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

We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain \begin{document}$Ω$ \end{document} . We take the coefficient \begin{document}$u(x)∈ L^∞(Ω)\cap BV(Ω)$ \end{document} in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix \begin{document}$A_{skew}∈ L^q(Ω;\mathbb{S}^N_{skew})$ \end{document} . We show that, in spite of unboundedness of the non-linear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable. At the same time, optimal solutions to such problem can inherit a singular character of the matrices \begin{document}$A^{skew}$ \end{document} . We indicate two types of optimal solutions to the above problem and show that one of them can be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP.


(Communicated by Suzanne Lenhart)
Abstract. We consider an optimal control problem associated to Dirichlet boundary value problem for non-linear elliptic equation on a bounded domain Ω. We take the coefficient u(x) ∈ L ∞ (Ω) ∩ BV (Ω) in the main part of the non-linear differential operator as a control and in the linear part of differential operator we consider coefficients to be unbounded skew-symmetric matrix A skew ∈ L q (Ω; S N skew ). We show that, in spite of unboundedness of the nonlinear differential operator, the considered Dirichlet problem admits at least one weak solution and the corresponding OCP is well-possed and solvable. 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 be attained by optimal solutions of regularized problems for coercive elliptic equations with bounded coefficients, using the two-parametric regularization of the initial OCP.
1. Introduction. Optimal control in coefficients for partial differential equations is a classical subject initiated by Lurie [25], Lions [23,24], and Pironneau [29]. 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 with unbounded and degenerate controls in coefficients, see [2,8,12,20,21]. The constrained optimal control problem in the coefficients of the principle part of elliptic operator was first 1364 OLHA P. KUPENKO AND ROSANNA MANZO discussed in detail by Casas [3] in case of the classical Laplace equation, where the scalar coefficient u in the div(u∇·) formulation was taken as control. The problem of existence and uniqueness of the underlying boundary value problem and the corresponding optimal control problem was treated and an optimality system has been derived and analyzed. Analogous results for the case of of the weighted p-Laplacian div(u|∇y| p−2 ∇y) were recently obtained by Casas, Kogut, and Leugering in [5]. However, the principle questions related with the study of optimal control problems in coefficients for the general case of quasilinear elliptic operator div(a(u, ∇·) remain open.
In this article we treat the case of the perturbed p-Laplacian div(u|∇y| p−2 ∇y + A skew ∇y) with unbounded matrix of coefficients A skew . Namely, we study the following optimal control problem (OCP) for a nonlinear elliptic equation containing generalized p-Laplacian with unbounded coefficients in the linear part of the elliptic operator: subject to constrains −div u|∇y| p−2 ∇y − div(A skew ∇y) = −divf in Ω, (2) u ∈ A ad ⊂ L ∞ (Ω) ∩ BV (Ω), y ∈ W 1,p 0 (Ω), where p ≥ 2, A ad is a class of admissible controls, y d ∈ L 2 (Ω) and f ∈ L 2 (Ω; R N ) are given distributions, A skew ∈ L q (Ω; R N ×N ) is a given skew-symmetric matrix, q = p/(p − 1). It is worth to emphasize that the first term in (1) is of the tracking-type, whereas the second term in (1) refers to the energy of a system, described by the given boundary value problem (for the details, we refer to the book [19]). So, the physical motivation of the cost functional is as follows: the state of the system is being driven as close to a target function y d as possible, while the energy term should be minimal as well.
The characteristic feature of such optimal control problem is the unboundedness of skew-symmetric matrix A skew ∈ L q (Ω; R N ×N ). As it was indicated in [12,13,14,17,37], this circumstance can lead to the existence of elements y ∈ W p 0 (Ω) such that y ∈ L ∞ (Ω), Ω ∇ϕ n , A skew ∇ϕ n R N dx = 0 ∀ n ∈ N, and lim n→∞ Ω ∇ϕ n , A skew ∇y where C ∞ 0 (Ω) ϕ n → y strongly in W p 0 (Ω). As a result, the existence, uniqueness, and variational properties of the weak solution to the Dirichlet problem (2)- (3) usually are drastically different from the corresponding properties of solutions to the elliptic equations with coercive L ∞ -matrices in coefficients (we refer to [6,30,31,32,34] for the details and other results in this field). However, it is worth to emphasize that under some special restrictions on the skew-symmetric matrix A skew that do not eliminate its local unboundedness property, the corresponding Dirichlet boundary value problem (2)-(3) can admit a unique solution for each u ∈ A ad (see, for instance, [7,26,28]).
