Homogenization of Boundary Optimal Control Problem Governed by Heat Equation in Domain With Rugose Boundary

Abstract

In this article, we investigate the boundary optimal control problem associated with Heat equation in a two-dimensional highly oscillating domain $Ω_{ϵ}$ in which the control is applied periodically via Neumann condition on the oscillating part of the boundary. We characterize the optimal control in terms of unfolding operator and then study the homogenization to obtain two limit optimal control problems depending on the scalar parameters. In the limit optimal control problem, we obtain three controls, namely interior control, an interfacial control and a boundary control.

Keywords

homogenization optimal control heat equation unfolding operator

1. Introduction

This article examines an optimal control problem (OCP) in a two-dimensional oscillating domain $Ω_{ϵ} .$ As seen in Figure 1(a), the domain $Ω_{ϵ}$ is composed of an oscillating upper part, $Ω_{ϵ}^{+}$ , and a fixed bottom part, $Ω^{-}$ . The oscillating portion of the boundary is subjected to periodic control with appropriate scaling using the Neumann condition. We consider a control function defined on a fixed boundary segment (denoted by $Γ$ ). This control is rescaled by a factor of $ε^{α}$ , where $α \geq 1$ , along the vertical part of $Γ$ , and is then periodically distributed over the entire oscillating portion of the boundary. There are two types of limiting problems corresponding to the cases $α > 1$ and $α = 1$ (the critical case). In the latter case, three distinct controls arise in the limiting system: an interior (distributed) control within the domain, a boundary control acting on the upper part of $Ω$ , and an interface control operating between the subdomains $Ω^{+}$ and $Ω^{-}$ . On the other hand, when $α > 1$ , no distributed (interior) control appears in the limiting problem. When modeling problems related to industrial applications, such as flows in channels with rough walls, electromagnetic waves in an area with a rough interface, heat transfer in winglets, elastic bodies with a rough interface, and so on (see, Amirat & Simon, 1997; Birnir et al., 2007; Casado-Díaz et al., 2003; D’Angelo et al., 2013; Jäger & Mikelić, 2003; Lenczner, 2007), challenges with rugose borders frequently emerge.

Figure 1.

(a) The domain $Ω_{ϵ}$ for $N = 6,$ (b) domain $Ω$ and (c) the fixed boundary $Γ = \cup_{i = 1}^{4} Γ_{i} .$

As for the literature on homogenizing the OCP associated with evolution equations in the domain with oscillating boundaries, the work presented in De Maio et al. (2004) investigates homogenizing an OCP associated with a parabolic equation in a domain that is defined by a set of cylinders positioned $ϵ$ -periodically over a fixed base. A limiting OCP with a alternative cost functional is derived by the authors after they analyze an interior OCP with a classical cost functional. Within an oscillating domain, the authors of Durante et al. (2007) have examined the homogenization of wave equations. Additionally, by extending control to the whole domain and accounting for the $L^{2} -$ cost functional, the authors have also examined related interior OCP. By first creating an adjoint problem and then articulating an optimal control, the authors have performed the convergence analysis for optimal control. A parabolic OCP with Dirichlet boundary control, specified on a rapidly oscillating cylindrical domain, is examined by the authors in D’Apice et al. (2008). They undertake asymptotic analysis with the aid of an extension operator and employ a traditional $L^{2}$ cost functional. Later, in De Maio et al. (2015), the homogeneity of interior OCP related to the wave equation with constant coefficients is examined in a three-dimensional oscillating domain. The authors have implemented control over the domain’s interior, and the Tartar’s approach of oscillating test function is employed to do the convergence study. The cost functional and state equation of the limit OCP that the authors have obtained differ significantly from those of the comparable non-homogeneous OCP. The asymptotic behavior of a linear parabolic problem having a pillar-kind of domain along with periodic branching was examined by the authors in Aiyappan and Nandakumaran (2017). Additionally, the authors examined the homogenization of an interior OCP in which a $L^{2} -$ cost functional is used to apply control to the oscillating portion of the domain. A multi-sheeted function, where each sheet represents the achieved limit of the confined solution to the various branches of the domain, is used by the authors to express the optimal control and limit problem leveraging the unfolding operator approach. In Faella et al. (2024), the authors homogenize a two-dimensional oscillating domain that resembles a pillar and is driven by the wave equation with oscillating coefficients. They use the unfolding operator approach to analyze convergence of a Dirichlet cost functional with oscillating coefficients. One can refer (Aiyappan et al., 2018; 2019; Aiyappan & Sardar, 2019; Brizzi & Chalot, 1997; De Maio et al., 2024; Gaudiello et al., 2017; Gaudiello & Lenczner, 2020; Gaudiello & Mel’nyk, 2018; Gaudiello & Mel’nyk, 2019; Mel’nyk & Sadovyi, 2019; Nandakumaran et al., 2015; Nandakumaran & Sufian, 2021; Sardar & Sufian, 2022) for the homogenization or OCPs on domains with oscillating boundaries.

In the literature on boundary OCPs in domain with oscillating boundaries, the work presented in Durante and Mel’nyk (2010) investigated the boundary OCP in a plane thick two-level junction with a large number of thin rods for Poisson equation and further provides a limit OCP. Also, the boundary OCP where controls act periodically on the oscillatory boundary is studied in Nandakumaran et al. (2015) where they characterize the limit control problem by three controls namely distributed, boundary and interfacial controls. Then, boundary OCP for stationary Stokes equation is done in Sardar and Sufian (2022) where limit problem can have one or two controls depending upon the choice of scaling parameters. In both theoretical analysis and practical applications, the study of boundary optimal control for the heat equation is essential because many real-world systems, including mass transport, heat conduction in plates with densely spaced stiffeners, and fluid structure interaction, frequently involve rough and oscillating domains where direct control within the medium is either impractical or expensive. Therefore, rather of relying on internal changes, we need to regulate through boundary techniques. Boundary conditions may be optimized to effectively control the system’s internal state, guaranteeing optimal performance while using the least amount of energy or material possible. In addition to improving modeling accuracy, an understanding of the interaction between boundary factors and the interior dynamics of the heat equation results in effective computational techniques for managing intricate physical processes. This article’s goal is to characterize the control by the unfolding operator using an adjoint equation and investigate the limiting behavior of the optimal solution, or optimal control, the associated state, and the related adjoint state. First presented in Cioranescu et al. (2002), the periodic unfolding approach was refined in Cioranescu et al. (2008). Even though there are other approaches in the literature, such as two-scale convergence, we choose to utilize the unfolding technique since it seems more appropriate in this situation because of the extra difficulty posed by the boundary oscillation. Thus, we believe that the unfolding approach is an appropriate panacea to the problem in this work. Although the present analysis is carried out in two dimensions for the sake of clarity and notational simplicity, the arguments and results can be extended to higher-dimensional settings without essential difficulty.

The framework of this article is as follows: the problem and domain are described in Section 2. In Section 3, the definition and properties of unfolding operator is recalled. Characterization of optimal control and uniform estimates are presented in Section 4. Section 5 introduces two homgenized OCP that rely on the scaling parameter $α$ , and finally, we discuss the accompanying convergence results in Section 6.

