matrices.Letα∈ℝ be a fixed positive value. Let ζad:Ω→ℝ¯ be a given function satisfying the properties
(6)ζad∈L1(Ω),ζad-1∈L1(Ω),ζad-1∉L∞(Ω).
Let Ψ* be a nonempty compact subset of L1(Ω) such that for any ζ*∈Ψ* the following conditions hold true:
(7)ζad(x)<ζ*(x)a.e.inΩ,ζ*:Ω⟶ℝ+isasmoothfunctionalongtheboundary∂Ω,ζ*=αon∂Ω.
By𝔐αβ(Ω), we denote the set of all matricesA(x)=[aij(x)]∈L1(Ω;𝕊N)such that
(8)A(x)≤β(x)Ia.e.inΩ,(9)∃ζ*∈Ψ*s.t.ζ*(x)I≤A(x)a.e.inΩ.
Here, β∈L1(Ω) is a given function such that β(x)>0 a.e. in Ω and such that β·ζad-1∈L∞(Ω),Iis the identity matrix in ℝN×N, and (8)-(9) should be considered in the sense of quadratic forms. Therefore, (8)-(9) imply the following inequalities:
(10)ifA∈𝔐αβ(Ω),then∥A(x)∥L1(Ω;𝕊N)≤∥β∥L1(Ω)<+∞,(11)ζ*(x)∥ξ∥ℝN2≤(A(x)ξ,ξ)ℝN≤β(x)∥ξ∥ℝN2a.e.inΩ∀ξ∈ℝN.Remark 1.
Since every measurable matrix-valued functionA:Ω→𝕊Ncan be associated with the collection of its eigenvalues{λ1A,…,λNA}, where eachλkA=λkA(x)is counted with its multiplicity, (9), in view of the properties of the classΨ*, means that eigenvalues of matricesA∈𝔐αβ(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∈𝔐αβ(Ω), we will associate two weighted Sobolev spaces:
(12)WA=W(Ω;Adx),HA=H(Ω;Adx),
whereWAis the set of functionsy∈W01,1(Ω)for which the norm, given by (4), is finite, andHAis the closure ofC0∞(Ω)inWA. Note that due to inequality (11) and estimates
(13)∫Ω‍|y|dx≤(∫Ω‍|y|2dx)1/2|Ω|1/2≤C∥y∥A,(14)∫Ω‍∥∇y∥ℝNdx≤(∫Ω‍∥∇y∥ℝN2ζ*dx)1/2(∫Ω‍ζ*-1dx)1/2≤C(∫Ω‍(∇y,A(x)∇y)ℝNdx)1/2≤C∥y∥A,
the space WA is complete with respect to the norm ∥·∥A. It is clear that HA⊂WA, and WA, HA are Hilbert spaces. If the eigenvalues {λ1A,…,λNA} of A:Ω→𝕊N are bounded between two positive constants, then it is easy to verify that WA=HA. However, for a “typical” matrix A∈𝔐αβ(Ω) the space of smooth functions C0∞(Ω) is not dense in WA. Hence, the identity WA=HA 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
L
1
(
Ω
;
𝕊
N
)
. Throughout the paper we will often use the concept of weak and strong convergence inL1(Ω;𝕊N). Let{Aε}ε>0be a bounded sequence of matrices inL1(Ω;𝕊N). We recall that{Aε}ε>0is called equi-integrable onΩ, if for anyδ>0there is aτ=τ(δ)such that∫S‍∥Aε∥𝕊Ndx<δfor every measurable subsetS⊂Ωof Lebesgue measure|S|<τ. Then the following assertions are equivalent forL1(Ω;𝕊N)-bounded sequences:(i)
the sequence{Ak}k∈ℕ is weakly compact in L1(Ω;𝕊N);(ii)
the sequence{Ak}k∈ℕ is equi-integrable.Lemma 2 (Lebesgue’s Theorem).
If a sequence{Ak}k∈ℕ⊂L1(Ω;𝕊N)is equi-integrable andAk→Aalmost everywhere in Ω, then Ak→A in L1(Ω;𝕊N).Functions with Bounded Variation. Letf:Ω→ℝbe a function ofL1(Ω). Define
(15)∫Ω‍|Df|=sup{∫Ω‍fdivφdx:φ=(φ1,…,φN)∈C01(Ω;ℝN),|φ(x)|≤1forx∈Ω{∫Ω‍fdivφdx:φ=(φ1,…,φN)∈C01(Ω;ℝN),},
wheredivφ=∑i=1N‍(∂φi/∂xi).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 Dsf, singular with respect to the N-dimensional Lebesgue measure ℒN⌊Ω restricted to Ω, such that
(16)Df=∇fℒN⌊Ω+Dsf.Definition 3.