The aim of this work is to study the existence of optimal controls to the problem (1)-(3) under rather general assumptions on matrix A skew ∈ L q (Ω; R N ×N ) and propose the scheme of their approximation. Using the direct method in the Calculus of variations, we show in Section 3 that the original OCP admits at least one solution (u 0 , y 0 ) ∈ L ∞ (Ω) × W 1,p 0 (Ω) such that y 0 is a weak solution in the sense of Minty to the corresponding Dirichlet problem (2)- (3). However, in this case it is unknown whether the optimal state y 0 is a weak solution to BVP (2)- (3) in the sense of distributions. Besides, an important aspect in the study of any optimization problem is deriving of the corresponding optimality conditions. In the case of OCP (1)- (3), one faces up a number of difficulties on this way concerned with degeneracies the weighted p-Laplacian −div u|∇y| p−2 ∇y + A skew ∇y as ∇y tends to zero and also if u approaches zero. Moreover, when the term u|∇y| p−2 I + A skew is regarded as the coefficient of the Laplace operator, we also have the case of unbounded coefficients. In order to avoid degeneracy and singularity in this case, we introduce in Section 4 the regularization of the differential operator in the left-hand side of equation (2) (see [5] for comparison) that leads to a sequence of monotone and bounded operators skew is an L ∞ -approximations of the unbounded skew-symmetric matrix A skew . As a result, we deal with a two-parameter family of optimal control problems in the coefficients for elliptic Dirichlet boundary value problems with coercive and strictly monotone operators. We show that such OCPs with regularized p-Laplacian are well-posed and have nonempty set of solutions in the classical space L ∞ (Ω) × H 1 0 (Ω) for each k ∈ N and ε > 0. The main question we study in Section 5 is about the asymptotic behaviour of optimal solutions to the approximate OCPs as k → ∞ and ε → 0. We show that any sequence of optimal pairs u 0 εn,kn , y 0 εn,kn n∈N to the approximate OCPs is compact with respect to the product of weak- * topology of BV (Ω) and the weak topology of H 1 0 (Ω). Moreover, all its cluster pairs belong to (L ∞ (Ω) ∩ BV (Ω)) × W 1,p 0 (Ω). As a main result of Section 5, we propose the sufficient conditions under which some optimal solutions of the original optimal control problem (1)-(3) can be attained through optimal solutions to regularized problems. In the last section (Section 6) we focus on the deriving of the first order optimality conditions to regularized OCPs.
2. Notation and preliminaries. Throughout the paper Ω is a bounded open subset of R N , N ≥ 2, for which Poincaré's inequality holds. For real numbers 2 ≤ p < +∞ and 1 < q < +∞ such that 1/p + 1/q = 1, let W 1,p 0 (Ω) be the closure of C ∞ 0 (Ω) in the Sobolev space W 1,p (Ω) and W 1,p 0 (Ω) is endowed with the norm . Let M N be the set of all N × N real matrices. We denote by S N skew the set of all skew-symmetric N × N -matrices C = [c ij ] N i,j=1 and by S N sym the set of all N × N symmetric matrices. Thus, if C ∈ S N skew then c ij = −c ji and, hence, c ii = 0. By matrix norm in M N (and for functions with values in S N skew as well) we mean a submultiplicative norm A := sup |Aξ| R N : ξ ∈ R N with |ξ| R N = 1 . It is worth to note that, in the case of Euclidean norm | · | R N , the norm A can be computed as the spectral norm A = λ max (A t A), where λ max (A t A) is the largest eigenvalue of the positive-semidefinite matrix A t A. Let L q Ω; S N skew be the normed space of measurable integrable with power q functions whose values are skew-symmetric matrices.
Let χ E be the characteristic function of a set E ⊂ R N and let |E| be its Ndimensional Lebesgue measure. For any vector field v ∈ L q (Ω; R N ), its divergence is an element of the space W −1, q (Ω) defined by the formula Here ·, · W −1,q (Ω);W 1,p 0 (Ω) denotes the duality pairing between spaces W −1,q (Ω) and W 1,p 0 (Ω), and (·, ·) R N denotes the scalar product of two vectors in R N . Functions with bounded variations. Let f : Ω → R be a function of L 1 (Ω). Define ∂ϕi ∂xi . According to the Radon-Nikodym theorem, if T V (f ) < +∞ then the distribution Df is a measure and there exist a vector-valued function ∇f ∈ L 1 (Ω; R N ) and a measure D s f , singular with respect to the N -dimensional Lebesgue measure L N Ω restricted to Ω, such that Df = ∇f L N Ω + D s f.
Under the norm f BV (Ω) = f L 1 (Ω) + T V (f ), BV (Ω) is a Banach space. For our further analysis, we need the following properties of BV -functions (see [10]): k=1 be a sequence in BV (Ω) strongly converging to some f in L 1 (Ω) and satisfying condition sup k∈N T V (f k ) < +∞. Then there exists a subsequence, still denoted by f k , and a function f ∈ BV (Ω) such that f k → f in L 1 (Ω).
Admissible Controls and Generalized p-Laplacian. Let α, β, γ, and m be given positive constants such that 0 < α ≤ β < +∞ and α|Ω| ≤ m ≤ β|Ω|. We define the class of admissible controls A ad as follows Remark 2.1. Typically the controls in the coefficients of the principal part of non-linear elliptic operator reflect the physical properties of materials or systems such as conductivity, elasticity, and etc. From this point of view it is reasonable to suppose that such coefficients are essentially bounded and strictly positive functions. So, L ∞ (Ω) arguably looks as the natural functional space for the controls in this case. However, the subtle point here is the choice of the appropriate topology with respect to which each minimizing sequence is convergent. Moreover, since the solvability of the considered optimal control problem strictly depends on properties of the mapping L ∞ (Ω) u → y(u) ∈ W 1,p 0 (Ω) (this mapping must be at least continuous), it follows from our further analysis that it would be reasonable to consider the mapping u → y(u) with respect to the strong topology of L p (Ω) for the controls and the weak topology of W 1,p 0 (Ω) for the corresponding states. This is the main reason why we choose the control set to be bounded in the space of BV -functions. In this case it can be shown (see Proposition 3.2 below) that the set of admissible controls A ad given by the rule (5) is a nonempty convex and compact subset of L p (Ω) with an empty topological interior. Moreover, as it can be seen further, in this case the the corresponding optimal control problem admits a nonempty set of solutions.