2. An Overview of the Domain and Problem

2.1. An Overview of the Domain

In this work, we employ an oscillating domain $Ω_{ϵ}$ , which is analogous to the one provided in Nandakumaran et al. (2016). See, for example, Aiyappan and Nandakumaran (2017), Aiyappan and Sardar (2019), and Nandakumaran et al. (2015) for numerous types of pillar type oscillating domains. We provide a brief overview of the oscillating domain in this section. Let $ϵ = 1 / N$ with $N \in N$ . Let $K : R \to R$ be a smooth periodic function with period 1 and $0 < p < q < 1$ . Let $η_{ϵ}$ be a periodic function defined over $[0, 1]$ , having periodicity $ϵ$ , defined over $[0, ϵ]$ by

\begin{aligned} η_{ϵ} (x_{1}) = {\begin{cases} h_{2} & if x_{1} \in (ϵ p, ϵ q), \\ h_{1} & if x_{1} \in [0, ϵ) / (ϵ p, ϵ q), \end{cases} \end{aligned}

where

h_{2} > h_{1} > {max}_{s \in [0, 1]} K (s)

. We define the domain

Ω_{ϵ}

\begin{aligned} Ω_{ϵ} := {(x_{1}, x_{2}) \in R^{2} : 0 < x_{1} < 1, K (x_{1}) < x_{2} < η_{ϵ} (x_{1})} . \end{aligned}

The two fixed boundary of

Ω_{ϵ}

, namely the bottom boundary

Γ_{b}

and the side boundary

Γ_{s}

are given as

\begin{aligned} Γ_{b} & := {(x_{1}, x_{2}) : x_{2} = K (x_{1}), x_{1} \in [0, 1]}, \end{aligned}

\begin{aligned} Γ_{s} & := {(0, x_{2}) : K (0) \leq x_{2} \leq h_{1}} \cup {(1, x_{2}) : K (1) \leq x_{2} \leq h_{1}} . \end{aligned}

The oscillating boundary

Γ_{ϵ}

is defined as

Γ_{ϵ} := \partial Ω_{ϵ} / (Γ_{b} \cup Γ_{s})

. The oscillating part of the domain

Ω_{ϵ}

is defined by

\begin{aligned} Ω_{ϵ}^{+} := \cup_{j = 0}^{N - 1} (j ϵ + ϵ p, j ϵ + ϵ q) \times (h_{1}, h_{2}) . \end{aligned}

The fixed part of the domain

Ω_{ϵ}

is defined as

\begin{aligned} Ω^{-} = {(x_{1}, x_{2}) : 0 < x_{1} < 1, K (x_{1}) < x_{2} < h_{1}} . \end{aligned}

The top boundary of

Ω^{-}

is denoted by

Γ_{c} = {(x_{1}, h_{1}) : 0 \leq x_{1} \leq 1} .

The domain

Ω_{ϵ}

can also be written as

Ω_{ϵ} = int (\bar{Ω_{ϵ}^{+} \cup Ω^{-}}) .

An example of domain

Ω_{ϵ}

for

N = 6

is shown in Figure 1(a).

We denote fix domain $Ω$ as $Ω = {(x_{1}, x_{2}) : 0 < x_{1} < 1, K (x_{1}) < x_{2} < h_{2}} .$ The top and vertical boundary of domain $Ω$ is denoted by $Γ_{u} = {(x_{1}, h_{2}) : 0 \leq x_{1} \leq 1}, Γ_{s^{'}} = {(0, x_{2}) : K (0) \leq x_{2} \leq h_{2}} \cup {(1, x_{2}) : K (1) \leq x_{2} \leq h_{2}}$ , respectively. Upper part of fixed domain $Ω$ is denoted by $Ω^{+} = {(x_{1}, x_{2}) : 0 < x_{1} < 1, h_{1} < x_{2} < h_{2}}$ . We can also write $Ω = int (\bar{Ω^{+} \cup Ω^{-}}) .$ The domain $Ω$ is illustrated in Figure 1(b).

The reference boundary denoted by $Γ$ (see Figure 1(c)) is defined by $Γ = Γ_{1} \cup Γ_{2} \cup Γ_{3} \cup Γ_{4},$ where

\begin{aligned} Γ_{1} = {(y_{1}, h_{1}) : 0 \leq y_{1} \leq p} \cup {(y_{1}, h_{1}) : q \leq y_{1} \leq 1}, \end{aligned}

\begin{aligned} Γ_{2} = {(y_{1}, h_{2}) : p \leq y_{1} \leq q}, \end{aligned}

\begin{aligned} Γ_{3} = {(p, y_{2}) : h_{1} \leq y_{2} \leq h_{2}}, \end{aligned}

\begin{aligned} Γ_{4} = {(q, y_{2}) : h_{1} \leq y_{2} \leq h_{2}} . \end{aligned}

We define

Γ_{1}^{ϵ} := Γ_{ϵ} \cap Γ_{c}, Γ_{2}^{ϵ} := Γ_{ϵ} \cap Γ_{u}

and the common boundary between

Ω_{ϵ}^{+}

and

Ω^{-}

is denoted by

Γ_{3}^{ϵ} := \cup_{j = 0}^{N - 1} (j ϵ + ϵ p, j ϵ + ϵ q) \times {h_{1}} .

We use the notation

A_{1} = Γ_{1}, A_{2} = Γ_{2}

and

A_{3} = (p, q) \times {h_{1}}

for our convenience. We denote by

\tilde{u}

extension of any function

u

from

Ω_{ϵ}^{+}

Ω^{+}

by zero.

2.2. Description of the Problem

Given $θ \in L^{2} (Γ \times (0, T))$ and $Γ -$ periodic, we define $θ_{ϵ} = (χ_{Γ_{1}} + ϵ^{α} χ_{Γ_{3}} + χ_{Γ_{2}} + ϵ^{α} χ_{Γ_{4}}) θ,$ where $α \geq 1.$ $χ_{D}$ denotes characteristic function of any set D. We define the periodic control function ${\hat{θ}}_{ϵ} \in L^{2} (Γ_{ϵ} \times (0, T))$ as

\begin{aligned} {\hat{θ}}_{ϵ} (x_{1}, x_{2}, t) = θ_{ϵ} (\frac{x_{1}}{ϵ}, x_{2}, t) . \end{aligned}

(2.1)

Consider the following boundary OCP governed by the heat equation:

\begin{aligned} inf_{θ \in L^{2} (Γ \times (0, T))} {J_{ϵ} (θ) = \frac{1}{2} \int_{Ω_{ϵ} \times (0, T)} | u_{ϵ} - u_{d} |^{2} + \frac{β}{2} \int_{Γ \times (0, T)} | θ |^{2}} \end{aligned}

(2.2)

subject to

\begin{aligned} {\begin{cases} u_{ϵ}^{'} - Δ u_{ϵ} + u_{ϵ} = f in Ω_{ϵ} \times (0, T), \\ \frac{\partial u_{ϵ}}{\partial ν} = {\hat{θ}}_{ϵ} on Γ_{ϵ} \times (0, T), \\ u_{ϵ} = 0 on Γ_{b} \times (0, T), \\ u_{ϵ} (x, 0) = 0 in Ω_{ϵ}, \\ u_{ϵ} is Γ_{s} periodic, \end{cases} \end{aligned}

(2.3)

where

β > 0

is a regularization parameter,

f \in L^{2} (Ω \times (0, T))

is a given function.

For $α < 1,$ unfortunately, we can not obtain a priori estimates needed for convergence analysis (see Proposition 4.3). Therefore $α = 1$ becomes the critical value. When $α < 1$ , the solution of the boundary control problem is not equi-bounded as $ε \to 0$ . In this case, one could consider a normalized formulation by the corresponding upper estimate to obtain a meaningful limiting behavior. However, this regime is not treated in the present paper and will be the subject of future investigation.

Our goal is to investigate the limiting behavior as $ϵ \to 0$ of the optimal control and the associated state of the problem (2.2). Let us introduce the space

\begin{aligned} W^{ϵ} = {u \in L^{2} (0, T; H^{1} (Ω_{ϵ})) : u^{'} \in L^{2} (0, T; (H^{1} (Ω_{ϵ}))^{'}), u |_{Γ_{b}} = 0}, \end{aligned}

W^{ϵ}

is a Banach space with the norm given by

\begin{aligned} | | u | |_{W^{ϵ}} = | | u_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} + | | u_{ϵ}^{'} | |_{L^{2} (0, T; (H^{1} (Ω_{ϵ}))^{'})} . \end{aligned}

The variational formulation of the state equation (2.3) is now defined.

