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

Abstract

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 coefficients $A = A^{sym} + A^{skew}$ of such system as a control in $L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N})$ . One of the important features of the class of admissible controls is the fact that the matrices $A (x)$ are unbounded on Ω and eigenvalues of the symmetric parts $A^{sym}$ may vanish in Ω. In spite of the fact that the equations of this type can exhibit non-uniqueness of weak solutions, the corresponding OCP is well-possed and admits at least one solution. At the same time, optimal solutions to such problem can inherit a singular character of the matrices $A^{skew}$ . We indicate two types of optimal solutions to the above problem and show that one of them can not be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the Steklov smoothing of matrix-valued controls A.

Keywords

degenerate elliptic equations control in coefficients weighted Sobolev spaces variational convergence

1. Introduction

Optimal control in coefficients for partial differential equations is a classical subject initiated by Lurie [19], Lions [17], and Pironneau [20]. 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 coefficients in the main part of elliptic operators, while only a few articles deal with unbounded and degenerate controls in coefficients, see [2,6,15,16].

The aim of this paper is to study the following optimal control problem (OCP) for a linear elliptic equation with unbounded coefficients in the main part of the elliptic operator $\begin{array}{l} (1.1) & Minimize I (A, y) = ‖ y - y_{d} ‖_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla y, A^{sym} \nabla y)}_{R^{N}} d x \\ subject to the constraints \\ (1.2) & - div (A (x) \nabla y) + a_{0} (x) y = - div f in Ω, \\ (1.3) & y = 0 on \partial Ω with A \in A_{ad}, \end{array}$ where the matrix $A = A^{sym} + A^{skew} \in L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N})$ is adopted as a control with $p ⩾ 1$ , $f \in D^{'} (Ω; R^{N})$ , $a_{0} \in L^{\infty} (Ω)$ , and $y_{d} \in L^{2} (Ω)$ are given distributions such that $a_{0} (x) ⩾ α > 0$ almost everywhere in Ω for some positive constant α. As a characteristic feature of such optimal control problem is a specification of the class of admissible controls. Namely, we define a class of admissible controls $A_{ad}$ as a nonempty subset of $L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N})$ such that for every $A \in A_{ad}$ we have $\begin{array}{l} A^{*} (x) ⪯ A^{skew} (x) ⪯ A^{* *} (x) a.e. in Ω, \\ ζ_{ad} (x) I ⩽ A^{sym} (x) ⩽ β (x) I a.e. in Ω, \int_{Ω} A^{sym} (x) d x = M, \end{array}$ where $M \in S_{sym}^{N}$ and $A^{*}, A^{* *} \in L^{2 p} (Ω; S_{skew}^{N})$ are given matrices, $β \in L^{1} (Ω)$ , $β ⩾ ζ_{ad} ⩾ 0$ , and $ζ_{ad}^{- 1} \in L^{2 q} (Ω)$ for $q = p / (p - 1)$ .

We note that these assumptions on the class of admissible controls together with $L^{2 p}$ -properties of the skew-symmetric parts are essentially weaker than they usually are in the literature. Thereby, for a “typical” symmetric part of admissible control we deal with the Dirichlet boundary value problem for degenerate anisotropic elliptic equation. Hence, this problem can exhibit the so-called Lavrentieff phenomenon, non-uniqueness of the weak solutions as well as other surprising consequences (see, for instance, [1,3,25,27]). On the other hand, the unboundedness of skew-symmetric part of admissible controls can lead to the existence of elements $y \in H_{0}^{1} (Ω)$ such that $y \notin L^{\infty} (Ω)$ , $\begin{matrix} \int_{Ω} {(\nabla φ_{n}, A^{skew} \nabla φ_{n})}_{R^{N}} d x = 0, \forall n \in N, and lim_{n \to \infty} \int_{Ω} {(\nabla φ_{n}, A^{skew} \nabla y)}_{R^{N}} d x \neq 0, \end{matrix}$ where $C_{0}^{\infty} (Ω) ∋ φ_{n} \to y$ strongly in $H_{0}^{1} (Ω)$ [9,10,26]. As a result, the existence, uniqueness, and variational properties of the weak solution to (1.2)–(1.3) usually are drastically different from the corresponding properties of solutions to the elliptic equations with coercive $L^{\infty}$ -matrices in coefficients (we refer to [4,21–24] for the details and other results in this field).

We will prove (see Theorem 5.10) that some weak solutions to this problem can be obtained as the limit of solutions of coercive problems with bounded coefficients, using the Steklov smoothing operator. As a result, this allows us to prove that the corresponding optimal control problem (1.1)–(1.3) has a nonempty set of solutions (see Theorem 4.8). However, even if the OCP (1.1)–(1.3) has a unique solution $(A^{0}, y^{0})$ , it does not ensure that this pair can be attained in such way (see Theorem 5.14 for the details). We conclude the paper with an example of OCP for which there exists a unique solution that can not be attained through the limit of optimal solutions to the regularized problems.

2. Notation and preliminaries

Let Ω be a bounded open connected subset of $R^{N}$ ( $N ⩾ 2$ ) with Lipschitz boundary $\partial Ω$ . Let $χ_{E}$ be the characteristic function of a subset $E \subset Ω$ , i.e. $χ_{E} (x) = 1$ if $x \in E$ , and $χ_{E} (x) = 0$ if $x \notin E$ .

Let $M^{N}$ be the set of all $N \times N$ real matrices. We denote by $S_{skew}^{N}$ the set of all skew-symmetric matrices $C = {[c_{i j}]}_{i, j = 1}^{N}$ . Thus, if $C \in S_{skew}^{N}$ then $c_{i j} = - c_{j i}$ and, hence, $c_{i i} = 0$ . Let $S_{sym}^{N}$ be the set of all $N \times N$ symmetric matrices. In what follows, we will always identify each matrix $B \in M^{N}$ with its decomposition $B = B_{sym} + B_{skew}$ , where $B_{sym} : = \frac{1}{2} (B + B^{t}) \in S_{sym}^{N}$ and $B_{skew} : = \frac{1}{2} (B - B^{t}) \in S_{skew}^{N}$ . By matrix norm in $M^{N}$ (and for functions with values in $S_{sym}^{N}$ and $S_{skew}^{N}$ as well) we mean a sub-multiplicative norm $‖ A ‖ : = sup {| A ξ |_{R^{N}} : ξ \in R^{N} with | ξ |_{R^{N}} = 1}$ . It worth to note that, in the case of Euclidean norm $| \cdot |_{R^{N}}$ , the norm $‖ A ‖$ can be computed as the spectral norm $‖ A ‖ = \sqrt{λ_{\max} (A^{t} A)}$ , where $λ_{\max} (A^{t} A)$ is the largest eigenvalue of the positive-semidefinite matrix $A^{t} A$ .

Let $p, q \in [1, \infty]$ be given real numbers such that $\frac{1}{p} + \frac{1}{q} = 1$ . Let $L^{2 p} (Ω; S_{skew}^{N})$ be the normed space of measurable $2 p$ -integrable functions whose values are skew-symmetric matrices.

Let $A (x)$ and $B (x)$ be given matrices such that $A, B \in L^{2 p} (Ω; S_{skew}^{N})$ . We say that these matrices are related by the binary relation ⪯ over the set $L^{2 p} (Ω; S_{skew}^{N})$ (in symbols, $A (x) ⪯ B (x)$ a.e. in Ω), if $\begin{matrix} (2.1) & L^{N} (⋃_{i = 1}^{N} ⋃_{j = i + 1}^{N} {x \in Ω : a_{i j} (x) > b_{i j} (x)}) = 0 . \end{matrix}$ Here, $L^{N} (E)$ denotes the N-dimensional Lebesgue measure of $E \subset R^{N}$ defined on the completed Borelian σ-algebra. As immediately follows from (2.1), the binary relation ⪯ is reflexive, antisymmetric, and transitive. Hence, ⪯ specifies a (non-strict) partial order over the set $L^{2 p} (Ω; S_{skew}^{N})$ .

Let $ζ_{ad}$ and β be given $L^{1} (Ω)$ -functions satisfying the properties $\begin{array}{l} (2.2) & β > ζ_{ad} ⩾ 0 a.e. in Ω, ζ_{ad}^{- 1} \in L^{2 q} (Ω) . \end{array}$

By $M_{ζ_{ad}}^{β} (Ω)$ we denote the set of all matrices $A = [a_{i j} (\cdot)] \in L^{1} (Ω; S_{sym}^{N})$ such that $\begin{matrix} (2.3) & ζ_{ad} (x) I ⩽ A (x) ⩽ β (x) I a.e. in Ω . \end{matrix}$ Here, I is the identity matrix in $R^{N \times N}$ , and (2.3) should be considered in the sense of quadratic forms. Therefore, condition (2.3) implies the following inequalities: $\begin{array}{l} (2.4) & if A \in M_{ζ_{ad}}^{β} (Ω), then ‖ A ‖_{L^{1} (Ω; S_{sym}^{N})} ⩽ ‖ β ‖_{L^{1} (Ω)} < + \infty, \\ (2.5) & ζ_{ad} (x) | ξ |_{R^{N}}^{2} ⩽ {(A (x) ξ, ξ)}_{R^{N}} a.e. in Ω, \forall ξ \in R^{N}, \\ (2.6) & {| A^{- 1 / 2} (x) ξ |}_{R^{N}}^{2} ⩽ ζ_{ad}^{- 1} (x) | ξ |_{R^{N}}^{2} a.e. in Ω, \forall ξ \in R^{N}, \end{array}$ and, therefore, $\begin{matrix} (2.7) & A^{- 1 / 2} \in L^{4 q} (Ω; S_{sym}^{N}) and {‖ A^{- 1 / 2} ‖}_{L^{4 q} (Ω; S_{sym}^{N})} ⩽ \sqrt{{‖ ζ_{ad}^{- 1} ‖}_{L^{2 q} (Ω)}} . \end{matrix}$

To each matrix $A \in M_{ζ_{ad}}^{β} (Ω) \subset L^{1} (Ω; S_{sym}^{N})$ we will associate two weighted Sobolev spaces: $W_{A} (Ω) = W (Ω; A d x)$ and $H_{A} (Ω) = H (Ω; A d x)$ , where $W_{A} (Ω)$ is the set of functions $y \in W_{0}^{1, 1} (Ω)$ for which the norm $\begin{matrix} (2.8) & ‖ y ‖_{A} = {(\int_{Ω} (y^{2} + {(\nabla y, A \nabla y)}_{R^{N}}) d x)}^{1 / 2} \end{matrix}$ is finite, and $H_{A} (Ω)$ is the closure of $C_{0}^{\infty} (Ω)$ in $W_{A} (Ω)$ -norm. It is well-known that due to the inequality (2.5) the space $W_{A} (Ω)$ is complete with respect to the norm $‖ \cdot ‖_{A}$ (see [14]). It is clear that $H_{A} (Ω) \subset W_{A} (Ω)$ , and $W_{A} (Ω)$ , $H_{A} (Ω)$ 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 $ν \in (\frac{N}{2}, + \infty)$ such that $u^{- ν} \in L^{1} (Ω)$ , then the expressions (for more details see [7, p. 46]): $\begin{array}{l} (2.9) & ‖ y ‖_{1, H_{u}} = {[\int_{Ω} u | \nabla y |^{2} d x]}^{1 / 2} and \\ (2.10) & ‖ y ‖_{2, H_{u}} = {(\int_{Ω} (y^{2} + u | \nabla y |^{2}) d x)}^{1 / 2} \end{array}$ can be considered as equivalent norms on $H_{u} : = {cl}_{‖ \cdot ‖_{2, H_{u}}} C_{0}^{\infty} (Ω)$ . Moreover, in this case the embedding $H_{u} ↪ L^{2} (Ω)$ is compact. Taking this fact and definition of the class $M_{ζ_{ad}}^{β} (Ω)$ into account, we deduce that the norm $‖ \cdot ‖_{A}$ , given by (2.8), is equivalent to the following one $\begin{matrix} (2.11) & ‖ y ‖_{1, A} = {(\int_{Ω} {(\nabla y, A \nabla y)}_{R^{N}} d x)}^{1 / 2} \end{matrix}$ on $H_{A} (Ω)$ provided $A \in M_{ζ_{ad}}^{β} (Ω)$ , $ζ_{ad}^{- 1} \in L^{2 q} (Ω)$ , where $\begin{matrix} (2.12) & p \in [1, \infty) if N \in {2, 3}, and 1 ⩽ p ⩽ \frac{N}{N - 4} if N ⩾ 4 . \end{matrix}$ Indeed, since the conditions (2.12) implies the fulfilment of inequality $q > N / 4$ , it follows that $ν : = 2 q \in (\frac{N}{2}, + \infty)$ and $ζ_{ad}^{- ν} \in L^{1} (Ω)$ .

Let $A = A_{sym} + A_{skew}$ be a given matrix such that $A_{skew} \in L^{2 p} (Ω; S_{skew}^{N})$ and $A_{sym} \in L^{1} (Ω; S_{sym}^{N})$ . In what follows, we associate with A the bilinear skew-symmetric form $\begin{matrix} (2.13) & Φ {(y, v)}_{A} = \int_{Ω} {(\nabla v, A_{skew} \nabla y)}_{R^{N}} d x, \forall y, v \in W_{A_{sym}} (Ω), \end{matrix}$ and introduce the matrix $C (x) \in S_{skew}^{N}$ following the rule $\begin{matrix} (2.14) & C (x) = A_{sym}^{- 1 / 2} (x) A_{skew} (x) A_{sym}^{- 1 / 2} (x) a.e. in Ω . \end{matrix}$ It is easy to see that $C \in L^{2} (Ω; S_{skew}^{N})$ . Indeed, by the Cauchy–Bunyakowsky inequality and estimate (2.7), we have $\begin{array}{l} ‖ C ‖_{L^{2} (Ω; S_{skew}^{N})}^{2} & ⩽ \int_{Ω} ‖ A_{skew} ‖_{L (R^{N}, R^{N})}^{2} {‖ A_{sym}^{- \frac{1}{2}} ‖}_{L (R^{N}, R^{N})}^{4} d x \\ ⩽ {(\int_{Ω} ‖ A_{skew} ‖_{L (R^{N}, R^{N})}^{2 p} d x)}^{1 / p} {(\int_{Ω} {‖ A_{sym}^{- \frac{1}{2}} ‖}_{L (R^{N}, R^{N})}^{4 q} d x)}^{1 / q} \\ = ‖ A_{skew} ‖_{L^{2 p} (Ω; S_{skew}^{N})}^{2} {‖ A_{sym}^{- \frac{1}{2}} ‖}_{L^{4 q} (Ω; S_{sym}^{N})}^{4} \\ (2.15) & ⩽ {‖ ζ_{ad}^{- 1} ‖}_{L^{2 q} (Ω)}^{2} ‖ A_{skew} ‖_{L^{2 p} (Ω; S_{skew}^{N})}^{2} < + \infty . \end{array}$ Hence, the form $Φ {(y, v)}_{A}$ is unbounded on $W_{A_{sym}} (Ω)$ , in general.

However, if we temporary assume that $C \in L^{\infty} (Ω; S_{skew}^{N})$ , then the bilinear form $Φ {(\cdot, \cdot)}_{A}$ is obviously bounded on $W_{A_{sym}} (Ω)$ . In order to deal with the case $C \notin L^{\infty} (Ω; S_{skew}^{N})$ , we notice that the value $Φ {(y, v)}_{A}$ is always finite provided $y \in W_{A_{sym}} (Ω)$ and $φ \in C_{0}^{\infty} (Ω)$ . Indeed, $\begin{array}{l} {| Φ {(y, v)}_{A} |}^{2} & : = {| \int_{Ω} {(\nabla φ, A_{skew} \nabla y)}_{R^{N}} d x |}^{2} \\ ⩽ ‖ φ ‖_{C^{1} (\overline{Ω})}^{2} {(\int_{Ω} | A_{skew} \nabla y |_{R^{N}} d x)}^{2} \\ ⩽ ‖ φ ‖_{C^{1} (\overline{Ω})}^{2} \int_{Ω} {‖ A_{skew} A_{sym}^{- \frac{1}{2}} ‖}_{L (R^{N}, R^{N})}^{2} d x \int_{Ω} {| A_{sym}^{\frac{1}{2}} \nabla y |}_{R^{N}}^{2} d x \\ ⩽ ‖ φ ‖_{C^{1} (\overline{Ω})}^{2} \int_{Ω} ‖ A_{skew} ‖_{L (R^{N}, R^{N})}^{2} ζ_{ad}^{- 1} d x \int_{Ω} {(\nabla y, A_{sym} \nabla y)}_{R^{N}} d x \\ ⩽ ‖ φ ‖_{C^{1} (\overline{Ω})}^{2} ‖ y ‖_{A}^{2} {(\int_{Ω} ‖ A_{skew} ‖_{L (R^{N}, R^{N})}^{2 p} d x)}^{1 / p} {(\int_{Ω} ζ_{ad}^{- q} d x)}^{1 / q} \\ ⩽ ‖ φ ‖_{C^{1} (\overline{Ω})}^{2} ‖ y ‖_{A}^{2} | Ω |^{1 / 2 q} {‖ ζ_{ad}^{- 1} ‖}_{L^{2 q} (Ω)} ‖ A_{skew} ‖_{L^{2 p} (Ω; S_{skew}^{N})}^{2} < + \infty . \end{array}$

Hence, if $C \in L^{2} (Ω; S_{skew}^{N})$ then the integral $\int_{Ω} {(\nabla φ, A_{skew} \nabla y)}_{R^{N}} d x$ is well defined for every $y \in W_{A_{sym}} (Ω)$ and $φ \in C_{0}^{\infty} (Ω)$ . In what follows, we set $\begin{array}{l} {[y, φ]}_{A} = \int_{Ω} {(\nabla φ, A_{skew} \nabla y)}_{R^{N}} d x \\ = \int_{Ω} {(A_{sym}^{1 / 2} \nabla φ, C A_{sym}^{1 / 2} \nabla y)}_{R^{N}} d x, \forall y \in W_{A_{sym}} (Ω), \forall φ \in C_{0}^{\infty} (Ω), \end{array}$ where the matrix C is defined by (2.14), and introduce of the following notion.

