Asymptotic analysis of an optimal boundary control problem for ill-posed elliptic equation in domains with rugous boundary

Abstract

We study an optimal control problem for the mixed Dirichlet–Neumann boundary value problem for the strongly non-linear elliptic equation with exponential nonlinearity in a domain with rugous boundary. A density of surface traction u acting on a part of rugous boundary is taken as a control. The optimal control problem is to minimize the discrepancy between a given distribution and the current system state. We deal with such case of nonlinearity when we cannot expect to have a solution of the state equation for a given control. After having defined a suitable class of the weak solutions, we provide asymptotic analysis of the above mentioned optimal control problem posed in a family of perturbed domains and give the characterization of the limiting behavior of its optimal solutions.

Keywords

Nonlinear elliptic equation boundary control boundary oscillation asymptotic analysis

1. Introduction and setting of the optimal control problem

Let ε be a small positive parameter. Let $Ω_{ε}$ and Ω be given bounded open subsets of $R^{N}$ with $C^{1, 1}$ -boundaries confined in a fixed bounded domain D. It is assumed that $N > 2$ . We assume that Ω lies locally on one side of $\partial Ω$ and the boundary $\partial Ω$ consists of two disjoint parts $\partial Ω = Γ_{D} \cup Γ_{N}$ such that $Γ_{D} \subset \partial D$ and the sets $Γ_{D}$ and $Γ_{N}$ have positive $(N - 1)$ -dimensional Hausdorff measures $H^{N - 1} (Γ_{D})$ and $H^{N - 1} (Γ_{N})$ , respectively. We make the similar assumptions with respect to the boundary of $Ω_{ε}$ and, in addition, we suppose that $Γ_{D} \cap Γ_{N, ε} = \emptyset$ and $Γ_{D}$ -part does not depend on $ε > 0$ .

Let $W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ be the Banach space which is defined as the closure of $C_{0}^{\infty} (R^{N}; Γ_{D}) = {φ \in C_{0}^{\infty} (R^{N}) : φ = 0 on Γ_{D}}$ with respect to the norm $\begin{matrix} ‖ y ‖_{W_{0}^{1, 2} (Ω_{ε}; Γ_{D})} = {(\int_{Ω_{ε}} | \nabla y |^{2} d x)}^{1 / 2} . \end{matrix}$ Let $W^{- 1, 2} (Ω_{ε}; Γ_{D}) : = {(W_{0}^{1, 2} (Ω_{ε}; Γ_{D}))}^{*}$ be the dual space to $W^{1, 2} (Ω_{ε}; Γ_{D})$ .

Let $y_{d} \in L^{2} (D)$ and $g_{d} \in L^{q} (D)$ , with $q ⩾ \frac{2 N}{N + 2}$ , be given distributions. Hereinafter, we consider $\begin{array}{l} (1.1) & G_{ad} = L^{q} (D), \\ (1.2) & A_{ad} (Γ_{N, ε}) = {v |_{Γ_{N, ε}} : v \in W^{1, r} (D), ‖ v ‖_{W^{1, r} (D)} ⩽ β} \end{array}$ as the sets of admissible controls, where $r > N$ is a given exponent, β is a given positive weight, and elements of $G_{ad}$ can be interpreted as fictitious controls that should be chosen as close as possible to the given distribution $g_{d} \in L^{q} (D)$ with respect to the norm of $L^{q} (D)$ .

We consider $Ω_{ε}$ as some perturbation of Ω that excludes the case of perforated domains. We suppose that the $Γ_{N}$ -part of the boundary $\partial Ω$ can be drastically changed in the perturbed domains $Ω_{ε}$ . In particular, we admit the situation where $Γ_{N, ε}$ presents a rugosity with a highly oscillatory behaviour as the parameter ε tends to zero (see Fig. 1 for illustration).

Fig. 1.

Example of domain $Ω_{ε}$ .

Definition 1.1.

We say that domain $Ω_{ε}$ with $\partial Ω_{ε} = Γ_{D} \cup Γ_{N, ε}$ has a rugous $Γ_{N, ε}$ -part of boundary if there exists a $δ > 0$ , a finite number of points ${ξ^{k}}_{k = 1}^{m} \subset Γ_{N}$ , and a collection of $C^{1, 1}$ -functions ${ϕ^{k}, ϕ_{ε}^{k} : R^{N - 1} \to R}_{k = 1}^{m}$ such that $\begin{array}{l} (1.3) & Γ_{N, ε} \subset ⋃_{k = 1}^{m} Π_{N} (ξ^{k}), \\ Π_{N} (ξ^{k}) = {x \in R^{N} : | x_{i} - ξ_{i}^{k} | < δ, i = 1, \dots, N}, \\ (1.4) & Ω \cap Π_{N} (ξ^{k}) = {x \in Π_{N} (ξ^{k}) : x_{N} < ϕ^{k} (x_{1}, \dots, x_{N - 1})}, \\ (1.5) & Ω_{ε} \cap Π_{N} (ξ^{k}) = {x \in Π_{N} (ξ^{k}) : x_{N} < ϕ_{ε}^{k} (x_{1}, \dots, x_{N - 1})}, \\ (1.6) & ϕ_{ε}^{k} \to ϕ^{k} in C (\overline{Π_{N - 1} (ξ^{k})}) as ε \to 0, \end{array}$ where $Π_{N - 1} (ξ^{k}) = {x \in R^{N - 1} : | x_{i} - ξ_{i}^{k} | < δ, i = 1, \dots, N - 1}$ .

For more details and the example of domains with a rugous boundary we refer to Section 2 (see also [1] for comparison).

In perturbed domain $Ω_{ε}$ we consider the following optimal control problem (OCP) for nonlinear elliptic equation $\begin{array}{l} (1.7) & Minimize J_{ε} (u, g, y) = \frac{1}{2} \int_{Ω_{ε}} | y - y_{d} |^{2} d x + \frac{1}{2} \int_{Γ_{N, ε}} | u |^{2} d H^{N - 1} + \frac{1}{q} \int_{Ω_{ε}} | g - g_{d} |^{q} d x, \\ subject to constraints \\ (1.8) & - Δ y = f (y) + g in Ω_{ε}, \\ (1.9) & y = 0 on Γ_{D}, \partial_{ν} y = u on Γ_{N, ε}, \\ (1.10) & g \in G_{ad}, y \in W_{0}^{1, 2} (Ω_{ε}; Γ_{D}), u \in A_{ad} (Γ_{N, ε}) . \end{array}$ It is assumed that $f (y) = F^{'} (y)$ , $f : R \to (0, \infty)$ is a strictly convex function, and $F \in C^{2} (K)$ for any compact set $K \subset R$ , where F is a non-decreasing positive function such that $\begin{matrix} (1.11) & F^{'} (z) ⩾ C_{F} F (z), \forall z \in R and | \int_{- \infty}^{0} z F^{'} (z) d z | < + \infty, \end{matrix}$ for some constant $C_{F} > 0$ .

One of the main characteristic feature of the indicated OCP is the fact that we have two different types of controls – distributed and boundary, and the control zone for one of them is supported along of a rough part of $\partial Ω_{ε}$ . The second feature is related with the specificity of non-linearity $f (y)$ and the fact that this term belongs to $L^{1} (Ω_{ε})$ and not to $L^{2} (Ω_{ε})$ as usual. The originality of the present paper arises from the conjunction of these two features when ε tends to zero. As was mentioned in [3], because of the specificity of non-linearity $f (y)$ , we can not obtain any a priori estimate for the weak solutions in the sense of distributions in the standard Sobolev space $W_{0}^{1, 2} (Ω)$ . Moreover, we cannot assert that BVP (1.8)–(1.10) admits at least one solution for given controls $u \in A_{ad} (Γ_{N, ε})$ and $g \in G_{ad}$ . The other characteristic features and physical motivation to OCP (1.7)–(1.8) can be found in [3] (see also [4,5,11,12] and our recent publications [7–9,13,16,17,19–21,23]).

Having assumed that a small positive parameter ε varying within a strictly decreasing sequence of positive real numbers which converges to zero, our main intention in this paper is to study the asymptotic behaviour of OCP (1.7)–(1.10) as ε tends to zero. In spite of the fact that asymptotic behaviour of solutions of elliptic partial differential equations in domains with rugous boundary is a subject that has been addressed in the literature by different authors (see, for instance [1,2,6,24–26] and recent paper [14], where the authors consider the homogenization of an evolution problem with right-hand side not in $L^{2}$ in domain with oscillating boundaries), the analogous results for the case of optimal control problems for essentially nonlinear elliptic equations (1.8) with mixed boundary conditions (1.9) in the presence of boundary oscillations along the control zone remain arguably open.

Working in the framework of special class of weak solutions that was indicated in our previous paper [8] (see Definition 1.2 below), and following in some aspects the technique proposed in [6] (see also [15]), we realize the limit passage in OCP (1.7)–(1.10), identify the limit optimization problem, show that the limit problem admits a unique optimal solution, and prove the main variational property: optimal solutions for the original problem (1.7)–(1.10) converge to the optimal solution of the limit problem. As follows from the asymptotic analysis that we provide for OCP (1.7)–(1.10), even if parameter $ε > 0$ is small enough, the rugosity of the control zones (i.e., $Γ_{N, ε}$ -boundaries of domains $Ω_{ε}$ ) affects not only the limit boundary value problem, but the limit cost functional and boundary control constraints set as well. Thus, the limit optimal control problem has a structure that drastically differs from the original one. It means that the rugosity effect can not be neglected under precise consideration of the corresponding optimal control problems.

The ill-posedness of strongly nonlinear elliptic equation with $L^{1}$ -type of nonlinearity motivates us to introduce the following notion (see [8] for the details).

Definition 1.2.

We say that $(u, g, y) \in L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ is a feasible solution to the problem (1.7)–(1.10) if

$(u, g)$ are admissible controls, i.e. $u \in A_{ad} (Γ_{N, ε})$ and $g \in G_{ad}$ ;

$f (y) \in L^{1} (Ω_{ε})$ ;

the integral identity $\begin{matrix} (1.12) & \int_{Ω_{ε}} (\nabla y, \nabla φ) d x = \int_{Ω_{ε}} f (y) φ d x + \int_{Γ_{N, ε}} u φ d H^{N - 1} + \int_{Ω_{ε}} g φ d x \end{matrix}$ holds for every test function $φ \in C_{0}^{\infty} (R^{N}; Γ_{D})$ ;

the following integral inequalities $\begin{array}{l} (1.13) & \begin{matrix} \int_{Ω_{ε}} | \nabla y |^{2} d x & ⩽ \frac{2 N}{N - 2} \int_{Ω_{ε}} F (y) d x - \frac{2}{N - 2} \int_{Ω_{ε}} g (x - x_{0}, \nabla y) d x \\ + \frac{3 diam D}{N - 2} \int_{Γ_{N, ε}} | u |^{2} d H^{N - 1}, \end{matrix} \\ (1.14) & \int_{Γ_{N, ε}} F (γ_{0} (y (σ))) (ν (σ), σ - x_{0}) d H^{N - 1} ⩾ - F (0) \int_{Γ_{D}} (ν (σ), σ - x_{0}) d H^{N - 1}, \end{array}$ hold true with some $x_{0} \in int Ω \cap Ω_{ε}$ , where $γ_{0} : W_{0}^{1, 2} (Ω_{ε}; Γ_{D}) \to W^{1 / 2, 2} (\partial Ω_{ε})$ stands for the trace operator (see [22, Theorem 8.3]), $\begin{matrix} γ_{0} (y) = y |_{\partial Ω_{ε}}, \forall y \in W_{0}^{1, 2} (Ω_{ε}; Γ_{D}) \cap C (\overline{Ω_{ε}}) . \end{matrix}$

We denote by $Ξ_{ε}$ the set of all feasible solutions to the problem (1.7)–(1.10). In spite of the fact that, for given $ε > 0$ , $u \in A_{ad} (Γ_{N, ε})$ , and $g \in L^{q} (Ω)$ , the existence of weak solutions to the original BVP in the sense of Definition 1.2 is an open question for nowadays, we say that the optimal control problem (1.7)–(1.10) is consistent if:

the set of feasible solutions $Ξ_{ε}$ is non-empty;

for any triplet $(u, g, y)$ , where $u \in A_{ad} (Γ_{N, ε})$ , $g \in G_{ad} = L^{q} (D)$ , and $y = y (u, g) \in W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ is a distributional solution to the problem (1.8)–(1.9), there exists a distribution z such that $z \in W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ and $(u, g, z) \in Ξ_{ε}$ .