Definition 2.1

A function $u_{ϵ} \in W^{ϵ},$ is a weak solution to (2.3) if

\begin{aligned} \int_{Ω_{ϵ} \times (0, T)} u_{ϵ}^{'} ψ + \int_{Ω_{ϵ} \times (0, T)} \nabla u_{ϵ} \cdot \nabla ψ + \int_{Ω_{ϵ} \times (0, T)} u_{ϵ} ψ = \int_{Ω_{ϵ} \times (0, T)} f ψ + \int_{Γ_{ϵ} \times (0, T)} {\hat{θ}}_{ϵ} ψ, \end{aligned}

(2.4)

for all

ψ \in L^{2} (0, T; H^{1} (Ω_{ϵ})),

with

\begin{aligned} u_{ϵ} (x, 0) = 0 in Ω_{ϵ} . \end{aligned}

Note that, as $W^{ϵ} \subset C ([0, T]; L^{2} (Ω_{ϵ})),$ the solution $u_{ϵ}$ of problem (2.3) is more regular, that is, $u_{ϵ} \in C ([0, T]; L^{2} (Ω_{ϵ}))$ (see Gajewski et al., 1974, Chapter IV). Therefore, the initial condition $u_{ϵ} (x, 0) = 0$ is meaningful.

Let us recall, for any fixed $ϵ,$ the existence and uniqueness result of the solution of problem (2.3) (see Showalter, 1997 or Mel’nik & Sivak, 2010).

Theorem 2.2

For every fixed $ϵ > 0$ , the problem (2.3) admits a unique weak solution $u_{ϵ} \in W^{ϵ} .$ Moreover

\begin{aligned} | | u_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} + | | u_{ϵ}^{'} | |_{L^{2} (0, T; (H^{1} (Ω_{ϵ}))^{'})} \leq C_{ϵ} (| | f | |_{L^{2} (Ω \times (0, T))} + | | {\hat{θ}}_{ϵ} | |_{L^{2} (Γ_{ϵ} \times (0, T))}) . \end{aligned}

(2.5)

Where the constant

C_{ϵ} > 0

may depend on

ϵ .

In order to facilitate the convergence analysis in the subsequent section, we will demonstrate the uniform estimate for the state and control variable and characterize optimal control in terms of the unfolding operator using an adjoint system in the following section.

3. Unfolding Operator and Its Properties

For problems on oscillating domain, the periodic unfolding approach is a highly beneficial tool in homogenization theory. We will employ the unfolding operator defined on the oscillating domain’s boundary for the analysis of the boundary. The definition and properties of unfolding operators on the oscillating domain $Ω_{ϵ}^{+}$ (represented by $T^{ϵ}$ ) and on the boundary $Γ_{i}^{ϵ}, i = 1, 2, 3$ (represented by $T_{i}^{ϵ}$ ) are briefly reviewed in this section. See Cioranescu et al. (2002, 2008) for additional details on the unfolding operator. We write $[x] = ([x_{1}], [x_{2}]),$ for each $x = (x_{1}, x_{2}),$ where $[x_{i}]$ represents the integer portion of $x_{i} .$

Now, we revisited the definition and the properties of Unfolding operator $T^{ϵ}$ on the oscillating domain $Ω_{ϵ}^{+} .$

Definition 3.1
The $ϵ$ - unfolding of a function $u : Ω_{ϵ}^{+} \times (0, T) \to R$ is the function
$\begin{aligned} T^{ϵ} : {u : Ω_{ϵ}^{+} \times (0, T) \to R} \to {v : Ω^{+} \times (p, q) \times (0, T) \to R} \end{aligned}$
defined by
$\begin{aligned} T^{ϵ} (u) (x_{1}, x_{2}, x_{3}, t) = u (ϵ [\frac{x_{1}}{ϵ}] + ϵ x_{3}, x_{2}, t) . \end{aligned}$

In the following proposition, we revisit the properties of the unfolding operator $T^{ϵ}$ (see Nandakumaran et al., 2013 for details).
Proposition 3.2
The unfolding operator $T^{ϵ}$ has following properties: P1.
$T^{ϵ}$ is linear and
$\begin{aligned} T^{ϵ} (u v) = T^{ϵ} (u) T^{ϵ} (v) . \end{aligned}$

P2.
Let $u \in L^{1} (Ω_{ϵ}^{+} \times (0, T))$ then
$\begin{aligned} \int_{Ω^{+} \times (p, q) \times (0, T)} T^{ϵ} (u) d x d x_{3} d t = \int_{Ω_{ϵ}^{+} \times (0, T)} u d x d t . \end{aligned}$

P3.
Let’s denote $G = L^{2} ((0, 1) \times (0, T); H^{1} ((0, 1) \times (p, q)))$ . Let $u \in L^{2} (0, T; H^{1} (Ω_{ϵ}^{+}))$ then $T^{ϵ} (u) \in G .$ Moreover
$\begin{aligned} \frac{\partial}{\partial x_{2}} T^{ϵ} (u) = T^{ϵ} (\frac{\partial u}{\partial x_{2}}), \end{aligned}$

$\begin{aligned} \frac{\partial}{\partial x_{3}} T^{ϵ} (u) = ϵ T^{ϵ} (\frac{\partial u}{\partial x_{1}}) . \end{aligned}$

Further $| | T^{ϵ} (u) | |_{G} \leq C | | u | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}^{+}))} .$
P4.
Let $u \in L^{2} (Ω_{ϵ}^{+} \times (0, T))$ , then $T^{ϵ} (u) \in L^{2} (Ω^{+} \times (p, q) \times (0, T))$ and
$\begin{aligned} \int_{Ω^{+} \times (p, q) \times (0, T)} | T^{ϵ} (u) |^{2} d x d x_{3} d t = \int_{Ω_{ϵ}^{+} \times (0, T)} | u |^{2} d x d t . \end{aligned}$

P5.
Let $u \in L^{2} (Ω^{+} \times (0, T)),$ then
$\begin{aligned} T^{ϵ} (u) \to u in L^{2} (Ω^{+} \times (p, q) \times (0, T)) . \end{aligned}$

P6.
Let $u_{ϵ} \to u in L^{2} (Ω^{+} \times (0, T)) .$ Then
$\begin{aligned} T^{ϵ} (u_{ϵ}) \to u in L^{2} (Ω^{+} \times (p, q) \times (0, T)) . \end{aligned}$

P7.
Let $u_{ϵ} \in L^{2} (0, T; H^{1} (Ω_{ϵ}^{+}))$ be such that $T^{ϵ} (u_{ϵ}) ⇀ u$ weakly in $G$ . Then
$\begin{aligned} \tilde{u_{ϵ}} ⇀ \int_{p}^{q} u d x_{3} in L^{2} (Ω^{+} \times (0, T)) . \end{aligned}$

For our analysis on the boundary, we define boundary unfolding operator on $Γ_{1}^{ϵ}, Γ_{2}^{ϵ}$ and $Γ_{3}^{ϵ}$ .
Definition 3.3
For $i = 1, 2, 3,$ the $ϵ$ - unfolding of a function $u : Γ_{i}^{ϵ} \times (0, T) \to R$ is the function
$\begin{aligned} T_{i}^{ϵ} : {u : Γ_{i}^{ϵ} \times (0, T) \to R} \to {v : (0, 1) \times A_{i} \times (0, T) \to R} \end{aligned}$
defined by
$\begin{aligned} T_{i}^{ϵ} (u) (x_{1}, x_{2}, x_{3}, t) = u (ϵ [\frac{x_{1}}{ϵ}] + ϵ x_{3}, x_{2}, t) . \end{aligned}$