Let $V_{A_{sym}} (Ω)$ be some intermediate space with $H_{A_{sym}} (Ω) \subseteq V_{A_{sym}} (Ω) \subseteq W_{A_{sym}} (Ω)$ .
Definition 2.1.
Let $A = A_{sym} + A_{skew} \in L^{1} (Ω; M^{N})$ be a given matrix such that $A_{skew} \in L^{2 p} (Ω; S_{skew}^{N})$ . We say that an element $y \in V_{A_{sym}} (Ω)$ belongs to the set $D (V_{A_{sym}})$ if $\begin{array}{l} | \int_{Ω} {(\nabla φ, A_{skew} \nabla y)}_{R^{N}} d x | & ⩽ c (y, A) {(\int_{Ω} φ^{2} d x + \int_{Ω} {(\nabla φ, A_{sym} \nabla φ)}_{R^{N}} d x)}^{1 / 2} \\ (2.16) & = c (y, A) ‖ φ ‖_{A_{sym}}, \forall φ \in C_{0}^{\infty} (Ω) \end{array}$ with some constant c depending on y and A.

As a result, if $y \in D (V_{A_{sym}})$ then the mapping $φ \mapsto {[y, φ]}_{A}$ can be defined for all $φ \in H_{A_{sym}} (Ω)$ using (2.16) and the standard rule $\begin{matrix} (2.17) & {[y, φ]}_{A} = lim_{ε \to 0} {[y, φ_{ε}]}_{A}, \end{matrix}$ where ${φ_{ε}}_{ε > 0} \subset C_{0}^{\infty} (Ω)$ and $φ_{ε} \to φ$ strongly in $H_{A_{sym}} (Ω)$ (it is the case where we essentially use the fact that $C_{0}^{\infty} (Ω)$ is dense in $H_{A_{sym}} (Ω)$ ). In particular, if $y \in D (H_{A_{sym}})$ , then we can define the value ${[y, y]}_{A}$ and this one is finite for every $y \in D (H_{A_{sym}})$ , although the “integrand” ${(\nabla y, A_{skew} \nabla y)}_{R^{N}}$ needs not be integrable on Ω, in general.

Let $f : Ω \to R$ be a function of $L^{1} (Ω)$ . We define $\begin{array}{rcl} TV (f) & : = & \int_{Ω} | D f | \\ = & sup {\int_{Ω} f {(\nabla, φ)}_{R^{N}} d x : φ = (φ_{1}, \dots, φ_{N}) \in C_{0}^{1} (Ω; R^{N}), | φ (x) | ⩽ 1 for x \in Ω}, \end{array}$ where ${(\nabla, φ)}_{R^{N}} = \sum_{i = 1}^{N} \frac{\partial φ_{i}}{\partial x_{i}}$ .
Definition 2.2.
A function $f \in L^{1} (Ω)$ is said to have a bounded variation in Ω if $TV (f) < + \infty$ . By $BV (Ω)$ we denote the space of all functions in $L^{1} (Ω)$ with bounded variation, i.e. $BV (Ω) = {f \in L^{1} (Ω) : TV (f) < + \infty}$ .

Under the norm $‖ f ‖_{BV (Ω)} = ‖ f ‖_{L^{1} (Ω)} + TV (f)$ , $BV (Ω)$ is a Banach space. For our further analysis, we need the following properties of BV-functions (see [8]):
Proposition 2.3.
Let ${f_{k}}_{k = 1}^{\infty}$ be a sequence in $BV (Ω)$ strongly converging to some f in $L^{1} (Ω)$ and satisfying condition ${sup}_{k \in N} TV (f_{k}) < + \infty$ . Then $\begin{matrix} f \in BV (Ω) and TV (f) ⩽ \underset{k \to \infty}{lim inf} TV (f_{k}) \end{matrix}$ and for every bounded sequence ${f_{k}}_{k = 1}^{\infty} \subset BV (Ω)$ there exists a subsequence, still denoted by $f_{k}$ , and a function $f \in BV (Ω)$ such that $f_{k} \to f$ in $L^{1} (Ω)$ .

Let $I_{k} : U_{k} \times Y_{k} \to \overline{R}$ be a cost functional, $Y_{k}$ be a space of states, and $U_{k}$ be a space of controls. Let $min {I_{k} (u, y) : (u, y) \in Ξ_{k}}$ be a parameterized OCP, where $\begin{matrix} Ξ_{k} \subset {(u_{k}, y_{k}) \in U_{k} \times Y_{k} : u_{k} \in U_{k}, I_{k} (u_{k}, y_{k}) < + \infty} \end{matrix}$ is a set of all admissible pairs linked by some state equation. Hereinafter we always associate to such OCP the corresponding constrained minimization problem: $\begin{matrix} (2.18) & ({CMP}_{k}) : ⟨ inf_{(u, y) \in Ξ_{k}} I_{k} (u, y) ⟩ . \end{matrix}$ Since each of constrained minimization problems (2.18) lives in variable spaces $U_{k} \times Y_{k}$ , we assume that there exists a Banach space $U \times Y$ with respect to which a convergence in the scale of spaces ${U_{k} \times Y_{k}}_{k \in N}$ is well defined (for the details, we refer to [12,25]). In the sequel, we use the following notation for this convergence $(u_{k}, y_{k}) \overset{μ}{⟶} (u, y)$ in $U_{k} \times Y_{k}$ . Moreover, we assume that every bounded sequence in variable space $U_{k} \times Y_{k}$ is sequentially compact with respect to the μ-convergence.

In order to study the asymptotic behaviour of a family of ( ${CMP}_{k}$ ), the passage to the limit in (2.18) as the parameter k tends to $+ \infty$ has to be realized. The expression “passing to the limit” means that we have to find a kind of “limit cost functional” I and “limit set of constraints” Ξ with a clearly defined structure such that the limit object $⟨ {inf}_{(u, y) \in Ξ} I (u, y) ⟩$ may be interpreted as some OCP.

Following the scheme of the direct variational convergence [12], we adopt the following definition for the convergence of minimization problems in variable spaces.
Definition 2.4.
A problem $⟨ {inf}_{(u, y) \in Ξ} I (u, y) ⟩$ is the variational μ-limit of (2.18) as $k \to \infty$ , if and only if the following conditions are satisfied:
If sequences ${k_{n}}_{n \in N}$ and ${(u_{n}, y_{n})}_{n \in N}$ are such that $k_{n} \to \infty$ as $n \to \infty$ , $(u_{n}, y_{n}) \in Ξ_{k_{n}}$ , $\forall n \in N$ , and $(u_{n}, y_{n}) \overset{μ}{⟶} (u, y)$ in $U_{k_{n}} \times Y_{k_{n}}$ , then $\begin{matrix} (2.19) & (u, y) \in Ξ; I (u, y) ⩽ \underset{n \to \infty}{lim inf} I_{k_{n}} (u_{n}, y_{n}) . \end{matrix}$

For every $(u, y) \in Ξ \subset U \times Y$ , there are an integer $k^{0} > 0$ and a sequence ${(u_{k}, y_{k})}_{k \in N}$ (called a Γ-realizing sequence) such that $\begin{array}{l} (2.20) & (u_{k}, y_{k}) \in Ξ_{k}, \forall k ⩾ k^{0}, (u_{k}, y_{k}) \overset{μ}{⟶} (\hat{u}, \hat{y}) in U_{k} \times Y_{k}, \\ (2.21) & I (u, y) ⩾ \underset{k \to \infty}{lim sup} I_{k} (u_{k}, y_{k}) . \end{array}$

Then the following result takes place [12]. Theorem 2.5.
Assume that the constrained minimization problem $\begin{matrix} (2.22) & ⟨ inf_{(u, y) \in Ξ_{0}} I_{0} (u, y) ⟩ \end{matrix}$ is the variational μ-limit of (2.18) in the sense of Definition 2.4 and this problem has a nonempty set of solutions $\begin{matrix} Ξ_{0}^{opt} : = {(u^{0}, y^{0}) \in Ξ_{0} : I_{0} (u^{0}, y^{0}) = inf_{(u, y) \in Ξ_{0}} I_{0} (u, y)} . \end{matrix}$ For every $k \in N$ , let $(u_{k}^{0}, y_{k}^{0}) \in Ξ_{k}$ be a minimizer of $I_{k}$ on the corresponding set $Ξ_{k}$ . If the sequence ${(u_{k}^{0}, y_{k}^{0})}_{k \in N}$ is relatively compact with respect to the μ-convergence in variable spaces $U_{k} \times Y_{k}$ , then there exists a pair $(u^{0}, y^{0}) \in Ξ_{0}^{opt}$ such that (up to a subsequence) $\begin{array}{l} (2.23) & (u_{k}^{0}, y_{k}^{0}) \overset{μ}{⟶} (u^{0}, y^{0}) in U_{k} \times Y_{k}, \\ (2.24) & inf_{(u, y) \in Ξ_{0}} I_{0} (u, y) = I_{0} (u^{0}, y^{0}) = lim_{k \to \infty} I_{k} (u_{k}^{0}, y_{k}^{0}) = lim_{k \to \infty} inf_{(u_{k}, y_{k}) \in Ξ_{k}} I_{k} (u_{k}, y_{k}) . \end{array}$

3. Weak convergence in variable $L^{2}$ -spaces associated with $S_{sym}^{N}$ -matrices

Let ${A_{k}}_{k \in N}$ and A be a given collection of $S_{sym}^{N}$ -matrices such that $\begin{matrix} (3.1) & A_{k}, A \in L^{1} (Ω; S_{sym}^{N}) and A_{k} \to A in L^{1} (Ω; S_{sym}^{N}) . \end{matrix}$ Let $L^{2} {(Ω, A d x)}^{N}$ be the Hilbert space of measurable vector-valued functions $f \in R^{N}$ on Ω such that $\begin{matrix} ‖ f ‖_{L^{2} {(Ω, A d x)}^{N}} = {(\int_{Ω} {(f, A f)}_{R^{N}} d x)}^{1 / 2} < + \infty . \end{matrix}$

We say that a sequence ${v_{k} \in L^{2} {(Ω, A_{k} d x)}^{N}}_{k \in N}$ is bounded if $\begin{matrix} \underset{k \to \infty}{lim sup} \int_{Ω} {(v_{k}, A_{k} v_{k})}_{R^{N}} d x < + \infty . \end{matrix}$

Definition 3.1.
A bounded sequence ${v_{k} \in L^{2} {(Ω, A_{k} d x)}^{N}}_{k \in N}$ is weakly convergent to a function $v \in L^{2} {(Ω, A d x)}^{N}$ in variable space $L^{2} {(Ω, A_{k} d x)}^{N}$ if $\begin{matrix} (3.2) & lim_{k \to \infty} \int_{Ω} {(φ, A_{k} v_{k})}_{R^{N}} d x = \int_{Ω} {(φ, A v)}_{R^{N}} d x, \forall φ \in C_{0}^{\infty} {(Ω)}^{N} . \end{matrix}$
Definition 3.2.
A sequence ${v_{k} \in L^{2} {(Ω, A_{k} d x)}^{N}}_{k \in N}$ is said to be strongly convergent to a function $v \in L^{2} {(Ω, A d x)}^{N}$ if $\begin{matrix} (3.3) & lim_{k \to \infty} \int_{Ω} {(b_{k}, A_{k} v_{k})}_{R^{N}} d x = \int_{Ω} {(b, A v)}_{R^{N}} d x \end{matrix}$ whenever $b_{k} ⇀ b$ in $L^{2} {(Ω, A_{k} d x)}^{N}$ as $k \to \infty$ .
Remark 3.1.
Note that in the case $A_{k} \equiv A$ , Definitions 3.1–3.2 leads to the usual notion of convergence in weighted Hilbert space $L^{2} {(Ω, A d x)}^{N}$ .

The main properties of the weak and strong convergences in $L^{p} {(Ω, A_{k} d x)}^{N}$ can be expressed as follows (see [11,13,14] for the details):
Proposition 3.3.
If a sequence ${v_{k} \in L^{2} {(Ω, A_{k} d x)}^{N}}_{k \in N}$ is bounded and the condition (3.1) holds true, then it contains a weakly convergent subsequence in $L^{2} {(Ω, A_{k} d x)}^{N}$ .
Proposition 3.4.
If the sequence ${v_{k} \in L^{2} {(Ω, A_{k} d x)}^{N}}_{k \in N}$ converges weakly to $v \in L^{2} {(Ω, A d x)}^{N}$ and the condition (3.1) holds true, then $\begin{matrix} (3.4) & \underset{k \to \infty}{lim inf} \int_{Ω} {(v_{k}, A_{k} v_{k})}_{R^{N}} d x ⩾ \int_{Ω} {(v, A v)}_{R^{N}} d x . \end{matrix}$
Proposition 3.5.
Assume the condition (3.1) holds true. Then the weak convergence of a sequence ${v_{k} \in L^{2} {(Ω, A_{k} d x)}^{N}}_{k \in N}$ to $v \in L^{2} {(Ω, A d x)}^{N}$ and $\begin{matrix} (3.5) & lim_{k \to \infty} \int_{Ω} {(v_{k}, A_{k} v_{k})}_{R^{N}} d x = \int_{Ω} {(v, A v)}_{R^{N}} d x \end{matrix}$ are equivalent to the strong convergence of ${v_{k}}_{k \in N}$ to $v \in L^{2} {(Ω, A d x)}^{N}$ in $L^{2} {(Ω, A_{k} d x)}^{N}$ .

In what follows, we make use of the following result. Lemma 3.6.
Let $A^{skew} \in L^{2 p} (Ω; S_{skew}^{N})$ be a given matrix, and let ${A_{k}^{sym}}_{k \in N} \subset M_{ζ_{ad}}^{β} (Ω)$ be a sequence such that $\begin{matrix} (3.6) & A_{k}^{sym} \to A_{0}^{sym} in L^{1} (Ω; S_{sym}^{N}) . \end{matrix}$ Let $φ \in C_{0}^{\infty} (Ω)$ and $f \in L^{4 p / (p + 1)} (Ω; R^{N})$ be arbitrary functions. Then $\begin{array}{l} v_{k} : = {(A_{k}^{sym})}^{- 1} f \to {(A_{0}^{sym})}^{- 1} f = : v_{0} and \\ w_{k} : = {(A_{k}^{sym})}^{- 1} A^{skew} \nabla φ \to {(A_{0}^{sym})}^{- 1} A^{skew} \nabla φ = : w_{0} \end{array}$ strongly in variable space $L^{2} {(Ω, A_{k}^{sym} d x)}^{N}$ as $k \to \infty$ .
Proof.
Since $\begin{array}{l} ‖ v_{k} ‖_{L^{2} {(Ω, A_{k}^{sym} d x)}^{N}}^{2} = \int_{Ω} {(v_{k}, A_{k}^{sym} v_{k})}_{R^{N}} d x = \int_{Ω} {(f, {(A_{k}^{sym})}^{- 1} f)}_{R^{N}} d x \\ ⩽ \int_{Ω} | f |_{R^{N}}^{2} ζ_{ad}^{- 1} d x ⩽ ‖ f ‖_{L^{4 p / (p + 1)} (Ω; R^{N})}^{2} {‖ ζ_{ad}^{- 1} ‖}_{L^{2 q} (Ω)} < + \infty, \\ ‖ w_{k} ‖_{L^{2} {(Ω, A_{k}^{sym} d x)}^{N}}^{2} = \int_{Ω} {(A^{skew} \nabla φ, {(A_{k}^{sym})}^{- 1} A^{skew} \nabla φ)}_{R^{N}} d x \\ ⩽ \int_{Ω} ζ_{ad}^{- 1} {| A^{skew} \nabla φ |}^{2} d x ⩽ ‖ φ ‖_{C^{1} (Ω)}^{2} {‖ ζ_{ad}^{- 1} ‖}_{L^{q} (Ω)} {‖ A^{skew} ‖}_{L^{2 p} (Ω; S_{skew}^{N})}^{2} < + \infty \end{array}$ and, for each $ψ \in C_{0}^{\infty} (Ω)$ , $\begin{array}{l} (3.7) & \begin{matrix} \int_{Ω} {(\nabla ψ, A_{k}^{sym} v_{k})}_{R^{N}} d x = \int_{Ω} {(\nabla ψ, A_{k}^{sym} {(A_{k}^{sym})}^{- 1} f)}_{R^{N}} d x \\ = \int_{Ω} {(\nabla ψ, f)}_{R^{N}} d x = \int_{Ω} {(\nabla ψ, A_{0}^{sym} v_{0})}_{R^{N}} d x, \\ \int_{Ω} {(\nabla ψ, A_{k}^{sym} w_{k})}_{R^{N}} d x = \int_{Ω} {(\nabla ψ, A_{k}^{sym} {(A_{k}^{sym})}^{- 1} A^{skew} \nabla φ)}_{R^{N}} d x \\ = \int_{Ω} {(\nabla ψ, A^{skew} \nabla φ)}_{R^{N}} d x = \int_{Ω} {(\nabla ψ, A_{0}^{sym} w_{0})}_{R^{N}} d x, \end{matrix} \end{array}$ it follows that the sequences ${v_{k}}_{k \in N}$ and ${w_{k}}_{k \in N}$ are bounded and weakly convergent in variable space $L^{2} {(Ω, A_{k}^{sym} d x)}^{N}$ to vector-valued functions $v_{0} \in L^{2} {(Ω, A_{0}^{sym} d x)}^{N}$ and $w_{0} \in L^{2} {(Ω, A_{0}^{sym} d x)}^{N}$ , respectively.