In [8] (see Theorems 2.9 and 4.2), the authors established the following result.

Theorem 1.3.

Let $f \in C_{loc}^{1} (R)$ be a strictly convex function, and let $y_{d} \in L^{2} (Ω)$ and $g_{d} \in L^{q} (D)$ with $q ⩾ 2$ be arbitrary distributions. If, for a given $ε > 0$ , the domain $Ω_{ε}$ is star-shaped with respect to some point $x_{0} \in int Ω_{ε}$ , i.e., there exists $x_{0} \in int Ω_{ε}$ such that $\begin{matrix} (σ - x_{0}, ν (σ)) ⩾ 0 for H^{N - 1} -a.a. σ \in \partial Ω_{ε} \end{matrix}$ then optimal control problem ( 1.7 )–( 1.10 ) is consistent and has a unique solution.

The idea pursued in this article is to represent of the original OCP in perturbed domain as a constrained minimization problem and realize then the concept of variational convergence in variable space as ε tends to zero (see [18] for the details). To this end we will need the following technical results that were established in [8].

Theorem 1.4.

Let $u \in L^{2} (Γ_{N, ε})$ , $g \in L^{q} (D)$ with $q ⩾ 2$ , and let $y = y (u) \in W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ be a distributional solution to BVP ( 1.8 )–( 1.9 ) such that y satisfies inequality ( 1.13 ). Then there exist positive constants $C_{i}$ , $1 ⩽ i ⩽ 6$ , independent of u, y, and ε, such that $\begin{array}{l} (1.15) & \int_{Ω_{ε}} y f (y) d x ⩽ C_{1} ‖ u ‖_{L^{2} (Γ_{N, ε})}^{2} + C_{2} ‖ g ‖_{L^{q} (D)}^{2} + C_{3}, \\ (1.16) & ‖ y ‖_{W_{0}^{1, 2} (Ω_{ε}; Γ_{D})} ⩽ C_{4} ‖ u ‖_{L^{2} (Γ_{N, ε})} + C_{5} ‖ g ‖_{L^{q} (D)} + C_{6} . \end{array}$

Proposition 1.5.

Let ${(u_{k}, g_{k}, y_{k})}_{k \in N} \subset L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ , with $q ⩾ 2$ , be a sequence such that $\begin{array}{l} (1.17) & u_{k} ⇀ u weakly in L^{2} (Γ_{N, ε}), \\ (1.18) & g_{k} ⇀ g weakly in L^{q} (D), \\ (1.19) & y_{k} \to y weakly in W_{0}^{1, 2} (Ω_{ε}; Γ_{D}) and a.e. in Ω_{ε}, \\ (1.20) & f (y_{k}) \to f (y) strongly in L^{1} (Ω_{ε}), \\ (1.21) & - Δ y_{k} = f (y_{k}) + g_{k} in {(C_{0}^{\infty} (R^{N}; Γ_{D}))}^{*}, \forall k \in N, \\ (1.22) & γ_{0} (y_{k}) = 0 and γ_{1} (y_{k}) = u_{k}, \forall k \in N, \end{array}$ where $γ_{1} (y) = \frac{\partial y}{\partial ν} |_{Γ_{N, ε}}$ for all $y \in C^{1} (\overline{Ω_{ε}}) \cap W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ . Then $\begin{matrix} (1.23) & \nabla y_{k} \to \nabla y strongly in L^{r} {(Ω_{ε})}^{N} for any 1 ⩽ r < 2 . \end{matrix}$

Proposition 1.6.

Let $q ⩾ 2$ and let ${(u_{k}, g_{k}, y_{k})}_{k \in N} \subset Ξ_{ε}$ be a sequence of feasible solutions to the optimal control problem ( 1.7 )–( 1.10 ) for a given value $ε > 0$ . Assume that ${g_{k}}_{k \in N}$ is a bounded sequence in $L^{q} (D)$ . Then there exist a triple $(u, g, y) \in L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ and a subsequence of ${(u_{k}, g_{k}, y_{k})}_{k \in N}$ , still denoted by the same index $k \in N$ , such that $\begin{array}{l} (1.24) & u_{k} \to u strongly in L^{2} (Γ_{N, ε}) as k \to \infty, \\ (1.25) & g_{k} ⇀ g weakly in L^{q} (D) as k \to \infty, \\ (1.26) & y_{k} ⇀ y weakly in W_{0}^{1, 2} (Ω_{ε}; Γ_{D}) as k \to \infty, \\ (1.27) & (u, g, y) \in Ξ_{ε}, \\ (1.28) & f (y_{k}) \to f (y) strongly in L^{1} (Ω_{ε}) as k \to \infty . \end{array}$

The reasonings that were applied for the proof of Theorem 1.3 allow us to justify Definition 1.2 making use of the following observation.

Corollary 1.7.

Assume that, instead of the star-shaped property of $Ω_{ε}$ , we have the following ones: $Ω_{ε}$ is just a $Γ_{D}$ -star-shaped domain and the inequalities $\begin{array}{l} \int_{Γ_{D}} (σ - x_{0}, ν (σ)) {| \nabla y (σ) |}^{2} d H^{N - 1} ⩾ 0, \\ \int_{Γ_{N, ε}} F (γ_{0} (y (σ))) (ν (σ), σ - x_{0}) d H^{N - 1} ⩾ - F (0) \int_{Γ_{D}} (ν (σ), σ - x_{0}) d H^{N - 1} \end{array}$ hold true. Then relation ( 1.13 ) remains valid for given $y = y (u, g) \in W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ , $u \in L^{2} (Γ_{N, ε})$ , $g \in L^{q} (D)$ with $q ⩾ 2$ , and $f (y) \in L^{2} (Ω_{ε})$ .

We also indicate the following estimate that can be easily deduced from Theorem 1.4. $\begin{array}{rcl} sup_{k \in N} ‖ y_{k} ‖_{W_{0}^{1, p} (Ω_{ε}; Γ_{D})} & ⩽ & C_{4} sup_{k \in N} ‖ u_{k} ‖_{L^{2} (Γ_{N, ε})} + C_{5} sup_{k \in N} ‖ g_{k} ‖_{L^{q} (D)} + C_{6} \\ (1.29) & ⩽ & C_{4} C_{r} β \sqrt{H^{N - 1} (Γ_{N, ε})} + C_{5} sup_{k \in N} ‖ g_{k} ‖_{L^{q} (D)} + C_{6} \end{array}$ with $β = 3 diam D$ .

2. Description of the domain perturbations

The main object of our concern in this section is a family of domains ${Ω_{ε}}_{0 < ε ⩽ ε_{0}}$ . We assume that each of the sets $Ω_{ε}$ is Lebesgue measurable and can be associated with its characteristic function $\begin{matrix} χ_{Ω_{ε}} (x) = 1 if x \in Ω_{ε} and χ_{Ω_{ε}} (x) = 0 if x \notin Ω_{ε} . \end{matrix}$

We further also assume that Ω satisfies a strong Lipschitz condition, that is, $\partial Ω$ is locally the graph of Lipschitz function in some coordinate system, and Ω lies locally on one side of $\partial Ω$ . We suppose that each of domains $Ω_{ε}$ may have rather rough part of the boundary $\partial Ω_{ε} = Γ_{D} \cup Γ_{N, ε}$ . Since we consider $Ω_{ε}$ as some perturbation of Ω that excludes the case of perforated domains and the family ${Ω_{ε}}_{0 < ε ⩽ ε_{0}}$ must approach (in some sense) the open bounded set Ω, we suppose that $Γ_{N, ε}$ is parametrized over $Γ_{N}$ and assume that the surface measure of $Γ_{N, ε}$ converges to a bounded measurable function on $Γ_{N}$ in a rather weak sense (see Definition 1.1). As a direct consequence of this definition, we have the following property.

Lemma 2.1.
Let $Ω_{ε}$ be a domain with a rugous $Γ_{N, ε}$ -part of boundary (see Definition 1.1 ). Then $Ω_{ε} \to Ω$ as $ε \to 0$ in the following sense $\begin{matrix} (2.1) & lim_{ε \to 0} (sup_{x \in Ω Δ Ω_{ε}} d (x, \partial Ω)) = lim_{ε \to 0} (sup_{x \in Ω Δ Ω_{ε}} inf_{y \in \partial Ω} | x - y |_{R^{N}}) = 0, \end{matrix}$ where $Ω Δ Ω_{ε} : = (Ω ∖ Ω_{ε}) \cup (Ω_{ε} ∖ Ω)$ .
Proof.
In view of the initial assumptions, we can establish the equalities $\begin{array}{l} Ω = (Ω ∖ ⋃_{k = 1}^{m} Π_{N} (ξ^{k})) \cup (⋃_{k = 1}^{m} Π_{N} (ξ^{k}) \cap Ω), \\ Ω_{ε} = (Ω_{ε} ∖ ⋃_{k = 1}^{m} Π_{N} (ξ^{k})) \cup (⋃_{k = 1}^{m} Π_{N} (ξ^{k}) \cap Ω_{ε}), \\ Ω ∖ ⋃_{k = 1}^{m} Π_{N} (ξ^{k}) = Ω_{ε} ∖ ⋃_{k = 1}^{m} Π_{N} (ξ^{k}), \forall ε > 0, \\ \begin{matrix} Ω Δ Ω_{ε} & = (⋃_{k = 1}^{m} Π_{N} (ξ^{k}) \cap Ω) Δ (⋃_{k = 1}^{m} Π_{N} (ξ^{k}) \cap Ω_{ε}) \\ = ⋃_{k = 1}^{m} (Π_{N} (ξ^{k}) \cap Ω) Δ (Π_{N} (ξ^{k}) \cap Ω_{ε}) \\ = ⋃_{k = 1}^{m} [(Π_{N} (ξ^{k}) \cap Ω) ∖ Ω_{ε} \cup (Π_{N} (ξ^{k}) \cap Ω_{ε}) ∖ Ω] . \end{matrix} \end{array}$ Then we see that $\begin{array}{l} sup_{x \in Ω ∖ Ω_{ε}} inf_{y \in \partial Ω} | x - y |_{R^{N}} & ⩽ \sum_{k = 1}^{m} sup_{x \in (Π_{N} (ξ^{k}) \cap Ω) ∖ Ω_{ε}} inf_{y \in \partial Ω \cap Π_{N} (ξ^{k})} | x - y |_{R^{N}} \\ ⩽ \sum_{k = 1}^{m} sup_{x \in (Π_{N} (ξ^{k}) \cap Ω) ∖ Ω_{ε}} | x_{N} - ϕ^{k} (x_{1}, \dots, x_{N - 1}) | \\ ⩽ \sum_{k = 1}^{m} sup_{\begin{array}{c} x = (x_{1}, \dots, x_{N - 1}, x_{N}) \in Π_{N} (ξ^{k}) \\ ϕ_{ε}^{k} (x_{1}, \dots, x_{N - 1}) < x_{N} < ϕ^{k} (x_{1}, \dots, x_{N - 1}) \end{array}} | x_{N} - ϕ^{k} (x_{1}, \dots, x_{N - 1}) | \\ ⩽ \sum_{k = 1}^{m} sup_{x^{'} \in Π_{N - 1} (ξ^{k})} | ϕ_{ε}^{k} (x^{'}) - ϕ^{k} (x^{'}) | ⩽ \sum_{k = 1}^{m} {‖ ϕ_{ε}^{k} - ϕ^{k} ‖}_{C (\overline{Π_{N - 1} (ξ^{k})})} . \end{array}$ By analogy we also deduce $\begin{array}{l} sup_{x \in Ω_{ε} ∖ Ω} inf_{y \in \partial Ω} | x - y |_{R^{N}} & ⩽ \sum_{k = 1}^{m} sup_{x^{'} \in Π_{N - 1} (ξ^{k})} | ϕ^{k} (x^{'}) - ϕ_{ε}^{k} (x^{'}) | \\ ⩽ \sum_{k = 1}^{m} {‖ ϕ_{ε}^{k} - ϕ^{k} ‖}_{C (\overline{Π_{N - 1} (ξ^{k})})} . \end{array}$ As a result, we finally obtain $\begin{array}{l} 0 & ⩽ lim_{ε \to 0} (sup_{x \in Ω Δ Ω_{ε}} d (x, \partial Ω)) = lim_{ε \to 0} (sup_{x \in Ω Δ Ω_{ε}} inf_{y \in \partial Ω} | x - y |_{R^{N}}) \\ = lim_{ε \to 0} (sup_{x \in Ω ∖ Ω_{ε}} inf_{y \in \partial Ω} | x - y |_{R^{N}} + sup_{x \in Ω_{ε} ∖ Ω} inf_{y \in \partial Ω} | x - y |_{R^{N}}) \\ ⩽ 2 \sum_{k = 1}^{m} lim_{ε \to 0} {‖ ϕ_{ε}^{k} - ϕ^{k} ‖}_{C (\overline{Π_{N - 1} (ξ^{k})})} = 0 . \end{array}$ □