We recall some important properties of boundary unfolding operator given in (Damlamian & Pettersson, 2009; Nandakumaran et al., 2013).
Proposition 3.4
For $i = 1, 2, 3,$ $\textit{(i)}$
$T_{i}^{ϵ}$ is linear and for any $u, v : Γ_{i}^{ϵ} \times (0, T) \to R$ , $T_{i}^{ϵ} (u v) = T_{i}^{ϵ} (u) T_{i}^{ϵ} (v)$ .
$\textit{(ii)}$
if $u \in L^{2} (Γ_{i}^{ϵ} \times (0, T))$ , then $T_{i}^{ϵ} (u) \in L^{2} ((0, 1) \times A_{i} \times (0, T))$ and $| | T_{i}^{ϵ} (u) | |_{L^{2} ((0, 1) \times A_{i} \times (0, T))} = | | u | |_{L^{2} (Γ_{i}^{ϵ} \times (0, T))}$ .
$\textit{(iii)}$
If $u_{ϵ} \to u$ strongly in $L^{2} (0, T; H^{1} ((0, 1) \times (h_{1}, h_{2})))$ then $T_{i}^{ϵ} (u_{ϵ}) \to u$ strongly in $L^{2} ((0, 1) \times A_{i} \times (0, T))$
$\textit{(iv)}$
Let $u_{ϵ}$ be a sequence in $L^{2} (Γ_{i}^{ϵ} \times (0, T))$ such that $T_{i}^{ϵ} (u_{ϵ}) ⇀ u$ weakly in $L^{2} ((0, 1) \times A_{i} \times (0, T))$ , then
$\begin{aligned} \tilde{u_{ϵ}} ⇀ \int_{A_{i}} u d x_{3} weakly in L^{2} ((0, 1) \times (0, T)) . \end{aligned}$

4. Characterization of Optimal Control and Uniform Estimates

This section showcases the characterization of optimal control in the form of the unfolding operator utilizing an adjoint equation. Also, we prove the uniform estimates for the state variable which will be useful for convergence analysis in a later section. With the standard method of Calculus of Variations (Lions, 1971, Chapter 2), the OCP (2.2) permits a single optimal solution for any $ϵ > 0$ , which is $({\bar{u}}_{ϵ}, {\bar{θ}}_{ϵ}) .$ We demonstrate the necessary and sufficient condition for optimality in the subsequent theorem.