In order to show that the sequence ${v_{k}}_{k \in N}$ is strongly convergent to $v_{0} : = {(A_{0}^{sym})}^{- 1} f$ , we make use of Proposition 3.5. Following this assertion, it is enough to prove the equality $\begin{array}{l} lim_{k \to \infty} \int_{Ω} {(v_{k}, A_{k}^{sym} v_{k})}_{R^{N}} d x & = lim_{k \to \infty} \int_{Ω} {(f, {(A_{k}^{sym})}^{- 1} f)}_{R^{N}} d x \\ (3.8) & = \int_{Ω} {(f, {(A_{0}^{sym})}^{- 1} f)}_{R^{N}} d x = \int_{Ω} {(v_{0}, A_{0}^{sym} v_{0})}_{R^{N}} d x . \end{array}$ In view of estimate ${(f, {(A_{k}^{sym})}^{- 1} f)}_{R^{N}} ⩽ | f |_{R^{N}}^{2} ζ_{ad}^{- 1}$ , $\forall k \in N$ , and the fact that $\begin{matrix} {‖ ζ_{ad}^{- 1} | f |_{R^{N}}^{2} ‖}_{L^{1} (Ω)} ⩽ ‖ f ‖_{L^{4 p / (p + 1)} (Ω; R^{N})}^{2} {‖ ζ_{ad}^{- 1} ‖}_{L^{2 q} (Ω)} < + \infty, \end{matrix}$ the sequence ${{(\nabla φ, {(A_{k}^{sym})}^{- 1} \nabla φ)}_{R^{N}}}_{k \in N}$ is equi-integrable. On the other hand, property (3.6) implies that, within a subsequence, ${(A_{k}^{sym})}^{- 1} \to {(A_{0}^{sym})}^{- 1}$ almost everywhere in Ω. Hence, up to a subsequence, $\begin{matrix} {(f, {(A_{k}^{sym})}^{- 1} f)}_{R^{N}} \to {(f, {(A_{0}^{sym})}^{- 1} f)}_{R^{N}} a.e. in Ω . \end{matrix}$ Thus, the equality (3.8) is a direct consequence of Lebesgue Dominated Theorem, and hence, $\begin{matrix} {(A_{k}^{sym})}^{- 1} f \to {(A_{0}^{sym})}^{- 1} f strongly in L^{2} {(Ω, A_{k}^{sym} d x)}^{N} . \end{matrix}$ Following the same arguments, it can be shown that $\begin{array}{l} lim_{k \to \infty} \int_{Ω} {(w_{k}, A_{k}^{sym} w_{k})}_{R^{N}} d x & = - lim_{k \to \infty} \int_{Ω} {(\nabla φ, A^{skew} {(A_{k}^{sym})}^{- 1} A^{skew} \nabla φ)}_{R^{N}} d x \\ = - \int_{Ω} {(\nabla φ, A^{skew} {(A_{0}^{sym})}^{- 1} A^{skew} \nabla φ)}_{R^{N}} d x = \int_{Ω} {(w_{0}, A_{0}^{sym} w_{0})}_{R^{N}} d x . \end{array}$ Combining this fact with relation (3.7), by Proposition 3.5 we have: $\begin{matrix} {(A_{k}^{sym})}^{- 1} A^{skew} \nabla φ \to {(A_{0}^{sym})}^{- 1} A^{skew} \nabla φ strongly in L^{2} {(Ω, A_{k}^{sym} d x)}^{N}, \end{matrix}$ $\forall φ \in C_{0}^{\infty} (Ω)$ . The proof is complete. □

4. Setting of the optimal control problem

Let $p ⩾ 1$ be a given exponent and let $f : Ω \to R^{N}$ be a vector-valued function such that $f \in L^{4 p / (p + 1)} (Ω; R^{N})$ . Let $M \in S_{sym}^{N}$ be a constant matrix satisfying the condition $\begin{matrix} {(M ξ, ξ)}_{R^{N}} ⩾ m ‖ ξ ‖_{R^{N}}^{2} for some m > 0 . \end{matrix}$ The optimal control problem (1.1)–(1.3) we consider in this paper is to minimize the discrepancy (tracking error) between a given distribution $y_{d} \in L^{2} (Ω)$ and a solution y of the Dirichlet boundary value problem (1.2)–(1.3) by choosing an appropriate matrix-valued control $A (x) = A_{sym} (x) + A_{skew} (x)$ .

In order to define the class of admissible controls $A_{ad}$ , we begin with some preliminaries. Let $A^{*}, A^{* *} \in L^{2 p} (Ω; S_{skew}^{N})$ be given matrices such that $A^{*} ⪯ A^{* *}$ a.e. in Ω, let c be a given positive constant, and let Q be a nonempty convex compact subset of $L^{2 p} (Ω; S_{skew}^{N})$ such that the null matrix $A \equiv [0]$ belongs to Q. Further we make use of the following sets $\begin{array}{l} (4.1) & U_{a, 1} = {A = [a_{i j}] \in L^{1} (Ω; S_{sym}^{N}) ∣ TV (a_{i j}) ⩽ c, 1 ⩽ i ⩽ j ⩽ N}, \\ (4.2) & U_{b, 1} = {A = [a_{i j}] \in L^{1} (Ω; S_{sym}^{N}) | A \in M_{ζ_{ad}}^{β} (Ω), \int_{Ω} A (x) d x = M}, \\ (4.3) & U_{a, 2} = {A = [a_{i j}] \in L^{2 p} (Ω; S_{skew}^{N}) ∣ A^{*} (x) ⪯ A (x) ⪯ A^{* *} (x) a.e. in Ω}, \\ (4.4) & U_{b, 2} = {A = [a_{i j}] \in L^{2 p} (Ω; S_{skew}^{N}) ∣ A \in Q} . \end{array}$

Remark 4.1.
Hereinafter we assume that $\begin{matrix} A_{ad, 1} : = U_{a, 1} \cap U_{b, 1} \neq \emptyset and A_{ad, 2} : = U_{a, 2} \cap U_{b, 2} \neq \emptyset, \end{matrix}$ and, hence, the set $A_{ad} : = A_{ad, 1} \oplus A_{ad, 2}$ is nonempty. Moreover, it is easy to see that for a given $A^{}, A^{ *} \in L^{2 p} (Ω; S_{skew}^{N})$ , we can always guarantee the fulfilment of condition $A_{ad} \neq \emptyset$ by an appropriate choice of functions $ζ_{ad} \in L^{1} (Ω)$ and $β \in L^{1} (Ω)$ , a matrix $M \in S_{sym}^{N}$ , and a compact subset Q.
Definition 4.1.
We say that a matrix $A = A_{sym} + A_{skew}$ is an admissible control to the Dirichlet boundary value problem (1.2)–(1.3) (it is written as $A \in A_{ad}$ ) if $A_{sym} \in A_{ad, 1}$ and $A_{skew} \in A_{ad, 2}$ .

For our further analysis, we make use of the following results.
Proposition 4.2.
The set $A_{ad}$ is convex and compact with respect to the strong topology of $L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N})$ .
Proof.
Let ${A_{k} = A_{k}^{sym} + A_{k}^{skew}}_{k \in N} \subset A_{ad}$ be an arbitrary sequence of admissible controls. Since $A_{ad} = A_{ad, 1} \oplus A_{ad, 2}$ , $A_{ad, 1} \subset BV (Ω; S_{sym}^{N})$ , $A_{ad, 2} \subset U_{b, 2}$ , and $U_{b, 2}$ is a compact in $L^{2 p} (Ω; S_{skew}^{N})$ , it follows by the compactness of BV-functions (see Proposition 2.3) that there exist matrices $A_{0}^{sym} \in BV (Ω; S_{sym}^{N})$ and $A_{0}^{skew} \in U_{b, 2} \subset L^{2 p} (Ω; S_{skew}^{N})$ such that within a subsequence $\begin{array}{l} (4.5) & A_{k}^{sym} \to A_{0}^{sym} in L^{1} (Ω; S_{sym}^{N}), \\ (4.6) & A_{k}^{skew} \to A_{0}^{skew} in L^{2 p} (Ω; S_{skew}^{N}), and \\ (4.7) & A_{k} \to A_{0} : = A_{0}^{sym} + A_{0}^{skew} almost everywhere in Ω . \end{array}$

Combining these facts with definition of the binary relation ⪯ (see (2.1)), we arrive at the conclusion: $A_{0}^{skew} \in U_{a, 2}$ , $A_{0}^{skew} \in U_{b, 2}$ , and $A_{0}^{sym} \in U_{a, 1}$ . Hence, it remains to show the condition $A_{0}^{sym} \in U_{b, 1}$ . With that in mind we make use of the following observation.

By the initial suppositions, we have $A_{k}^{sym} \in M_{ζ_{ad}}^{β} (Ω)$ for all $k \in N$ . Hence, in view of $L^{1}$ -convergence $A_{k}^{sym} \to A_{0}^{sym}$ , we may assume that, up to a subsequence, $A_{k}^{sym} \to A_{0}^{sym}$ almost everywhere in Ω. Since $A_{k} (x) ⩾ ζ_{ad} I$ a.e. in Ω, it follows that $\begin{array}{l} M = lim_{k \to \infty} \int_{Ω} A_{k}^{sym} (x) d x = \int_{Ω} A_{0}^{sym} (x) d x, \\ A_{0}^{sym} (x) = lim_{k \to \infty} A_{k}^{sym} (x) ⩽ β (x) I a.e. in Ω, \\ ζ_{ad} (x) I ⩽ lim_{k \to \infty} A_{k}^{sym} (x) = A_{0}^{sym} (x) a.e. in Ω . \end{array}$ Thus, $A_{0}^{sym} \in U_{b, 1}$ , and, therefore, $A_{0} \in A_{ad}$ . Since the convexity of $A_{ad}$ is obviously valid, this concludes the proof. □
Definition 4.3.
We say that a function $y = y (A, f)$ is a weak solution to the boundary value problem (1.2)–(1.3) for a fixed admissible control $A = A_{sym} + A_{skew} \in A_{ad}$ and given distribution $f \in L^{4 p / (p + 1)} (Ω; R^{N})$ if $y \in W_{A_{sym}} (Ω)$ and the integral identity $\begin{array}{l} \int_{Ω} [{(\nabla φ, A_{sym} \nabla y)}_{R^{N}} + a_{0} y φ] d x + \int_{Ω} {(\nabla φ, A_{skew} \nabla y)}_{R^{N}} d x \\ (4.8) & = \int_{Ω} {(f, \nabla φ)}_{R^{N}} d x \end{array}$ holds for each $φ \in C_{0}^{\infty} (Ω)$ .

We note that by the initial assumptions and Hölder’s inequality, this definition makes a sense because $(A_{skew} \nabla y) \in L^{1} (Ω; R^{N})$ for each $y \in W_{A_{sym}} (Ω)$ . Indeed, $\begin{array}{rcl} \int_{Ω} | A_{skew} \nabla y |_{R^{N}} d x & ⩽ & \int_{Ω} {‖ A_{skew} A_{sym}^{- 1 / 2} ‖}_{L (R^{N}, R^{N})} {| A_{sym}^{1 / 2} \nabla y |}_{R^{N}} d x \\ ⩽ & {(\int_{Ω} ‖ A_{skew} ‖_{L (R^{N}, R^{N})}^{2} {‖ A_{sym}^{- 1 / 2} ‖}_{L (R^{N}, R^{N})}^{2} d x)}^{1 / 2} \\ \times {(\int_{Ω} {(\nabla y, A_{sym} \nabla y)}_{R^{N}} d x)}^{1 / 2} \\ ⩽ & {(\int_{Ω} ‖ A_{skew} ‖_{L (R^{N}, R^{N})}^{2 p} d x)}^{1 / 2 p} \\ \times {(\int_{Ω} {‖ A_{sym}^{- 1 / 2} ‖}_{L (R^{N}, R^{N})}^{2 q} d x)}^{1 / 2 q} ‖ y ‖_{A_{sym}} \\ ⩽ & | Ω |^{1 / 4 q} ‖ A_{skew} ‖_{L^{2 p} (Ω; S_{skew}^{N})} {‖ A_{sym}^{- 1 / 2} ‖}_{L^{4 q} (Ω; S_{skew}^{N})} ‖ y ‖_{A_{sym}} . \end{array}$

On the other hand, Definition 4.3 gives another motivation to introduce the set $D (W_{A_{sym}})$ (see for comparison [9,10]).
Proposition 4.4.
Let $A = A_{sym} + A_{skew} \in A_{ad}$ and $f \in L^{4 p / (p + 1)} (Ω; R^{N})$ be given distributions. Let $y \in V_{A_{sym}} (Ω)$ be a weak solution to the boundary value problem (1.2)–(1.3) for some intermediate space $V_{A_{sym}} (Ω)$ with $H_{A_{sym}} (Ω) \subseteq V_{A_{sym}} (Ω) \subseteq W_{A_{sym}} (Ω)$ . Then $y \in D (V_{A_{sym}})$ .
Proof.
In order to prove this assertion it is enough to rewrite the integral identity (4.8) in the form $\begin{matrix} (4.9) & {[y, φ]}_{A} = - \int_{Ω} {(A_{sym} \nabla y, \nabla φ)}_{R^{N}} d x - \int_{Ω} a_{0} y φ d x + \int_{Ω} {(f, \nabla φ)}_{R^{N}} d x \end{matrix}$ and apply Hölder’s inequality to the right-hand side of (3.3). As a result, we have $\begin{array}{l} | \int_{Ω} {(A_{sym} \nabla y, \nabla φ)}_{R^{N}} d x | ⩽ {(\int_{Ω} {| A_{sym}^{1 / 2} \nabla y |}_{R^{N}}^{2} d x)}^{1 / 2} {(\int_{Ω} {| A_{sym}^{1 / 2} \nabla φ |}_{R^{N}}^{2} d x)}^{1 / 2} \\ ⩽ ‖ y ‖_{A_{sym}} ‖ φ ‖_{A_{sym}}, \\ | \int_{Ω} a_{0} y φ d x | ⩽ ‖ a_{0} ‖_{L^{\infty} (Ω)} ‖ y ‖_{L^{2} (Ω)} {(‖ φ ‖_{L^{2} (Ω)}^{2} + \int_{Ω} {| A_{sym}^{1 / 2} \nabla φ |}_{R^{N}}^{2} d x)}^{1 / 2} \\ ⩽ ‖ a_{0} ‖_{L^{\infty} (Ω)} ‖ y ‖_{A_{sym}} ‖ φ ‖_{A_{sym}}, \\ | \int_{Ω} {(f, \nabla φ)}_{R^{N}} d x | ⩽ {(\int_{Ω} {| A_{sym}^{- 1 / 2} f |}_{R^{N}}^{2} d x)}^{1 / 2} {(\int_{Ω} {| A_{sym}^{1 / 2} \nabla φ |}_{R^{N}}^{2} d x)}^{1 / 2} \\ = {(\int_{Ω} {| A_{sym}^{- 1 / 2} f |}_{R^{N}}^{2} d x)}^{1 / 2} {(\int_{Ω} {(\nabla φ, A_{sym} \nabla φ)}_{R^{N}} d x)}^{1 / 2} \\ ⩽ {(\int_{Ω} {‖ A_{sym}^{- 1 / 2} ‖}_{L (R^{N}, R^{N})}^{4 q} d x)}^{1 / 4 q} {(\int_{Ω} | f |_{R^{N}}^{4 p / p + 1} d x)}^{(p + 1) / 4 p} \\ \times {(\int_{Ω} {(\nabla φ, A_{sym} \nabla φ)}_{R^{N}} d x)}^{1 / 2} \\ (4.10) & ⩽ \sqrt{{‖ ζ_{ad}^{- 1} ‖}_{L^{2 q} (Ω)}} ‖ f ‖_{L^{4 p / p + 1} (Ω; R^{N})} ‖ φ ‖_{A_{sym}}, \end{array}$ and, hence, $\begin{array}{rcl} | {[y, φ]}_{A} | & ⩽ & ((1 + ‖ a_{0} ‖_{L^{\infty} (Ω)}) ‖ y ‖_{A_{sym}} + \sqrt{{‖ ζ_{ad}^{- 1} ‖}_{L^{2 q} (Ω)}} ‖ f ‖_{L^{4 p / p + 1} (Ω; R^{N})}) \\ \times {(\int_{Ω} {(\nabla φ, A_{sym} \nabla φ)}_{R^{N}} d x)}^{1 / 2} ⩽ c (y, A) ‖ φ ‖_{A_{sym}}, \forall φ \in C_{0}^{\infty} (Ω) . \end{array}$ □