We recall that the distance $μ (Ω_{1}, Ω_{2})$ between two open sets $Ω_{1} \subset D$ and $Ω_{2} \subset D$ which is defined as the Lebesgue measure of the symmetric difference $\begin{matrix} μ (Ω_{1}, Ω_{2}) = L^{N} (Ω_{1} Δ Ω_{2}), \end{matrix}$ coincides with the well known Ekeland metric in $L^{\infty} (D)$ applied to the characteristic functions (see [10,18]): $\begin{matrix} (2.2) & d_{E} (χ_{Ω_{1}}, χ_{Ω_{2}}) = L^{N} {x \in D | χ_{Ω_{1}} (x) \neq χ_{Ω_{2}} (x)} = ‖ χ_{Ω_{1}} - χ_{Ω_{2}} ‖_{L^{1} (D)} = μ (Ω_{1}, Ω_{2}) . \end{matrix}$ In view of these observations it is east to deduce the equivalence of the metrics $‖ χ_{Ω_{ε}} - χ_{Ω} ‖_{L^{1} (D)}$ and $‖ χ_{Ω_{ε}} - χ_{Ω} ‖_{L^{p} (D)}$ for $p \in [1, \infty)$ . Namely, we have the following result.
Proposition 2.2.
Let ${Ω_{ε}}_{0 < ε ⩽ ε_{0}}$ be a given sequence of open subdomains of D such that each of $Ω_{ε}$ has a rugous $Γ_{N, ε}$ -part of boundary. Then the corresponding sequence of characteristic functions ${χ_{Ω_{ε}}}_{0 < ε ⩽ ε_{0}}$ converges strongly to $χ_{Ω}$ in $L^{p} (D)$ for every $p \in [1, \infty)$ , i.e. the sequence ${Ω_{ε}}_{0 < ε ⩽ ε_{0}}$ strongly χ-converges to Ω as $ε \to 0$ .

The next point we are going to discuss here is about the shape-star property of the sets ${Ω_{ε}}_{0 < ε ⩽ ε_{0}}$ which plays a crucial role in substantiation of consistency of the original control problem (1.7)–(1.10) and uniqueness of its solution. The main question is to find out the conditions that should be imposed on the $Γ_{N, ε}$ -part of boundary such that the corresponding perturbed domain would be star-shaped. With that in mind we assume that the domain Ω satisfies the so-called enhanced star-shaped property with respect to some point $x_{0} \in int Ω$ , i.e. there exists a constant $γ > 0$ such that $\begin{matrix} (2.3) & \begin{matrix} (σ - x_{0}, ν (σ)) ⩾ 0 for H^{N - 1} -a.a. σ \in Γ_{D} and \\ (σ - x_{0}, ν (σ)) ⩾ γ for H^{N - 1} -a.a. σ \in Γ_{N} . \end{matrix} \end{matrix}$ Taking into account the representation (1.4), it follows from (2.3) that $\begin{matrix} inf_{σ \in Γ_{N}} min_{1 ⩽ k ⩽ m} (σ - x_{0}, \nabla Φ^{k} (σ)) ⩾ γ, \end{matrix}$ where $Φ^{k} (x) = x_{N} - ϕ^{k} (x_{1}, \dots, x_{N - 1})$ . Hence, we can suppose that there exists a constant $γ^{} > 0$ such that $\begin{matrix} (2.4) & cos (σ - x_{0}, ν (σ)) ⩾ inf_{σ \in Γ_{N}} min_{1 ⩽ k ⩽ m} \frac{(σ - x_{0}, \nabla Φ^{k} (σ))}{| σ - x_{0} | J ϕ^{k} (σ)} ⩾ γ^{}, for H^{N - 1} -a.a σ \in Γ_{N} . \end{matrix}$ Here, $\begin{matrix} (2.5) & J ϕ^{k} = \sqrt{1 + {(\frac{\partial ϕ^{k}}{\partial x_{1}})}^{2} + \dots + {(\frac{\partial ϕ^{k}}{\partial x_{N - 1}})}^{2}} . \end{matrix}$ As a consequence of the above reasoning, we arrive at the following result.
Proposition 2.3.
Let $Ω \subset D$ be an open domain satisfying the enhanced star-shaped property with respect to some point $x_{0} \in int Ω$ . Assume that for given small enough $ε > 0$ , the $Γ_{N, ε}$ -part of $\partial Ω_{ε}$ is rugous in the sense of Definition 1.1 . Then $Ω_{ε} \subset D$ is a star-shaped perturbation of Ω provided the following conditions hold true: $\begin{matrix} (2.6) & min_{1 ⩽ k ⩽ m} inf_{σ^{'} \in Π_{N - 1} (ξ^{k})} [\frac{(\nabla Φ_{ε}^{k} (σ_{ε}), \nabla Φ^{k} (σ))}{J ϕ^{k} (σ) J ϕ_{ε}^{k} (σ_{ε})} \frac{| σ_{ε} - x_{0} | | σ - x_{0} |}{(σ_{ε} - x_{0}, σ - x_{0})}] ⩾ \sqrt{1 - {(γ^{})}^{2}}, \end{matrix}$ where* $γ^{}$ is defined in (* 2.4 ), $\begin{matrix} σ = (σ^{'}, ϕ^{k} (σ^{'})) \in Γ_{N}, σ_{ε} = (σ^{'}, ϕ_{ε}^{k} (σ^{'})) \in Γ_{N, ε}, \end{matrix}$ and $\begin{array}{l} (2.7) & J ϕ_{ε}^{k} = \sqrt{1 + {(\frac{\partial ϕ_{ε}^{k}}{\partial x_{1}})}^{2} + \dots + {(\frac{\partial ϕ_{ε}^{k}}{\partial x_{N - 1}})}^{2}} = {| \nabla Φ_{ε}^{k} |}_{R^{N}}, \\ (2.8) & Φ_{ε}^{k} (x) = x_{N} - ϕ_{ε}^{k} (x_{1}, \dots, x_{N - 1}) . \end{array}$
Proof.
For given $ε > 0$ and each $k \in {1, \dots, m}$ and $σ^{'} \in Π_{N - 1} (ξ^{k})$ , we can define values $α_{i} \in [0, π]$ , $i = 1, \dots, 4$ , such that $\begin{array}{l} cos α_{1} = \frac{(σ_{ε} - x_{0}, \nabla Φ_{ε}^{k} (σ_{ε}))}{| σ_{ε} - x_{0} | J ϕ_{ε}^{k}}, cos α_{2} = \frac{(σ_{ε} - x_{0}, σ - x_{0})}{| σ_{ε} - x_{0} | | σ - x_{0} |}, \\ cos α_{3} = \frac{(\nabla Φ_{ε}^{k} (σ_{ε}), \nabla Φ^{k} (σ))}{J ϕ^{k} J ϕ_{ε}^{k}}, cos α_{4} = \frac{(σ - x_{0}, \nabla Φ^{k} (σ))}{| σ - x_{0} | J ϕ^{k}} . \end{array}$

Then, in view of the initial assumptions, we have: $α_{4} \in [0, arccos (γ^{})]$ by (2.4), and ${lim}_{ε \to 0} α_{2} = 0$ by Lemma 2.1. Hence, we can suppose that $α_{2} + α_{4} \in [0, \frac{π}{2}]$ for $ε > 0$ small enough. Since $α_{1} ⩽ α_{2} + α_{3} + α_{4}$ and, for each $σ^{'} \in Π_{N - 1} (ξ^{k})$ , there can be found an index $k \in {1, \dots, m}$ such that $(σ_{ε} - x_{0}, ν (σ_{ε})) = | σ_{ε} - x_{0} | J ϕ_{ε}^{k} cos α_{1}$ , it follows that the star-shaped property for the domain $Ω_{ε}$ with respect to the point $x_{0} \in int Ω$ holds true if only $α_{3} ⩽ \frac{π}{2} - α_{2} - α_{4}$ for each $σ^{'} \in Π_{N - 1} (ξ^{k})$ . From this, we finally deduce $\begin{matrix} \frac{(\nabla Φ_{ε}^{k} (σ_{ε}), \nabla Φ^{k} (σ))}{J ϕ^{k} (σ) J ϕ_{ε}^{k} (σ_{ε})} = cos α_{3} ⩾ sin (α_{2} + α_{4}) ⩾ \frac{(σ_{ε} - x_{0}, σ - x_{0})}{| σ_{ε} - x_{0} | | σ - x_{0} |} \sqrt{1 - {(γ^{})}^{2}} . \end{matrix}$ Then (2.6) is a straightforward consequence of the last inequality. □
Remark 2.1.
As follows from (2.6), the star-shaped property of the domains ${Ω_{ε}}_{0 < ε ⩽ ε_{0}}$ with $ε_{0} > 0$ small enough can be induced by the enhanced star-shaped property of non-perturbed domain Ω provided the surface area of the graph $Γ_{N, ε}$ over $Γ_{N}$ does not go to infinity locally in measure as ε tends to zero. Otherwise, we come into conflict with the inequality (2.6).

Taking this observation into account, we restrict our further consideration by the following type of “rugosity”along $Γ_{N, ε}$ -part of the boundary $\partial Ω_{ε}$ .
Definition 2.4.
We say that a sequence ${Ω_{ε}}_{0 < ε ⩽ ε_{0}} \subset D$ has a mild rugosity along $Γ_{N, ε}$ -part of the boundary if these domains admit the description (1.4)–(1.5) with properties (1.6) and ${ϕ_{ε}^{k} : R^{N - 1} \to R}_{0 < ε ⩽ ε_{0}}$ are the bounded sequences in $W^{1, \infty} (Π_{N - 1} (ξ^{k}))$ for each value $k \in {1, \dots, m}$ .
Remark 2.2.
It is clear that the mild rugosity property of parametrized domains ${Ω_{ε}}_{0 < ε ⩽ ε_{0}}$ implies the following characteristic feature $\begin{matrix} (2.9) & max_{1 ⩽ k ⩽ m} sup_{0 < ε ⩽ ε_{0}} {‖ \nabla Φ_{ε}^{k} ‖}_{L^{\infty} {(Π_{N} (ξ^{k}))}^{N}} < + \infty, \end{matrix}$ where the functions $Φ_{ε}^{k} : Π_{N} (ξ^{k}) \to R$ are defined by (2.8).