Theorem 4.1
Let $({\bar{u}}_{ϵ}, {\bar{θ}}_{ϵ})$ be the optimal solution of (2.2). Then the optimal control is given by
$\begin{aligned} {\bar{θ}}_{ϵ} (y_{1}, y_{2}, t) & = - \frac{1}{β} [χ_{Γ_{1}} \int_{0}^{1} T_{1}^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, h_{1}, y_{1}, t) d x_{1} + χ_{Γ_{3}} ϵ^{α - 1} \int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, y_{2}, p, t) d x_{1} \\ + χ_{Γ_{2}} \int_{0}^{1} T_{2}^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, h_{2}, y_{1}, t) d x_{1} + χ_{Γ_{4}} ϵ^{α - 1} \int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, y_{2}, q, t) d x_{1}], \end{aligned}$
(4.1)
where ${\bar{v}}_{ϵ}$ is the unique solution of the adjoint equation
$\begin{aligned} {\begin{cases} {\bar{v}}_{ϵ}^{'} - Δ {\bar{v}}_{ϵ} + {\bar{v}}_{ϵ} = {\bar{u}}_{ϵ} - u_{d} in Ω_{ϵ} \times (0, T), \\ \frac{\partial {\bar{v}}_{ϵ}}{\partial ν} = 0 on Γ_{ϵ} \times (0, T), \\ {\bar{v}}_{ϵ} = 0 on Γ_{b} \times (0, T), \\ {\bar{v}}_{ϵ} (x, T) = 0 in Ω_{ϵ}, \\ {\bar{v}}_{ϵ} is Γ_{s} periodic . \end{cases} \end{aligned}$
(4.2)
Conversly, if $({\bar{u}}_{ϵ}, {\bar{v}}_{ϵ})$ satisfy the following system:
$\begin{aligned} {\begin{cases} {\bar{u}}_{ϵ}^{'} - Δ {\bar{u}}_{ϵ} + {\bar{u}}_{ϵ} = f in Ω_{ϵ} \times (0, T), \\ {\bar{v}}_{ϵ}^{'} - Δ {\bar{v}}_{ϵ} + {\bar{v}}_{ϵ} = {\bar{u}}_{ϵ} - u_{d} in Ω_{ϵ} \times (0, T), \\ \frac{\partial {\bar{u}}_{ϵ}}{\partial ν} = \hat{{\bar{θ}}_{ϵ}}, \frac{\partial {\bar{v}}_{ϵ}}{\partial ν} = 0 on Γ_{ϵ} \times (0, T), \\ {\bar{u}}_{ϵ}, {\bar{v}}_{ϵ} = 0 on Γ_{b} \times (0, T), \\ {\bar{u}}_{ϵ} (x, 0) = 0, {\bar{v}}_{ϵ} (x, T) = 0 in Ω_{ϵ}, \\ {\bar{u}}_{ϵ}, {\bar{v}}_{ϵ} is Γ_{s} periodic, \end{cases} \end{aligned}$
(4.3)
where
$\begin{aligned} {\bar{θ}}_{ϵ} (y_{1}, y_{2}, t) & = - \frac{1}{β} [χ_{Γ_{1}} \int_{0}^{1} T_{1}^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, h_{1}, y_{1}, t) d x_{1} + χ_{Γ_{3}} ϵ^{α - 1} \int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, y_{2}, p, t) d x_{1} \\ + χ_{Γ_{2}} \int_{0}^{1} T_{2}^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, h_{2}, y_{1}, t) d x_{1} + χ_{Γ_{4}} ϵ^{α - 1} \int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, y_{2}, q, t) d x_{1}], \end{aligned}$
Then $({\bar{u}}_{ϵ}, {\bar{θ}}_{ϵ})$ is the optimal solution of (2.2).
Proof.
Let $({\bar{u}}_{ϵ}, {\bar{θ}}_{ϵ})$ be the optimal solution of (2.2). For $θ \in L^{2} (Γ \times (0, T)),$ let $F (θ) = J_{ϵ} (u_{ϵ} (f, {\hat{θ}}_{ϵ}), θ) .$ Since ${\bar{θ}}_{ϵ}$ is optimal, for any $λ > 0,$ we have
$\begin{aligned} \frac{1}{λ} (F ({\bar{θ}}_{ϵ} + λ θ) - F ({\bar{θ}}_{ϵ})) \geq 0. \end{aligned}$
Note that
$\begin{aligned} F ({\bar{θ}}_{ϵ} + λ θ) - F ({\bar{θ}}_{ϵ}) & = \frac{1}{2} \int_{Ω_{ϵ} \times (0, T)} | u_{ϵ, λ} - u_{d} |^{2} + \frac{β}{2} \int_{Γ \times (0, T)} | {\bar{θ}}_{ϵ} + λ θ |^{2} \\ - \frac{1}{2} \int_{Ω_{ϵ} \times (0, T)} | {\bar{u}}_{ϵ} - u_{d} |^{2} - \frac{β}{2} \int_{Γ \times (0, T)} | {\bar{θ}}_{ϵ} |^{2} \\ = \frac{1}{2} \int_{Ω_{ϵ} \times (0, T)} (u_{ϵ, λ} - {\bar{u}}_{ϵ}) (u_{ϵ, λ} + {\bar{u}}_{ϵ} - 2 u_{d}) + \frac{β}{2} \int_{Γ \times (0, T)} (2 λ {\bar{θ}}_{ϵ} θ + λ^{2} θ^{2}), \end{aligned}$
where $u_{ϵ, λ}$ is the solution of (2.3) with boundary term ${\hat{\bar{θ}}}_{ϵ} + λ {\hat{θ}}_{ϵ} .$ Note that $w_{ϵ, λ} = u_{ϵ, λ} - {\bar{u}}_{ϵ}$ solves the following equation:
$\begin{aligned} {\begin{cases} w_{ϵ, λ}^{'} - Δ w_{ϵ, λ} + w_{ϵ, λ} = 0 in Ω_{ϵ} \times (0, T), \\ \frac{\partial w_{ϵ, λ}}{\partial ν} = λ {\hat{θ}}_{ϵ} on Γ_{ϵ} \times (0, T), \\ w_{ϵ, λ} = 0 on Γ_{b} \times (0, T), \\ w_{ϵ, λ} (x, 0) = 0 in Ω_{ϵ}, \\ w_{ϵ, λ} is Γ_{s} periodic. \end{cases} \end{aligned}$
From the continuity of the solution operator, we get $| | w_{ϵ, λ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} \leq C_{ϵ} λ | | {\hat{θ}}_{ϵ} | |_{L^{2} (Γ_{ϵ} \times (0, T))}$ . Hence $w_{ϵ, λ} \to 0$ in $L^{2} (0, T; H^{1} (Ω_{ϵ}))$ as $λ \to 0.$ Hence the sequence $u_{ϵ, λ} \to {\bar{u}}_{ϵ}$ in $L^{2} (0, T; H^{1} (Ω_{ϵ}))$ . Let $w_{{\hat{θ}}_{ϵ}} = \frac{1}{λ} w_{ϵ, λ} .$ Note that $w_{{\hat{θ}}_{ϵ}}$ solves the following equation:
$\begin{aligned} {\begin{cases} w_{{\hat{θ}}_{ϵ}}^{'} - Δ w_{{\hat{θ}}_{ϵ}} + w_{{\hat{θ}}_{ϵ}} = 0 in Ω_{ϵ} \times (0, T), \\ \frac{\partial w_{{\hat{θ}}_{ϵ}}}{\partial ν} = {\hat{θ}}_{ϵ} on Γ_{ϵ} \times (0, T), \\ w_{{\hat{θ}}_{ϵ}} = 0 on Γ_{b} \times (0, T), \\ w_{{\hat{θ}}_{ϵ}} (x, 0) = 0 in Ω_{ϵ}, \\ w_{{\hat{θ}}_{ϵ}} is Γ_{s} periodic. \end{cases} \end{aligned}$
(4.4)
Thus $w_{{\hat{θ}}_{ϵ}}$ is independent of $λ$ , so
$\begin{aligned} 0 \leq lim_{λ \to 0} \frac{1}{λ} (F ({\bar{θ}}_{ϵ} + λ θ) - F ({\bar{θ}}_{ϵ})) = \int_{Ω_{ϵ} \times (0, T)} ({\bar{u}}_{ϵ} - u_{d}) w_{{\hat{θ}}_{ϵ}} + β \int_{Γ \times (0, T)} {\bar{θ}}_{ϵ} θ . \end{aligned}$
(4.5)
Hence $F^{'} ({\bar{θ}}_{ϵ}) θ \geq 0$ for all $θ \in L^{2} (Γ \times (0, T)),$ which gives $F^{'} ({\bar{θ}}_{ϵ}) θ = 0$ for all $θ \in L^{2} (Γ \times (0, T))$ . Hence, from (4.5), we obtain
$\begin{aligned} \int_{Ω_{ϵ} \times (0, T)} ({\bar{u}}_{ϵ} - u_{d}) w_{{\hat{θ}}_{ϵ}} = - β \int_{Γ \times (0, T)} {\bar{θ}}_{ϵ} θ . \end{aligned}$
(4.6)
Since, ${\bar{v}}_{ϵ}$ is the solution of equation (4.2) and $w_{{\hat{θ}}_{ϵ}}$ solves equation (4.4), we get
$\begin{aligned} \int_{Γ_{ϵ} \times (0, T)} {\bar{v}}_{ϵ} {\hat{θ}}_{ϵ} = \int_{Ω_{ϵ} \times (0, T)} ({\bar{u}}_{ϵ} - u_{d}) w_{{\hat{θ}}_{ϵ}} = - β \int_{Γ \times (0, T)} {\bar{θ}}_{ϵ} θ . \end{aligned}$
(4.7)
Also, we know that
$\begin{aligned} \int_{Γ_{ϵ} \times (0, T)} {\bar{v}}_{ϵ} {\hat{θ}}_{ϵ} = \int_{Γ_{1}^{ϵ} \times (0, T)} {\bar{v}}_{ϵ} {\hat{θ}}_{ϵ} + \int_{Γ_{2}^{ϵ} \times (0, T)} {\bar{v}}_{ϵ} {\hat{θ}}_{ϵ} + \int_{Γ_{ϵ} / (Γ_{1}^{ϵ} \cup Γ_{2}^{ϵ}) \times (0, T)} {\bar{v}}_{ϵ} {\hat{θ}}_{ϵ} . \end{aligned}$
(4.8)
Applying the unfolding operator, we get
$\begin{aligned} \int_{Γ_{1}^{ϵ} \times (0, T)} {\bar{v}}_{ϵ} {\hat{θ}}_{ϵ} & = \int_{(0, 1) \times A_{1} \times (0, T)} T_{1}^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, h_{1}, x_{3}, t) T_{1}^{ϵ} ({\hat{θ}}_{ϵ}) (x_{1}, h_{1}, x_{3}, t) d x_{1} d x_{3} d t \\ = \int_{(0, 1) \times A_{1} \times (0, T)} T_{1}^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, h_{1}, x_{3}, t) {\hat{θ}}_{ϵ} (ϵ [\frac{x_{1}}{ϵ}] + ϵ x_{3}, h_{1}, t) d x_{1} d x_{3} d t \\ = \int_{(0, 1) \times A_{1} \times (0, T)} T_{1}^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, h_{1}, x_{3}, t) θ (x_{3}, h_{1}, t) d x_{1} d x_{3} d t \\ = \int_{A_{1} \times (0, T)} [\int_{0}^{1} T_{1}^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, h_{1}, x_{3}, t) d x_{1}] θ (x_{3}, h_{1}, t) d x_{3} d t . \end{aligned}$
Similarly
$\begin{aligned} \int_{Γ_{2}^{ϵ} \times (0, T)} {\bar{v}}_{ϵ} {\hat{θ}}_{ϵ} = \int_{A_{2} \times (0, T)} [\int_{0}^{1} T_{2}^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, h_{2}, x_{3}, t) d x_{1}] θ (x_{3}, h_{2}, t) d x_{3} d t, \end{aligned}$
and
$\begin{aligned} \int_{Γ_{ϵ} / (Γ_{1}^{ϵ} \cup Γ_{2}^{ϵ}) \times (0, T)} {\bar{v}}_{ϵ} {\hat{θ}}_{ϵ} & = \sum_{j = 0}^{N - 1} [\int_{h_{1}}^{h_{2}} \int_{0}^{T} {\bar{v}}_{ϵ} (j ϵ + ϵ p, x_{2}, t) {\hat{θ}}_{ϵ} (j ϵ + ϵ p, x_{2}, t) d x_{2} d t \\ + \int_{h_{1}}^{h_{2}} \int_{0}^{T} {\bar{v}}_{ϵ} (j ϵ + ϵ q, x_{2}, t) {\hat{θ}}_{ϵ} (j ϵ + ϵ q, x_{2}, t)] d x_{2} d t \\ = ϵ^{α - 1} \sum_{j = 0}^{N - 1} [\int_{h_{1}}^{h_{2}} \int_{0}^{T} (\int_{j ϵ}^{(j + 1) ϵ} {\bar{v}}_{ϵ} (ϵ [\frac{x_{1}}{ϵ}] + ϵ p, x_{2}, t) d x_{1}) θ (p, x_{2}, t) d x_{2} d t \\ + \int_{h_{1}}^{h_{2}} \int_{0}^{T} (\int_{j ϵ}^{(j + 1) ϵ} {\bar{v}}_{ϵ} (ϵ [\frac{x_{1}}{ϵ}] + ϵ q, x_{2}, t) d x_{1}) θ (q, x_{2}, t) d x_{2} d t] \\ = ϵ^{α - 1} [\int_{h_{1}}^{h_{2}} \int_{0}^{T} (\int_{0}^{1} T^{ϵ} {\bar{v}}_{ϵ} (x_{1}, x_{2}, p, t) d x_{1}) θ (p, x_{2}, t) d x_{2} d t \\ + \int_{h_{1}}^{h_{2}} \int_{0}^{T} (\int_{0}^{1} T^{ϵ} {\bar{v}}_{ϵ} (x_{1}, x_{2}, q, t) d x_{1}) θ (q, x_{2}, t) d x_{2} d t] . \end{aligned}$
Since $θ$ is arbitrary, using (4.7) and (4.8), we arrive at the characterization of optimal control ${\bar{θ}}_{ϵ}$ as given in the theorem. Now, we prove the converse of the theorem. Let $({\bar{u}}_{ϵ}, {\bar{v}}_{ϵ}) \in W^{ϵ} \times W^{ϵ}$ be the solution of optimality system (4.3). For $θ \in L^{2} (Γ \times (0, T)),$ we have
$\begin{aligned} F ({\bar{θ}}_{ϵ} + θ) - F ({\bar{θ}}_{ϵ}) & = \frac{1}{2} \int_{Ω_{ϵ} \times (0, T)} | u_{ϵ}^{1} - {\bar{u}}_{ϵ} |^{2} + \frac{β}{2} \int_{Γ \times (0, T)} | {\bar{θ}}_{ϵ} |^{2} \\ + \int_{Ω_{ϵ} \times (0, T)} (u_{ϵ}^{1} - {\bar{u}}_{ϵ}) ({\bar{u}}_{ϵ} - u_{d}) + β \int_{Γ \times (0, T)} {\bar{θ}}_{ϵ} θ, \end{aligned}$
where $u_{ϵ}^{1}$ is the solution of (2.3) with boundary term ${\hat{\bar{θ}}}_{ϵ} + {\hat{θ}}_{ϵ} .$ Note that
$\begin{aligned} \int_{Ω_{ϵ} \times (0, T)} (u_{ϵ}^{1} - {\bar{u}}_{ϵ}) ({\bar{u}}_{ϵ} - u_{d}) & = \int_{Ω_{ϵ} \times (0, T)} \nabla (u_{ϵ}^{1} - {\bar{u}}_{ϵ}) \cdot \nabla {\bar{v}}_{ϵ} + \int_{Ω_{ϵ} \times (0, T)} (u_{ϵ}^{1} - {\bar{u}}_{ϵ}) {\bar{v}}_{ϵ} \\ + \int_{\partial Ω_{ϵ} \times (0, T)} \frac{\partial {\bar{v}}_{ϵ}}{\partial ν} (u_{ϵ}^{1} - {\bar{u}}_{ϵ}) = \int_{Γ_{ϵ} \times (0, T)} {\bar{v}}_{ϵ} {\hat{θ}}_{ϵ} = - β \int_{Γ \times (0, T)} {\bar{θ}}_{ϵ} θ . \end{aligned}$
Thus, $F ({\bar{θ}}_{ϵ} + θ) - F ({\bar{θ}}_{ϵ}) \geq 0.$ Hence $({\bar{u}}_{ϵ}, {\bar{v}}_{ϵ})$ is the optimal solution of (2.2). This complete the proof.
Now, we prove some preliminary results which will be useful for proving the uniform estimates. Let $({\bar{u}}_{ϵ}, {\bar{θ}}_{ϵ})$ be the optimal solution of (2.2). Let $u_{ϵ} (0)$ be the solution of (2.3) corresponding to $θ = 0,$ then, we obtain
$\begin{aligned} | | u_{ϵ} (0) | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} \leq C, \end{aligned}$
(4.9)
where $C > 0$ is the constant independent of $ϵ .$ Using the optimality of the solution $({\bar{u}}_{ϵ}, {\bar{θ}}_{ϵ}),$ we obtain
$\begin{aligned} \int_{Ω_{ϵ} \times (0, T)} | {\bar{u}}_{ϵ} - u_{d} |^{2} + \frac{β}{2} \int_{Γ \times (0, T)} | {\bar{θ}}_{ϵ} |^{2} \leq \int_{Ω_{ϵ} \times (0, T)} | {\bar{u}}_{ϵ} (0) - u_{d} |^{2} \leq C . \end{aligned}$
(4.10)
Thus
$\begin{aligned} | | {\bar{θ}}_{ϵ} | |_{L^{2} (Γ \times (0, T))} \leq C and | | {\bar{u}}_{ϵ} | |_{L^{2} (Ω_{ϵ} \times (0, T))} \leq C . \end{aligned}$
(4.11)
From the weak formulation of adjoint equation (4.2), we get
$\begin{aligned} | | {\bar{v}}_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} \leq C, \end{aligned}$
(4.12)
where $C > 0$ is independent of $ϵ .$
Lemma 4.2
For any $u_{ϵ} \in L^{2} (0, T; H^{1} (Ω_{ϵ})),$ there exist a constant $C > 0$ (independent of $ϵ$ ) such that
$\begin{aligned} | | u_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{c}) \times (0, T))} \leq C | | u_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))}, \\ | | u_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{u}) \times (0, T))} \leq C | | u_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} . \end{aligned}$