As estimate (4.10) obviously indicates, Proposition 4.4 can be specified as follows.
Corollary 4.5.
Let $A = A_{sym} + A_{skew} \in A_{ad}$ be an arbitrary admissible control, and let f be a distribution such that $A_{sym}^{- 1 / 2} f \in L^{2} (Ω; R^{N})$ . Let $y \in V_{A_{sym}} (Ω)$ be a weak solution to the boundary value problem (1.2)–(1.3). Then $y \in D (V_{A_{sym}})$ .
Remark 4.2.
Due to Proposition 4.4, Definition 4.3 can be reformulated as follows: y is a weak solution to the problem (1.2)–(1.3) for a given control $A = A_{sym} + A_{skew} \in A_{ad}$ , if and only if $y \in D (W_{A_{sym}})$ and $\begin{matrix} (4.11) & \int_{Ω} [{(\nabla φ, A_{sym} \nabla y)}_{R^{N}} + a_{0} y φ] d x + {[y, φ]}_{A} = \int_{Ω} {(f, \nabla φ)}_{R^{N}} d x, \forall φ \in C_{0}^{\infty} (Ω) . \end{matrix}$ Moreover, as follows from (2.17), (4.11), and (4.10), if a weak solution to the problem (1.2)–(1.3) belongs to the space $H_{A_{sym}} (Ω)$ then it satisfies the energy equality $\begin{matrix} (4.12) & \int_{Ω} {(\nabla y, A_{sym} \nabla y)}_{R^{N}} d x + \int_{Ω} a_{0} y^{2} d x + {[y, y]}_{A} = \int_{Ω} {(f, \nabla y)}_{R^{N}} d x . \end{matrix}$

It is worth to notice that the original boundary value problem (1.2)–(1.3) is ill-possed, in general. In view of definition of the set $M_{ζ_{ad}}^{β} (Ω)$ and the fact that the skew-symmetric form (2.13) can be unbounded on $W_{A_{sym}} (Ω)$ , the existence of a weak solution to (1.2)–(1.3) for fixed $A \in A_{ad}$ and $f \in L^{4 p / (p + 1)} (Ω; R^{N})$ seems to be an open question. Moreover, even if a weak solution to the above problem exists, the question about its uniqueness remains open. Indeed, because of the properties of function $ζ_{ad}$ , we face with the problem of density of smooth functions $C_{0}^{\infty} (Ω)$ in $W_{A_{sym}} (Ω)$ . As was indicated in [27], there exists a diagonal matrix-valued function $A (x) = ρ (x) I$ with $ρ ⩾ ζ_{ad}$ such that the subspace $C_{0}^{\infty} (Ω)$ is not dense in $W_{A_{sym}} (Ω)$ . Therefore, if we assume that we have two weak solutions $y_{1} (A, f), y_{2} (A, f) \in W_{A_{sym}} (Ω)$ such that $y_{1} (A, f) \neq y_{2} (A, f)$ , $\int_{Ω} {(\nabla y_{k}, A_{skew} \nabla y_{k})}_{R^{N}} d x = 0$ , $k = 1, 2$ (this is always true for $A_{skew} \in L^{\infty} (Ω; S_{skew}^{N})$ ), and each of these solutions satisfies the corresponding energy equality $\begin{matrix} (4.13) & \int_{Ω} {(\nabla y_{k}, A_{sym} \nabla y_{k})}_{R^{N}} d x + \int_{Ω} y_{k}^{2} d x = \int_{Ω} {(f, \nabla y_{k})}_{R^{N}} d x, k = 1, 2, \end{matrix}$ then the element $y = (y_{1} (A, f) + y_{2} (A, f)) / 2$ is a weak solution to (1.2)–(1.3) too, but it does not satisfy (4.13) in general. Thus, the degenerate boundary value problem (1.2)–(1.3) can admit weak solutions which do not satisfy energy equality. For more details and other types of solutions to degenerate equations we refer to [25,27].

On the other hand, as it follows from the definition of the bilinear form ${[y, φ]}_{A}$ , the value ${[y, y]}_{A}$ is not equal to zero, in general, and does not preserve a constant sign for all $y \in D (W_{A_{sym}})$ . Hence, even if the relation $H_{A_{sym}} (Ω) = W_{A_{sym}} (Ω)$ is valid, the energy equality (4.12) does not allow us to derive a reasonable a priory estimate in $‖ \cdot ‖_{A}$ -norm for the weak solutions. Thus, in general, the mapping $A \mapsto y (A, f)$ can be multivalued (see [9] for the details).

Taking these observations into account, we restrict of our analysis to the following set of admissible solutions for the original optimal control problem. Namely, we indicate the set $\begin{matrix} (4.14) & Ξ = {(A, y) ∣ A \in A_{ad}, y \in W_{A_{sym}} (Ω), (A, y) are related by (4.11)} . \end{matrix}$

The characteristic feature of this set is the fact that for different admissible controls $A \in A_{ad}$ the “corresponding” weak solutions y belong to different weighted spaces. Moreover, we adopt the following hypothesis, which is mainly motivated by the previous reasonings.
Hypothesis A.
The set of admissible solutions Ξ is nonempty.

We say that a pair $(A^{0}, y^{0}) \in L^{1} (Ω; M^{N}) \times W_{A_{sym}^{0}} (Ω)$ is a weak optimal solution to the problem (1.1)–(1.3) on the set Ξ, if $\begin{matrix} (4.15) & (A^{0}, y^{0}) \in Ξ and I (A^{0}, y^{0}) = inf_{(A, y) \in Ξ} I (A, y) . \end{matrix}$

Our next observation deals with some specification of the set of admissible controls $A_{ad}$ . With that in mind we give a few auxiliary results.
Lemma 4.6.
Let ${A_{k} = A_{k}^{sym} + A_{k}^{skew}}_{k \in N} \subset A_{ad}$ and ${y_{k} \in W_{A_{k}^{sym}}}_{k \in N}$ be sequences such that $\begin{array}{l} (4.16) & A_{k}^{sym} + A_{k}^{skew} \to A_{0}^{sym} + A_{0}^{skew} in L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N}), \\ (4.17) & y_{k} ⇀ y in L^{2} (Ω), \nabla y_{k} ⇀ v in L^{2} {(Ω, A_{k}^{sym} d x)}^{N} . \end{array}$ Then $A_{0} : = A_{0}^{sym} + A_{0}^{skew} \in A_{ad}$ , $y \in W_{A_{0}^{sym}} (Ω)$ , and $\nabla y = v$ .
Proof.
In view of Proposition 4.2, it is enough to prove the equality $\nabla y = v$ . Taking into account the estimates $\begin{array}{l} \begin{matrix} \int_{Ω} | \nabla y_{k} |_{R^{N}} d x & = & \int_{Ω} {| {(A_{k}^{sym})}^{- 1 / 2} {(A_{k}^{sym})}^{1 / 2} \nabla y_{k} |}_{R^{N}} d x \\ ⩽ & {(\int_{Ω} {‖ {(A_{k}^{sym})}^{- 1 / 2} ‖}_{L (R^{N}, R^{N})}^{2} d x)}^{1 / 2} {(\int_{Ω} {(\nabla y_{k}, A_{k}^{sym} \nabla y_{k})}_{R^{N}} d x)}^{1 / 2} \\ \overset{{by (4.17)}_{2} and (2.6)}{⩽} & C {(\int_{Ω} ζ_{ad}^{- 1} d x)}^{1 / 2} ⩽ C | Ω |^{\frac{p + 1}{4 p}} {‖ ζ_{ad}^{- 1} ‖}_{L^{2 q} (Ω)}^{\frac{1}{2}} < + \infty, \end{matrix} \\ \int_{Ω} | v |_{R^{N}} d x ⩽ ‖ v ‖_{L^{2} {(Ω, A_{0}^{sym} d x)}^{N}} | Ω |^{\frac{p + 1}{4 p}} {‖ ζ_{ad}^{- 1} ‖}_{L^{2 q} (Ω)}^{\frac{1}{2}} < + \infty, \end{array}$ we conclude that $\nabla y_{k}, v \in L^{1} {(Ω)}^{N}$ for all $k \in N$ .

Further, we make use of Lemma 3.6. Following this result, for each test function $φ \in C_{0}^{\infty} (Ω)$ , we have ${(A_{k}^{sym})}^{- 1} \nabla φ \to {(A_{0}^{sym})}^{- 1} \nabla φ$ strongly in variable space $L^{2} {(Ω, A_{k}^{sym} d x)}^{N}$ . Then, the definition of the strong convergence in variable spaces implies $\forall φ \in C_{0}^{\infty} (Ω)$ $\begin{array}{rcl} lim_{k \to \infty} \int_{Ω} {(\nabla φ, \nabla y_{k})}_{R^{N}} d x & = & lim_{k \to \infty} \int_{Ω} {({(A_{k}^{sym})}^{- 1} \nabla φ, A_{k}^{sym} \nabla y_{k})}_{R^{N}} d x \\ \overset{{by (4.17)}_{2}, (3.3)}{=} & \int_{Ω} {({(A_{0}^{sym})}^{- 1} \nabla φ, A_{0}^{sym} v)}_{R^{N}} d x = \int_{Ω} {(\nabla φ, v)}_{R^{N}} d x . \end{array}$ Combining this fact with relation $\begin{matrix} lim_{k \to \infty} \int_{Ω} y_{k} φ d x = \int_{Ω} y φ d x, \forall φ \in C_{0}^{\infty} (Ω), \end{matrix}$ we finally conclude: $y_{k} ⇀ y$ in $L^{1} (Ω)$ , $\nabla y_{k} ⇀ v$ in $L^{1} {(Ω)}^{N}$ , and therefore, $\nabla y = v$ and $y \in W_{0}^{1, 1} (Ω)$ by completeness of the Sobolev space $W_{0}^{1, 1} (Ω)$ . To end the proof, it remains to observe that $y \in L^{2} (Ω)$ and $\nabla y \in L^{2} {(Ω, A_{0}^{sym} d x)}^{N}$ . □

For our further analysis we temporary assume that the functions β and $ζ_{ad}$ are extended to the whole space of $R^{N}$ , i.e. $\begin{matrix} β, ζ_{ad} \in L_{loc}^{1} (R^{N}), 0 ⩽ ζ_{ad} (x) ⩽ β (x) a.e. x \in R^{N}, and ζ_{ad}^{- 1} \in L_{loc}^{2 q} (R^{N}), \end{matrix}$ and there exists a constant $C > 0$ such that $\begin{matrix} (4.18) & sup_{B \in R^{N}} (\frac{1}{| B |} \int_{B} β d x) (\frac{1}{| B |} \int_{B} ζ_{ad}^{- 1} d x) ⩽ C, \end{matrix}$ where B is a ball in $R^{N}$ .
Theorem 4.7 ([25]).

Assume the condition (4.18) holds true for some constant $C > 0$ . Then for each admissible control $A = A_{sym} + A_{skew} \in A_{ad}$ , we have $H_{A_{sym}} (Ω) = W_{A_{sym}} (Ω)$ and, hence, every weak solution to the boundary value problem (1.2)–(1.3) satisfies the energy equality (4.12) .

We are now in a position to establish the main result of this section.

Theorem 4.8.
Assume that, for given threshold matrices $A^{}, A^{ *} \in L^{2 p} (Ω; S_{skew}^{N})$ , Hypothesis A is valid. Then the optimal control problem (1.1)–(1.3) admits at least one solution for all distributions $f \in L^{4 p / (p + 1)} (Ω; R^{N})$ and $y_{d} \in L^{2} (Ω)$ .
Proof.
Since the original problem is regular and the cost functional for the given problem is bounded below on Ξ, it follows that there exists a minimizing sequence ${(A_{k}, y_{k})}_{k \in N} \subset Ξ$ such that $I (A_{k}, y_{k}) \underset{k \to \infty}{⟶} I_{\min} \equiv {inf}_{(A, y) \in Ξ} I (A, y) ⩾ 0$ . Hence, ${sup}_{k \in N} I (A_{k}, y_{k}) ⩽ C$ , where the constant C is independent of k. Since $\begin{array}{rcl} sup_{k \in N} ‖ y_{k} ‖_{A_{k}^{sym}}^{2} & = & \int_{Ω} (y_{k}^{2} + {(\nabla y_{k}, A_{k}^{sym} \nabla y_{k})}_{R^{N}}) d x \\ (4.19) & ⩽ & 2 sup_{k \in N} I (A_{k}, y_{k}) + 2 ‖ y_{d} ‖_{L^{2} (Ω)}^{2} ⩽ 2 (C + ‖ y_{d} ‖_{L^{2} (Ω)}^{2}), \end{array}$ in view of Propositions 3.3, 4.2, and Lemma 3.6, it follows that passing to a subsequence if necessary, we may assume the existence of a pair $(A_{0}, y_{0}) \in A_{ad} \times W_{A_{0}^{sym}} (Ω)$ such that $\begin{array}{l} (4.20) & A_{k}^{sym} \to A_{0}^{sym} in L^{1} (Ω; S_{sym}^{N}), \\ (4.21) & A_{k}^{skew} \to A_{0}^{skew} in L^{2 p} (Ω; S_{skew}^{N}), \\ (4.22) & y_{k} ⇀ y_{0} in L^{2} (Ω), \\ (4.23) & \nabla y_{k} ⇀ \nabla y_{0} in L^{2} {(Ω, A_{k}^{sym} d x)}^{N}, \\ (4.24) & I (A_{0}, y_{0}) < + \infty . \end{array}$ Since $(A_{k}, y_{k}) \in Ξ$ for every $k \in N$ , it follows that the integral identity $\begin{array}{l} \int_{Ω} [{(\nabla φ, A_{k}^{sym} \nabla y_{k})}_{R^{N}} + a_{0} φ y_{k}] d x + \int_{Ω} {(\nabla φ, A_{k}^{skew} \nabla y_{k})}_{R^{N}} d x \\ (4.25) & = \int_{Ω} {(f, \nabla φ)}_{R^{N}} d x \end{array}$ holds true for all $φ \in C_{0}^{\infty} (Ω)$ . In order to pass to the limit in (4.25), we note that $\begin{array}{rcl} \int_{Ω} (\nabla φ, A_{k}^{skew} \nabla y_{k}) d x & = & - \int_{Ω} ({(A_{k}^{sym})}^{- \frac{1}{2}} (A_{k}^{skew} - A_{0}^{skew}) \nabla φ, {(A_{k}^{sym})}^{\frac{1}{2}} \nabla y_{k}) d x \\ - \int_{Ω} ({(A_{k}^{sym})}^{- \frac{1}{2}} A_{0}^{skew} \nabla φ, {(A_{k}^{sym})}^{\frac{1}{2}} \nabla y_{k}) d x = I_{1, k} + I_{2, k} \end{array}$ by the skew-symmetry property of $A_{k}^{skew}$ and $A_{0}^{skew}$ . Since $\begin{array}{rcl} lim_{k \to \infty} | I_{1, k} | & ⩽ & lim_{k \to \infty} \int_{Ω} | {(A_{k}^{sym})}^{- \frac{1}{2}} (A_{k}^{skew} - A_{0}^{skew}) \nabla φ | | {(A_{k}^{sym})}^{\frac{1}{2}} \nabla y_{k} | d x \\ ⩽ & lim_{k \to \infty} {(\int_{Ω} {‖ A_{k}^{skew} - A_{0}^{skew} ‖}^{2} {‖ {(A_{k}^{sym})}^{- \frac{1}{2}} ‖}^{2} | \nabla φ |^{2} d x)}^{\frac{1}{2}} \\ \times lim_{k \to \infty} {(\int_{Ω} {| {(A_{k}^{sym})}^{\frac{1}{2}} \nabla y_{k} |}^{2} d x)}^{\frac{1}{2}} \\ ⩽ & ‖ φ ‖_{C^{1} (Ω)} sup_{k \in N} ‖ y_{k} ‖_{A_{k}^{sym}} lim_{k \to \infty} {(\int_{Ω} {‖ A_{k}^{skew} - A_{0}^{skew} ‖}^{2 p} d x)}^{\frac{1}{2 p}} {(\int_{Ω} ζ_{ad}^{- q} d x)}^{\frac{1}{2 q}} \\ \overset{by (4.19)}{⩽} & \sqrt{2 (C + ‖ y_{d} ‖_{L^{2} (Ω)}^{2})} ‖ φ ‖_{C^{1} (Ω)} | Ω |^{\frac{1}{4 q}} {‖ ζ_{ad}^{- 1} ‖}_{L^{2 q} (Ω)}^{\frac{1}{2}} \\ (4.26) & \times lim_{k \to \infty} {‖ A_{k}^{skew} - A_{0}^{skew} ‖}_{L^{2 p} (Ω; S_{skew}^{N})} \overset{by (4.21)}{=} 0, \end{array}$ and $\begin{array}{l} lim_{k \to \infty} I_{2, k} & = - lim_{k \to \infty} \int_{Ω} {({(A_{k}^{sym})}^{- 1} A_{0}^{skew} \nabla φ, A_{k}^{sym} \nabla y_{k})}_{R^{N}} d x \\ (4.27) & = - \int_{Ω} {({(A_{0}^{sym})}^{- 1} A_{0}^{skew} \nabla φ, A_{0}^{sym} \nabla y_{0})}_{R^{N}} d x = \int_{Ω} {(\nabla φ, A_{0}^{skew} \nabla y_{0})}_{R^{N}} d x \end{array}$ by (4.23), Lemma 3.6, and definition of the strong convergence in variable spaces, it follows that $\begin{matrix} \int_{Ω} {(\nabla φ, A_{k}^{skew} \nabla y_{k})}_{R^{N}} d x ⟶ \int_{Ω} {(\nabla φ, A_{0}^{skew} \nabla y_{0})}_{R^{N}} d x as k \to \infty . \end{matrix}$