The optimal control problem we consider in this paper is to minimize the cost functional (1), where y d ∈ L 2 (Ω) and y ∈ W 1,p 0 (Ω) is a weak solution to the boundary value problem (2) by choosing an appropriate function u ∈ A ad as control. Here by ∆ p : A ad × W 1,p 0 (Ω) → W −1,q (Ω) we denote the generalized p-Laplacian which can be defined by the rule , or via the paring It is easy to see that for every admissible control u ∈ A ad , the operator ∆ p (u, ·) turns out to be strictly monotone, coercive, and semi-continuous, where the above mentioned properties for the operator A : X → X * acting in a Banach space X have the following meaning (see [11,23,33]): Ay, y X * ;X y X → +∞ provided y X→∞ ; R t → A(y + tv), w X * ;X is continuous ∀ y, v, w ∈ X, respectively. In what follows, we associate with A skew ∈ L q (Ω; S N skew ) the bilinear skew-symmetric form It is easy to see, that the form Φ(y, v) is unbounded on W 1,p 0 (Ω), since there is no reason to assume that ∇v, A skew ∇y R N ∈ L 1 (Ω) for all y, v ∈ W 1,p 0 (Ω), in general. However, if we temporary assume that A skew ∈ L ∞ (Ω; S N skew ), then the bilinear form Φ(·, ·) becomes bounded for each y, v ∈ W 1,p 0 (Ω). In order to deal with the case A skew ∈ L ∞ (Ω; S N skew ), we notice that if A skew ∈ L q Ω; S N skew then the integral Ω ∇ϕ, A skew ∇y R N dx is well defined for every y ∈ W 1,p 0 (Ω) and ϕ ∈ C ∞ 0 (Ω). Indeed, Let I k : U k × Y k → 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) ∈ Ξ k } be a parameterized OCP, where is a set of all feasible pairs linked by some state equation. In what follows we make a difference between the notations inf (u,y)∈ Ξ k I k (u, y) and inf (u,y)∈ Ξ k I k (u, y) . The first one means the infimum m = inf {I k (u, y) : (u, y) ∈ Ξ k } of I k over the set Ξ k , whereas by the second one, we mean the constrained minimization problem as an object being defined by the triplet (I k , Ξ k , U k ×Y k ). Hereinafter we always associate to such OCP the corresponding constrained minimization problem: Since each of constrained minimization problems (11) lives in variable spaces U k × Y k , we assume that there exists a Banach space U × Y with respect to which a convergence in the scale of spaces {U k × Y k } k∈N is well defined (for the details, we refer to [19,36]). In the sequel, we use the following notation for this convergence (u k , y k ) τ −→ (u, y) in U k × Y k . Moreover, we assume that every bounded sequence in variable space U k ×Y k is sequentially compact with respect to the τ -convergence.
In order to study the asymptotic behaviour of a family of (CMP k ), the passage to the limit in (11) as the parameter k tends to +∞ 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)∈Ξ I(u, y) could be interpreted as some OCP.
Following the scheme of the direct variational convergence [19], we adopt the following definition for the convergence of minimization problems in variable spaces. Definition 2.2. A problem inf (u,y)∈Ξ I(u, y) is the variational τ -limit of (11) as k → ∞, if and only if the following conditions are satisfied: (d) If sequences {k n } n∈N and {(u n , y n )} n∈N are such that k n → ∞ as n → ∞, (u n , y n ) ∈ Ξ kn ∀ n ∈ N, and (u n , y n ) τ −→ (u, y) in U kn × Y kn , then (u, y) ∈ Ξ; I(u, y) ≤ lim inf n→∞ I kn (u n , y n ); (dd) For every (u, y) ∈ Ξ ⊂ U × Y, there are an integer k 0 > 0 and a sequence {(u k , y k )} k∈N (called a Γ-realizing sequence) such that Then the following result takes place [19].
is the variational τ -limit of (11) in the sense of Definition 2.2 and this problem has a nonempty set of solutions For every k ∈ N, let (u 0 k , y 0 k ) ∈ Ξ k be a minimizer of I k on the corresponding set Ξ k . If the sequence {(u 0 k , y 0 k )} k∈N is relatively compact with respect to the τ -convergence in variable spaces U k × Y k , then there exists a pair (u 0 , y 0 ) ∈ Ξ opt 0 such that (up to a subsequence) inf (u,y)∈ Ξ0 (17) 3. Setting of the optimal control problem. Let f : Ω → R N be a vectorvalued function such that f ∈ L 2 (Ω; R N ). The optimal control problem (1)-(3) we consider in this paper is to minimize the discrepancy (tracking error) between a given distribution y d ∈ L 2 (Ω) and a solution y of the Dirichlet boundary value problem (2)-(3) by choosing an appropriate control function u(x) ∈ A ad .
, and f ∈ L 2 (Ω; R N ) be given distributions. We say that a function y = y(u, A skew , f ) is a weak solution in the Minty sense (weak Minty solution) to boundary value problem (2),(3) if y ∈ W 1,p 0 (Ω) and the inequality (18) holds for any ϕ ∈ C ∞ 0 (Ω). We note that by the initial assumptions and Hölder's inequality, this definition makes a sense because (A skew ∇y) ∈ L 1 (Ω; R N ) for each y ∈ W 1,p 0 (Ω). It is worth to notice that the original boundary value problem (2)-(3) is illpossed, in general. Moreover, since the skew-symmetric form (9) can be unbounded on W 1,p 0 (Ω), the existence of a weak solution in the Minty sense to (2),(3) for fixed u ∈ A ad , A skew ∈ L q (Ω; S N skew ), and f ∈ L 2 (Ω; R N ) seems to be an open question. Further, we restrict our analysis to the following set of feasible solutions for the original optimal control problem. Namely, we indicate the set Ξ = (u, y) u ∈ A ad , y ∈ W 1,p 0 (Ω), (u, y) are related by (18) .