A functionf∈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{fk}k=1∞⊂BV(Ω)weakly converges to somef∈BV(Ω), and we writefk⇀fif and only if the following two conditions hold:fk→fstrongly inL1(Ω), andDfk⇀Dfweakly-*in the space of Radon measuresM(Ω;ℝN); that is,
(17)limk→∞∫Ω‍(φ→,Dfk)ℝN=∫Ω‍(φ→,Df)ℝN∀φ→∈C0(ℝN)N.In the following proposition, we give a compactness result related to this convergence, together with lower semicontinuity (see [32]).Proposition 6.
Let{fk}k=1∞be a sequence inBV(Ω)strongly converging to somefinL1(Ω)and satisfyingsupk∈ℕ∫Ω‍|Dfk|<+∞. Then (i)
f∈BV(Ω)and∫Ω‍|Df|≤liminfk→∞∫Ω‍|Dfk|;(ii)
fk⇀finBV(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
(18)〈By,v-y〉V≥〈f,v-y〉V∀v∈K
admits a unique solution provided operator B is corcive in the following sense: there exists an element v0∈K such that
(19)〈By,y-v0〉V∥y∥V⟶+∞as∥y∥V⟶∞,y∈K.


## 3.𝕊N-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;𝕊N)denote the space of all𝕊N-valued nonnegative Radon measures. Thenμ→=[μij]∈M(Ω;𝕊N)⇔μij∈(C0(Ω))*,i,j=1,…,N.Letμ→ 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;𝕊N) if
(20)limk→∞∫Ω‍φ→·dμ→k=∫Ω‍φ→·dμ→∀φ→∈C0(Ω;𝕊N).
A typical example of such measures is
(21)dμ→k=Ak(x)dx,dμ→=A(x)dx,
where
(22)Ak,A∈𝔐αβ(Ω)∩L1(Ω;𝕊N),Ak⇀AinL1(Ω;𝕊N),
or
(23)Ak,A∈𝔐αβ(Ω)∩L∞(Ω;𝕊N),Ak⇀*AinL∞(Ω;𝕊N).
As we will see later (see Lemma 15), the sets𝔐αβ(Ω)∩L1(Ω;𝕊N)are sequentially closed with respect to strong convergence inL1(Ω;𝕊N).In this section, we suppose that the measuresμ→ and {μ→k}k∈ℕ are defined by (21) and μ→k⇀*μ→ in M(Ω;𝕊N). Further, we will use L2(Ω,Adx)N to denote the set of measurable vector-valued functions f∈ℝN onΩ such that
(24)∥f∥L2(Ω,Adx)N=(∫Ω‍(f,A(x)f)ℝNdx)1/2<+∞.
As follows from estimate (14), any vector-valued function ofL2(Ω,Adx)Nis Lebesgue integrable onΩ.We say that a sequence{vk∈L2(Ω,Akdx)N}k∈ℕ is bounded if
(25)limsupk→∞∫Ω‍(vk,Ak(x)vk)ℝNdx<+∞.Definition 8.
LetA and {Ak}k∈ℕ be matrices satisfying conditions (22). A bounded sequence {vk∈L2(Ω,Akdx)N}k∈ℕ is weakly convergent to a function v∈L2(Ω,Adx)N in the variable space L2(Ω,Akdx)Nif
(26)limk→∞∫Ω‍(φ→,Ak(x)vk)ℝNdx=∫Ω‍(φ→,A(x)v)ℝNdx∀φ→∈C0∞(Ω)N.The main property concerning the weak convergence inL2(Ω,Akdx)Ncan be expressed as follows (see for comparison [33]).Proposition 9 (see [29]).
LetA and {Ak}k∈ℕ be matrices satisfying conditions (22). If a sequence {vk∈L2(Ω,Akdx)N}k∈ℕ is bounded, then it is compact in the sense of weak convergence in L2(Ω,Akdx)N.The next property of weak convergence inL2(Ω,Akdx)Nis the lower semicontinuity of the variableL2-norm.Proposition 10 (see [29]).
If the sequence{vk∈L2(Ω,Akdx)N}k∈ℕ converges weakly to v∈L2(Ω,Adx)N, then
(27)liminfk→∞∫Ω‍(vk,Ak(x)vk)ℝNdx≥∫Ω‍(v,A(x)v)ℝNdx.Definition 11.
A sequence{vk∈L2(Ω,Akdx)N}k∈ℕ is said to be strongly convergent to a function v∈L2(Ω,Adx)N if
(28)limk→∞∫Ω‍(bk,Ak(x)vk)ℝNdx=∫Ω‍(b,A(x)v)ℝNdx
whenever bk⇀b in L2(Ω,Akdx)N as k→∞. We have the following property of strong convergence in the variable L2(Ω,Akdx)N-spaces.Proposition 12 (see [29]).