Proof.
Using Trace theorem, there exist a positive constant $C > 0$ (independent of $ϵ$ ) such that
$\begin{aligned} | | u_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{c}) \times (0, T))}^{2} = \int_{(Γ_{ϵ} \cap Γ_{c}) \times (0, T)} | u_{ϵ} |^{2} \leq \int_{Γ_{c} \times (0, T)} | u_{ϵ} |^{2} \leq | | u_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω^{-}))} \leq | | u_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} . \end{aligned}$
Again, using Trace theorem and Holders inequality, we get
$\begin{aligned} | | u_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{u}) \times (0, T))} & = \int_{(Γ_{ϵ} \cap Γ_{u}) \times (0, T)} | u_{ϵ} |^{2} = \sum_{j = 0}^{N - 1} \int_{j ϵ + ϵ p}^{j ϵ + ϵ q} \int_{0}^{T} (u_{ϵ} (x_{1}, h_{2}, t))^{2} d x_{1} d t \\ = \sum_{j = 0}^{N - 1} \int_{j ϵ + ϵ p}^{j ϵ + ϵ q} \int_{0}^{T} {(\int_{K (x_{1})}^{h_{2}} \frac{\partial u_{ϵ}}{\partial x_{2}} (x_{1}, x_{2}, t) d x_{2} + u_{ϵ} (x_{1}, K (x_{1}), t))}^{2} d x_{1} d t \\ \leq C \sum_{j = 0}^{N - 1} \int_{j ϵ + ϵ p}^{j ϵ + ϵ q} \int_{0}^{T} ({(\int_{K (x_{1})}^{h_{2}} \frac{\partial u_{ϵ}}{\partial x_{2}} (x_{1}, x_{2}, t) d x_{2})}^{2} + u_{ϵ}^{2} (x_{1}, K (x_{1}), t)) d x_{1} d t \\ \leq C (\sum_{j = 0}^{N - 1} \int_{j ϵ + ϵ p}^{j ϵ + ϵ q} \int_{0}^{T} \int_{K (x_{1})}^{h_{2}} | \frac{\partial u_{ϵ}}{\partial x_{2}} |^{2} + | | u_{ϵ} | |_{L^{2} (Γ_{b} \times (0, T))}) \\ \leq C (| | \frac{\partial u_{ϵ}}{\partial x_{2}} | |_{L^{2} (Ω_{ϵ} \times (0, T))}^{2} + | | u_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω^{-}))}^{2}) . \end{aligned}$