As an example of domains $Ω_{ε}$ with a mild rugosity of their boundaries, we can consider the following case. Let $Γ_{N}$ -part of the boundary of Ω be given locally as the graph of the smooth function, i.e. there exist a $δ > 0$ , a finite number of points ${ξ^{k}}_{k = 1}^{m} \subset Γ_{N}$ , and a collection of smooth functions ${ϕ^{k} : R^{N - 1} \to R}_{k = 1}^{m}$ such that $\begin{array}{l} Γ_{N} \subset ⋃_{k = 1}^{m} Π_{N} (ξ^{k}) and \\ Γ_{N} \cap Π_{N} (ξ^{k}) = {x \in Π_{N} (ξ^{k}) : x_{N} = ϕ^{k} (x_{1}, \dots, x_{N - 1})} \forall k = 1, \dots, m . \end{array}$ Then the domains $Ω_{ε}$ with above mentioned property can be represented as follows: $\partial Ω_{ε} = Γ_{D} \cup Γ_{N, ε}$ , $Γ_{N, ε} \subset ⋃_{k = 1}^{m} Π_{N} (ξ^{k})$ , and the following representation $\begin{matrix} Γ_{N, ε} \cap Π_{N} (ξ^{k}) = {x \in Π_{N} (ξ^{k}) : x_{N} = ϕ^{k} (x_{1}, \dots, x_{N - 1}) + ε \prod_{i = 1}^{N} sin (\frac{x_{i}}{ε^{α}}) φ (x_{1}, \dots, x_{N - 1})} \end{matrix}$ holds for all $k = 1, \dots, m$ , some positive smooth function $φ \in C_{0}^{\infty} (R^{N - 1})$ , and $0 < α ⩽ 1$ .
3. Asymptotic analysis of OCP (1.7)–(1.10)

The aim of this section is to provide asymptotic analysis of optimal control problem (1.7)–(1.10) as $ε \to 0$ and find out how the rugosity of the boundaries of domains $Ω_{ε}$ affects the optimal solutions $(u_{ε}^{0}, g_{ε}^{0}, y_{ε}^{0})$ in the limit. Note that if the small parameter $ε > 0$ is changed, then all components of the original control problem, including the ε-perturbed domain $Ω_{ε}$ , the control constraints set $A_{ad} (Γ_{N, ε})$ , the cost functional $J_{ε}$ , and the set of feasible solutions $Ξ_{ε}$ , where we seek its infimum, are changed as well. Moreover, the original problem (in a domain with a rugous control zone) for different values of the parameter ε “lives” in different functional spaces.

Let ${(u_{ε}, g_{ε}, y_{ε}) \in Ξ_{ε}}_{0 < ε ⩽ ε_{0}}$ be an arbitrary sequence of feasible solutions to the corresponding problem (1.7)–(1.10). By default, we suppose that each element of the sequence ${g_{ε} \in L^{q} (Ω_{ε})}_{0 < ε ⩽ ε_{0}}$ is extended outside of $Ω_{ε}$ by zero, so $g_{ε} \in L^{q} (D)$ for all $ε > 0$ . Since $u_{ε} \in A_{ad} (Γ_{N, ε})$ , it follows from (1.10) that there exists a prototype $v_{ε} \in W^{1, r} (D)$ , with $‖ v_{ε} ‖_{W^{1, r} (D)} ⩽ \hat{β}$ and $r > N$ , such that $v_{ε} \in C (\overline{D})$ and $u_{ε} = v_{ε} |_{Γ_{N, ε}}$ . In view of compactness of the embedding $W^{1, r} (D) ↪ C (\overline{D})$ , it follows that $\begin{array}{l} ‖ u_{ε} ‖_{L^{2} (Γ_{N, ε})} & ⩽ {‖ γ_{0} (v_{ε}) ‖}_{C (Γ_{N, ε})} {(\int_{Γ_{N, ε}} d H^{N - 1})}^{1 / 2} \\ ⩽ ‖ v_{ε} ‖_{C (D)} {(\sum_{k = 1}^{m} \int_{\partial Ω_{ε} \cap Π_{N} (ξ^{k})} d H^{N - 1})}^{1 / 2} \\ ⩽ C ‖ v_{ε} ‖_{W^{1, r} (D)} \sqrt{\sum_{k = 1}^{m} \int_{Π_{N - 1} (ξ^{k})} \sqrt{1 + {(\frac{\partial ϕ_{ε}^{k}}{\partial s_{1}})}^{2} + \dots + {(\frac{\partial ϕ_{ε}^{k}}{\partial s_{N - 1}})}^{2}} d s} \\ ⩽ C \hat{β} {(\sum_{k = 1}^{m} {‖ \nabla Φ_{ε}^{k} ‖}_{L^{\infty} {(Π_{N} (ξ^{k}))}^{N}} | Π_{N - 1} (ξ^{k}) |)}^{1 / 2} \\ = C \hat{β} {(\sum_{k = 1}^{m} {‖ \nabla Φ_{ε}^{k} ‖}_{L^{\infty} {(Π_{N} (ξ^{k}))}^{N}} δ^{N - 1})}^{1 / 2} \\ ⩽ C \hat{β} {(m δ^{N - 1})}^{1 / 2} max_{1 ⩽ k ⩽ m} sup_{0 < ε ⩽ ε_{0}} {‖ \nabla Φ_{ε}^{k} ‖}_{L^{\infty} {(Π_{N} (ξ^{k}))}^{N}}^{\frac{1}{2}} \\ (3.1) & = C^{*} < + \infty . \end{array}$ Then we can deduce from Theorem 1.4 the following a priori estimate $\begin{array}{l} ‖ y_{ε} ‖_{W_{0}^{1, p} (Ω_{ε}; Γ_{D})} & ⩽ C_{1} ‖ u_{ε} ‖_{L^{2} (Γ_{N, ε})} + C_{2} ‖ g_{ε} ‖_{L^{q} (D)} + C_{3} \\ (3.2) & ⩽ C_{1} C^{*} + C_{2} sup_{0 < ε ⩽ ε_{0}} ‖ g_{ε} ‖_{L^{q} (D)} + C_{3}, \end{array}$ where the constants $C_{i} > 0$ , $i = 1, 2, 3$ , are independent of $ε > 0$ .

Let us show that estimate (3.2) implies some compactness properties of the sequence ${y_{ε} \in W_{0}^{1, 2} (Ω_{ε}; Γ_{D})}_{0 < ε ⩽ ε_{0}}$ . With that in mind we make use of the following property. We say that a subset $K \subset D$ is semi-compact with respect to Ω if $\begin{matrix} (3.3) & Γ_{D} \subset \partial K, K \subset Ω, and Γ_{D} \cup K is a compact set . \end{matrix}$ It is clear that if the sequence of parametrized domains ${Ω_{ε}}_{0 < ε ⩽ ε_{0}} \subset D$ has a mild rugosity along $Γ_{N, ε}$ -part and $Ω_{ε} \to Ω$ as $ε \to 0$ in the sense of equality (2.1), then whatever semi-compact subset $K \subset Ω$ , we have: $K \subset Ω_{ε}$ for ε small enough (see Section 5.10 in [6] for the details).

Lemma 3.1.

Let ${(u_{ε}, g_{ε}, y_{ε}) \in Ξ_{ε}}_{0 < ε ⩽ ε_{0}}$ be an arbitrary sequence of feasible solution to the corresponding problem ( 1.7 )–( 1.10 ). Assume that the sequence of distributed controls ${g_{ε}}_{0 < ε ⩽ ε_{0}}$ is bounded in $L^{q} (D)$ , and the sequence ${Ω_{ε}}_{0 < ε ⩽ ε_{0}} \subset D$ has a mild rugosity along $Γ_{N, ε}$ -part of the boundary. Then there exist a sequence of indices ${ε_{k}}_{k \in N}$ with $ε_{k} \to 0$ as $k \to \infty$ , a sequence of prototypes ${v_{k}}_{k \in N} \subset W^{1, r} (D)$ with $‖ v_{k} ‖_{W^{1, r} (D)} ⩽ \hat{β}$ for all $k \in N$ , and a triplet $(u, g, y) \in A_{ad} (Γ_{N}) \times G_{ad} \times W_{0}^{1, 2} (Ω; Γ_{D})$ such that $\begin{array}{l} (3.4) & u_{ε_{k}} = v_{k} |_{Γ_{N, ε_{k}}}, v_{k} \to v in C (\overline{D}) as k \to \infty, u = v |_{Γ_{N}}, \\ (3.5) & g_{ε_{k}} ⇀ g in L^{q} (D) as k \to \infty, \\ (3.6) & f (y_{ε_{k}}) \to f (y) in L_{loc}^{1} (Ω), f (y) \in L^{1} (Ω), \\ (3.7) & \nabla y_{ε_{k}} ⇀ \nabla y in L_{loc}^{2} {(Ω)}^{N}, y_{ε_{k}} \to y in L_{loc}^{2} (Ω) as k \to \infty, \\ (3.8) & \nabla y_{ε_{k}} \to \nabla y strongly in L_{loc}^{r} {(Ω)}^{N} for any 1 ⩽ r < 2 as k \to \infty, \end{array}$ where the “locality” effect in ( 3.6 )–( 3.7 ) has the following sense: $\begin{array}{l} f (y_{ε_{k}}) \to f (y) in L^{1} (K), \nabla y_{ε_{k}} ⇀ \nabla y in L^{2} {(K)}^{N}, \\ y_{ε_{k}} \to y in L^{2} (K), and \nabla y_{ε_{k}} \to \nabla y in L^{r} {(K)}^{N} \end{array}$ for any semi-compact subset K of Ω.

Proof.

The assertion (3.4) is a direct consequence of the definition of the set $A_{ad} (Γ_{N, ε})$ and compactness of the embedding $W^{1, r} (D) ↪ C (\overline{D})$ for $r > N$ , whereas (3.5) immediately follows from the reflexivity of the space $L^{q} (D)$ . As for the properties (3.7)–(3.8), we fix any semi-compact set $K \subset Ω$ . Then it is easy to deduce from (3.2) that the sequence ${y_{ε}}_{0 < ε ⩽ ε_{0}}$ is bounded in $W_{0}^{1, 2} (K; Γ_{D})$ . Hence, there exists a weakly converging subsequence in $W_{0}^{1, 2} (K; Γ_{D})$ which converges strongly in $L^{2} (K)$ and almost everywhere in K. Since this conclusion remains valid for any semi-compact set $K \subset Ω$ , by a diagonalizing procedure we can construct a subsequence ${y_{ε_{k}}}_{k \in N}$ with $ε_{k} \to 0$ as $k \to \infty$ satisfying properties (3.7). Since a priori estimate (3.2) does not depend on K, we can suppose that $\begin{matrix} (3.9) & y \in L^{2} (Ω) and \nabla y \in L^{2} {(Ω)}^{N} . \end{matrix}$ Combining this fact with the trace property $y = 0$ on $Γ_{D}$ (it is the main reason why we involve the semi-compact approximations of Ω, see (3.3)), we deduce from (3.9) that $y \in W_{0}^{1, 2} (Ω; Γ_{D})$ . As for the property (3.7), we can utilize similar arguments to the proof of Proposition 1.5.

It remains to check the assertion (3.6). With that in mind we make use of the estimates (3.1) and (1.15). As a result, we have $\begin{array}{l} \int_{Ω_{ε}} y_{ε_{k}} f (y_{ε_{k}}) d x & ⩽ C_{1} ‖ u_{ε_{k}} ‖_{L^{2} (Γ_{N, ε})}^{2} + C_{2} ‖ g_{ε_{k}} ‖_{L^{q} (D)}^{2} + C_{3} \\ (3.10) & ⩽ C_{1} {(C^{*})}^{2} + C_{3} + C_{2} {(sup_{0 < ε ⩽ ε_{0}} ‖ g_{ε} ‖_{L^{q} (D)})}^{2} < + \infty, \end{array}$ where the constants $C_{i}$ are independent of $ε_{k}$ . Fixing again any semi-compact set $K \subset Ω$ , we see that $y_{ε_{k}} (x) \to y (x)$ almost everywhere in K. As a result, we have the pointwise convergence: $f (y_{ε_{k}}) \to f (y)$ almost everywhere in K. In order to deduce the strong convergence $f (y_{ε_{k}}) \to f (y)$ in $L^{1} (K)$ , it is enough to apply estimate (3.10) and Lebesgue’s Convergence Theorem (see the proof of Proposition 1.6 in [8]). Since this conclusion remains valid for any semi-compact subset $K \subset Ω$ , the expected property (3.6) immediately follows from the diagonalizing procedure. □

For our further analysis, we assume that there exists a collection of functions $\begin{matrix} {J_{0}^{k} \in L^{\infty} (Π_{N - 1} (ξ^{k}))}_{k = 1}^{m} \end{matrix}$ such that $\begin{matrix} (3.11) & J ϕ_{ε}^{k} : = \sqrt{1 + {(\frac{\partial ϕ_{ε}^{k}}{\partial x_{1}})}^{2} + \dots + {(\frac{\partial ϕ_{ε}^{k}}{\partial x_{N - 1}})}^{2}} \overset{*}{⇀} J_{0}^{k} in L^{\infty} (Π_{N - 1} (ξ^{k})) as ε \to 0 . \end{matrix}$

Since $ϕ_{ε}^{k} : R^{N - 1} \to R \in C^{1, 1} (R^{N - 1})$ for each $k \in {1, \dots, m}$ , it obviously follows from condition (1.6) that the sequence ${J ϕ_{ε}^{k}}_{0 < ε ⩽ ε_{0}}$ is bounded in $L^{\infty} (Π_{N - 1} (ξ^{k}))$ provided the domains ${Ω_{ε}}_{0 < ε ⩽ ε_{0}}$ possess the mild rugosity property in the sense of Definition 2.4 (see also Remark 2.2). Hence, the existence of above mentioned functions ${J_{0}^{k}}_{k = 1}^{m}$ is a direct consequence of Banach-Alaoglu theorem.