Weak convergence of a sequence{vk∈L2(Ω,Akdx)N}k∈ℕ to v∈L2(Ω,Adx)N and
(29)limk→∞∫Ω‍(vk,Ak(x)vk)ℝNdx=∫Ω‍(v,A(x)v)ℝNdx
is equivalent to strong convergence of {vk}k∈ℕ in L2(Ω,Akdx)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
(30)ζ*,n⟶ζ*inL1(Ω),ζ*∈Ψ*,ζ*,n-1⟶ζ*-1inL1(Ω),ζ*,n-1⟶ζ*-1invariablespaceL2(Ω,ζ*,ndx).For our further analysis, we make use of the following concept.Definition 14.
We say that a bounded sequence(31){(An,yn)∈L1(Ω;𝕊N)×H(Ω;Andx)}n∈ℕw-converges to(A,y)∈L1(Ω;𝕊N)×W01,1(Ω)asn→∞if
(32)An⟶AinL1(Ω;𝕊N),(33)yn⇀yinL2(Ω),(34)∇yn⇀∇yinthevariablespaceL2(Ω,Andx)N;
therefore,
(35)limn→∞∫Ω‍An·η→dx=∫Ω‍A·η→dx∀η→∈L∞(Ω;𝕊N),limn→∞∫Ω‍ynλdx=∫Ω‍yλdx∀λ∈L2(Ω),limn→∞∫Ω‍(ξ→,An∇yn)ℝNdx=∫Ω‍(ξ→,A∇y)ℝNdx∀ξ→∈C0∞(Ω)N.In order to motivate this definition, we give the following result.Lemma 15.
Let{(An,yn)∈L1(Ω;𝕊N)×H(Ω;Andx)}n∈ℕ be a sequence such that(i)
the sequence{yn∈H(Ω;Andx)}n∈ℕ is bounded; that is,
(36)supn∈ℕ∫Ω‍(yn2+(∇yn,An∇yn))dx<+∞;(ii)
{An}n∈ℕ⊂𝔐αβ(Ω)and there exists a matrix-valued functionA(x)∈𝕊Nsuch that
(37)An⟶A,An-1⟶A-1inL1(Ω;𝕊N)asn⟶∞.Then, A∈𝔐αβ(Ω)∩L1(Ω;𝕊N); and the original sequence is relatively compact with respect tow-convergence. Moreover, eachw-limit pair(A,y)belongs to the spaceL1(Ω;𝕊N)×H(Ω;Adx).Proof.
We note that (36)-(37), (4), and (13)-(14) immediately imply the boundedness of the original sequence inL1(Ω;𝕊N)×W1,1(Ω). Moreover, due to (37), we have
(38)dμ→n:=Andx⇀*Adx=:dμ→inℳ(Ω;𝕊N).
Thus, the compactness criterium for weak convergence in variable spaces (see Proposition9) and (36) imply the existence of a pair (y,v)∈L2(Ω)×L2(Ω,Adx)N such that, within a subsequence of{yn}n∈ℕ,
(39)yn⇀yinL2(Ω),(40)∇yn⇀vinvariablespaceL2(Ω,Andx)N.
Our aim is to show thatA∈𝔐αβ(Ω),v=∇y, andy∈H(Ω;Adx). It is clear thatA(x)∈𝕊Nand this matrix satisfies (8). SinceAn∈𝔐αβ(Ω)∩L1(Ω;𝕊N)for alln∈ℕ, it follows that there is a sequence{ζ*,n}n∈ℕinΨ*such that
(41)ζ*,n(x)I≤An(x)I≤β(x)Ia.e.inΩ∀k∈{1,…,N}.
Then, byL1-compactness of the setΨ*, there exists an elementζ*∈Ψ*such thatζ*,n→ζ*inL1(Ω)asn→∞. Moreover, Lemma 13 implies strong convergence
(42)ζ*,n-1⟶ζ*-1inL1(Ω)
and (7). Hence, passing to the limit in (41) asn→∞, we come to (9). Thus,A∈𝔐αβ(Ω)and the limit matrixA(x)∈𝕊Nsatisfies (10)-(11).