Hence, we have $| | u_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{u}) \times (0, T))} \leq C | | u_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} .$
Proposition 4.3
Let $({\bar{u}}_{ϵ}, {\bar{θ}}_{ϵ})$ be the optimal solution of (2.2). For $α \geq 1$ , there exist a constant $C > 0$ (independent of $ϵ$ ) such that $| | {\bar{u}}_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} \leq C .$
Proof.
Using $ψ = {\bar{u}}_{ϵ}$ in the weak formulation (2.4), we get
$\begin{aligned} \frac{1}{2} \int_{Ω_{ϵ}} {\bar{u}}_{ϵ} (x, T) |^{2} + | | {\bar{u}}_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} = \int_{Ω_{ϵ} \times (0, T)} f {\bar{u}}_{ϵ} + \int_{Γ_{ϵ} \times (0, T)} {\hat{θ}}_{ϵ} {\bar{u}}_{ϵ} . \end{aligned}$
(4.13)
By Cauchy–Schwarz inequality, we have
$\begin{aligned} \int_{Ω_{ϵ} \times (0, T)} f {\bar{u}}_{ϵ} \leq | | f | |_{L^{2} (Ω_{ϵ} \times (0, T))} | | {\bar{u}}_{ϵ} | |_{L^{2} (Ω_{ϵ} \times (0, T))} . \end{aligned}$
(4.14)
Note that
$\begin{aligned} \int_{Γ_{ϵ} \times (0, T)} {\hat{θ}}_{ϵ} {\bar{u}}_{ϵ} \leq \int_{Γ_{ϵ} \times (0, T)} | {\hat{θ}}_{ϵ} | {\bar{u}}_{ϵ} | & = \sum_{j = 0}^{N - 1} [\int_{j ϵ}^{j ϵ + ϵ p} \int_{0}^{T} | {\hat{θ}}_{ϵ} (x_{1}, h_{1}, t) | {\bar{u}}_{ϵ} (x_{1}, h_{1}, t) | d x_{1} d t \\ + \int_{h_{1}}^{h_{2}} \int_{0}^{T} | {\hat{θ}}_{ϵ} (j ϵ + ϵ p, x_{2}, t) | {\bar{u}}_{ϵ} (j ϵ + ϵ p, x_{2}, t) | d x_{2} d t \\ + \int_{j ϵ + ϵ p}^{j ϵ + ϵ q} \int_{0}^{T} | {\hat{θ}}_{ϵ} (x_{1}, h_{1}, t) | {\bar{u}}_{ϵ} (x_{1}, h_{1}, t) | d x_{1} d t \\ + \int_{h_{1}}^{h_{2}} \int_{0}^{T} | {\hat{θ}}_{ϵ} (j ϵ + ϵ q, x_{2}, t) | {\bar{u}}_{ϵ} (j ϵ + ϵ q, x_{2}, t) | d x_{2} d t \\ + \int_{j ϵ + ϵ q}^{j ϵ + ϵ} \int_{0}^{T} | {\hat{θ}}_{ϵ} (x_{1}, h_{2}, t) | {\bar{u}}_{ϵ} (x_{1}, h_{2}, t) | d x_{1} d t] . \end{aligned}$
Applying Cauchy–Schwarz inequality, we obtain
$\begin{aligned} \int_{Γ_{ϵ} \times (0, T)} {\hat{θ}}_{ϵ} {\bar{u}}_{ϵ} & \leq \sum_{j = 0}^{N - 1} [ϵ^{\frac{1}{2}} {(\int_{0}^{p} \int_{0}^{T} | {\bar{θ}}_{ϵ} (y_{1}, h_{1}, t) |^{2} d y_{1} d t)}^{\frac{1}{2}} {(\int_{j ϵ}^{j ϵ + ϵ p} \int_{0}^{T} | {\bar{u}}_{ϵ} (x_{1}, h_{1}, t) |^{2} d x_{1} d t)}^{\frac{1}{2}} \\ + ϵ^{α} {(\int_{h_{1}}^{h_{2}} \int_{0}^{T} | {\bar{θ}}_{ϵ} (p, y_{2}, t) |^{2} d y_{2} d t)}^{\frac{1}{2}} {(\int_{h_{1}}^{h_{2}} \int_{0}^{T} | {\bar{u}}_{ϵ} (j ϵ + ϵ p, x_{2}, t) |^{2} d x_{2} d t)}^{\frac{1}{2}} \\ + ϵ^{\frac{1}{2}} {(\int_{p}^{q} \int_{0}^{T} | {\bar{θ}}_{ϵ} (y_{1}, h_{2}, t) |^{2} d y_{1} d t)}^{\frac{1}{2}} {(\int_{j ϵ + ϵ p}^{j ϵ + ϵ q} \int_{0}^{T} | {\bar{u}}_{ϵ} (x_{1}, h_{2}, t) |^{2} d x_{1} d t)}^{\frac{1}{2}} \\ + ϵ^{α} {(\int_{h_{1}}^{h_{2}} \int_{0}^{T} | {\bar{θ}}_{ϵ} (q, y_{2}, t) |^{2} d y_{2} d t)}^{\frac{1}{2}} {(\int_{h_{1}}^{h_{2}} \int_{0}^{T} | {\bar{u}}_{ϵ} (j ϵ + ϵ q, x_{2}, t) |^{2} d x_{2} d t)}^{\frac{1}{2}} \\ + ϵ^{\frac{1}{2}} {(\int_{q}^{1} \int_{0}^{T} | {\bar{θ}}_{ϵ} (y_{1}, h_{1}, t) |^{2} d y_{1} d t)}^{\frac{1}{2}} {(\int_{j ϵ + ϵ q}^{j ϵ + ϵ} \int_{0}^{T} | {\bar{u}}_{ϵ} (x_{1}, h_{1}, t) |^{2} d x_{1} d t)}^{\frac{1}{2}}] \end{aligned}$

$\begin{aligned} \leq ϵ^{\frac{1}{2}} | | {\bar{θ}}_{ϵ} | |_{L^{2} (Γ \times (0, T))} \sum_{j = 0}^{N - 1} [{(\int_{j ϵ}^{j ϵ + ϵ p} \int_{0}^{T} | {\bar{u}}_{ϵ} (x_{1}, h_{1}, t) |^{2} d x_{1} d t)}^{\frac{1}{2}} \\ + ϵ^{α - \frac{1}{2}} {(\int_{h_{1}}^{h_{2}} \int_{0}^{T} | {\bar{u}}_{ϵ} (j ϵ + ϵ p, x_{2}, t) |^{2} d x_{2} d t)}^{\frac{1}{2}} \\ + {(\int_{j ϵ + ϵ p}^{j ϵ + ϵ q} \int_{0}^{T} | {\bar{u}}_{ϵ} (x_{1}, h_{2}, t) |^{2} d x_{1} d t)}^{\frac{1}{2}} \\ + ϵ^{α - \frac{1}{2}} {(\int_{h_{1}}^{h_{2}} \int_{0}^{T} | {\bar{u}}_{ϵ} (j ϵ + ϵ q, x_{2}, t) |^{2} d x_{2} d t)}^{\frac{1}{2}} \\ + {(\int_{j ϵ + ϵ q}^{j ϵ + ϵ} \int_{0}^{T} | {\bar{u}}_{ϵ} (x_{1}, h_{1}, t) |^{2} d x_{1} d t)}^{\frac{1}{2}}] \end{aligned}$