Definition 3.2. We say that an element y ∈ W 1,p 0 (Ω) belongs to the set D ⊂ with some constant c(y) depending on y.

OLHA P. KUPENKO AND ROSANNA MANZO
As a result, if y ∈ D then the mapping ϕ → [y, ϕ] A skew can be defined for all ϕ ∈ W 1,p 0 (Ω) using (10) and the standard rule where {ϕ ε } ε>0 ⊂ C ∞ 0 (Ω) and ϕ ε → ϕ strongly in W 1,p 0 (Ω). In particular, if y ∈ D, then we can define the value [y, y] A skew and this one is finite for every y ∈ D, although the "integrand" ∇y, A skew ∇y R N needs not be integrable on Ω, in general. We adopt the following hypothesis.
Hypothesis A. For each u ∈ A ad the following implication holds: (22) holds for all ϕ ∈ C ∞ 0 (Ω). Then, obviously, y ∈ D. Indeed, However, this property is not clear in the case of Minty weak solutions.
As it follows from the definition of the bilinear form [y, ϕ] A skew , in general, the value [y, y] A skew is not equal to zero and does not preserve a constant sign for all y ∈ D. In particular, in the case p = q = 2 there can be found an element y 0 ∈ D and a matrix A skew ∈ L 2 (Ω; S N skew ) such that ∇y(x), A skew (x)∇y(x) R N is identical to zero for a.e. x ∈ Ω whereas [y d , y d ] A skew < 0 (for the corresponding examples, we refer to [37,12,18]). This fact does not allow us to derive a reasonable a priory estimate in · W 1,p 0 (Ω) -norm for Minty weak solutions. Moreover, if A skew ∈ L q (Ω; S N skew ), then the differential operator in the left-hand side of equation (2) is neither monotone nor coercive, so two definitions of weak solutions, given by (18) and (22) are not equivalent [27]. Besides, the mapping u → y(u, f ) can be multivalued (see [13] for the details).
Here we make one more hypothesis, which is mainly motivated by the previous reasonings.
Hypothesis B. The set of feasible solutions Ξ is nonempty.
Remark 3.2. It should be mentioned here that Hypotheses A and B are satisfied provided the matrix A skew possesses some special BMO-properties (for the details we refer to [7,28]).
For our further analysis, we make use of some auxiliary results. We begin with the following property.
Proof. Since the estimate holds true for any r ∈ [1, +∞), it follows that u k → u in L r (Ω).
To deduce the weak- * convergence property u k * → u in L ∞ (Ω), it is enough to note that the strong convergence u k → u in L 1 (Ω) implies, up to a subsequence, the convergence u k (x) → u(x) almost everywhere in Ω. Hence, by Lebesgue Theorem, we have . Since this conclusion is true for any weakly- * convergent subsequence of {u k } k∈N , it follows that u is the weak- * limit for the whole sequence {u k } k∈N .
As for the inclusion u ∈ A ad , this fact immediately follows from definition of the set A ad , the pointwise convergence u k → u in Ω, and Proposition 2.1(i).

Proposition 3.2.
A ad is a sequentially compact subset of L r (Ω) for any r ∈ [1, +∞), and it is a sequentially weakly- * compact subset of L ∞ (Ω).
. As a result, the statement immediately follows from Propositions 3.1 and 2.1(iii). Proposition 3.3. Assume Hypothesis B holds true. Then the set Ξ is sequentially closed in the following sense: for any sequence {(u k , y k ) ∈ Ξ} k∈N such that we have (u 0 , y 0 ) ∈ Ξ, where y 0 is a Minty weak solution to the Dirichlet boundary value problem (2)-(3) with u = u 0 .
Proof. Since u k ∈ A ad for every k ∈ N then in view of Proposition 3.1, we immediately get u 0 ∈ A ad . Moreover, the Minty inequality (24) holds true for all ϕ ∈ C ∞ 0 (Ω) and k ∈ N. Since u k |∇ϕ| p−2 ∇ϕ → u 0 |∇ϕ| p−2 ∇ϕ strongly in L q (Ω; R N ) (see Proposition 3.1) and ∇y k ∇y in L p (Ω; R N ), we can pass to the limit in (24) as k → ∞. As a result, we have Therefore (u 0 , y) ∈ Ξ.
We are now in a position to establish the main result of this section.
Proof. Since Ξ = ∅ and the cost functional is bounded from below on Ξ, it follows that there exists a minimizing sequence Therefore, passing to a subsequence if necessary and taking into account Proposition 3.2, we may assume the existence of a pair (u 0 , In remains to show that (u 0 , y 0 ) is an optimal pair. Since u k → u strongly in L r (Ω) for every 1 ≤ r < +∞, {u k } k∈N is bounded in L ∞ (Ω) and u k (x) ≥ α for almost all x ∈ Ω, using conditions (26)- (28), it is easy to check that ∇y k u Taking into account the property of lower semicontinuity of the norms · L 2 (Ω) and · L p (Ω;R N ) with respect to the the weak topologies of L 2 (Ω) and L p (Ω; R N ), respectively, we get Thus, and, hence, the pair (u 0 , y 0 ) is optimal for problem (1)-(3). The proof is complete. (1)-(3). In this section we introduce the two-parameter regularization of the considered optimization problem for the case when the term [∇y] 2 may grow large or tend to zero. Moreover, in the approximating OCP we consider an L ∞ -regularization of the unbounded matrix A skew ∈ L q (Ω; S N skew ). As a result, we show that in suitable topologies optimal solutions of regularized problems tend to some optimal solutions of the initial problem.