For our further analysis, we fix any test functionφ→∈C0∞(Ω)N and make use of the following equality:
(43)∫Ω‍(An-1φ→,Anψ→)ℝNdx=∫Ω‍(φ→,ψ→)ℝNdx=∫Ω‍(A-1φ→,Aψ→)ℝNdx,
which is obviously true for eachψ→∈C0∞(Ω)Nand for alln∈ℕ. Since
(44)limsupn→∞∫Ω‍(An-1φ→,AnAn-1φ→)ℝNdx=limsupn→∞∫Ω‍(φ→,An-1φ→)ℝNdx≤limsupn→∞∫Ω‍ζ*,n-1∥φ→∥ℝN2dx=by(42)∫Ω‍ζ*-1∥φ→∥ℝN2dx≤∥φ→∥C(Ω)N2∥ζ*-1∥L1(Ω)<+∞,
it follows that the sequence {An-1φ→∈L2(Ω,Andx)N}n∈ℕ is bounded. Consequently, combining this fact with (43), we conclude that An-1φ→⇀A-1φ→ in the variable space L2(Ω,Andx)N (see Definition 8). At the same time, strong convergence in (37) implies the relation
(45)limn→∞∫Ω‍(An-1φ→,AnAn-1φ→)ℝNdx=limn→∞∫Ω‍(φ→,An-1φ→)ℝNdx=∫Ω‍(φ→,A-1φ→)ℝNdx=∫Ω‍(A-1φ→,AA-1φ→)ℝNdx.
Hence (see Proposition 12),
(46)An-1φ→⟶A-1φ→stronglyinL2(Ω,Andx)N∀φ→∈C0∞(Ω)N.
Further, we note that for every measurable subset S⊂Ω, the estimate
(47)∫S‍∥∇yn∥ℝNdx≤(∫S‍∥∇yn∥ℝN2ζ*,ndx)1/2(∫S‍ζ*,n-1dx)1/2≤(∫Ω‍(∇yn,An(x)∇yn)ℝNdx)1/2(∫S‍ζ*,n-1dx)1/2≤C(∫S‍ζ*,n-1dx)1/2
implies equi-integrability of the family {∥∇yn∥ℝN}n∈ℕ. Hence, {∥∇yn∥ℝN}n∈ℕ is weakly compact in L1(Ω), which means the weak compactness of the vector-valued sequence {∇yn}n∈ℕ in L1(Ω)N. As a result, by the properties of the strong convergence in variable spaces, we obtain
(48)∫Ω‍(ξ→,∇yn)ℝNdx=∫Ω‍(An-1ξ→,An∇yn)ℝNdx⟶by(28),(40),and(46)∫Ω‍(A-1ξ→,Av)ℝNdx=∫Ω‍(ξ→,v)ℝNdx∀ξ→∈C0∞(Ω)N.
Thus, in view of the weak compactness property of{∇yn}n∈ℕinL1(Ω)N, we conclude that
(49)∇yn⇀vinL1(Ω;ℝN)asn⟶∞.
Since yn∈W01,1(Ω) for all n∈ℕ and the Sobolev space W01,1(Ω) is complete, (39) and (49) imply ∇y=v, and consequently y∈W01,1(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 C0∞(Ω) strongly converging in W(Ω;Adx) to this element.
Since for alln∈ℕ, yn∈H(Ω;Andx), there exist sequences {φkn}k∈ℕ⊂C0∞(Ω), such that φkn→yn strongly in H(Ω;Andx) as k→∞. Let us show that the sequence {φnn}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 φnn⇀y in L2(Ω) and ∇φnn⇀∇y in L2(Ω;Adx)N.
Indeed, for an arbitraryψ∈C0∞(Ω), we have
(50)|∫Ω‍(y-φnn)ψdx|≤|∫Ω‍(y-yn)ψdx|+|∫Ω‍(yn-φnn)ψdx|⟶0,asn⟶∞,
since yn⇀y in L2(Ω), as n→∞ and for any ε>0 there exists N0∈ℕ such that |∫Ω‍(yn-φnn)ψdx|<ε for all n>N0. Similarly, for an arbitrary ψ→∈C0∞(Ω)N, we have
(51)|∫Ω‍(A(∇y-∇φnn),ψ→)ℝNdx|≤|∫Ω‍(A∇y,ψ→)ℝNdx-∫Ω‍(An∇yn,ψ→)ℝNdx|+|∫Ω‍(An(∇yn-∇φnn),ψ→)ℝNdx|+|∫Ω‍((An-A)∇φnn,ψ→)ℝNdx|⟶0,asn⟶∞.
It is only the convergence to zero of the third summand that should be explained here:
(52)|∫Ω‍((An-A)∇φnn,ψ→)ℝNdx|≤(∫Ω‍(|An-A|∇φnn,∇φnn)ℝNdx)1/2×(∫Ω‍(|An-A|ψ→,ψ→)ℝNdx)1/2=J1·J2.