Taking this fact and property (4.24) into account, we can pass to the limit in (4.25). As a result, we obtain $\begin{matrix} \int_{Ω} [(\nabla φ, A_{0}^{sym} \nabla y_{0}) + a_{0} φ y_{0}] d x + \int_{Ω} (\nabla φ, A_{0}^{skew} \nabla y_{0}) d x = \int_{Ω} (f, \nabla φ) d x \end{matrix}$ that is, a function $y_{0} = y (A_{0}, f)$ is a weak solution to the boundary value problem (1.2)–(1.3) for admissible control $A_{0} = A_{0}^{sym} + A_{0}^{skew} \in A_{ad}$ . Hence, $y_{0} \in D (W_{A_{0}^{sym}})$ by Proposition 4.4, and, therefore, $(A_{0}, y_{0})$ is an admissible pair to problem (1.1)–(1.3).

In remains to show that $(A_{0}, y_{0})$ is an optimal pair. Using conditions (4.22)–(4.24) and the property of lower semicontinuity of the norms $‖ \cdot ‖_{L^{2} {(Ω, A d x)}^{N}}$ and $‖ \cdot ‖_{L^{2} (Ω)}$ with respect to the weak topologies of $L^{2} {(Ω, A d x)}^{N}$ and $L^{2} (Ω)$ , respectively (see Proposition 3.4), we get $\begin{array}{l} \underset{k \to \infty}{lim inf} ‖ y_{k} - y_{d} ‖_{L^{2} (Ω)}^{2} ⩾ ‖ y_{0} - y_{d} ‖_{L^{2} (Ω)}^{2}, \\ \underset{k \to \infty}{lim inf} \int_{Ω} {(\nabla y_{k}, A_{k}^{sym} \nabla y_{k})}_{R^{N}} d x ⩾ \int_{Ω} {(\nabla y_{0}, A_{0}^{sym} \nabla y_{0})}_{R^{N}} d x . \end{array}$ Thus, $\begin{array}{rcl} I (A_{0}, y_{0}) & ⩾ & inf_{(A, y) \in Ξ} I (A, y) = lim_{k \to \infty} I (A_{k}, y_{k}) ⩾ \underset{k \to \infty}{lim inf} I (A_{k}, y_{k}) \\ ⩾ & ‖ y_{0} - y_{d} ‖_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla y_{0}, A_{0}^{sym} \nabla y_{0})}_{R^{N}} d x = I (A_{0}, y_{0}), \end{array}$ and hence, the pair $(A_{0}, y_{0})$ is optimal for problem (1.1)–(1.3). The proof is complete. □

5. On variational solutions to OCP (1.1)–(1.3) and their approximation

The question we are going to discuss in this section is about some pathological properties that can be inherited by optimal pair to the problem (1.1)–(1.3) and other unexpected surprises concerning the approximation of the original OCP and its solutions.

To begin with, we show that the main assumption on the regularity property of OCP (1.1)–(1.3) in Theorem 4.8 (see Hypothesis A) can be eliminated due to the approximation approach. For instance, the conditions $ζ_{ad} \in L^{1} (Ω)$ and $ζ_{ad}^{- 1} \in L^{2 q} (Ω)$ ensure the existence of a sequence of scalar positive functions ${ζ_{k}}_{k \in N}$ such that $ζ_{k} \in L^{\infty} (Ω)$ for all $k \in N$ , and $\begin{matrix} (5.1) & ζ_{k} \to ζ_{ad} strongly in L^{1} (Ω), L^{\infty} (Ω) ∋ ζ_{k}^{- 1} \to ζ_{ad}^{- 1} strongly in L^{2 q} (Ω) . \end{matrix}$ The simplest way to construct such sequences is to apply the procedure of the direct smoothing (5.2)–(5.3).

Lemma 5.1.

If $ζ_{ad} \in L^{1} (Ω)$ and $ζ_{ad}^{- 1} \in L^{2 q} (Ω)$ , then the direct smoothing $ζ_{k} : = {(ζ_{ad})}_{k}$ possesses the properties (5.1), where $\begin{matrix} (5.2) & {(ζ_{ad})}_{k} (x) = k^{N} \int_{R^{N}} K (k (x - z)) \tilde{ζ_{ad}} (z) d z, \end{matrix}$ K is a positive compactly supported smooth function such that $\begin{matrix} (5.3) & K \in C_{0}^{\infty} (R^{N}), \int_{R^{N}} K (x) d x = 1, and K (x) = K (- x), \end{matrix}$ and $\tilde{}$ is zero extension operator outside of Ω .

Proof.

Indeed, the property (5.1)₁ is the direct consequence of the classical properties of smoothing. In order to prove the property (5.1)₂, we note that $\begin{matrix} {(ζ_{ad})}_{k} (x) ⩽ C \int_{Π} \tilde{ζ_{ad}} (x + k^{- 1} z) d z, a.e. in R^{N}, \forall k \in N, \end{matrix}$ where Π is the support of the smoothing kernel K and $K (x) ⩽ C$ by (5.3). Hence, ${(ζ_{ad})}_{k} \in L_{loc}^{1} (R^{N})$ (or more precise ${(ζ_{ad})}_{k} \in C_{0}^{\infty} (R^{N})$ ) for all $k \in N$ . For a given $k \in N$ and $x \in Ω$ , we set $Ω_{k} (x) = {z \in R^{N} ∣ x + k^{- 1} z \subset Ω}$ . It is clear that $| supp (K) ∖ Ω_{k} (x) | \to 0$ as $k \to \infty$ . Then, using the Cauchy inequality, $\begin{array}{l} C_{Ω_{k} (x)} & = 1 - \int_{R^{N} ∖ Ω_{k} (x)} K (z) d z = \int_{R^{N}} K (z) d z - \int_{R^{N} ∖ Ω_{k} (x)} K (z) d z \\ = \int_{Ω_{k} (x)} K (z) d z = \int_{Ω_{k} (x)} {[K (z) ζ_{ad} (x + \frac{z}{k})]}^{\frac{1}{2}} {[K (z) ζ_{ad}^{- 1} (x + \frac{z}{k})]}^{\frac{1}{2}} d z \\ = \int_{R^{N}} {[K (z) \tilde{ζ_{ad}} (x + \frac{z}{k})]}^{\frac{1}{2}} {[K (z) \tilde{ζ_{ad}^{- 1}} (x + \frac{z}{k})]}^{\frac{1}{2}} d z \\ ⩽ {(\int_{R^{N}} K (z) \tilde{ζ_{ad}} (x + \frac{z}{k}) d z)}^{\frac{1}{2}} {(\int_{R^{N}} K (z) \tilde{ζ_{ad}^{- 1}} (x + \frac{z}{k}) d z)}^{\frac{1}{2}} \\ = {[{(ζ_{ad})}_{k} {(ζ_{ad}^{- 1})}_{k}]}^{\frac{1}{2}}, \end{array}$ we see that $\begin{matrix} (5.4) & {(ζ_{ad})}_{k}^{- 1} ⩽ C_{Ω_{k} (x)}^{- 2} {(ζ_{ad}^{- 1})}_{k} in R^{N} . \end{matrix}$ Since $C_{Ω_{k}} \to 1 - 0$ as $k \to \infty$ , it follows that there exists a constant $C_{*} \in (0; 1)$ independent of k such that $C_{Ω_{k}} ⩾ C_{*}$ for all $k \in N$ large enough. Therefore, $\begin{array}{l} C_{*}^{4 q} {(ζ_{ad})}_{k}^{- 2 q} & ⩽ {(ζ_{ad}^{- 1})}_{k}^{2 q} = {(\int_{R^{N}} K (z) \tilde{ζ_{ad}^{- 1}} (x + \frac{z}{k}) d z)}^{2 q} \\ = {(\int_{R^{N}} K^{\frac{2 q - 1}{2 q}} (z) K^{\frac{1}{2 q}} (z) \tilde{ζ_{ad}^{- 1}} (x + \frac{z}{k}) d z)}^{2 q} \\ ⩽ {(\int_{R^{N}} K (z) d z)}^{2 q - 1} (\int_{R^{N}} K (z) {(\tilde{ζ_{ad}^{- 1}} (x + \frac{z}{k}))}^{2 q} d z) \\ = \int_{R^{N}} K (z) \tilde{ζ_{ad}^{- 2 q}} (x + \frac{z}{k}) d z = {(ζ_{ad}^{- 2 q})}_{k} \\ (5.5) & ⩽ C \int_{Q} \tilde{ζ_{ad}^{- 2 q}} (x + \frac{z}{k}) d z . \end{array}$ Hence, ${(ζ_{ad})}_{k}^{- 1} \in L_{loc}^{2 q} (R^{N})$ for all $k \in N$ .

Since ${(ζ_{ad})}_{k} \to ζ_{ad}$ in $L^{1} (Ω)$ as $k \to \infty$ by the classical properties of smoothing, we can suppose that ${(ζ_{ad})}_{k}^{- 2 q} (x) \to ζ_{ad}^{- 2 q} (x)$ almost everywhere in Ω. In the meantime the inequality (5.5) guarantees the equi-integrability of ${{(ζ_{ad})}_{k}^{- 2 q}}_{k \in N}$ because the sequence ${{(ζ_{ad}^{- 2 q})}_{k}}_{k \in N}$ , converging to $ζ_{ad}^{- 2 q}$ strongly in $L^{1} (Ω)$ , possesses this property. As a result, Lebesgue’s Theorem implies that ${(ζ_{ad})}_{k}^{- 2 q} \to ζ_{ad}^{- 2 q}$ in $L^{1} (Ω)$ as $k \to \infty$ , and so the proof of property (5.1)₂ is complete. □

By analogy, we can approximate the components of $A^{*}, A^{* *} \in L^{2 p} (Ω; S_{skew}^{N})$ and $β \in L^{1} (Ω)$ . Before proceeding further, we give a few auxiliary results the validity of which immediately follows from the Young inequality and the classical properties of Steklov smoothing.

Lemma 5.2.

Let $f \in L^{2 p} (Ω)$ and ${f_{n}}_{n \in N} \subset L^{2 p} (Ω)$ be such that $f_{n} \to f$ in $L^{2 p} (Ω)$ as $n \to \infty$ . Then, for each positive integer $k \in N$ , we have $\begin{matrix} (5.6) & {(f_{n})}_{k} \to {(f)}_{k} in L^{2 p} (Ω) as n \to \infty . \end{matrix}$

Lemma 5.3.

Let $f \in L^{2 r} (Ω)$ and ${f_{n}}_{n \in N} \subset L^{2 r} (Ω)$ , $r ⩾ 1 / 2$ , be such that $f_{n} \to f$ in $L^{2 r} (Ω)$ as $n \to \infty$ . Let ${k_{n}}_{n \in N}$ be a sequence of positive integers converging to $+ \infty$ as $n \to \infty$ . Then ${(f_{n})}_{k_{n}} \to f$ in $L^{2 r} (Ω)$ as $n \to \infty$ .

Taking these results into account, we bring into consideration the following sequence of constrained minimization problems associated with the Steklov smoothing operator ${(\cdot)}_{k}$ : $\begin{matrix} (5.7) & {⟨ inf_{(A, y) \in Ξ_{k}} I (A, y) ⟩, k \to \infty} . \end{matrix}$ Here, $\begin{array}{l} (5.8) & Ξ_{k} = \{(A, y) |\begin{matrix} - div (A^{sym} \nabla y + A^{skew} \nabla y) = - div f in Ω, \\ y = 0 on \partial Ω, y \in W_{A^{sym}} (Ω), \\ A = A^{sym} + A^{skew} \in A_{ad}^{k} = A_{ad, 1}^{k} \oplus A_{ad, 2}^{k}, \\ A^{sym} \in A_{ad, 1}^{k} iff \exists C^{sym} \in A_{ad, 1} s.t. A^{sym} = {(C^{sym})}_{k}, \\ A^{skew} \in A_{ad, 2}^{k} iff \exists C^{skew} \in A_{ad, 2} s.t. A^{skew} = {(C^{skew})}_{k} \end{matrix}\} . \end{array}$

Before we will provide an accurate analysis of the optimal control problems (5.7), we describe in more details some topological properties of the sets $A_{ad, 1}^{k}$ and $A_{ad, 2}^{k}$ . We begin with the following observation.

Remark 5.1.

In view of definition of the sets $A_{ad, 1}^{k}$ and $A_{ad, 2}^{k}$ , the condition $A \in A_{ad, 1} = A_{ad, 1}^{k} \oplus A_{ad, 2}^{k}$ implies the existence of a certain matrix $C (x) = C^{sym} (x) + C^{skew} (x)$ (the so-called “prototype” of A) such that $C^{sym} \in A_{ad, 1}$ , $C^{skew} \in A_{ad, 2}$ , and $A = {(C^{sym})}_{k} + {(C^{skew})}_{k}$ whatever matrix A was chosen.

Lemma 5.4.

For every $k \in N$ there exist positive constants $α_{k}$ and $γ_{k}$ such that $γ_{k} > α_{k}$ and $\begin{array}{l} (5.9) & α_{k} ‖ ξ ‖_{R^{N}}^{2} ⩽ {(A^{sym} (x) ξ, ξ)}_{R^{N}} ⩽ γ_{k} ‖ ξ ‖_{R^{N}}^{2} in Ω, \forall ξ \in R^{N}, \\ (5.10) & TV (a_{i j}^{sym}) ⩽ c, 1 ⩽ i ⩽ j ⩽ N, \\ (5.11) & \int_{R^{N}} A^{sym} (x) d x = M \end{array}$ for each $A^{sym} \in A_{ad, 1}^{k}$ .

Proof.

Let $C^{sym}$ be an arbitrary element of the set $A_{ad, 1}$ . Since, $A_{ad, 1} \subset M_{ζ_{ad}}^{β} (Ω)$ , it follows that $ζ_{ad} (x) ‖ ξ ‖_{R^{N}}^{2} ⩽ {(C^{sym} (x) ξ, ξ)}_{R^{N}} ⩽ β (x) ‖ ξ ‖_{R^{N}}^{2}$ a.e. in Ω, $\forall ξ \in R^{N}$ . Hence, for any $k \in N$ , we have $\begin{matrix} {(ζ_{ad})}_{k} ‖ ξ ‖_{R^{N}}^{2} ⩽ {({(C^{sym})}_{k} ξ, ξ)}_{R^{N}} ⩽ {(β)}_{k} ‖ ξ ‖_{R^{N}}^{2} in Ω, \forall ξ \in R^{N}, \end{matrix}$ and, therefore, the constants $α_{k}$ and $γ_{k}$ in (5.9) can be defined as follows $\begin{matrix} α_{k} = inf_{x \in Ω} {(ζ_{ad})}_{k} (x), γ_{k} = sup_{x \in Ω} {(β)}_{k} (x) . \end{matrix}$ In view of the initial assumptions (2.4)–(2.6) and definition of the Steklov smoothing operator ${(\cdot)}_{k}$ , we have ${(β)}_{k} \in L^{\infty} (Ω)$ and ${(ζ_{ad})}_{k}^{- 1} \in L^{\infty} (Ω)$ (see (5.4)). Hence, $α_{k}$ is a positive constant, and $γ_{k} < + \infty$ .