Regularization of OCP
As was pointed out in [33], the p-Laplacian ∆ p (u, y) provides an example of a quasi-linear operator in divergence form with a so-called degenerate nonlinearity for p > 2. In this context we have non-differentiability of the state y with respect to the control u. As follows from Theorem 3.3, this fact is not an obstacle to prove existence of considered optimal controls in the coefficients, but it causes certain difficulties when deriving the optimality conditions for the considered problem. To overcome this difficulty, we introduce a special family of approximating control problems (see, for comparison, the approach of Casas and Fernandez [4] for quasilinear elliptic equations with a distributed control in the right hand side).
Let A k skew ∈ L ∞ (Ω; S N skew ) be the sequence such that A k skew → A skew strongly in L q (Ω; S N skew ) as k → ∞. Existence of such sequence is a well-known fact of functional analysis as well as methods of its construction. Indeed, let for every k ∈ N, T k : R → R be the truncation function defined by Then, for an arbitrary g ∈ L q (Ω), we have (see, for example [16]) So, for a given matrix In what follows we associate with the initial optimal control problem (1)-(3) the following sequence of optimization problems (see, for comparison, [4]) subject to constrains Here, A ad is defined in (5), k ∈ N, ε is a small parameter, which varies within a strictly decreasing sequence of positive numbers converging to 0, and F k : and the constants δ and δ * are independent of k ∈ N. In particular, if then δ = 4/27 and δ * = 4/3 satisfy (34). It is clear that the effect of such perturbations of ∆ p (u, y) is in its regularization around critical points and points where |∇y(x)| becomes unbounded. In particular, if y ∈ W 1,p 0 (Ω) and Ω k (y) := x ∈ Ω : |∇y(x)| > √ k 2 + 1 , then the following chain of inequalities shows that the Lebesgue measure of the set Ω k (y) satisfies the estimate i.e. the approximation F k (|∇y| 2 ) is essential on sets with small Lebesgue measure.
The main goal of this section is to show that for each ε > 0 and k ∈ N, the perturbed optimal control problem (31)-(33) is well posed and its solutions can be considered as a reasonable approximation of optimal pairs to the original problem (1)-(3). To begin with, we establish a few auxiliary results concerning monotonicity and growth conditions for the regularized operator A ε,k,u : where u ∈ A ad is a fixed control. For our further analysis, we make use of the following the notation Remark 4.1. For an arbitrary element y * ∈ H 1 0 (Ω) let us consider the level set Ω k (y * ) := x ∈ Ω : |∇y * (x)| > √ k 2 + 1 . Then Hence, the Lebesgue measure of the set Ω k (y * ) satisfies the estimate Now, we establish the following results.
Proposition 4.1. For every u ∈ A ad , k ∈ N and ε > 0, the operator A ε,k,u : (36) is bounded, strictly monotone, coercive and semi-continuous in the sense of relations (6)- (8) and Proof. Boundedness. Following the definition of F k , A k skew , and the boundedness of u, we obtain A ε,k,u = sup Strict monotonicity. We make use of the following algebraic inequality, which is proved in [22,Proposition 4.4]: With this, having put a := ∇y, b := ∇v we obtain it follows that the strict monotonicity property (6) holds true for each u ∈ A ad , k ∈ N, and ε > 0.
Coercivity. The coercivity property obviously follows from the estimate it is enough to observe that as t → 0 almost everywhere in Ω, and make use of Lebesgue's dominated convergence theorem.
The existence of a unique solution to the boundary value problem (32)-(33) follows from the following abstract theorem on monotone operators (see, for instance, [23] or [35,§II.2]). Indeed, applying the above theorem to the equation A ε,k,u y = divf with divf ∈ H −1 (Ω), we arrive at the following assertion.
Moreover, this solution satisfies the Minty inequality and the energy equality Thus, as follows from Theorem 4.2, for every ε > 0 and k ∈ N, the set of feasible pairs to problem (31)- (33) Ξ ε,k = (u, y) u ∈ A ad , y ∈ H 1 0 (Ω), (u, y) are related by equality (39) is always nonempty. Now we are in a position to prove the existence result for regularized OCPs (31)- (33). Proof. Indeed, we have already established that the set Ξ ε,k is not empty. Let {(u i , y i )} i∈N ⊂ Ξ ε,k be a minimizing sequence , i.e.
Following the proof of Theorem 3.3 it is easy to see that there exists u 0 ∈ A ad such that u i * u 0 in BV (Ω) and u i → u 0 strongly in L 1 (Ω). Also we have and, therefore, ∃ y 0 ∈ H 1 0 (Ω) such that, within a subsequence, y i y 0 in H 1 0 (Ω) as i → ∞. Similarly to the proof of Proposition 3.3, we pass to the limit in the Minty relation (see (40)) as i → ∞ and get the assertion (u 0 , y 0 ) ∈ Ξ ε,k . As follows from the direct method of the Calculus of Variations, the lower semicontinuous property of the cost functional with respect to suitable topologies in 'control-state' space plays a crucial role in the proof of solvability of optimization problems. However, in our case we make use of a weaker property, namely, we establish the fulfilment of the inequality I ε,k (u 0 , y 0 ) ≤ lim inf i→∞ I ε,k (u i , y i ) only for elements (u i , y i ) ∈ Ξ ε,k of minimizing sequence. To this end, we proceed as in the proof of Theorem 3.3. We have to show that the following relation is valid. In fact, this is an equality and it can be proved as follows. Since pairs (u i , y i ) and (u 0 , y 0 ) satisfy (41), we infer from (41) that

Remark 4.2.