Obviously,J2→0asn→∞. It is left to prove, thatJ1≤C. Indeed,
(53)∫Ω‍(|An-A|∇φnn,∇φnn)ℝNdx≤∫Ω‍(An∇φnn,∇φnn)ℝNdx+∫Ω‍(A∇φnn,∇φnn)ℝNdx≤∫Ω‍(An∇φnn,∇φnn)ℝNdx+∫Ω‍βζad-1(An∇φnn,∇φnn)ℝNdx≤(1+∥βζad-1∥L∞(Ω))(∫Ω‍(An∇φnn,∇φnn)ℝNdx±∫Ω‍(An∇yn,∇yn)ℝNdx)≤C,
since ∥∇φnn∥L2(Ω;Andx)2-∥∇yn∥L2(Ω;Andx)2→0 as n→∞ and {∥∇yn∥L2(Ω;Andx)2}n∈ℕ is a bounded sequence in ℝ.
Hence, we proved the existence of the sequence fromC0∞(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 {φnn}n∈ℕ. Linear combinations of smooth functions are smooth as well. Hence, as the strong limit of the sequence from C0∞(Ω), the element y belongs to the space H(Ω;Adx).


## 5. Setting of the Optimal Control Problem

LetM∈𝕊Nbe a given constant matrix satisfying the condition
(54)(Mξ,ξ)ℝN≥m∥ξ∥ℝN2forsomem>0,∀ξ∈ℝN.LetKbe a convex closed subset of the spaceW01,1(Ω) such thatK∩C0∞(Ω)≠∅, and letf∈L2(Ω)be a given distribution. Having fixedA(x)∈L1(Ω;𝕊N), we denote byKAthe closure of{K∩H(Ω;Adx)}with respect to the norm∥·∥A, and consider the following elliptic variational inequality: to find an elementy∈KAsuch that
(55)〈-div(A(x)∇y)+y,v-y〉HA≥〈f,v-y〉HA,∀v∈KA.Remark 16.
The distinctive feature of this variational problem is the following fact: choosing a matrix of coefficientsA(x)∈L1(Ω;𝕊N), 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;𝕊N)is an admissible control for variational inequality (55) (it is written asA∈𝔄ad) if
(56)A∈BV(Ω;𝕊N),A∈𝔐αβ(Ω),∫Ω‍A(x)dx=M.
Hereinafter, we assume that the set𝔄adis nonempty.The optimal control problem we consider here is to minimize the discrepancy (tracking error) between a given distributionyd∈L2(Ω) and the solution of elliptic variational inequality (55)-(56) by choosing an appropriate coefficients matrixA∈𝔄ad. More precisely, we are concerned with the following optimal control problem:
(57)Minimize{+∑i,j=1N‍∫Ω‍|Daij(x)|I(A,y)=∫Ω‍|y(x)-yd(x)|2dx+∫Ω‍(∇y(x),A(x)∇y(x))ℝNdx+∑i,j=1N‍∫Ω‍|Daij(x)|}
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 theHA-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:(58)Ξ={(A,y)∣A∈𝔄ad,y∈HA,(A,y)arerelatedby(55)}.Remark 19.
Note that due to (13)-(14), we have the following obvious inclusion for the set of admissible solutions: Ξ⊂L1(Ω;𝕊N)×W01,1(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∈𝔄ad, the operatorBA:KA→H(Ω;Adx), defined as
(59)〈BA(y),v〉HA=∑i,j=1N‍∫Ω‍(aij(x)∂y∂xj)∂v∂xidx+∫Ω‍yvdx,∀v∈H(Ω;Adx),
is strictly monotone and coercive.Proof.
LetA∈𝔄adbe a fixed matrix. Then the relation
(60)〈BA(y),y〉HA=∫Ω‍(∇y,A∇y)ℝNdx+∫Ω‍y2dx=∥y∥A2
implies the strict monotonicity and coercivity property of the operatorBA. Indeed, by setting in (60)y=u-v,u,v∈HA, we immediately obtain
(61)〈BA(u)-BA(v),u-v〉HA=∥u-v∥A2>0∀u≠v.Proposition 21.
For every controlA∈𝔄adand every f∈L2(Ω), there exists a unique solution to degenerate elliptic variational inequality (55)-(56).Proof.
Taking into account Lemmas7 and 20, it is enough to show that there exists an elementv0∈KAsuch that
(62)〈-div(A(x)∇y)+y,y-v0〉HA∥y∥A⟶+∞as∥y∥A⟶∞,y∈KA.