Further we associate with the non-perturbed domain Ω the following optimal control problem $\begin{matrix} (3.12) & Minimize J_{0} (u, g, y) = \frac{1}{2} \int_{Ω} | y - y_{d} |^{2} d x + \frac{1}{2} \int_{Γ_{N}} | u |^{2} Γ_{0} d H^{N - 1} + \frac{1}{q} \int_{Ω} | g - g_{d} |^{q} d x, \end{matrix}$ subject to constraints $\begin{array}{l} (3.13) & - Δ y = f (y) + g in Ω, \\ (3.14) & y = 0 on Γ_{D}, \partial_{ν} y = Γ_{0} u on Γ_{N}, \\ (3.15) & g \in G_{ad} = L^{q} (D), y \in W_{0}^{1, 2} (Ω; Γ_{D}), \\ (3.16) & u \in A_{ad} (Γ_{N}) = {v |_{Γ_{N}} : v \in W^{1, r} (D), ‖ v ‖_{W^{1, r} (D)} ⩽ \hat{β}}, \end{array}$ where the function $Γ_{0} : Γ_{N} \to R$ is defined as follows $\begin{matrix} (3.17) & \begin{matrix} Γ_{0} (σ) = Γ_{0} (x^{'}, ϕ^{k} (x^{'})) on Π_{N} (ξ^{k}) \cap Γ_{N}, \forall k \in {1, \dots, m}, \\ Γ_{0} (x^{'}, ϕ^{k} (x^{'})) : = \frac{J_{0}^{k} (x^{'})}{J ϕ^{k} (x^{'})} = \frac{J_{0}^{k} (x^{'})}{| \nabla Φ^{k} (x^{'}) |_{R^{N}}} . \end{matrix} \end{matrix}$

By analogy with the parametrized problem (1.7)–(1.10) (see Definition 1.2), we say that $(u, g, y) \in L^{2} (Γ_{N}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω; Γ_{D})$ is a feasible solution to the optimal control problem (3.12)–(3.16) if:

$(u, g)$ are admissible controls, i.e. $u \in A_{ad} (Γ_{N})$ and $g \in G_{ad}$ ;

$f (y) \in L^{1} (Ω)$ ;

the integral identity $\begin{matrix} (3.18) & \int_{Ω} (\nabla y, \nabla φ) d x = \int_{Ω} f (y) φ d x + \int_{Γ_{N}} u Γ_{0} φ d H^{N - 1} + \int_{Ω} g φ d x \end{matrix}$ holds for every test function $φ \in C_{0}^{\infty} (R^{N}; Γ_{D})$ ;

the following integral inequality $\begin{array}{l} \int_{Ω} | \nabla y |^{2} d x & ⩽ \frac{2 N}{N - 2} \int_{Ω} F (y) d x - \frac{2}{N - 2} \int_{Ω} g (x - x_{0}, \nabla y) d x \\ (3.19) & + \frac{3 diam D}{N - 2} \int_{Γ_{N}} | u |^{2} Γ_{0} d H^{N - 1}, \end{array}$ holds true with some $x_{0} \in int Ω$ .

We denote by $Ξ_{0}$ the set of all feasible solutions to the problem (3.12)–(3.16).

It is worth to indicate the following result that can be established by analogy with the proof of Theorem 1.3 in [8].

Theorem 3.2.

Let $y_{d} \in L^{2} (Ω)$ , $f \in C_{loc}^{1} (R)$ , and $g_{d} \in L^{q} (D)$ with $q ⩾ 2$ be arbitrary distributions. If the domain Ω is star-shaped with respect to some point $x_{0} \in int Ω$ , then the set of feasible solutions $Ξ_{0}$ is nonempty, optimal control problem ( 3.12 )–( 3.16 ) is consistent and has at least one solution $(u^{0}, g^{0}, y^{0}) \in Ξ_{0}$ , $J_{0} (u^{0}, g^{0}, y^{0}) = {inf}_{(u, g, y) \in Ξ_{0}} J_{0} (u, g, y)$ . If in addition to the above mentioned conditions, the function $f : R \to (0, \infty)$ is strictly convex, then the optimal solution to the problem ( 3.12 )–( 3.16 ) is unique.

Hereinafter we associate with the optimal control problems (3.12)–(3.16) and (1.7)–(1.10) the following constrained minimization problems $(CM P_{ε})$ : $\begin{matrix} (3.20) & ({CMP}_{ε}) : ⟨ inf_{(u, g, y) \in Ξ_{ε}} J_{ε} (u, g, y) ⟩ \end{matrix}$ and $\begin{matrix} (3.21) & ({CMP}_{0}) : ⟨ inf_{(u, g, y) \in Ξ_{0}} J_{0} (u, g, y) ⟩, \end{matrix}$ respectively.

In order to study the asymptotic behavior of a family of $(CM P_{ε})$ , the passage to the limit in (3.20) as the small parameter ε tends to zero has to be realized. However, the sequence of constrained minimization problems (3.20) lives in variable spaces $L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ . In order to formalize the convergence in the scale of spaces $L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ , we make use of the conditions (3.4)–(3.7).

Definition 3.3.

We say that a sequence $\begin{matrix} {(u_{ε}, g_{ε}, y_{ε}) \in A_{ad} (Γ_{N, ε}) \times G_{ad} \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D})}_{0 < ε ⩽ ε_{0}}, \end{matrix}$ μ-converges to a triplet $(u, g, y)$ in the scale of spaces $L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ if (in the sequel, we use the following notation for this convergence $(u_{ε}, g_{ε}, y_{ε}) \overset{μ}{⟶} (u, g, y)$ in $L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D})$ ):

$(u, g, y) \in L^{2} (Γ_{N}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω; Γ_{D})$ ;

The domains ${Ω_{ε}}_{0 < ε ⩽ ε_{0}}$ possess the mild rugosity property in the sense of Definition 2.4 and, hence, $Ω_{ε} \to Ω$ as $ε \to 0$ in the sense of relation (2.1);

there exists a sequence of prototypes $\begin{matrix} {v_{ε}}_{0 < ε ⩽ ε_{0}} \subset W^{1, r} (D) with ‖ v_{ε} ‖_{W^{1, r} (D)} ⩽ \hat{β} for all 0 < ε ⩽ ε_{0} \end{matrix}$ such that $\begin{array}{l} (3.22) & u_{ε} = v_{ε} |_{Γ_{N, ε}}, v_{ε} \to v in C (\overline{D}) as ε \to 0, u = v |_{Γ_{N}}, \\ (3.23) & g_{ε} ⇀ g in L^{q} (D) as ε \to 0, \\ (3.24) & f (y_{ε}) \to f (y) in L_{loc}^{1} (Ω), f (y) \in L^{1} (Ω), \\ (3.25) & \nabla y_{ε} ⇀ \nabla y in L_{loc}^{2} {(Ω)}^{N} and y_{ε} \to y in L_{loc}^{2} (Ω), \\ (3.26) & \nabla y_{ε} \to \nabla y strongly in L_{loc}^{r} {(Ω)}^{N} for any 1 ⩽ r < 2 . \end{array}$

In the next definition we emphasize the fact that the boundary value problem (3.13)–(3.15) is ill-posed and we cannot guarantee that for a given admissible controls $(u, g)$ there exists a unique feasible solution $(u, g, y) \in Ξ_{0}$ that can be attainable in the limit by some feasible solutions ${(u_{ε}, g_{ε}, y_{ε}) \in Ξ_{ε}}$ to the parametrized problems (1.7)–(1.10).

Definition 3.4.

We say that a feasible solution $(u, g, y) \in Ξ_{0}$ is regular for optimal control problem (3.12)–(3.16) (or, what is equivalent, for the constrained minimization problem (3.21)) if there exists a sequence $\begin{matrix} {(v_{ε}, y_{ε}) \in W^{1, r} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D})}_{0 < ε ⩽ ε_{0}} \end{matrix}$ such that $\begin{matrix} (3.27) & \begin{matrix} v_{ε} \to v in C (\overline{D}) as ε \to 0, ‖ v_{ε} ‖_{W^{1, r} (D)} ⩽ \hat{β} for all 0 < ε ⩽ ε_{0}, \\ y_{ε} ⇀ y in W_{loc}^{1, 2} (Ω) as ε \to 0, \\ (u_{ε}, g, y_{ε}) \in Ξ_{ε} \forall 0 < ε ⩽ ε_{0} with u_{ε} = v_{ε} |_{Γ_{N, ε}}, \end{matrix} \end{matrix}$ where $v \in W^{1, r} (D)$ is such that $v |_{Γ_{N}} = u$ and $‖ v ‖_{W^{1, r} (D)} ⩽ β$ .

Remark 3.1.

As immediately follows from Theorem 3.2 and Lemma 3.1, the set $Ξ_{0}$ contains a non-empty subset of regular feasible solutions provided Ω is a star-shaped domain. However, it is unknown whether all feasible solutions $(u, g, y) \in Ξ_{0}$ possess this property.

Another characteristic feature of regular feasible solutions is related with property (3.27). Indeed, if $(u, g, y) \in Ξ_{0}$ and ${y_{ε} \in W_{0}^{1, 2} (Ω_{ε}; Γ_{D})}_{0 < ε ⩽ ε_{0}}$ are such that $(u_{ε}, g, y_{ε}) \in Ξ_{ε}$ $\forall 0 < ε ⩽ ε_{0}$ with $u_{ε} = v |_{Γ_{N, ε}}$ , then estimate (3.2) implies that $\begin{array}{rcl} sup_{ε > 0} ‖ y_{ε} ‖_{W_{0}^{1, p} (Ω_{ε}; Γ_{D})} & ⩽ & C_{1} ‖ v ‖_{L^{2} (Γ_{N, ε})} + C_{2} ‖ g ‖_{L^{q} (D)} + C_{3} \\ (3.28) & \overset{by (3.1)}{⩽} & C_{1} C^{*} + C_{2} ‖ g ‖_{L^{q} (D)} + C_{3}, \end{array}$ where the constants $C_{i} > 0$ , $i = 1, 2, 3$ , are independent of $ε > 0$ .

Utilizing this fact and proceeding exactly as in the proof of Lemma 3.1, we can deduce the existence of a subsequence ${y_{ε_{k}}}_{k \in N}$ of ${y_{ε}}_{0 < ε ⩽ ε_{0}}$ and element $\hat{y} \in W_{0}^{1, 2} (Ω; Γ_{D})$ (in general, $\hat{y} \neq y$ ) such that $\begin{matrix} (3.29) & \begin{matrix} \nabla y_{ε_{k}} ⇀ \nabla \hat{y} in L_{loc}^{2} {(Ω)}^{N}, y_{ε_{k}} \to \hat{y} in L_{loc}^{2} (Ω), \\ \nabla y_{ε_{k}} \to \nabla \hat{y} strongly in L_{loc}^{r} {(Ω)}^{N} for any 1 ⩽ r < 2 \end{matrix} \end{matrix}$ as $ε_{k} \to 0$ . Hence, $y_{ε} ⇀ \hat{y}$ in $W_{loc}^{1, 2} (Ω)$ , but not to y. Thus, condition (3.27) can be omitted in Definition 3.4 provided the mentioned feasible solution $(u, g, y) \in Ξ_{0}$ is unique for given controls $(u, g)$ .