As for the estimate (5.10), for an arbitrary $φ = (φ_{1}, \dots, φ_{N}) \in C_{0}^{1} (Ω; R^{N})$ such that $| φ (x) | ⩽ 1$ in Ω, and arbitrary matrix $A^{sym} = {[a_{i j}^{sym}]}_{i, j = 1}^{N} \in A_{ad, 1}^{k}$ , we have $\begin{array}{l} TV (a_{i j}^{sym}) & = sup_{| φ | ⩽ 1} {\int_{R^{N}} a_{i j}^{sym} (x) {(\nabla, φ)}_{R^{N}} d x} = sup_{| φ | ⩽ 1} {\int_{R^{N}} {(c_{i j}^{sym})}_{k} (x) {(\nabla, φ)}_{R^{N}} d x} \\ = sup_{| φ | ⩽ 1} {\int_{R^{N}} \int_{R^{N}} K (z) \tilde{c_{i j}^{sym}} (x + k^{- 1} z) d z {(\nabla, φ)}_{R^{N}} d x} \\ ⩽ \int_{R^{N}} K (z) sup_{| φ | ⩽ 1} {\int_{R^{N}} \tilde{c_{i j}^{sym}} (x + k^{- 1} z) {(\nabla, φ)}_{R^{N}} d x} d z \\ = \int_{R^{N}} K (z) TV (\tilde{c_{i j}^{sym}}) (x + k^{- 1} z) d z ⩽ c \int_{R^{N}} K (z) d z = c . \end{array}$ Having applied the similar arguments, namely, $\begin{array}{l} \int_{R^{N}} A^{sym} (x) d x & = \int_{R^{N}} {(C^{sym})}_{k} (x) d x = \int_{R^{N}} \int_{R^{N}} K (z) {\tilde{C}}^{sym} (x + k^{- 1} z) d z d x \\ = k^{N} \int_{R^{N}} \int_{R^{N}} K (k (x - y)) {\tilde{C}}^{sym} (y) d y d x \\ = \int_{R^{N}} (k^{N} \int_{R^{N}} K (k (x - y))) d x {\tilde{C}}^{sym} (y) d y = \int_{Ω} {\tilde{C}}^{sym} (y) d y = M, \end{array}$ we arrive at the control constraint (5.11). □

Lemma 5.5.

$A_{ad, 1}^{k}$ is a convex and compact set with respect to the strong topology of $L^{1} (Ω; S_{sym}^{N})$ for each $k \in N$ .

Proof.

Since the convexity of $A_{ad, 1}^{k}$ immediately follows from the linearity of the smoothing operator ${(\cdot)}_{k}$ , we concentrate on the compactness property of this set. Let ${A_{n}^{sym}}_{n \in N}$ be an arbitrary sequence in $A_{ad, 1}^{k}$ , and let ${C_{n}^{sym}}_{n \in N} \subset A_{ad, 1}$ be a sequence of its prototypes, that is, $A_{n}^{sym} (x) = {(C_{n}^{sym})}_{k} (x)$ for all $n \in N$ . By Proposition 4.2, there exists a matrix $C_{0}^{sym} \in A_{ad, 1}$ such that, within a subsequence, $C_{n}^{sym} \to C_{0}^{sym}$ in $L^{1} (Ω; S_{sym}^{N})$ . As a result, Lemma 5.3, specified for the case $L^{1} (Ω)$ , implies the strong convergence $A_{n}^{sym} \to A_{0}^{sym}$ in $L^{1} (Ω; S_{sym}^{N})$ , where $A_{0}^{sym} = {(C_{0}^{sym})}_{k}$ for a given $k \in N$ . □

We recall here that a sequence ${A_{ad, 1}^{k}}_{k \in N}$ of the subsets of $L^{1} (Ω; S_{sym}^{N})$ is said to be convergent to a closed set S in the sense of Kuratowski with respect to the strong topology of $L^{1} (Ω; S_{sym}^{N})$ , if the following two properties hold:

for every $A \in S$ , there exists a sequence of matrices ${A_{k} \in A_{ad, 1}^{k}}_{k \in N}$ such that $A_{k} \to A$ in $L^{1} (Ω; S_{sym}^{N})$ as $k \to \infty$ ;

if ${k_{n}}_{n \in N}$ is a sequence of indices converging to $+ \infty$ , ${A_{n}}_{n \in N}$ is a sequence of symmetric matrices such that $A_{n} \in A_{ad, 1}^{k_{n}}$ for each $n \in N$ , and ${A_{n}}_{n \in N}$ strongly converges in $L^{1} (Ω; S_{sym}^{N})$ to some matrix A, then $A \in S$ .

For the details we refer to [12]. As a result, we have the following result concerning asymptotic behaviour of the sequence

{A_{ad, 1}^{k}}_{k \in N}

Lemma 5.6.

The sequence of sets ${A_{ad, 1}^{k}}_{k \in N}$ converges to $A_{ad, 1}$ as $k \to \infty$ in the sense of Kuratowski with respect to the strong topology of $L^{1} (Ω; S_{sym}^{N})$ .

Proof.

In order to show that $S = A_{ad, 1}$ , we begin with the verification of $(K_{2})$ -item. Let ${k_{n}}_{n \in N}$ be a given sequence of indices such that $k_{n} \to \infty$ , and let ${A_{n} \in A_{ad, 1}^{k_{n}}}_{n \in N}$ be a sequence satisfying the property $A_{n} \to A$ in $L^{1} (Ω; S_{sym}^{N})$ as $n \to \infty$ . By definition of the sets $A_{ad, 1}^{k}$ and Proposition 4.2, there exists a sequence of prototypes ${C_{n}^{sym}}_{n \in N} \subset A_{ad, 1}$ and matrix $C_{0}^{sym} \in A_{ad, 1}$ such that $A_{n} = {(C_{n}^{sym})}_{k_{n}}$ for all $n \in N$ and, within a subsequence, $C_{n}^{sym} \to C_{0}^{sym}$ in $L^{1} (Ω; S_{sym}^{N})$ . Then Lemma 5.3 guarantees the strong convergence $A_{n} \to C_{0}^{sym}$ in $L^{1} (Ω; S_{sym}^{N})$ . As a result, we have $A = C_{0}^{sym}$ and, therefore, $A \in A_{ad, 1}$ . Since this assertion is valid for each $L^{1}$ -converging subsequence of ${C_{n}^{sym}}_{n \in N} \subset A_{ad, 1}$ , we finally get: the symmetric matrix A is $L^{1}$ -limit for the entire sequence ${C_{n}^{sym}}_{n \in N} \subset A_{ad, 1}$ .

It remains to verify the $(K_{1})$ -item. To this end, we fix an arbitrary symmetric matrix $A \in A_{ad, 1}$ and construct the sequence ${A_{k} \in A_{ad, 1}^{k}}_{k \in N}$ as follows: $A_{k} = {(A)}_{k}$ for all $k \in N$ . Then $A_{k} \to A$ in $L^{1} (Ω; S_{sym}^{N})$ as $k \to \infty$ by main properties of the smoothing operator, and inclusions $A_{k} \in A_{ad, 1}^{k}$ , for each $k \in N$ , hold true by definition of the sets $A_{ad, 1}^{k}$ . □

Our next intention is to study the topological and asymptotic properties of the sets $A_{ad, 2}^{k}$ .

Lemma 5.7.

For every $k \in N$ each of the sets $A_{ad, 2}^{k}$ is convex, compact with respect to the strong topology of $L^{2 p} (Ω; S_{sym}^{N})$ , and such that $\begin{matrix} (5.12) & {(A^{*})}_{k} (x) ⪯ A (x) ⪯ {(A^{* *})}_{k} (x) in Ω, \forall A \in A_{ad, 2}^{k} . \end{matrix}$

Proof.

The convexity of $A_{ad, 2}^{k}$ is a direct consequence of definition of the set $A_{ad, 2}$ and the rule (5.8)₅. To prove the compactness property of this set let us consider an arbitrary sequence ${A_{n}}_{n \in N}$ in $A_{ad, 2}^{k}$ . Let ${C_{n}^{skew}}_{n \in N} \subset A_{ad, 2}$ be their prototypes, that is, $A_{n} (x) = {(C_{n}^{skew})}_{k} (x)$ for all $n \in N$ . Since ${C_{n}^{skew}}_{n \in N} \subset Q$ , where Q is a nonempty convex compact subset of $L^{2 p} (Ω; S_{skew}^{N})$ , it follows that there exists a skew-symmetric matrix $C_{0}^{skew} \in Q$ such that, up to a subsequence, $C_{n}^{skew} \to C_{0}^{skew}$ in $L^{2 p} (Ω; S_{skew}^{N})$ . Then Lemma 5.2 implies the strong convergence $A_{n} \to A_{0} : = {(C_{0}^{skew})}_{k}$ in $L^{2 p} (Ω; S_{skew}^{N})$ for every $k \in N$ . It remains to note that in view of the definition of binary relation ⪯ (see (2.1)), for every $A = [a_{i j}] \in A_{ad, 2}$ , $i \in {1, \dots, N}$ , $j \in {i + 1, \dots, N}$ , and $x \in R^{N}$ , we have $\begin{array}{l} {(a_{i j})}_{k} (x) & : = k^{N} \int_{R^{N}} K (k (x - y)) \tilde{a_{i j}} (y) d y \\ ⩽ k^{N} \int_{R^{N}} K (k (x - y)) \tilde{a_{i j}^{* *}} (y) d y = {(a_{i j}^{* *})}_{k} (x), \forall k \in N . \end{array}$ By analogy it can be shown that ${(a_{i j}^{*})}_{k} (x) ⩽ {(a_{i j})}_{k} (x)$ in Ω. Hence, the restriction (5.12) holds true for each $k \in N$ and $A \in A_{ad, 2}^{k}$ . The proof is complete. □

Lemma 5.8.

The sequence of sets ${A_{ad, 2}^{k}}_{k \in N}$ converges to $A_{ad, 2}$ as $k \to \infty$ in the sense of Kuratowski with respect to the strong topology of $L^{2 p} (Ω; S_{skew}^{N})$ .

Proof.

We begin with the verification of $(K_{2})$ -property of the set $A_{ad, 2}$ in the framework of definition of Kuratowski limit set with respect to the strong topology of $L^{2 p} (Ω; S_{skew}^{N})$ . Let ${k_{n}}_{n \in N}$ be an arbitrary sequence of indices such that $k_{n} \to \infty$ , and let ${B_{n} \in A_{ad, 2}^{k_{n}}}_{n \in N}$ be a sequence satisfying the property $B_{n} \to B$ in $L^{2 p} (Ω; S_{skew}^{N})$ and, hence, up to a subsequence, $B_{n} (x) \to B (x)$ almost everywhere in Ω as $n \to \infty$ . By Lemma 5.7, we have $\begin{matrix} (5.13) & {(A^{*})}_{k_{n}} (x) ⪯ B_{n} (x) ⪯ {(A^{* *})}_{k_{n}} (x) a.e. in Ω, \end{matrix}$ where ${(A^{*})}_{k} \to A^{*}$ and ${(A^{* *})}_{k} \to A^{* *}$ strongly in $L^{2 p} (Ω; S_{skew}^{N})$ . Taking into account the fact that the binary relation ⪯ is a partial order, we can pass to the limit in relation (5.13) as $n \to \infty$ (in the sense of almost everywhere) and get $A^{*} (x) ⪯ B (x) ⪯ A^{* *} (x)$ almost everywhere in Ω. In the meantime, closely following the arguments of the proof of Lemma 5.6 (see also Lemma 5.3), it can be shown that the matrix B is $L^{2 p}$ -limit of the corresponding sequence of prototypes ${C_{n}^{skew}}_{n \in N} \subset Q \subset A_{ad, 2}$ , where $B_{n} = {(C_{n}^{skew})}_{k_{n}}$ for all $n \in N$ . Since, Q is a compact set, it follows that $B \in Q$ , and, therefore, $B \in A_{ad, 2}$ .

To verify the $(K_{1})$ -property, we fix an arbitrary skew-symmetric matrix $B \in A_{ad, 2}$ and construct the sequence ${B_{k} \in A_{ad, 2}^{k}}_{k \in N}$ as follows: $B_{k} = {(B)}_{k}$ for all $k \in N$ . Then $B_{k} \to B$ in $L^{2 p} (Ω; S_{skew}^{N})$ as $k \to \infty$ by main properties of the smoothing operator, and inclusions $B_{k} \in A_{ad, 2}^{k}$ , for each $k \in N$ , hold true by definition of the sets $A_{ad, 2}^{k}$ and Lemma 5.12. The proof is complete. □

In what follows, we make use the following concept.

Definition 5.9.

We say that a sequence of pairs $\begin{matrix} {(A_{k}, y_{k}) = (A_{k}^{sym} + A_{k}^{skew}, y_{k}) \in [L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N})] \times W_{A_{k}^{sym}} (Ω)}_{k \in N} \end{matrix}$ τ-converges to a pair $(A, y) = (A^{sym} + A^{skew}, y) \in [L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N})] \times W_{A^{sym}} (Ω)$ if $\begin{array}{l} A_{k}^{sym} \to A^{sym} in L^{1} (Ω; S_{sym}^{N}), A_{k}^{skew} \to A^{skew} in L^{2 p} (Ω; S_{skew}^{N}), \\ y_{k} ⇀ y in L^{2} (Ω), \nabla y_{k} ⇀ \nabla y in variable space L^{2} {(Ω, A_{k}^{sym} d x)}^{N} . \end{array}$

We are now in a position to study the optimal control problems (5.7).

Theorem 5.10.

Let $f \in L^{4 p / (p + 1)} (Ω; R^{N})$ and $y_{d} \in L^{2} (Ω)$ be given distributions. Assume that the original OCP (1.1)–(1.3) has a nonempty set of admissible controls. Then OCPs (5.7) are regular for each $k \in N$ (i.e. the corresponding sets of admissible solutions $Ξ_{k}$ are nonempty), and for every $k \in N$ there exists a minimizer $(A_{k}^{0}, y_{k}^{0}) \in Ξ_{k}$ to the corresponding minimization problems (5.7) such that the sequence of pairs ${(A_{k}^{0}, y_{k}^{0}) \in Ξ_{k}}_{k \in N}$ is relatively compact with respect to the τ-convergence and each of its τ-cluster pairs $(\hat{A}, \hat{y})$ possesses the properties: $\begin{array}{l} (5.14) & (\hat{A}, \hat{y}) \in Ξ, \\ (5.15) & \int_{Ω} {(\nabla \hat{y}, {\hat{A}}^{sym} \nabla \hat{y})}_{R^{N}} d x + \int_{Ω} a_{0} {(\hat{y})}^{2} d x ⩽ \int_{Ω} {(f, \nabla \hat{y})}_{R^{N}} d x . \end{array}$

Proof.

Since $A_{ad} \neq \emptyset$ , it follows that $A_{ad}^{k} \neq \emptyset$ for every $k \in N$ , and $A_{ad}^{k} \subset L^{\infty} (Ω; M^{N})$ . Hence, for any admissible control $A_{k} = A_{k}^{sym} + A_{k}^{skew} \in A_{ad}^{k}$ , we can claim that $A_{k}^{skew} \in L^{\infty} (Ω; S_{skew}^{N})$ , $A_{k}^{sym} \in L^{\infty} (Ω; S_{sym}^{N})$ , and, therefore, the corresponding bilinear form $\begin{array}{l} {[y, φ]}_{A_{k}} & = \int_{Ω} {(\nabla φ, A_{k}^{skew} \nabla y)}_{R^{N}} d x \\ = \int_{Ω} {({(A_{k}^{sym})}^{1 / 2} \nabla φ, \underset{C_{k}}{\underset{︸}{[{(A_{k}^{sym})}^{- 1 / 2} A_{k}^{skew} {(A_{k}^{sym})}^{- 1 / 2}]}} {(A_{k}^{sym})}^{1 / 2} \nabla y)}_{R^{N}} d x \\ ⩽ ‖ C_{k} ‖_{L^{\infty} (Ω; S_{skew}^{N})} ‖ y ‖_{A_{k}^{sym}} ‖ φ ‖_{A_{k}^{sym}} is bounded on W_{A_{k}^{sym}} (Ω) \end{array}$ and satisfies the identity $\int_{Ω} {(\nabla φ, A_{k}^{skew} \nabla y)}_{R^{N}} d x = - \int_{Ω} {(\nabla y, A_{k}^{skew} \nabla φ)}_{R^{N}} d x$ . Therefore, $\begin{matrix} (5.16) & \int_{Ω} {(\nabla v, A_{k}^{skew} \nabla v)}_{R^{N}} d x = 0, \forall v \in W_{A_{k}^{sym}} (Ω) \end{matrix}$ and, hence, boundary value problem (5.8) has a unique solution $y_{k} \in W_{A_{k}^{sym}} (Ω)$ for each $A_{k} \in A_{ad}^{k} \subset L^{\infty} (Ω; M^{N})$ by the Lax–Milgram lemma. Thus, $Ξ_{k} \neq \emptyset$ for every $k \in N$ . It is worth to note that in view of the definition of the class of admissible controls $A_{ad}^{k}$ (see Lemma 5.4), the norms $‖ \cdot ‖_{H_{0}^{1} (Ω)}$ and $‖ \cdot ‖_{A_{k}^{sym}}$ are equivalent, therefore, we can identify $H_{0}^{1} (Ω)$ with the weighted Sobolev space $W_{A_{k}^{sym}} (Ω)$ .