In order to do this, we fix an arbitrary elementv0∈KAand a matrixA∈𝔄ad. Then for ally∈KA, in view of estimate (60), we have
(63)〈-div(A(x)∇y)+y,y-v0〉HA=∥y∥A2-〈-div(A(x)∇y)+y,v0〉HA≥∥y∥A2-|〈-div(A(x)∇y)+y,v0〉HA|,
where
(64)|〈-div(A(x)∇y)+y,v0〉HA|=|∫Ω‍(A(x)∇y,∇v0)ℝNdx+∫Ω‍yv0dx|≤(∫Ω‍|A1/2∇y|ℝN2dx)1/2(∫Ω‍|A1/2∇v0|ℝN2dx)1/2+∥y∥L2(Ω)∥v0∥L2(Ω)=(∫Ω‍(∇y,A∇y)ℝNdx)1/2(∫Ω‍(∇v0,A∇v0)ℝNdx)1/2+∥y∥L2(Ω)∥v0∥L2(Ω)=∥∇y∥L2(Ω;Adx)N∥v0∥L2(Ω;Adx)N+∥y∥L2(Ω)∥v0∥L2(Ω)≤∥y∥A∥v0∥A.
Combining (63) and (64) we come to the inequality
(65)〈-div(A(x)∇y)+y,y-v0〉HA∥y∥A≥∥y∥A-∥v0∥A⟶+∞,as∥y∥A⟶∞.
Hence, problem (55)-(56) satisfies all conditions of Lemma 7. Therefore, for every control A(x)∈𝔄ad and every f∈L2(Ω), the considered problem has a unique solution.Thus, as follows from Proposition21, the set of admissible solutions Ξ is nonempty. Hence, the minimization problem inf(A,y)∈ΞI(A,y) is regular. We say that a pair (A0,y0)∈L1(Ω;𝕊N)×HA is H-optimal for problem (55)-(56), (57) if
(66)(A0,y0)∈Ξ,I(A0,y0)=inf(A,y)∈ΞI(A,y).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 setK⊂W01,1(Ω), we have the following: for all A∈𝔄ad, for all y∈KA, and for all ε=ε(A)>0there exists an element yε∈KA such that
(67)BA(yε)=-div(A∇yε)+yε∈L2(Ω),yε+εBA(yε)=y.
Then, as a direct consequence of Theorem  8.7 in [24] and Proposition 21, we come to the following result.Theorem 22.
Assume Hypothesis1 holds true. LetA∈𝔄adbe any admissible control and lety=y(A)be the corresponding solution of variational inequality (55). ThenBA(y)∈L2(Ω)providedf∈L2(Ω).As an example of the setK∈W01,1(Ω)satisfying the Hypothesis 1, we propose the following one.Example 23.
Letψ∈C0∞(Ω)be a given function such that
(68)-div(A(x)∇ψ)+ψ≤0∀A∈𝔄ad.
Let us define the closed convex setKas follows. 							(69)K={v∣v∈W01,1(Ω),v≥ψa.e.inΩ}.
Then, for an arbitrary matrixA∈𝔄ad, we have
(70)KA=cl{K∩H(Ω;Adx)}HA={v∣v∈H(Ω;Adx),v≥ψa.e.inΩ}.
Let us show that for all y∈KA, for all ε>0 there exists an element yε∈KA satisfying the following conditions:
(71)-div(A∇yε)+yε∈L2(Ω),yε+εBA(yε)=y.
As a result, we would have
(72)BA(yε)-y-yεε=0,
and having set yε-y=wε∈L2(Ω), we come to the equation
(73)BA(wε)+wεε=-BA(y).
Obviously, the mappingH(Ω;Adx)∋wε→BA(wε)+wε/εis strictly monotone and coercive (see Proposition 21). Hence, there exists a unique solutionwεto (73). Moreover, in this caseBA(yε)=(y-yε)/ε∈L2(Ω).
Let us show thatyε∈KA, namely,yε≥ψalmost everywhere inΩ. In order to do this, we setzε=sup{ψ-yε,0}. Then it is enough to prove thatzε=0. From (72) it follows that
(74)BA(ψ-yε)-BA(ψ)=yε-yε.
Having noticed〈BA(ψ-yε),zε〉HA=〈BA(zε),zε〉HA≥0, we conclude that
(75)〈-BA(ψ),zε〉HA≤〈yε-yε,zε〉HA.
Sincezε≥0andBA(ψ)≤0by the initial assumptions, we have 〈(yε-y)/ε,zε〉HA≥0; that is,
(76)∫E‍yε-yεzεdx≥0,E={x∣x∈Ω,ψ(x)≥yε(x)}.
At the same time, sincey∈KA, theny≥ψand, hence,y≥yεonE. As a result, we haveyε-y≤0onE. Therefore,
(77)∫E‍yε-yεzεdx≤0⟹∫E‍yε-yεzεdx=0.
Thus, we arrive at the following conclusion: eitheryε=yorzε=0a.e. onE. Hence,zε=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{(An,yn)∈Ξ}n∈ℕis bounded if
(78)supn∈ℕ[∥An∥BV(Ω;𝕊N)+∥yn∥An]<+∞.Lemma 25.