Since the main goal of this section is to study the asymptotic behaviour of optimal solutions to the parametrized problems (1.7)–(1.10) as small parameter ε tends to zero, we begin with the study of limit passage in the sequence of constrained minimization problems (3.20). To do so, we note that the expression “passing to the limit in (3.20) as $ε \to 0$ ” means that we have to find a kind of “limit cost functional” I and “limit set of feasible solutions” Ξ with a clearly defined structure such that the limit object $⟨ {inf}_{(u, g, y) \in Ξ} I (u, g, y) ⟩$ can be interpreted as some optimal control problem. Following the scheme of the direct variational convergence [18], we adopt the following concept for the convergence of minimization problems (3.20) in variable spaces.

Definition 3.5.

A constrained minimization problem $⟨ {inf}_{(u, g, y) \in Ξ} J (u, g, y) ⟩$ is the variational limit of the sequence (3.20) as $ε \to 0$ , $\begin{matrix} in symbols, ⟨ inf_{(u, g, y) \in Ξ_{ε}} J_{ε} (u, g, y) ⟩ \overset{Var}{\to_{ε \to 0}^{}} ⟨ inf_{(u, g, y) \in Ξ} J (u, g, y) ⟩, \end{matrix}$ if the following conditions are satisfied:

If the sequences ${ε_{k}}_{k \in N}$ and ${(u_{k}, g_{k}, y_{k})}_{k \in N}$ are such that $ε_{k} \to 0$ as $k \to \infty$ , $(u_{k}, g_{k}, y_{k}) \in Ξ_{ε_{k}}$ $\forall k \in N$ , and $(u_{k}, g_{k}, y_{k}) \overset{μ}{⟶} (u, g, y)$ in $L^{2} (Γ_{N, ε_{k}}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε_{k}}; Γ_{D})$ , then $\begin{matrix} (3.30) & (u, g, y) \in Ξ; J (u, g, y) ⩽ \underset{k \to \infty}{lim inf} J_{ε_{k}} (u_{k}, g_{k}, y_{k}); \end{matrix}$

For every regular feasible solution $(u, g, y) \in Ξ \subset L^{2} (Γ_{N}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω; Γ_{D})$ there is a sequence ${(u_{ε}, g_{ε}, y_{ε}) \in Ξ_{ε}}_{0 < ε ⩽ ε_{0}}$ (called a realizing sequence) such that $\begin{array}{l} (3.31) & J (u, g, y) ⩾ \underset{ε \to 0}{lim sup} J_{ε} (u_{ε}, g_{ε}, y_{ε}), \\ (3.32) & (u_{ε}, g_{ε}, y_{ε}) \overset{μ}{⟶} (u, g, y) in L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D}) . \end{array}$

Our next intension is to show that $\begin{matrix} (3.33) & ⟨ inf_{(u, g, y) \in Ξ_{ε}} J_{ε} (u, g, y) ⟩ \overset{Var}{\to_{ε \to 0}^{}} ⟨ inf_{(u, g, y) \in Ξ_{0}} J_{0} (u, g, y) ⟩ . \end{matrix}$ We begin with some auxiliary results.

Lemma 3.6.

Let ${(u_{j}, g_{j}, y_{j})}_{j \in N}$ be a sequence such that $(u_{j}, g_{j}, y_{j}) \in Ξ_{ε_{j}}$ $\forall j \in N$ , $ε_{j} \to 0$ as $j \to \infty$ , and $(u_{j}, g_{j}, y_{j}) \overset{μ}{⟶} (u, g, y)$ in $L^{2} (Γ_{N, ε_{j}}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε_{j}}; Γ_{D})$ as $j \to \infty$ . Let $Ψ \in C (\overline{D} \times R)$ be an arbitrary function. Then $\begin{matrix} (3.34) & lim_{j \to \infty} \int_{Γ_{N, ε_{j}}} Ψ (x, u_{j} (x)) d H^{N - 1} = \int_{Γ_{N}} Ψ (x, u (x)) Γ_{0} d H^{N - 1}, \end{matrix}$ where the function $Γ_{0} : Γ_{N} \to R$ is defined in ( 3.17 ).

Proof.

Using the standard partition of unity and localization arguments, the boundary integral over $Γ_{N, ε}$ in the left-hand side of (3.34) can be expressed as a sum of boundary integrals over $Γ_{N, ε} \cap Π_{N} (ξ^{j})$ . Then, using the area formula, we get $\begin{array}{l} \int_{Γ_{N, ε_{j}}} Ψ (x, u_{j} (x)) d H^{N - 1} & = \sum_{k = 1}^{m} \int_{Γ_{N, ε_{j}} \cap Π_{N} (ξ^{k})} Ψ (x, u_{j} (x)) d H^{N - 1} \\ = \sum_{k = 1}^{m} \int_{Γ_{N, ε_{j}} \cap Π_{N} (ξ^{k})} Ψ (x, v_{j} (x)) d H^{N - 1} \\ (3.35) & = \sum_{k = 1}^{m} \int_{Π_{N - 1} (ξ^{k})} Ψ (x^{'}, ϕ_{ε_{j}}^{k} (x^{'}), v_{j} (x^{'}, ϕ_{ε_{j}}^{k} (x^{'}))) J ϕ_{ε_{j}}^{k} (x^{'}) d x^{'}, \end{array}$ where the Jacobian $J ϕ_{ε_{j}}^{k}$ is defined in (2.7). Since $\begin{array}{l} ϕ_{ε_{j}}^{k} \to ϕ^{k} in C (\overline{Π_{N - 1} (ξ^{k})}) as ε_{j} \to 0 (see (1.6)), \\ v_{j} \to v in C (\overline{D}) as j \to 0, \end{array}$ and Ψ is a continuous function on $\overline{Π_{N} (ξ^{k})} \times R$ , it follows that the convergence $\begin{matrix} Ψ (x^{'}, ϕ_{ε_{j}}^{k} (x^{'}), v_{j} (x^{'}, ϕ_{ε_{j}}^{k} (x^{'}))) \to Ψ (x^{'}, ϕ^{k} (x^{'}), v (x^{'}, ϕ^{k} (x^{'}))) as j \to \infty \end{matrix}$ is strong in $C (\overline{Π_{N - 1} (ξ^{k})})$ and, hence, it is strong in $L^{1} (Π_{N - 1} (ξ^{k}))$ . Then property (3.11) and the area formula imply that $\begin{array}{l} lim_{j \to \infty} \int_{Π_{N - 1} (ξ^{k})} Ψ (x^{'}, ϕ_{ε_{j}}^{k} (x^{'}), v_{j} (x^{'}, ϕ_{ε_{j}}^{k} (x^{'}))) J ϕ_{ε_{j}}^{k} (x^{'}) d x^{'} \\ = \int_{Π_{N - 1} (ξ^{k})} Ψ (x^{'}, ϕ^{k} (x^{'}), v (x^{'}, ϕ^{k} (x^{'}))) J_{0}^{k} (x^{'}) d x^{'} \\ = \int_{Π_{N - 1} (ξ^{k})} Ψ (x^{'}, ϕ^{k} (x^{'}), v (x^{'}, ϕ^{k} (x^{'}))) Γ_{0} (x^{'}, ϕ^{k} (x^{'})) J ϕ^{k} (x^{'}) d x^{'} \\ = \int_{Γ_{N} \cap Π_{N} (ξ^{k})} Ψ (x, v (x)) Γ_{0} (x) d H^{N - 1} \\ (3.36) & = \int_{Γ_{N} \cap Π_{N} (ξ^{k})} Ψ (x, u (x)) Γ_{0} (x) d H^{N - 1} . \end{array}$ As a result, assertion (3.34) is a direct consequence of (3.36) and representation (3.35). □

Lemma 3.7.

Let ${(u_{j}, g_{j}, y_{j})}_{j \in N}$ be a sequence with properties indicated in Lemma 3.6 . Then, for every test function $φ \in C_{0}^{\infty} (R^{N}; Γ_{D})$ , we have the following equalities $\begin{array}{l} (3.37) & lim_{j \to \infty} \int_{Ω_{ε_{j}}} (\nabla y_{j}, \nabla φ) d x = \int_{Ω} (\nabla y, \nabla φ) d x, \\ (3.38) & lim_{j \to \infty} \int_{Ω_{ε_{j}}} f (y_{j}) φ d x = \int_{Ω} f (y) φ d x, \\ (3.39) & lim_{j \to \infty} \int_{Ω_{ε_{j}}} | y_{j} - y_{d} |^{2} d x = \int_{Ω} | y - y_{d} |^{2} d x . \end{array}$

Proof.

Let $Ω^{'}$ be any semi-compact subset of Ω. Then for $j \in N$ large enough, we have $Ω^{'} \subset Ω$ and $Ω^{'} \subset Ω_{ε_{j}}$ . Hence, $\begin{array}{l} | \int_{Ω_{ε_{j}}} (\nabla y_{j}, \nabla φ) d x - \int_{Ω} (\nabla y, \nabla φ) d x | \\ ⩽ | \int_{Ω^{'}} (\nabla y_{j}, \nabla φ) d x - \int_{Ω^{'}} (\nabla y, \nabla φ) d x | \\ + | \int_{Ω_{ε_{j}} ∖ Ω^{'}} (\nabla y_{j}, \nabla φ) d x | + | \int_{Ω ∖ Ω^{'}} (\nabla y, \nabla φ) d x | \\ ⩽ \int_{Ω^{'}} (\nabla y_{j} - \nabla y, \nabla φ) d x + ‖ y_{j} ‖_{W_{0}^{1, 2} (Ω_{ε_{j}}; Γ_{D})} ‖ \nabla φ ‖_{L^{2} (Ω_{ε_{j}} ∖ Ω^{'})} \\ + ‖ y ‖_{W_{0}^{1, 2} (Ω; Γ_{D})} ‖ \nabla φ ‖_{L^{2} (Ω ∖ Ω^{'})} \\ (3.40) & = I_{1} + I_{2} + I_{3} . \end{array}$ As follows from Lemma 3.1 and definition of the μ-convergence, the condition $\nabla y_{ε_{j}} ⇀ \nabla y$ in $L_{loc}^{2} {(Ω)}^{N}$ implies that $\nabla y_{ε_{j}} ⇀ \nabla y$ in $L^{2} {(Ω^{'})}^{N}$ . Hence, the term $I_{1}$ tends to zero as $j \to \infty$ . As for the terms $I_{2}$ and $I_{3}$ , we note that $\begin{matrix} I_{2} ⩽ sup_{j \to 0} ‖ y_{j} ‖_{W_{0}^{1, p} (Ω_{ε_{j}}; Γ_{D})} ‖ \nabla φ ‖_{L^{2} (Ω_{ε_{j}} ∖ Ω^{'})} \overset{be (3.2)}{⩽} C ‖ \nabla φ ‖_{L^{2} (Ω_{ε_{j}} ∖ Ω^{'})}, \end{matrix}$ and $‖ y ‖_{W_{0}^{1, 2} (Ω; Γ_{D})} < + \infty$ by properties of the μ-convergence (see item (i) in Definition 3.3). Since the choice of $Ω^{'}$ is arbitrary, it follows that the terms $I_{2}$ and $I_{3}$ can be made as small as we want. Thus, relation (3.37) immediately follows from (3.40).

In order to prove equality (3.38), we can apply similar arguments. Namely, for an arbitrary semi-compact subset $Ω^{'}$ of Ω such that $Ω^{'} \subset Ω$ and $Ω^{'} \subset Ω_{ε_{j}}$ for $j \in N$ large enough, we have $\begin{array}{l} | \int_{Ω_{ε_{j}}} f (y_{j}) φ d x - \int_{Ω} f (y) φ d x | & ⩽ | \int_{Ω^{'}} (f (y_{j}) - f (y)) φ d x | \\ + | \int_{Ω_{ε_{j}} ∖ Ω^{'}} f (y_{j}) φ d x | + | \int_{Ω ∖ Ω^{'}} f (y) φ d x | \\ ⩽ \int_{Ω^{'}} | f (y_{j}) - f (y) | d x ‖ φ ‖_{C (\overline{D})} \\ (3.41) & + | \int_{Ω_{ε_{j}} ∖ Ω^{'}} f (y_{j}) φ d x | + \int_{Ω ∖ Ω^{'}} | f (y) | d x ‖ φ ‖_{C (\overline{D})} . \end{array}$ Let us show that for a given $δ > 0$ (which is assumed to be small enough), there are $j_{0} \in N$ and $Ω^{'}$ such that $\begin{array}{l} \int_{Ω^{'}} | f (y_{j}) - f (y) | d x < \frac{δ}{3 ‖ φ ‖_{C (\overline{D})}}, \\ \int_{Ω ∖ Ω^{'}} | f (y) | d x < \frac{δ}{3 ‖ φ ‖_{C (\overline{D})}}, | \int_{Ω_{ε_{j}} ∖ Ω^{'}} f (y_{j}) φ d x | < \frac{δ}{3} \end{array}$ for all $j > j_{0}$ .