It is worth to note that the cost functional (31) does not possess the lower semicontinuity property on BV (Ω) × H 1 0 (Ω) with respect to the weak- * convergence in BV (Ω) and the weak convergence in H 1 0 (Ω), in general. As follows from Theorem 4.3, this property is valid only on the set of feasible pairs Ξ ε,k ⊂ BV (Ω) × H 1 0 (Ω). For our further analysis, we need to obtain some appropriate a priory estimates for the weak solutions to the problem (32)- (33). With that in mind, we make use of the following auxiliary results (see for the comparison Proposition 4.2 in [5]). Proposition 4.2. Let u ∈ A ad , k ∈ N, and ε > 0 be given. Then, for arbitrary g ∈ L 2 (Ω; R N ) and y ∈ H 1 0 (Ω), we have Proof. Let us fix an arbitrary element y of H 1 0 (Ω). We associate with this element the set Ω k (y), where Ω k (y) := x ∈ Ω : |∇y(x)| > k 2 + 1 .
To conclude this section, let us show that for every u ∈ A ad and f ∈ L 2 (Ω; R N ), the sequence of weak solutions to problem (32)-(33) y ε,k = y ε,k (u, A k skew , f ) ε>0 k∈N is bounded with respect to the · ε,k,u -quasi-seminorm in the sense of Definition 4.4. Indeed, the integral identity (39) together with estimate (43) (for g = f ) immediately lead us to the relation y ε,k ε,k,u + α − 1 2 y ε,k p 2 ε,k,u .
As a result, it follows from (48) that where . Moreover, taking g = ∇y = ∇y ε,k in (43) and using (49), we also have ∀ ε > 0, ∀ k ∈ N, 5. Asymptotic analysis of the approximate OCP (31)- (33). Our main intention in this section is to show that some optimal solutions to the original OCP (1)-(3) can be attained (in certain sense) by optimal solutions to the approximate problems (31)- (33). With that in mind, we make use of the concept of variational convergence of constrained minimization problems (see Definition 2.2) and study the asymptotic behaviour of a family of OCPs (31)-(33) as ε → 0 and k → ∞.
We begin with some auxiliary results concerning the weak compactness in H 1 0 (Ω) of · ε,k,u -bounded sequences.
Proof. In view of Propositions 2.1 and 3.2, the sequence {u n } nN ⊂ A ad is compact with respect to the weak- * convergence in BV (Ω) and the strong convergence in L 1 (Ω). Hence, as far as sup n∈N y n H 1 0 (Ω) ≤ const (see (50)), using Lemma 5.1, we can claim that there exists a subsequence {(u i , y i )} i∈N ⊂ {(u n , y n )} n∈N and a pair (u, y) ∈ A ad × W 1,p 0 (Ω) such that (u i , y i ) τ −→ (u, y) in BV (Ω) × H 1 0 (Ω). The rest of proof is divided into three steps.
Step 1. On this step we prove that χ Ω\Ω k (yi) ∇y i ∇y in L p (Ω; R N ). Following the definition of the sets Ω ki (y i ) and using (49), we obtain Hence, taking a new subsequence if necessary, we infer the existence of a vectorvalued function g ∈ L p (Ω; On the other hand, in view of the weak convergence ∇y i ∇y in L 2 (Ω; R N ),
Step 2. On this step we prove assertion (54). Since u i → u in L r (Ω) for every 1 ≤ r < +∞, {u i } i∈N is bounded in L ∞ (Ω), and u i (x) ≥ α for a. a. x ∈ Ω, due to the result of the previous step, it is easy to check that χ Ω\Ω k i (yi) ∇y i u 1/p i ∇yu 1/p in L p (Ω; R N ). Using this convergence and lower semi-continuity of norm in L p (Ω; R N ) we get Step 3. Here we show that the pair (u, y) is admissible to the initial OCP (1)-(3). Let us prove that y is a solution of (2)-(3) corresponding to the control function u ∈ A ad . Let us fix an arbitrary test function ϕ ∈ C ∞ 0 (Ω) and pass to the limit in the Minty inequality as i → ∞. Taking into account that In view of the weak convergence ∇y i ∇y in L 2 (Ω; R N ) and the strong convergence u i → u in L r (Ω), for all r < ∞, the boundedness of {u i } i∈N in L ∞ (Ω) and the fact that A ki skew → A skew strongly in L q (Ω; S N skew ), we obtain Let us explain two last relations in details. Indeed, we see that In view of (60) and the weak convergence χ Ω\Ω k i (yi) ∇y i → ∇y in L p (Ω; R N ), we immediately get I 1 → 0. As for I 2 , we have Hence, this integrand is obviously equi-integrable. Equi-integrability property implies that I 2 → 0 as far as |Ω ki (y i )| → 0. To get (62), the similar argumentation can be used together with the estimate (37) and (49) ≤ C 2 y i H 1 0 (Ω k i (yi)) → 0 as i → ∞. Thus, passing to the limit in relation (59) as i → ∞, we arrive at the inequality (18) for every ϕ ∈ C ∞ 0 (Ω).