Let{(An,yn)∈Ξ}n∈ℕbe a bounded sequence in the sense of Definition 24. Then there exists a pair(A,y)∈L1(Ω;𝕊N)×W01,1(Ω)such that, up to a subsequence,
(79)(An,yn)⟶w(A,y),A∈𝔄ad,y∈H(Ω;Adx).Proof.
By the compactness ofBV-functions (see Proposition 6), there exist a subsequence of {An}n∈ℕ, still denoted by the same indices, and a matrix A∈BV(Ω;𝕊N) such that An→A in L1(Ω;𝕊N). Thus,
(80)A∈BV(Ω;𝕊N),∫Ω‍A(x)dx=limn→∞∫Ω‍An(x)dx=M,
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∈𝔄ad, we make use of the following observation.
We have(An,yn)∈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*,n-1→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 An→A, we may assume that An-1→A-1 almost everywhere in Ω. Since An(x)≥ζ*,nI a.e. in Ω, it follows that
(81)∫S‍(ξ→,An-1ξ→)ℝNdx≤∫S‍ζ*,n-1dx∥ξ→∥ℝN2∀n∈ℕ
for any subsetS⊂Ω. Hence, due to the strongL1-convergenceζ*,n-1→ζ*-1, the sequence{An-1}n∈ℕis equi-integrable. Then, by Lebesgue’s Theorem (see Lemma 2) we obtainAn-1→A-1inL1(Ω;𝕊N)asn→∞. As a result,
(82)A∈𝔐αβ(Ω)∩L1(Ω;𝕊N)
by Lemma 15. Combining this fact with properties (80), we concludeA∈𝔄ad.
It remains to notice that conditions (33)-(34) of Definition 14 and y∈HA for the w-limiting component (A,y) of the sequence {(An,yn)}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{(An,yn)∈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∈𝔄ad and y∈H(Ω;Adx).
Further, let us show thaty∈KA. Indeed, the condition yn∈KAn implies yn∈K for all n∈ℕ, whereas relations (33)-(34) imply the weak convergence of {yn}n∈ℕ to y in W01,1(Ω) within a subsequence (see (13)-(14)). Since the set K is closed in W01,1(Ω) and convex, then, in view of Mazur’s lemma, K is weakly closed in W01,1(Ω). Therefore, y∈K and y∈H(Ω;Adx). Combining these facts, we conclude that y∈KA.
It remains to show that the pair(A,y) satisfies relation (55) for all v∈KA. In order to do this, we divide the rest of the proof into three steps.
Step   1. Let us show that for everyv∈KA, there exists a sequence{vn∈KAn}n∈ℕ, such that
(83)vn⟶vinL2(Ω),∇vn⇀∇vinvariablespaceL2(Ω;Andx)N.
Indeed, sincev∈H(Ω;Adx), it follows that there exists a sequence{φn}n∈ℕinC0∞(Ω)such that
(84)φn⟶vinL2(Ω),∇φn⟶∇vinL2(Ω;Adx)Nasn⟶∞.
Moreover, by the initial assumptions, KA contains only smooth functions and their limits with respect to the strong convergence in H(Ω;Adx). Hence, φn∈KA for all indices large enough. Let us show that {∇φn}n∈ℕ converges to ∇v weakly in L2(Ω;Andx)N. To begin with, we show the boundedness of the sequence {∇φn}n∈ℕ in L(Ω;Andx)N. Indeed,
(85)limsupn→∞∫Ω‍(An∇φn,∇φn)ℝNdx≤limsupn→∞∫Ω‍β(x)∥∇φn(x)∥ℝN2dx≤limsupn→∞∫Ω‍β(x)ζ*,n-1(x)(A∇φn,∇φn)ℝNdx<limsupn→∞∫Ω‍β(x)ζad-1(x)(A∇φn,∇φn)ℝNdx≤∥βζad-1∥L∞(Ω)limsupn→∞∫Ω‍(A∇φn,∇φn)ℝNdx<∞.
In view of Proposition 9, the sequence {∇φn}n∈ℕ, within a subsequence, is weakly convergent in the variable space L(Ω;Andx)N. Let us show that ∇v is its weak limit. Let ψ∈C0∞(Ω)N be an arbitrary smooth function. Then
(86)|∫Ω‍(An∇φn,ψ)ℝNdx-∫Ω‍(A∇v,ψ)ℝNdx|≤|∫Ω‍((An-A)∇φn,ψ)ℝNdx|+|∫Ω‍(A(∇φn-∇v),ψ)ℝNdx|=I1+I2.