Since $f (y_{j}) \to f (y)$ strongly in $L^{1} (Ω^{'})$ (see property (3.6)), it follows that the first estimate holds true for all $j > j_{1} (δ)$ with some $j_{1} (δ) \in N$ . The property (3.6) also indicates that $f (y) \in L^{1} (Ω)$ . Then by the absolute continuity of the Lebesgue integral, there exists a semi-compact subset $Ω^{'} (δ)$ of Ω such that the integral $\int_{Ω ∖ Ω^{'} (δ)} | f (y) | d x$ is less than $δ / (3 ‖ φ ‖_{C (\overline{D})})$ . As for the last estimate, it remains to make use of the equi-integrability property of the sequence ${f (y_{j})}_{j \in N}$ . As a result, utilizing these estimates in (3.41), we deduce that $\begin{matrix} | \int_{Ω_{ε_{j}}} f (y_{j}) φ d x - \int_{Ω} f (y) φ d x | < δ . \end{matrix}$ From this equality (3.38) easily follows. Since relation (3.39) can be established in the similar manner as we did it with (3.37), this concludes the proof. □

We are now in a position to show that optimal control problem (3.12)–(3.16) is the variational limit of optimization problems (1.7)–(1.10) as $ε \to 0$ in the sense of relation (3.33).

Theorem 3.8.

Let $Ω \subset D$ be an open domain satisfying the enhanced star-shaped property with respect to some point $x_{0} \in int Ω$ . Then assertion (d) of Definition 3.5 is valid for $Ξ = Ξ_{0}$ and $I = I_{0}$ .

Proof.

Let ${(u_{k}, g_{k}, y_{k})}_{k \in N}$ be an arbitrary sequence with properties: $ε_{k} \to 0$ as $k \to \infty$ , $(u_{k}, g_{k}, y_{k}) \in Ξ_{ε_{k}}$ $\forall k \in N$ , and $(u_{k}, g_{k}, y_{k}) \overset{μ}{⟶} (u, g, y)$ in $L^{2} (Γ_{N, ε_{k}}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε_{k}}; Γ_{D})$ . To begin with, let us show that in this case the inequality (3.30)₂ is valid. Let $Ω^{'}$ be an arbitrary semi-compact subset of Ω such that $Ω^{'} \subset Ω$ and $Ω^{'} \subset Ω_{ε_{k}}$ for $k \in N$ large enough. Since $\begin{array}{l} J_{ε_{k}} (u_{k}, g_{k}, y_{k}) & = \frac{1}{2} \int_{Ω_{ε_{k}}} | y_{k} - y_{d} |^{2} d x + \frac{1}{2} \int_{Γ_{N, ε_{k}}} | u_{k} |^{2} d H^{N - 1} \\ + \frac{1}{q} \int_{Ω_{ε_{k}}} | g_{k} - g_{d} |^{q} d x = J_{1, k} + J_{2, k} + J_{3, k}, \end{array}$ where $\begin{array}{l} J_{1, k} = \frac{1}{2} \int_{Ω^{'}} | y_{k} - y_{d} |^{2} d x + \frac{1}{q} \int_{Ω^{'}} | g_{k} - g_{d} |^{q} d x, \\ J_{2, k} = \frac{1}{2} \int_{Γ_{N, ε_{k}}} | u_{k} |^{2} d H^{N - 1}, \\ J_{3, k} = \int_{Ω_{ε_{k}} ∖ Ω^{'}} [\frac{1}{2} | y_{k} - y_{d} |^{2} + \frac{1}{q} | g_{k} - g_{d} |^{q}] d x, \end{array}$ we see that the relation $\begin{array}{l} \underset{k \to \infty}{lim inf} J_{1, k} & = \underset{k \to \infty}{lim inf} [\frac{1}{2} \int_{Ω^{'}} | y_{k} - y_{d} |^{2} d x + \frac{1}{q} \int_{Ω^{'}} | g_{k} - g_{d} |^{q} d x] \\ ⩾ \frac{1}{2} \int_{Ω^{'}} | y - y_{d} |^{2} d x + \frac{1}{q} \int_{Ω^{'}} | g - g_{d} |^{q} d x \\ (3.42) & = \frac{1}{2} \int_{Ω} | y - y_{d} |^{2} d x + \frac{1}{q} \int_{Ω} | g - g_{d} |^{q} d x - J_{4}, \end{array}$ with $\begin{matrix} J_{4} = \frac{1}{2} \int_{Ω ∖ Ω^{'}} | y - y_{d} |^{2} d x + \frac{1}{q} \int_{Ω ∖ Ω^{'}} | g - g_{d} |^{q} d x, \end{matrix}$ is a direct consequence of μ-convergence and lower semi-continuity of $L^{2}$ -norms with respect to the weak convergence, and $\begin{matrix} (3.43) & lim_{k \to \infty} \int_{Γ_{N, ε_{k}}} | u_{k} |^{2} d H^{N - 1} = \int_{Γ_{N}} | u |^{2} Γ_{0} d H^{N - 1} \end{matrix}$ by Lemma 3.6.

Now we look at the terms $J_{3, k}$ and $J_{4}$ . Proceeding exactly as in the corresponding part of the proof of Lemma 3.7, it can be shown that for an arbitrary given $δ > 0$ there exist $k^{0} \in N$ and a semi-compact subset $Ω^{'} (δ)$ of Ω such that $\begin{matrix} (3.44) & | J_{3, k} - J_{4} | < δ, \forall k > k^{0} . \end{matrix}$ Utilizing relations (3.42)–(3.43), we finally obtain $\begin{array}{l} (3.45) & \underset{k \to \infty}{lim inf} J_{ε_{k}} (u_{k}, g_{k}, y_{k}) ⩾ J_{0} (u, g, y) + \underset{k \to \infty}{lim inf} (J_{3, k} - J_{4}) . \end{array}$ Thus, in view of (3.44), the inequality (3.30)₂ follows.

It remains to show that $(u, g, y) \in Ξ$ . Proceeding much as in the proof of Lemma 3.1, we deduce that $(u, g)$ are admissible controls for the limit problem, i.e. $u \in A_{ad} (Γ_{N})$ , $g \in G_{ad}$ , and, moreover, $f (y) \in L^{1} (Ω)$ . In order to establish the integral identity (3.18), it is enough to fix an arbitrary test function $φ \in C_{0}^{\infty} (R^{N}; Γ_{D})$ and realize the limit passage in relation $\begin{array}{l} (3.46) & \int_{Ω_{ε_{k}}} (\nabla y_{k}, \nabla φ) d x & = \int_{Ω_{ε_{k}}} φ f (y_{k}) d x + \int_{Γ_{N, ε_{k}}} φ u_{k} d H^{N - 1} + \int_{Ω_{ε_{k}}} g_{k} φ d x \end{array}$ as $k \to \infty$ , using Lemmas 3.6 and 3.7.

To conclude the proof, we have to show that the limit triple $(u, g, y)$ satisfies the integral inequality (3.19). To begin with, we note that in view of Proposition 2.3, we may suppose that each element of the given sequence of parametrized domains ${Ω_{ε_{k}}}_{k \in N}$ possesses the star-shaped property with respect to some point $x_{0} \in int Ω$ that does not depend on k. Hence, for each $k \in N$ , we have $\begin{array}{l} \int_{Ω_{ε_{k}}} | \nabla y_{k} |^{2} d x & ⩽ \frac{2 N}{N - 2} \int_{Ω_{ε_{k}}} F (y_{k}) d x - \frac{2}{N - 2} \int_{Ω_{ε_{k}}} g_{k} (x - x_{0}, \nabla y_{k}) d x \\ (3.47) & + \frac{3 diam D}{N - 2} \int_{Γ_{N, ε_{k}}} | u_{k} |^{2} d H^{N - 1} . \end{array}$

Let $Ω^{'}$ be an arbitrary semi-compact subset of Ω such that $Ω^{'} \subset Ω$ and $Ω^{'} \subset Ω_{ε_{k}}$ for $k \in N$ large enough. Then the sequence ${\nabla y_{k}}_{k \in N}$ is bounded in $L^{2} {(Ω^{'})}^{N}$ . Therefore, it follows from (1.29) and Proposition 1.5 that $\begin{array}{l} (3.48) & \nabla y_{k_{n}} \to \nabla y strongly in L^{q^{'}} {(Ω^{'})}^{N} with q^{'} = q / (q - 1) . \end{array}$ Proceeding much as in the proof of Lemma 3.7 (see also the proof of inequality (3.45)), it can be shown that $\begin{array}{l} (3.49) & lim_{k \to \infty} \int_{Ω_{ε_{k}}} g_{k} (x - x_{0}, \nabla y_{k}) d x = \int_{Ω} g (x - x_{0}, \nabla y) d x, \\ (3.50) & lim_{k \to \infty} \int_{Γ_{N, ε_{k}}} | \nabla u_{k} |^{2} d H^{N - 1} = \int_{Γ_{N}} | \nabla u |^{2} Γ_{0} d H^{N - 1}, \\ (3.51) & lim_{k \to \infty} \int_{Ω_{ε_{k}}} F (y_{k}) d x = \int_{Ω} F (y) d x ⩽ C_{F}^{- 1} \int_{Ω} f (y) d x < + \infty, \\ (3.52) & \underset{k \to \infty}{lim inf} \int_{Ω_{ε_{k}}} | \nabla y_{k} |^{2} d x ⩾ \int_{Ω} | \nabla y |^{2} d x . \end{array}$ As a result, the expected integral inequality (3.19) immediately follows from (3.47) and properties (3.49)–(3.52). The proof is complete. □

Theorem 3.9.

Let $Ω \subset D$ be an open domain satisfying the enhanced star-shaped property with respect to some point $x_{0} \in int Ω$ , and the sequence ${Ω_{ε}}_{0 < ε ⩽ ε_{0}} \subset D$ has a mild rugosity along $Γ_{N, ε}$ -part of the boundary. Then assertion (dd) of Definition 3.5 is valid for $Ξ = Ξ_{0}$ and $I = I_{0}$ .

Proof.

Let $(u, g, y) \in Ξ_{0} \subset L^{2} (Γ_{N}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω; Γ_{D})$ be a regular feasible solution to optimal control problem (3.12)–(3.16). Then there exists a sequence $\begin{matrix} {(v_{ε}, y_{ε}) \in W^{1, r} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D})}_{0 < ε ⩽ ε_{0}} \end{matrix}$ such that $y_{ε} ⇀ y$ in $W_{loc}^{1, 2} (Ω)$ as $ε \to 0$ , and $(u_{ε}, g, y_{ε}) \in Ξ_{ε}$ for all $0 < ε ⩽ ε_{0}$ with $u_{ε} = v_{ε} |_{Γ_{N, ε}}$ , where ${v_{ε} \in W^{1, r} (D)}$ possesses the properties indicated in Definition 3.4. Let us show that the sequence ${(u_{ε}, g, y_{ε}) \in Ξ_{ε}}_{0 < ε ⩽ ε_{0}}$ is realizing in the sense of property (dd). Indeed, as was mentioned in the proof of Theorem 3.8, in view of Proposition 2.3, we may suppose that each element of the given sequence of parametrized domains ${Ω_{ε_{k}}}_{k \in N}$ possesses the star-shaped property with respect to some point $x_{0} \in int Ω$ that does not depend on k. Hence, due to the initial assumptions, we deduce that $\begin{matrix} (u_{ε}, g, y_{ε}) \overset{μ}{⟶} (u, g, y) in L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D}) . \end{matrix}$ Let us show that in this case we can guarantee the following equality $\begin{matrix} J_{0} (u, g, y) = lim_{ε \to 0} J_{ε} (u_{ε}, g, y_{ε}) . \end{matrix}$ As immediately follows from (3.27) and specific choice of the boundary controls ${u_{ε} = v_{ε} |_{Γ_{N, ε}}}$ , we see that $\begin{array}{l} lim_{ε \to 0} \int_{Ω^{'}} | y_{ε} - y_{d} |^{2} d x \overset{by (3.29)}{=} \int_{Ω^{'}} | y - y_{d} |^{2} d x, \\ lim_{ε \to 0} \int_{Γ_{N, ε}} | v_{ε} |^{2} d H^{N - 1} \overset{by Lemma 3.6}{=} \int_{Γ_{N}} | v |^{2} Γ_{0} d H^{N - 1}, \\ lim_{ε \to 0} \int_{D} | g - g_{d} |^{q} χ_{Ω_{ε}} d x \overset{by Proposition 2.2}{=} \int_{D} | g - g_{d} |^{q} χ_{Ω} d x, \end{array}$ where $Ω^{'}$ is an arbitrary semi-compact subset of Ω such that $Ω^{'} \subset Ω$ and $Ω^{'} \subset Ω_{ε}$ for $ε > 0$ small enough. Arguing as in the proof of Theorem 3.8, it can be shown that for an arbitrary given $δ > 0$ there exist $ε^{0} > 0$ and a semi-compact subset $Ω^{'} (δ)$ of Ω and $Ω_{ε}$ such that $\begin{matrix} | \int_{Ω ∖ Ω^{'} (δ)} | y - y_{d} |^{2} d x | + | \int_{Ω_{ε} ∖ Ω^{'} (δ)} | y_{ε} - y_{d} |^{2} d x | < δ, \forall ε < ε^{0} . \end{matrix}$ As a result, utilizing this fact and the above equalities, we arrive at the desired relation $\begin{matrix} (3.53) & lim_{ε \to 0} J_{ε} (u_{ε}, g, y_{ε}) = J_{0} (u, g, y) . \end{matrix}$ □

Remark 3.2.