Step 4. On this step we prove energy inequality (55). For each i ∈ N, we have the energy equalities Taking into account the weak convergence y i y in H 1 0 (Ω) and (54), after passing to the limit in (63), we immediately arrive at the desired assertion.
As follows from the statement of Theorem 5.2, Hypothesis B can be eliminated from Proposition 3.3 and Theorem 3.3. Corollary 1. If A ad = ∅, then the set of feasible solutions Ξ to OCP (1)- (3) is nonempty for every f ∈ L 2 (Ω; R N ) and y d ∈ L 2 (Ω). Remark 5.1. As follows from Theorem 5.2, feasible solutions to the regularized OCPs (31)-(33) always lead in the limit to some admissible solution (u, y) of the original OCP (1)-(3). It is reasonable 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)-(3) can be attainable in such way.
Taking these observations into account, we make use of the following notion.
and ( u, y ) is related by energy equality As a consequence of Theorem 5.2 and properties of the variational limits of constrained minimization problems (see Theorem 2.3), we have the following result. to the corresponding regularized OCPs. Then this sequence is relatively compact with respect to the τ -convergence and each its τ -cluster pair ( u, y ) ∈ BV (Ω) × W 1,p 0 (Ω) is a variational solution to OCP (1)-(3) in the sense of Definition 5.3. Moreover, up to a subsequence, we have y 0 ε,k → y in H 1 0 (Ω) and as ε → 0, k → ∞, Proof. To begin with, we note that the τ -compactness property of the sequence (u 0 ε,k , y 0 ε,k ) ∈ Ξ ε,k ε>0 k∈N is a direct consequence of Theorem 5.2. In order to prove the strong convergence (66), we make use of the main properties of the variational convergence. Following Theorems 2.3, 3.3, and 5.2 (see also Corollary 1), we can claim that OCP (1)-(3) is solvable and there exists an optimal pair ( u, y ) ∈ Ξ to this problem such that inf (u,y)∈ Ξ I(u, y) = I ( u, y ) = y − y d 2 However, because of the lower semicontinuity of · L 2 (Ω) and · L p (Ω;R N ) with respect to the weak convergence, the convergence (u 0 ε,k , y 0 ε,k ) τ → ( u, y ) implies that |∇y 0 ε,k | 2 dx ≥ Ω u|∇ y| p dx (see (58) and Step 2 of the proof of Theorem 5.2), and, hence, inf (u,y)∈ Ξ I(u, y) Since the pair ( u, y ) is feasible for the problem (1)-(3) (see Theorem 5.2), it follows that ( u, y ) is an optimal pair. Therefore, the last relation in (69) becomes equality. The weak convergence y 0 ε,k y in H 1 0 (Ω) implies that y ε,k → y strongly in L 2 (Ω) and, therefore, lim ε→0 . Whence, we immediately get (65).
To prove the energy equality (65) it is enough to pass to the limit in relation which holds for each ε and k. As a result, using (67), we finally have In fact, from (67) it follows that Using the weak convergence χ Ω\Ω k (y 0 ε,k ) ∇y 0 ε,k u 0 ε,k 1/p ∇ y u 1/p in L p (Ω; R N ) (see the proof of Theorem 5.2, Step 2 for similar arguments), and (70), we get The weak convergence χ Ω\Ω k (y 0 ε,k ) ∇y 0 ε,k u 0 ε,k 1/p ∇ y u 1/p and the convergence of their norms χ Ω\Ω k (y 0 ε,k ) ∇y 0 ε,k u 0 ε,k 1/p L p (Ω;R N ) → ∇ y u 1/p L p (Ω;R N ) imply the strong convergence in L p (Ω; R N ). Now, it is a simple exercise to check the strong convergence χ Ω\Ω k (y 0 ε,k ) ∇y 0 ε,k → ∇ y in L p (Ω; R N ). Further, from here and (70) we obtain lim ε>0 k∈N Applying (71), it is easy to deduce from this that lim ε→0 k→∞ Combining this estimate and the strong convergence χ Ω\Ω k (y 0 ε,k ) ∇y 0 ε,k → ∇ y in L p (Ω; R N ), we conclude that We are now in a position to discuss the existence of variational solutions to the OCP (1)-(3).
Proof. To begin with, we note that in this case every weak solution in the Minty sense to the boundary value problem (2)-(3) satisfies energy equality (65). Indeed, since, by Hypothesis A, every weak solution y = y(u) ∈ W 1,p 0 (Ω) such that (u, y) ∈ Ξ, belongs to the set D, the energy inequality can be obtained, using the argumentation similar to the so-called Minty trick (see Pastukhova [27]). Starting from Minty inequality Ω u|∇ϕ| p−2 (∇ϕ, ∇ϕ − ∇y) R N dx as a test functions in (73) we can take ϕ = y ± tv, where t ∈ R N , v ∈ D. Taking into account, that (A skew ∇ϕ, ∇ϕ) R N = 0 and (A skew ∇ϕ, ∇ϕ − ∇y) R N = −(A skew ∇ϕ, ∇y) R N = (∇ϕ, A skew ∇y) R N , and using the semi-continuity of ∆ p (u, ·), after passing to the limit as t → 0 we have which yields in view of (72) the validity of the integral identity and, taking v = y we get (65). Further, our aim is to show that OCP (1)-(3) can be interpreted as the variational limit of the sequence of constrained minimization problems (31)-(33) for ε > 0 and k ∈ N. To do so, we have to verify the fulfilment of all conditions of Definition 2.2. Let {(u ε,k , y ε,k ) ⊂ Ξ ε,k } ε>0 k∈N be a sequence in BV (Ω) × H 1 0 (Ω) with the following properties: (a) (u n , y n ) ∈ Ξ εn,kn for every n ∈ N, where {ε n , k n } n∈N is a subsequence of {ε, k} such that ε n → 0, k n → ∞ as n → ∞; (aa) (u n , y n ) which holds true for any sequence (u n , y n ) ∈ A ad × H 1 0 (Ω) n∈N with properties (a)-(aa).