Obviously,I2→0asn→∞provided by the strong convergenceφn→vinH(Ω;Adx). Further, using the Cauchy-Bunjakowsky inequality, we have
(87)|∫Ω‍((An-A)∇φn,ψ)ℝNdx|≤(∫Ω‍((An-A)∇φn,∇φn)ℝNdx)1/2·(∫Ω‍((An-A)∇ψ,∇ψ)ℝNdx)1/2⟶0,asn⟶∞
as the product of a bounded sequence and a sequence converging to zero. It follows that
(88)φn⟶vstronglyinL2(Ω),∇φn⇀∇vweaklyinL2(Ω;Andx)asn⟶∞.
Step   2. Here we have to substantiate the correctness of the limit passage in the following variational inequality:
(89)〈-div(An(x)∇yn)+yn,vn-yn〉HAn≥〈f,vn-yn〉HAn∀vn∈KAn,
whereyn∈KAnis its solution. To do so, it is enough to show that
(90)div(An(x)∇yn)⇀div(A(x)∇y)inL2(Ω)asn⟶∞.
Since for every n∈ℕ relation (89) holds true for everyvn∈KAn, we may consider only those elements{vn∈KAn}n∈ℕwhich form convergent sequences to elementsv∈KAin a way that was established in Step 1.
Taking into account Hypothesis1 and Theorem 22, we obtain
(91)div(An(x)∇yn)∈L2(Ω)∀n∈ℕ,
and, moreover, for an arbitrary ψ∈C0∞(Ω), we have
(92)〈div(An(x)∇yn),ψ〉L2(Ω)=∫Ω‍(An∇yn,∇ψ)ℝNdx⟶∫Ω‍(A∇y,∇ψ)ℝNdx=〈div(A(x)∇y),ψ〉L2(Ω)asn⟶∞.
Hence, the assertion (90) is valid.
Step   3. In view of the initial assumptions, we have(1)
An→Astrongly inL1(Ω;𝕊N);(2)
yn⇀yinL2(Ω);(3)
∇yn⇀∇yinL2(Ω;Andx);(4)
div(An(x)∇yn)⇀div(A(x)∇y)inL2(Ω);(5)
forallv∈KA there exists a sequence {vn∈KAn}n∈ℕ, such that vn→v strongly in L2(Ω) and ∇vn⇀∇v in L2(Ω;Andx).Hence, using this, Proposition 10 and the lower semicontinuity property ofL2-norm with respect to the weak convergence, we obtain
(93)〈-div(A(x)∇y)+y,v-y〉HA=〈-div(A(x)∇y)+y,v〉L2(Ω)-∫Ω‍((A∇y,∇y)ℝN+y2)dx≥limsupn→∞〈-div(An(x)∇yn)+yn,vn〉L2(Ω)-liminfn→∞∫Ω‍((An∇yn,∇yn)ℝN+yn2)dx≥limsupn→∞(∫Ω‍((An∇yn,∇yn)ℝN+yn2)dx〈-div(An(x)∇yn)+yn,vn〉HAn-∫Ω‍((An∇yn,∇yn)ℝN+yn2)dx)≥limn→∞〈f,vn-yn〉HAn=limn→∞〈f,vn-yn〉L2(Ω)=〈f,v-y〉HA∀v∈KA.
As a result, we come to the desired relation:
(94)〈-div(A(x)∇y)+y,v-y〉HA≥〈f,v-y〉HA∀v∈KA,
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;𝕊N)×H(Ω;Andx), 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.
Letf∈L2(Ω) and yd∈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;𝕊N)×H(Ω;A0dx).Proof.
Since the cost functionalI=I(A,y) is bounded below and Ξ≠∅, it provides the existence of a minimizing sequence {(An,yn)∈Ξ}n∈ℕ to the problem (66). Then,
(95)inf(A,y)∈ΞI(A,y)=limn→∞I(An,yn)=limn→∞[+∑i,j=1N‍∫Ω∖Q‍|Daijn(x)|∫Ω‍|yn(x)-yd(x)|2dx+∫Ω‍(∇yn(x),A(x)∇yn(x))ℝNdx+∑i,j=1N‍∫Ω∖Q‍|Daijn(x)|]<+∞
implies the existence of a constant C>0 such that
(96)supn∈ℕ∥∇yn∥L2(Ω,Andx)N≤C,supn∈ℕ∥yn∥L2(Ω)≤C,supn∈ℕ∥An∥BV(Ω∖Q;𝕊N)≤C.
Hence, the minimizing sequence {(An,yn)∈Ξ}n∈ℕ is bounded in the sense of Definition 24. Then, by Lemma 25, there exist functions A0∈L1(Ω;𝕊N) and y0∈H(Ω;A0dx) such that, up to a subsequence, (An,yn)→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,
(97)I(A0,y0)≤liminfn→∞I(An,yn)=inf(A,y)∈ΞI(A,y);
that is, (A0,y0) is an optimal solution. The proof is complete.



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*Source: 821964-2013-07-24.xml*