Combining the results given by Theorems 3.8 and 3.9, we can summarize them as follows:

If an open domain $Ω \subset D$ satisfies the enhanced star-shaped property with respect to some point $x_{0} \in int Ω$ , and the sequence ${Ω_{ε}}_{0 < ε ⩽ ε_{0}} \subset D$ has a mild rugosity along $Γ_{N, ε}$ -part of the boundary, then optimal control problem (3.12)–(3.16) is the variational limit (in the sense of Definition 3.5) of the problems (1.7)–(1.10) as $ε \to 0$ . Moreover, as follows the current description of the limit optimization problem, even if parameter $ε > 0$ is small enough, the rugosity of the control zones ( $Γ_{N, ε}$ -boundaries of domains $Ω_{ε}$ ) affects not only the boundary value problem (3.13)–(3.14), but the limit cost functional (3.12) and boundary control constraints set $A_{ad} (Γ_{N})$ as well. Thus, the control through a rugous boundary leads in the limit to an optimal control problem with different structure. It means that the indicated rugosity effect can not be neglected under precise consideration of the corresponding optimal control problem.

To conclude this section, we specify some variational properties of optimal solutions to the original problem (1.7)–(1.10) as rugosity parameter ε tends to zero.

Theorem 3.10.

Let $Ω \subset D$ be an open domain satisfying the enhanced star-shaped property with respect to some point $x_{0} \in int Ω$ , and the sequence ${Ω_{ε}}_{0 < ε ⩽ ε_{0}} \subset D$ has a mild rugosity along $Γ_{N, ε}$ -part of the boundary. Let ${(u_{ε}^{0}, g_{ε}^{0}, y_{ε}^{0}) \in Ξ_{ε}}_{0 < ε ⩽ ε_{0}}$ be a sequence of optimal solutions to the problem ( 1.7 )–( 1.10 ) for the corresponding $ε > 0$ . Assume that the set of feasible solutions $Ξ_{0}$ contains at least one $(u, g, y)$ which is regular in the sense of Definition 3.4 . Then there exists a triplet $(u^{*}, g^{*}, y^{*})$ such that $\begin{array}{l} (3.54) & (u^{*}, g^{*}, y^{*}) \in Ξ_{0}, \\ (3.55) & (u_{ε}^{0}, g_{ε}^{0}, y_{ε}^{0}) \overset{μ}{⟶} (u^{*}, g^{*}, y^{*}) in L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D}), \\ (3.56) & J_{0} (u^{*}, g^{*}, y^{*}) = lim_{ε \to 0} J_{ε} (u_{ε}^{0}, g_{ε}^{0}, y_{ε}^{0}) = inf_{\begin{array}{c} (u, g, y) \in Ξ_{0} \\ (u, g, y) is regular \end{array}} J_{0} (u, g, y) . \end{array}$

Proof.

Let $(u, g, y) \in Ξ_{0} \subset L^{2} (Γ_{N}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω; Γ_{D})$ be a regular feasible solution to optimal control problem (3.12)–(3.16). Then proceeding exactly as in the proof of Theorem 3.9, it can be shown that there exists a sequence ${(u_{ε}, g, y_{ε}) \in Ξ_{ε}}_{0 < ε ⩽ ε_{0}}$ such that $\begin{matrix} (u_{ε}, g, y_{ε}) \overset{μ}{⟶} (u, g, y) in L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D}), \end{matrix}$ and, therefore, $\begin{matrix} (3.57) & sup_{ε > 0} [‖ u_{ε} ‖_{L^{2} (Γ_{N, ε})}^{2} + ‖ g ‖_{L^{q} (Ω_{ε})}^{q} + ‖ y_{ε} ‖_{L^{2} (Ω_{ε})}^{2}] ⩽ C^{*} \end{matrix}$ for some $C^{*} > 0$ . Since $\begin{matrix} J_{ε} (u_{ε}^{0}, g_{ε}^{0}, y_{ε}^{0}) ⩽ J_{ε} (u_{ε}, g, y_{ε}), \forall ε \in (0, ε_{0}), \end{matrix}$ it follows from (3.57) that $\begin{array}{l} {‖ g_{ε}^{0} ‖}_{L^{q} (Ω_{ε})}^{q} & ⩽ 2^{q - 1} q [J_{ε} (u_{ε}^{0}, g_{ε}^{0}, y_{ε}^{0}) + \frac{1}{q} ‖ g_{d} ‖_{L^{q} (D)}^{q}] \\ ⩽ 2^{q - 1} q J_{ε} (u_{ε}, g, y_{ε}) + 2^{q - 1} ‖ g_{d} ‖_{L^{q} (D)}^{q} \\ ⩽ {[max {q, 2^{q - 1}}]}^{2} (‖ u_{ε} ‖_{L^{2} (Γ_{N, ε})}^{2} + ‖ g ‖_{L^{q} (Ω_{ε})}^{q} + ‖ y_{ε} ‖_{L^{2} (Ω_{ε})}^{2}) \\ + 2^{q - 1} (q ‖ y_{d} ‖_{L^{2} (Ω_{ε})}^{2} + 2^{q - 1} ‖ g_{d} ‖_{L^{q} (D)}^{q}) + 2^{q - 1} ‖ g_{d} ‖_{L^{q} (D)}^{q} \overset{by (3.57)}{<} + \infty . \end{array}$ Taking into account the a priori estimates (3.1)–(3.2), we deduce that the sequence of optimal solutions is bounded in the following sense: there exists a constant $C^{0} > 0$ such that $\begin{matrix} (3.58) & sup_{ε > 0} [{‖ u_{ε}^{0} ‖}_{L^{2} (Γ_{N, ε})}^{2} + {‖ g_{ε}^{0} ‖}_{L^{q} (Ω_{ε})}^{q} + {‖ y_{ε}^{0} ‖}_{L^{2} (Ω_{ε})}^{2}] ⩽ C^{0} . \end{matrix}$ Then Lemma 3.1 implies its compactness with respect to the μ-convergence. Hence, there exists a sequence of indices ${ε_{k}}_{k \in N}$ with $ε_{k} \to 0$ as $k \to \infty$ , and a triplet $(u^{*}, g^{*}, y^{*}) \in A_{ad} (Γ_{N}) \times G_{ad} \times W_{0}^{1, 2} (Ω; Γ_{D})$ such that $\begin{matrix} (u_{ε_{k}}^{0}, g_{ε_{k}}^{0}, y_{ε_{k}}^{0}) \overset{μ}{⟶} (u^{*}, g^{*}, y^{*}) in L^{2} (Γ_{N, ε}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω_{ε}; Γ_{D}) as k \to \infty . \end{matrix}$ Arguing then as in Theorem 3.9, it can be shown that $(u^{*}, g^{*}, y^{*})$ is a feasible solution to the limit problem (3.12)–(3.16).

Let us prove that $(u^{*}, g^{*}, y^{*})$ is the μ-limit for the whole sequence of optimal triplets ${(u_{ε}^{0}, g_{ε}^{0}, y_{ε}^{0}) \in Ξ_{ε}}_{0 < ε ⩽ ε_{0}}$ and $(u^{*}, g^{*}, y^{*})$ is a regular optimal solution to optimization problem (3.12)–(3.16). Indeed, since $(u, g, y) \in Ξ_{0} \subset L^{2} (Γ_{N}) \times L^{q} (D) \times W_{0}^{1, 2} (Ω; Γ_{D})$ is a regular feasible solution to optimal control problem (3.12)–(3.16), by analogy with the proof of Theorem 3.9, it can be shown that $\begin{matrix} J_{0} (u, g, y) = lim_{ε \to 0} J_{ε} (u_{ε}, g, y_{ε}), where (u_{ε}, g, y_{ε}) \in Ξ_{ε} for all ε . \end{matrix}$ Then, using now the above identity, the μ-lower semi-continuity property (3.30), the fact that $(u_{ε}^{0}, g_{ε}^{0}, y_{ε}^{0}) \in Ξ_{ε}$ is a solution to the parametrized problem (1.7)–(1.10), we get $\begin{array}{rcl} J_{0} (u^{*}, g^{*}, y^{*}) & \overset{by Theorem 3.8}{⩽} & \underset{k \to \infty}{lim inf} J_{ε_{k}} (u_{ε_{k}}^{0}, g_{ε_{k}}^{0}, y_{ε_{k}}^{0}) \\ ⩽ & \underset{k \to \infty}{lim sup} J_{ε_{k}} (u_{ε_{k}}^{0}, g_{ε_{k}}^{0}, y_{ε_{k}}^{0}) \\ (3.59) & ⩽ & \underset{k \to \infty}{lim sup} J_{ε_{k}} (u_{ε_{k}}, g, y_{ε_{k}}) = J_{0} (u, g, y) . \end{array}$ Since $(u, g, y) \in Ξ_{0}$ is an arbitrary regular feasible solution to the limit problem, it follows from (3.59) that $\begin{matrix} J_{0} (u^{*}, g^{*}, y^{*}) = inf_{\begin{array}{c} (u, g, y) \in Ξ_{0} \\ (u, g, y) is regular \end{array}} J_{0} (u, g, y), \end{matrix}$ that is $(u^{*}, g^{*}, y^{*})$ is a minimizer for the problem (3.12)–(3.16) on the class of regular feasible solutions. Moreover, taking $(u, g, y) = (u^{*}, g^{*}, y^{*})$ in the above inequalities (it is possible because of regularity of $(u^{*}, g^{*}, y^{*})$ ), (3.59) implies the relation $\begin{matrix} (3.60) & J_{0} (u^{*}, g^{*}, y^{*}) = lim_{k \to \infty} J_{ε_{k}} (u_{ε_{k}}^{0}, g_{ε_{k}}^{0}, y_{ε_{k}}^{0}) = inf_{\begin{array}{c} (u, g, y) \in Ξ_{0} \\ (u, g, y) is regular \end{array}} J_{0} (u, g, y) . \end{matrix}$ It remains to notice that since relation (3.60) is valid for any μ-converging subsequence of ${(u_{ε}, g, y_{ε}) \in Ξ_{ε}}_{0 < ε ⩽ ε_{0}}$ , it follows that $(u^{*}, g^{*}, y^{*})$ is the μ-limit for the whole sequence. As a result, (3.56) is a direct consequence of (3.60). □

Remark 3.3.

As follows from Theorem 3.10, an optimal solution to the limit problem (3.12)–(3.16) in a non-perturbed domain Ω can be used as a basis for the construction of suboptimal controls for the original control problem in domain with a rough boundary. For the details we refer to the bo [18].

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