We focus now on the verification of condition (dd) of Definition 2.2. Let (u , y ) be an arbitrary admissible pair to the original problem. We set {u ε,k ≡ u } ε>0 k∈N and y ε,k = y(u ε,k , f, A k skew ) ε>0 k∈N we choose as corresponding solutions to the regularized boundary value problems (32)- (33). Then having applied the arguments of the proof of Theorem 5.2, it can be shown that the sequence {y ε,k } ε>0 k∈N is uniformly bounded in H 1 0 (Ω) and there exists an element y ∈ W 1,p 0 (Ω) such that y ∈ D, (u , y) ∈ Ξ, and, up to a subsequence, Our aim is to show that y = y and the following identity I(u , y ) = lim sup ε→0 k→∞ holds true. Indeed, since (u , y ) ∈ Ξ and (u , y) ∈ Ξ, in view of preconditions of the theorem, it follows that this pairs satisfy the integral identity (74) After subtraction and setting v = y − y (since the form [y, ϕ] A skew is bilinear, it is clear that the set D is a linear manifold), we get The strict monotonicity of ∆ p (u , ·) and (77) immediately imply y = y. Moreover, in view of preconditions of the theorem, for the limit pair (u , y) instead of the energy inequality (55) i.e. desired condition (dd) obviously holds: 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)- (3). With that in mind, we adopt the following concept.
Lemma 5.2. Assume that there exists an element u 0 ∈ A ad , A skew ∈ L q (Ω; S N skew ) and an element v ∈ D ⊂ W 1,p 0 (Ω) with property [v, v] A skew > 0 and such that |∇v| p−2 ∇v + A skew ∇v ∈ L 2 (Ω; R N ). Then there are distributions f ∈ L 2 (Ω; R N ) and y d ∈ W 1,p 0 (Ω) ⊂ L 2 (Ω) such that the optimal control problem Minimize I(u, y) = y − y d Proof. Further we put Since v ∈ D, it follows that y d ∈ W 1,p 0 (Ω), and in view of preconditions of the lemma, f ∈ L 2 (Ω; R N ). Then, taking into account monotonicity of ∆ p (u 0 , ·) and the fact, that (A skew ∇ϕ, ∇v) Hence, the distribution y d is a weak Minty solution to boundary value problem (2)-(3). Moreover, using the fact that I(u 0 , y d ) = 0, we can conclude: (u 0 , y d ) is the unique optimal pair to the above OCP. It remains to observe that in view of takes place and, hence, (u 0 , y d ) is a non-variational solution to the above problem. The proof is complete.
6. Optimality conditions for perturbed OCPs (31)- (33). In view of Proposition 5.1 and Theorem 5.4 under fulfillment of Hypothesis A and condition (72), some optimal solutions to initial OCP (1)-(3) can be attained through optimal solutions to optimization problems (31)- (33). Mainly motivated by these reasoning and for the sake of numerical computations, we focus on establishment of optimality conditions to OCPs (31)- (33). Before deriving the optimality conditions for OCP (31)-(33), we recall the well-known Ioffe & Tikhomirov theorem (see [15]) justifying the Lagrange principle for well-posed boundary value problems.
Theorem 6.1. Let Y , U and V be Banach spaces. Let I : U × Y → R ∪ {+∞} be a cost functional, F : U × Y → V a mapping, and U ∂ a nonempty subset of U . Let (u 0 , y 0 ) ∈ U × Y be a solution to the following extremal problem Assume that for every u ∈ U ∂ , the mappings y → I(u, y) and y → F (u, y) are continuously differentiable for y ∈ O(y 0 ), where O(y 0 ) is some neighborhood of the point y 0 . Let Im F y (u 0 , y 0 ) be a closed subset of V and has a finite codimension in V . Assume that for each y ∈ O(y 0 ) the function u → I(u, y) is convex, and the mapping u → F (u, y) is continuous from U to V and affine, i.e., Then there exists a pair (λ, µ) ∈ (R + × V * )\{0} such that for the Lagrange function to the problem (83), that is defined by the equality Λ(u, y, λ, µ) = λI(u, y) + µ, F (u, y) V * ;V , we have Moreover, if Im F y (u 0 , y 0 ) = V , then we can assume that λ = 1 in (86)-(87).
It follows that the mapping y → A ε,k,u (y) has a Gâteaux derivative and this derivative satisfies the following relation As follows from (90), the Gâteaux derivative (A ε,k,u (y)) G is well defined in any neighborhood of a given point y ∈ H 1 0 (Ω) and is continuous at this point (for the technical details we refer to [1]). As a result, the Gâteaux derivative implies the existence of the Fréchet derivative and these derivatives coincide [9].
As an obvious consequence of this result and the fact that Fréchet differentiability of operator y → A ε,k,u (y) implies the existence of the gradient for the functional ϕ : is given by (90). Similarly, for I ε,k (u, ·) : H 1 0 (Ω) → R, we have As a result, we arrive at the following obvious result.
As a result, for every v ∈ H 1 0 (Ω), we have We are now in a position to establish the main result of this section.