As obvious consequence of this observation and the property of τ-lower semicontinuity of the cost functional I, we conclude that the corresponding minimization problem (5.7) admits at least one solution $(A_{k}^{0}, y_{k}^{0}) \in Ξ_{k}$ [18]. Moreover, having fixed a control $A_{k} \in A_{ad}^{k}$ , condition (5.16) implies the fulfilment of the following identities for every $k \in N$ $\begin{array}{l} (5.17) & \int_{Ω} [{(\nabla φ, A_{k} \nabla y_{k})}_{R^{N}} + a_{0} φ y_{k}] d x = \int_{Ω} {(f, \nabla φ)}_{R^{N}} d x, \forall φ \in C_{0}^{\infty} (Ω), \\ (5.18) & \int_{Ω} {(\nabla y_{k}, A_{k}^{sym} \nabla y_{k})}_{R^{N}} d x + \int_{Ω} a_{0} y_{k}^{2} d x = \int_{Ω} {(f, \nabla y_{k})}_{R^{N}} d x, \end{array}$ where $y_{k} = y_{k} (A_{k}, f) \in W_{A_{k}^{sym}} (Ω)$ are the corresponding solutions to the boundary value problems (5.8). Taking into account estimate (4.10), the equality (5.18) implies that $\begin{array}{l} ‖ y_{k} ‖_{A_{k}^{sym}}^{2} & : = \int_{Ω} [{(\nabla y_{k}, A_{k}^{sym} \nabla y_{k})}_{R^{N}} + y_{k}^{2}] d x \\ ⩽ \frac{1}{min {1, α}} \int_{Ω} [{(\nabla y_{k}, A_{k}^{sym} \nabla y_{k})}_{R^{N}} + a_{0} y_{k}^{2}] d x \\ ⩽ \frac{\sqrt{‖ ζ_{k}^{- 1} ‖_{L^{2 q} (Ω)}}}{min {1, α}} ‖ f ‖_{L^{4 p / (p + 1)} {(Ω; R)}^{N}} ‖ y_{k} ‖_{A_{k}^{sym}} . \end{array}$ Hence, $\begin{matrix} (5.19) & ‖ y_{k} ‖_{A_{k}^{sym}} ⩽ \frac{\sqrt{‖ ζ_{k}^{- 1} ‖_{L^{2 q} (Ω)}}}{min {1, α}} ‖ f ‖_{L^{4 p / p + 1} {(Ω; R)}^{N}} = C_{1} ‖ f ‖_{L^{4 p / p + 1} {(Ω; R)}^{N}}, \forall k \in N \end{matrix}$ and, therefore, the sequence ${y_{k}}_{k \in N}$ is bounded in variable space $W_{A_{k}^{sym}} (Ω)$ . As a result, we arrive at the relation $\begin{array}{l} I (A_{k}^{0}, y_{k}^{0}) & = inf_{(A, y) \in Ξ_{k}} I (A, y) ⩽ I (A_{k}, y_{k}) ⩽ 2 ‖ y_{d} ‖_{L^{2} (Ω)}^{2} + 2 ‖ y_{k} ‖_{A_{k}^{sym}}^{2} \\ (5.20) & ⩽ 2 ‖ y_{d} ‖_{L^{2} (Ω)}^{2} + C_{1}^{2} ‖ f ‖_{L^{4 p / p + 1} (Ω; R^{N})}^{2} ⩽ C, \forall k \in N . \end{array}$ Thus, the sequence of minimal values for the problems (5.7) is uniformly bounded, $\begin{matrix} (5.21) & sup_{k \in N} inf_{(u, y) \in Ξ_{k}} I (u, y) ⩽ C for some C > 0 . \end{matrix}$ Hence, ${sup}_{k \in N} ‖ y_{k}^{0} ‖_{A_{k}^{0, sym}}^{2} < + \infty$ .

In the meantime, due to the definition of the sets $A_{ad}^{k}$ , it is easy to see that the corresponding sequence of optimal controls ${A_{k}^{0}}_{k \in N}$ belongs to $A_{ad, 1}^{k} \oplus A_{ad, 2}^{k}$ . Hence, by Lemmas 5.5 and 5.7, we get: there exists a matrix $\hat{A}$ such that $\begin{array}{l} (5.22) & \hat{A} : = {\hat{A}}^{sym} + {\hat{A}}^{skew}, \\ (5.23) & A_{k}^{0, sym} \to {\hat{A}}^{sym} in L^{1} (Ω; S_{sym}^{N}), \\ (5.24) & A_{k}^{0, skew} \to {\hat{A}}^{skew} in L^{2 p} (Ω; S_{skew}^{N}) . \end{array}$ Therefore, taking into account Lemmas 5.6 and 5.8, we conclude: $\hat{A} \in A_{ad}$ .

Since ${sup}_{k \in N} ‖ y_{k}^{0} ‖_{A_{k}^{0, sym}}^{2} < + \infty$ , it follows by Lemma 4.6 that there exists an element $\hat{y} \in W_{{\hat{A}}^{sym}} (Ω)$ such that, up to a subsequence, we have $\begin{matrix} (5.25) & y_{k}^{0} ⇀ \hat{y} in L^{2} (Ω), \nabla y_{k}^{0} ⇀ \nabla \hat{y} in L^{2} {(Ω, A_{k}^{0, sym} d x)}^{N} . \end{matrix}$

As a result, summing up the above properties of the sequences ${y_{k}^{0}}_{k \in N}$ and ${A_{k}^{0}}_{k \in N}$ , we obtain $(A_{k}^{0}, y_{k}^{0}) \overset{τ}{\to} (\hat{A}, \hat{y})$ .

The next step is to show that $(\hat{A}, \hat{y}) \in Ξ$ . With that in mind, we pass to the limit in (5.17) with $A = A_{k}^{0}$ and $y = y_{k}^{0}$ as $k \to \infty$ using the properties (5.22)–(5.25). Having fixed a test function $φ \in C_{0}^{\infty} (Ω)$ , we get (see definition of the weak convergence in variable spaces) $\begin{array}{l} \int_{Ω} {(f, \nabla φ)}_{R^{N}} d x & = lim_{k \to \infty} \int_{Ω} [{(\nabla φ, A_{k}^{0} \nabla y_{k}^{0})}_{R^{N}} + a_{0} φ y_{k}^{0}] d x (by (5.23), (5.25)) \\ = lim_{k \to \infty} \int_{Ω} {(\nabla φ, A_{k}^{0, skew} \nabla y_{k}^{0})}_{R^{N}} d x + \int_{Ω} {(\nabla φ, {\hat{A}}^{sym} \nabla \hat{y})}_{R^{N}} d x + \int_{Ω} a_{0} φ \hat{y} d x . \end{array}$ Since ${lim}_{k \to \infty} \int_{Ω} {(\nabla φ, A_{k}^{0, skew} \nabla y_{k}^{0})}_{R^{N}} d x = {lim}_{k \to \infty} I_{1, k} + {lim}_{k \to \infty} I_{2, k}$ , where $\begin{array}{l} I_{1, k} = - \int_{Ω} {({(A_{k}^{0, sym})}^{- \frac{1}{2}} (A_{k}^{0, skew} - {\hat{A}}^{skew}) \nabla φ, {(A_{k}^{0, sym})}^{\frac{1}{2}} \nabla y_{k}^{0})}_{R^{N}} d x, \\ I_{2, k} = - \int_{Ω} {({(A_{k}^{0, sym})}^{- 1} {\hat{A}}^{skew} \nabla φ, A_{k}^{0, sym} \nabla y_{k}^{0})}_{R^{N}} d x, \end{array}$ it follows from (4.26)–(4.27) that ${lim}_{k \to \infty} I_{2, k} = \int_{Ω} {(\nabla φ, {\hat{A}}^{skew} \nabla \hat{y})}_{R^{N}} d x$ and ${lim}_{k \to \infty} I_{1, k} = 0$ . Thus, the τ-limit pair $(\hat{A}, \hat{y})$ is related by integral identity $\begin{matrix} \int_{Ω} [{(\nabla φ, \hat{A} \nabla \hat{y})}_{R^{N}} + a_{0} φ \hat{y}] d x = \int_{Ω} {(f, \nabla φ)}_{R^{N}} d x, \forall φ \in C_{0}^{\infty} (Ω), \end{matrix}$ and, hence, $\hat{y}$ is a weak solution to the boundary value problem (1.2)–(1.3) under $A = \hat{A}$ . Thus, $\hat{y} \in D (\hat{A})$ and, therefore, this pair is admissible for the original OCP (1.1)–(1.3), i.e. $(\hat{A}, \hat{y}) \in Ξ$ .

It remains to prove the energy inequality (5.15). To this end, we pass to the limit in the energy equality (5.18) using the lower semicontinuity of the norms $‖ \cdot ‖_{L^{2} (Ω)}$ and $‖ \cdot ‖_{L^{2} {(Ω; A_{k}^{0, sym} d x)}^{N}}$ with respect to the weak convergence (5.25), and the fact that ${(A_{k}^{0, sym})}^{- 1} f \to {({\hat{A}}^{sym})}^{- 1} f$ strongly in variable space $L^{2} {(Ω, A_{k}^{sym} d x)}^{N}$ as $k \to \infty$ (see Lemma 3.6). As a result, we have $\begin{array}{rcl} \int_{Ω} {(f, \nabla \hat{y})}_{R^{N}} d x & = & \int_{Ω} {({({\hat{A}}^{sym})}^{- 1} f, {\hat{A}}^{sym} \nabla \hat{y})}_{R^{N}} d x \\ \overset{by Definition 3.2}{=} & lim_{k \to \infty} \int_{Ω} {({(A_{k}^{0, sym})}^{- 1} f, A_{k}^{0, sym} \nabla y_{k}^{0})}_{R^{N}} d x \\ = & lim_{k \to \infty} \int_{Ω} {(f, \nabla y_{k}^{0})}_{R^{N}} d x \\ = & lim_{k \to \infty} \int_{Ω} {(\nabla y_{k}^{0}, A_{k}^{0, sym} \nabla y_{k}^{0})}_{R^{N}} d x + lim_{k \to \infty} \int_{Ω} a_{0} {(y_{k}^{0})}^{2} d x \\ (5.26) & \overset{by Proposition 3.4}{⩾} & \int_{Ω} {(\nabla \hat{y}, {\hat{A}}^{sym} \nabla \hat{y})}_{R^{N}} d x + \int_{Ω} a_{0} {\hat{y}}^{2} d x . \end{array}$ The proof is complete. □

Remark 5.2.

As energy inequality (5.15) indicates, if $(\hat{A}, \hat{y}) \in Ξ$ is a τ-cluster pair of the sequence ${(A_{k}^{0}, y_{k}^{0}) \in Ξ_{k}}_{k \in N}$ and $\hat{y} \in H_{{\hat{A}}_{sym}} (Ω)$ , then the direct comparison of (5.15) and (4.12) implies that ${[\hat{y}, \hat{y}]}_{\hat{A}} ⩾ 0$ .

As immediately follows from this theorem, Hypothesis A can be eliminated from Theorem 4.8.

Corollary 5.11.

If $A_{ad} \neq \emptyset$ , then the set of admissible solutions Ξ to OCP (1.1)–(1.3) is nonempty for every $f \in L^{4 p / (p + 1)} (Ω; R^{N})$ and $y_{d} \in L^{2} (Ω)$ .

Remark 5.3.

As follows from Theorem 5.10, for any positive compactly supported smooth function K satisfying conditions (5.3), optimal solutions to the regularized OCPs (5.7) always lead in the limit to some admissible (but not optimal in general) solution $(\hat{A}, \hat{y})$ of the original OCP (1.1)–(1.3). Moreover, in general, this limit pair depends on the choice of smoothing kernel K. It is reasonably to call such pair attainable. However, up to now the structure of the entire set of all attainable pairs remains unclear. For instance, it is unknown whether this set is convex and closed in Ξ. It is also unknown whether all optimal solutions to OCP (1.1)–(1.3) can be attainable in such way.

Taking these observations into account, we make use of the following notion.

Definition 5.12.

We say that a pair $(\hat{A}, \hat{y}) \in [L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N})] \times W_{{\hat{A}}_{sym}} (Ω)$ is a variational solution to OCP (1.1)–(1.3) if $\begin{matrix} (5.27) & I (\hat{A}, \hat{y}) = inf_{(A, y) \in Ξ} I (A, y), (\hat{A}, \hat{y}) \in Ξ, \end{matrix}$ and $(\hat{A}, \hat{y})$ is related by energy equality $\begin{matrix} (5.28) & \int_{Ω} {(\nabla \hat{y}, {\hat{A}}^{sym} \nabla \hat{y})}_{R^{N}} d x + \int_{Ω} a_{0} {(\hat{y})}^{2} d x = \int_{Ω} {(f, \nabla \hat{y})}_{R^{N}} d x . \end{matrix}$

As a consequence of Theorem 5.10 and properties of the variational limits of constrained minimization problems (see Theorem 2.5), we have the following result.

Proposition 5.13.

Let K be a smoothing kernel with properties (5.3). Assume that the sequence of minimization problems (5.7) defined by the rules (5.8) is such that $\begin{matrix} (5.29) & ⟨ inf_{(A, y) \in Ξ_{k}} I (A, y) ⟩ \overset{Var (τ)}{\underset{k \to \infty}{⟶}} ⟨ inf_{(A, y) \in Ξ} I (A, y) ⟩ (see Definition 2.4) . \end{matrix}$ Let ${(A_{k}^{0}, y_{k}^{0}) \in Ξ_{k}}_{k \in N}$ be a sequence of optimal solutions to the corresponding regularized OCPs. Then this sequence is relatively compact with respect to the τ-convergence and each its τ-cluster pair $(\hat{A}, \hat{y}) \in [L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N})] \times W_{{\hat{A}}_{sym}} (Ω)$ is a variational solution to OCP (1.1)–(1.3) in the sense of Definition 5.12. Moreover, up to a subsequence, we have $\begin{matrix} (5.30) & y_{k}^{0} \to \hat{y} in L^{2} (Ω) and \nabla y_{k}^{0} \to \nabla \hat{y} in L^{2} {(Ω, A_{k}^{0, sym} d x)}^{N} as k \to \infty . \end{matrix}$

Proof.

Indeed, the τ-compactness of the sequence ${(A_{k}^{0}, y_{k}^{0}) \in Ξ_{k}}_{k \in N}$ is a direct consequence of a priori estimate (5.19), Lemma 4.6, and properties (5.22)–(5.24). In order to prove the strong convergence (5.30), we make use of the main properties of the variational convergence. Following Theorems 2.5, 5.10, and 4.8 (see also Corollary 5.11), we can claim that OCP (1.1)–(1.3) is solvable and there exists an optimal pair $(\hat{A}, \hat{y}) \in Ξ$ to this problem such that $\begin{array}{l} inf_{(A, y) \in Ξ} I (A, y) & = I (\hat{A}, \hat{y}) \\ : = ‖ \hat{y} - y_{d} ‖_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla \hat{y}, {\hat{A}}^{sym} \nabla \hat{y})}_{R^{N}} d x \\ = lim_{k \to \infty} inf_{(A_{k}, y_{k}) \in Ξ_{k}} I (A_{k}, y_{k}) = lim_{k \to \infty} I (A_{k}^{0}, y_{k}^{0}) \\ (5.31) & = lim_{k \to \infty} [{‖ y_{k}^{0} - y_{d} ‖}_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla y_{k}^{0}, A_{k}^{0, sym} \nabla y_{k}^{0})}_{R^{N}} d x] . \end{array}$ However, because of the lower semicontinuity of $‖ \cdot ‖_{L^{2} (Ω)}$ and $‖ \cdot ‖_{L^{2} {(Ω, A_{k}^{0, sym} d x)}^{N}}$ with respect to the weak convergence, the convergence $(A_{k}^{0}, y_{k}^{0}) \overset{τ}{\to} (\hat{A}, \hat{y})$ implies that $\begin{array}{rcl} inf_{(A, y) \in Ξ} I (A, y) & \overset{by (5.31)}{=} & lim_{k \to \infty} [{‖ y_{k}^{0} - y_{d} ‖}_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla y_{k}^{0}, A_{k}^{0, sym} \nabla y_{k}^{0})}_{R^{N}} d x] \\ ⩾ & ‖ \hat{y} - y_{d} ‖_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla \hat{y}, {\hat{A}}^{sym} \nabla \hat{y})}_{R^{N}} d x . \end{array}$ Since the pair $(\hat{A}, \hat{y})$ is admissible for the problem (1.1)–(1.3) (see Theorem 5.10), it follows that $(\hat{A}, \hat{y})$ is an optimal pair. Therefore, in view of (5.31), it gives $\begin{array}{l} inf_{(A, y) \in Ξ} I (A, y) & = I (\hat{A}, \hat{y}) \\ : = ‖ \hat{y} - y_{d} ‖_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla \hat{y}, {\hat{A}}^{sym} \nabla \hat{y})}_{R^{N}} d x \\ = lim_{k \to \infty} inf_{(A_{k}, y_{k}) \in Ξ_{k}} I (A_{k}, y_{k}) = lim_{k \to \infty} I (A_{k}^{0}, y_{k}^{0}) \\ (5.32) & = lim_{k \to \infty} [{‖ y_{k}^{0} - y_{d} ‖}_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla y_{k}^{0}, A_{k}^{0, sym} \nabla y_{k}^{0})}_{R^{N}} d x] . \end{array}$ Hence, the validity of (5.30) is a direct consequence of properties (5.31)–(5.32) and Proposition 3.5. It remains to prove the energy equality (5.28). To this end, it is enough to note that each of the pair $(A_{k}^{0}, y_{k}^{0})$ is related by energy equality (5.18). As a result, passing to the limit in (5.18) as $k \to \infty$ , we finally have $\begin{array}{rcl} 0 & \overset{by (5.16)}{=} & lim_{k \to \infty} {[y_{k}^{0}, y_{k}^{0}]}_{A_{k}^{0}} \\ \overset{by (5.18)}{=} & - lim_{k \to \infty} \int_{Ω} {(\nabla y_{k}^{0}, A_{k}^{0, sym} \nabla y_{k}^{0})}_{R^{N}} d x \\ - lim_{k \to \infty} \int_{Ω} a_{0} {(y_{k}^{0})}^{2} d x + lim_{k \to \infty} \int_{Ω} {(f, \nabla y_{k}^{0})}_{R^{N}} d x \\ \overset{by (5.30) and (5.32)}{=} & - \int_{Ω} {(\nabla \hat{y}, {\hat{A}}^{sym} \nabla \hat{y})}_{R^{N}} d x - \int_{Ω} a_{0} {(\hat{y})}^{2} d x + \int_{Ω} {(f, \nabla \hat{y})}_{R^{N}} d x . \end{array}$ □

Remark 5.4.

As follows from Proposition 5.13 and Theorem 5.10, even if the OCP (1.1)–(1.3) has a unique solution $(A^{0}, y^{0})$ , it does not ensure that this pair is the variational solution to the above problem. The matter is that the existence at least one the smoothing kernel K such that the approximated OCPs (5.7)–(5.8) would lead to the pair $(A^{0}, y^{0})$ in the sense of conditions (2.23)–(2.24) is an open problem. In other words, the existence of $(Γ, δ)$ -realizing sequence for the pair $(A^{0}, y^{0}) \in Ξ$ (see Definition 2.4) is not established.

We are now in a position to discuss the existence of variational solutions to the OCP (1.1)–(1.3).

Theorem 5.14.

Assume that

the function $ζ_{ad}$ in definition of the set $M_{ζ_{ad}}^{β} (Ω)$ satisfies conditions $ζ_{ad}^{- 1} \in L^{2 q} (Ω)$ with $q = p / (p - 1)$ , where p is defined by (2.12) ;

condition (4.18) holds true for some constant $C > 0$ ;

for every admissible control $A \in A_{ad} \subset L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N})$ , we have $\begin{matrix} (5.33) & {[y, y]}_{A} = 0, \forall y \in D (H_{A_{sym}}) . \end{matrix}$

Then the OCP (1.1)–(1.3) has variational solutions for every

f \in L^{4 p / (p + 1)} (Ω; R^{N})

and

y_{d} \in L^{2} (Ω)

Proof.

To begin with, we note that as follows from Theorem 4.7, condition (4.18) guarantees the fulfilment of equality $H_{A_{sym}} (Ω) = W_{A_{sym}} (Ω)$ for every admissible control $A = A_{sym} + A_{skew} \in A_{ad}$ . Moreover, in this case, every weak solution to the boundary value problem (1.2)–(1.3) satisfies the energy equality (4.12). Let K be an arbitrary positive compactly supported smooth function satisfying conditions (5.3). We associate with this function the sequence of constrained minimization problems (5.7), where the set $Ξ_{k}$ is defined by (5.8).

Let ${(A_{k}, y_{k})}_{k \in N}$ be a sequence in $[L^{1} (Ω; S_{sym}^{N}) \oplus L^{2 p} (Ω; S_{skew}^{N})] \times W_{A_{k}^{sym}} (Ω)$ with the following properties:

$(A_{k}, y_{k}) \in Ξ_{n_{k}}$ for every $k \in N$ , where ${n_{k}}_{k \in N}$ is a subsequence converging to ∞ as k tends to ∞;

$(A_{k}, y_{k}) \overset{τ}{\to} (A, y)$ in the sense of Definition 5.9.

Then proceeding as in the proof of Theorem 5.10, it can be shown that the limit pair

(A, y)

is admissible to the original OCP (1.1)–(1.3). Hence, this problem is regular and, therefore, it is solvable by Theorem 4.8. Our aim is to show that this problem can be interpreted as the variational limit of the sequence of constrained minimization problems (5.7). To do so, we have to verify the fulfilment of all conditions of Definition 2.4.

As for the property (d), it immediately follows from the following relation $\begin{array}{rcl} \underset{k \to \infty}{lim inf} I (A_{k}, y_{k}) & = & \underset{k \to \infty}{lim inf} [‖ y_{k} - y_{d} ‖_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla y_{k}, A_{k}^{sym} \nabla y_{k})}_{R^{N}} d x] \\ \overset{by (3.4)}{⩾} & ‖ y - y_{d} ‖_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla y, A^{sym} \nabla y)}_{R^{N}} d x \\ = & I (A, y), \end{array}$ which holds true for any sequence ${(A_{k}, y_{k}) \in A_{ad} \times W_{A_{k}^{sym}} (Ω)}_{k \in N}$ with properties (a)–(aa).

We focus now on the verification of condition (dd) of Definition 2.4. Let $(A^{♯}, y^{♯})$ be an arbitrary admissible pair to the original problem. Since $A^{♯} \in A_{ad}$ , it follows from Lemmas 5.6 and 5.8 that the sequence of smoothed matrices $\begin{matrix} {A_{k}^{♯} : = {(A^{♯})}_{k} = \int_{R^{N}} K (z) {\tilde{A}}^{♯} (x + k^{- 1} z) d z}_{k \in N} \end{matrix}$ such that $A_{k}^{♯} \in A_{ad}^{k}$ for all $k \in N$ , and $A_{k}^{♯} \to A^{♯} : = A^{♯, sym} + A^{♯, skew}$ as $k \to \infty$ in the sense of Definition 5.9, i.e. $\begin{array}{l} (5.34) & A_{k}^{♯, sym} \to A^{♯, sym} in L^{1} (Ω; S_{sym}^{N}), \\ (5.35) & A_{k}^{♯, skew} \to A^{♯, skew} in L^{2 p} (Ω; S_{skew}^{N}) . \end{array}$ Let ${y_{k} = y (A_{k}^{♯}, f)}_{k \in N}$ be the corresponding solutions to the regularized boundary value problems (5.7). Then having applied the arguments of the proof of Theorem 5.10, it can be shown that the sequence ${y_{k}}_{k \in N}$ is uniformly bounded in variable Sobolev space $W_{A_{k}^{♯, sym}} (Ω)$ and there exists an element $\hat{y} \in W_{A^{♯, sym}} (Ω)$ such that $\hat{y} \in D (W_{A^{♯, sym}})$ , $(A^{♯}, \hat{y}) \in Ξ$ , and, up to a subsequence, $\begin{matrix} (5.36) & y_{k} ⇀ \hat{y} in L^{2} (Ω), \nabla y_{k} ⇀ \nabla \hat{y} in L^{2} {(Ω, A_{k}^{♯, sym} d x)}^{N} . \end{matrix}$ Our aim is to show that $\hat{y} = y^{♯}$ and the following identity $\begin{matrix} (5.37) & I (A^{♯}, y^{♯}) = \underset{k \to \infty}{lim sup} I (A_{k}^{♯}, y_{k}) \end{matrix}$ holds true.

Indeed, since $(A^{♯}, y^{♯}) \in Ξ$ and $(A^{♯}, \hat{y}) \in Ξ$ , it follows that $y^{*} = y^{♯} - \hat{y}$ is a solution of the homogeneous problem $\begin{matrix} (5.38) & - div (A^{♯} \nabla y^{*}) + a_{0} y^{*} = 0 in Ω, y^{*} = 0 on \partial Ω . \end{matrix}$ Following the initial assumptions, we have $W_{A^{sym}} (Ω) = H_{A^{sym}} (Ω)$ and ${[y, y]}_{A} = 0$ , $\forall y \in D (W_{A^{sym}})$ and for each matrix $A \in A_{ad}$ . Hence, $\begin{matrix} 0 \overset{by (5.33)}{=} - {[y^{*}, y^{*}]}_{A^{♯}} \overset{by (4.12)}{=} \int_{Ω} [{(\nabla y^{*}, A^{♯, sym} \nabla y^{*})}_{R^{N}} + a_{0} {(y^{*})}^{2}] d x \end{matrix}$ and, therefore, problem (5.38) has the trivial solution only. Thus, $y^{♯} = \hat{y}$ .

To prove the equality (5.37), we use of the idea of D. Cioranescu and F. Murat (see [5]). In view of the initial assumptions and Remark 2.1, the embedding $H_{A^{♯, sym}} (Ω) ↪ L^{2} (Ω)$ is compact. Taking into account this fact, the properties (5.34)–(5.35), and the energy equalities (5.18) and (4.12), we get $\begin{array}{rcl} lim_{k \to \infty} I (A_{k}^{♯}, y_{k}) & = & lim_{k \to \infty} [‖ y_{k} - y_{d} ‖_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla y_{k}, A_{k}^{♯, sym} \nabla y_{k})}_{R^{N}} d x] \\ = & {‖ y^{♯} - y_{d} ‖}_{L^{2} (Ω)}^{2} + lim_{k \to \infty} \int_{Ω} {(\nabla y_{k}, A_{k}^{♯, sym} \nabla y_{k})}_{R^{N}} d x \\ \overset{by (5.18)}{=} & {‖ y^{♯} - y_{d} ‖}_{L^{2} (Ω)}^{2} + lim_{k \to \infty} [- \int_{Ω} a_{0} y_{k}^{2} d x + \int_{Ω} {(f, \nabla y_{k})}_{R^{N}} d x] \\ \overset{by (5.36) and (5.26)}{=} & {‖ y^{♯} - y_{d} ‖}_{L^{2} (Ω)}^{2} + \int_{Ω} {(f, \nabla y^{♯})}_{R^{N}} d x - \int_{Ω} a_{0} {(y^{♯})}^{2} d x \\ \overset{by (4.12)}{=} & {‖ y^{♯} - y_{d} ‖}_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla y^{♯}, A^{♯, sym} \nabla y^{♯})}_{R^{N}} d x + {[y^{♯}, y^{♯}]}_{A^{♯}} \\ \overset{by (5.33)}{=} & I (A^{♯}, y^{♯}) . \end{array}$ This concludes the proof. □

Our next observation shows that variational solutions do not exhaust the entire set of all possible solutions to the original OCP (1.1)–(1.3). With that in mind, we adopt the following concept.

Definition 5.15.

We say that a pair $(A_{0}, y_{0}) \in Ξ$ is a non-variational solution to OCP (1.1)–(1.3) if $\begin{array}{l} (5.39) & I (A_{0}, y_{0}) = inf_{(A, y) \in Ξ} I (A, y), (A_{0}, y_{0}) \in Ξ, and \\ (5.40) & \int_{Ω} {(\nabla y_{0}, A_{0}^{sym} \nabla y_{0})}_{R^{N}} d x + \int_{Ω} a_{0} y_{0}^{2} d x \neq \int_{Ω} {(f, \nabla y_{0})}_{R^{N}} d x . \end{array}$

Lemma 5.16.

Assume that there exists a matrix $A_{0} \in A_{ad}$ and an element $v \in D (H_{A_{0}^{sym}}) \subset H_{A_{0}^{sym}} (Ω)$ with property ${[v, v]}_{A_{0}} \neq 0$ . Then there are distributions $f \in D^{'} (Ω; R^{N})$ and $y_{d} \in L^{2} (Ω)$ such that the optimal control problem $\begin{array}{l} (5.41) & Minimize I (A, y) = ‖ y - y_{d} ‖_{L^{2} (Ω)}^{2} + \int_{Ω} {(\nabla y - \nabla y_{d}, A^{sym} (\nabla y - \nabla y_{d}))}_{R^{N}} d x \\ (5.42) & subject to the constraints (1.2) – (1.3) and A \in A_{ad} \subset L^{1} (Ω; M^{N}) \end{array}$ has a non-variational solution in the sense of Definition 5.15 .

Proof.

To begin with we note that for a given vector-valued function $f \in L^{2} (Ω; R^{N})$ the distribution $- div f$ can be always represented as follows $- div f = - div \hat{f} + a_{0} v$ for some $\hat{f} \in L^{2} (Ω; R^{N})$ . Taking this observation into account, we consider the OCP (5.41)–(5.42) with the right-hand side $- div f = - div \hat{f} + g$ , where $\begin{matrix} (5.43) & y_{d} = v, \hat{f} = A_{0} \nabla v, and g = a_{0} v . \end{matrix}$ Since $v \in D (H_{A_{0}^{sym}})$ , it follows that $y_{d} \in L^{2} (Ω)$ , $\nabla v \in L^{2} {(Ω, A_{0}^{sym} d x)}^{N}$ , and, therefore, ${(A_{0}^{sym})}^{- 1 / 2} \hat{f} \in L^{2} {(Ω)}^{N}$ . Indeed, as follows from (5.43)₂, we have ${(A_{0}^{sym})}^{- 1 / 2} \hat{f} = {\hat{f}}_{1} + {\hat{f}}_{2}$ , where ${\hat{f}}_{1} = {(A_{0}^{sym})}^{1 / 2} \nabla v$ and ${\hat{f}}_{2} = {(A_{0}^{sym})}^{- 1 / 2} A_{0}^{skew} \nabla v$ . Then $\begin{array}{l} \int_{Ω} | {\hat{f}}_{1} |^{2} d x = \int_{Ω} {(\nabla v, A_{0}^{sym} \nabla v)}_{R^{N}} d x \overset{v \in H_{A_{0}^{sym}} (Ω)}{<} + \infty, \\ \int_{Ω} | {\hat{f}}_{2} |^{2} d x = \int_{Ω} | \underset{C (x)}{\underset{︸}{{(A_{0}^{sym})}^{- 1 / 2} A_{0}^{skew} {(A_{0}^{sym})}^{- 1 / 2}}} {(A_{0}^{sym})}^{1 / 2} \nabla v | d x \\ ⩽ ‖ C ‖_{L^{2} (Ω; S_{skew}^{N})} ‖ v ‖_{L^{2} {(Ω, A_{0}^{sym} d x)}^{N}} \overset{by (2.15)}{<} + \infty . \end{array}$ Hence, ${(A_{0}^{sym})}^{- 1 / 2} \hat{f} \in L^{2} {(Ω)}^{N}$ and by Corollary 4.5 we conclude that $y_{d}$ is a weak solution to the boundary value problem (1.2)–(1.3) under $A = A_{0}$ . Since $v \in D (H_{A_{0}^{sym}}) \subset H_{A_{0}^{sym}} (Ω)$ and $- div f = - div \hat{f} + g$ , it follows that (see Remark 4.2) the distribution $y_{d}$ satisfies the energy equality $\begin{matrix} (5.44) & \int_{Ω} {(\nabla y_{d}, A_{0}^{sym} \nabla y_{d})}_{R^{N}} d x + {[y_{d}, y_{d}]}_{A_{0}} = \int_{Ω} {(\hat{f}, \nabla y_{d})}_{R^{N}} d x . \end{matrix}$ Moreover, using the fact that $I (A_{0}, y_{d}) = 0$ , we can conclude: $(A_{0}, y_{d})$ is the unique optimal pair to the above OCP. It remains to observe that in view of condition ${[y_{d}, y_{d}]}_{A_{0}} : = {[v, v]}_{A_{0}} \neq 0$ , the energy equality (5.44) leads to the strict inequality $\begin{matrix} \int_{Ω} {(\nabla y_{d}, A_{0}^{sym} \nabla y_{d})}_{R^{N}} d x \neq \int_{Ω} {(\hat{f}, \nabla y_{d})}_{R^{N}} d x \end{matrix}$ and, hence, $(A_{0}, y_{d})$ is a non-variational solution to the above problem. The proof is complete. □

Remark 5.5.

As follows from Theorem 5.10, if $(A_{0}, y_{0}) \in A_{ad} \times H_{A_{0}^{sym}} (Ω)$ is a non-variational solution such that ${[y_{0}, y_{0}]}_{A_{0}} < 0$ , then this solutions can not be attainable through the limit of optimal solutions to the regularized problems (5.7)–(5.8).

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