$\begin{aligned} \leq ϵ^{\frac{1}{2}} | | {\bar{θ}}_{ϵ} | |_{L^{2} (Γ \times (0, T))} (ϵ^{\frac{- 1}{2}} | | {\bar{u}}_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{u}) \times (0, T))} + ϵ^{\frac{- 1}{2}} | | {\bar{u}}_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{c}) \times (0, T))}) \\ + ϵ^{α - \frac{1}{2}} | | {\bar{θ}}_{ϵ} | |_{L^{2} (Γ \times (0, T))} \sum_{j = 0}^{N - 1} [{(\int_{j ϵ}^{j ϵ + ϵ} \int_{h_{1}}^{h_{2}} \int_{0}^{T} | T^{ϵ} {\bar{u}}_{ϵ} (x_{1}, x_{2}, p, t) |^{2} d x_{1} d x_{2} d t)}^{\frac{1}{2}} \\ + {(\int_{j ϵ}^{j ϵ + ϵ} \int_{h_{1}}^{h_{2}} \int_{0}^{T} | T^{ϵ} {\bar{u}}_{ϵ} (x_{1}, x_{2}, q, t) |^{2} d x_{1} d x_{2} d t)}^{\frac{1}{2}}] \\ \leq | | {\bar{θ}}_{ϵ} | |_{L^{2} (Γ \times (0, T))} (| | {\bar{u}}_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{u}) \times (0, T))} + | | {\bar{u}}_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{c}) \times (0, T))}) \\ + ϵ^{α - 1} | | {\bar{θ}}_{ϵ} | |_{L^{2} (Γ \times (0, T))} (| | T^{ϵ} {\bar{u}}_{ϵ} |_{x_{3} = p} | |_{L^{2} (Ω^{+} \times (0, T))} + | | T^{ϵ} {\bar{u}}_{ϵ} |_{x_{3} = q} | |_{L^{2} (Ω^{+} \times (0, T))}) . \end{aligned}$
Using Trace theorem and Proposition (3.2)( $P 3$ )
$\begin{aligned} \int_{Γ_{ϵ} \times (0, T)} {\hat{θ}}_{ϵ} {\bar{u}}_{ϵ} \leq | | {\bar{θ}}_{ϵ} | |_{L^{2} (Γ \times (0, T))} & [| | {\bar{u}}_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{u}) \times (0, T))} + | | {\bar{u}}_{ϵ} | |_{L^{2} ((Γ_{ϵ} \cap Γ_{c}) \times (0, T))} + ϵ^{α - 1} | | {\bar{u}}_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}^{+}))}] . \end{aligned}$
(4.15)
Thus, by combining (4.11), (4.13)–(4.15), Lemma (4.2), we obtain $| | {\bar{u}}_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}))} \leq C .$
5. Homogenized Optimal Control Problem

Let’s introduce the space

\begin{aligned} V (Ω) = {u \in L^{2} (Ω) : \frac{\partial u}{\partial x_{2}} \in L^{2} (Ω^{+}), u \in H^{1} (Ω^{-}), u |_{Γ_{b}} = 0} . \end{aligned}

(5.1)

Note that $V (Ω)$ is a Hilbert space with the norm given by

\begin{aligned} | | u | |_{V (Ω)}^{2} = | | u | |_{L^{2} (Ω^{+})}^{2} + | | \frac{\partial u}{\partial x_{2}} | |_{L^{2} (Ω^{+})}^{2} + | | u | |_{H^{1} (Ω^{-})}^{2} . \end{aligned}

We consider the space

\begin{aligned} W = {u \in L^{2} (0, T; V (Ω)) : u^{'} \in L^{2} (0, T; L^{2} (Ω))} . \end{aligned}

(5.2)

where

V (Ω)

is defined in (5.1).

Now, we consider the homogenized optimality system corresponding to parameter $α > 1$ and $α = 1.$ For given $f \in L^{2} (Ω \times (0, T)), θ \in L^{2} ((h_{1}, h_{2}) \times (0, T)), C_{1}$ and $C_{2}$ in $R,$ consider following two system for $j = 0, 1 :$

\begin{aligned} {\begin{cases} (u^{+})^{'} - \frac{\partial^{2} u^{+}}{\partial x_{2}^{2}} + u^{+} = f - j θ χ_{Ω^{+}} in Ω^{+} \times (0, T), \\ (u^{-})^{'} - Δ u^{-} + u^{-} = f in Ω^{-} \times (0, T), \\ \frac{\partial u^{+}}{\partial x_{2}} = C_{2} on Γ_{u} \times (0, T), \\ u^{+} = u^{-}, \frac{\partial u^{-}}{\partial x_{2}} - (q - p) \frac{\partial u^{+}}{\partial x_{2}} = C_{1} on Γ_{c} \times (0, T), \\ u^{-} = 0 on Γ_{b} \times (0, T), \\ u (x, 0) = 0, in Ω^{+}, \\ u is Γ_{s^{'}} periodic, \end{cases} \end{aligned}

(5.3)

where, we write

u = u^{+} χ_{Ω^{+}} + u^{-} χ_{Ω^{-}} .

Remark 5.1

Note that the triple $(V (Ω), L^{2} (Ω), (V (Ω))^{'})$ is an evolution triple (Durante et al., 2007; Lemma 3.7). Also, $W \subset C ([0, T]; L^{2} (Ω)) .$ Thus, we have $u \in C ([0, T]; L^{2} (Ω)) .$ Hence, initial conditions $u (0)$ make sense.

Existence and uniqueness of solution $u \in W$ of problem (5.3) follows in the standard way. Now, we consider the $L^{2}$ - cost functional $J_{1}$ and $J_{2}$ given by

\begin{aligned} J_{1} (u, θ, C_{1}, C_{2}) & = \frac{1}{2} \int_{Ω \times (0, T)} ((q - p) χ_{Ω^{+}} + χ_{Ω^{-}}) | u - u_{d} |^{2} + \frac{(q - p)^{2}}{4} \int_{h_{1}}^{h_{2}} \int_{0}^{T} | θ |^{2} + \frac{β}{2 (1 - (q - p))} | C_{1} |^{2} + \frac{β}{2} | C_{2} |^{2} . \end{aligned}

and

\begin{aligned} J_{2} (u, C_{1}, C_{2}) = \frac{1}{2} \int_{Ω \times (0, T)} ((q - p) χ_{Ω^{+}} + χ_{Ω^{-}}) | u - u_{d} |^{2} + \frac{β}{2 (1 - (q - p))} | C_{1} |^{2} + \frac{β}{2} | C_{2} |^{2} . \end{aligned}

Associated with these cost functionals, we introduce following optimal control problems:

\begin{aligned} inf {J_{1} (u, θ, C_{1}, C_{2}) : θ \in L^{2} ((h_{1}, h_{2}) \times (0, T)) C_{1}, C_{2} \in R, (u, θ, C_{1}, C_{2}) solves (5.3) for j = 1} \end{aligned}

(5.4)

and

\begin{aligned} inf {J_{2} (u, C_{1}, C_{2}) : C_{1}, C_{2} \in R, (u, C_{1}, C_{2}) solves (5.3) for j = 0} \end{aligned}

(5.5)

Later on, we will see that (5.4) corresponds to the case

α = 1

and (5.5) corresponds to

α > 1.

By using standard method of Calculus of Variations, the OCP (5.4) and (5.5) admits a unique solution. In the next two theorems, we give characterization of optimal controls. Proof of these theorems can be obtain following the line of proofs given for the Theorem 4.1, therefore we omit the proof of these theorems.

Theorem 5.2

Let $(\bar{u}, \bar{θ}, \bar{C_{1}}, \bar{C_{2}})$ be the optimal solution of (5.4), then

\begin{aligned} \bar{θ} (x_{2}, t) & = \frac{2}{(q - p)} \int_{0}^{1} \bar{v} (x_{1}, x_{2}, t) d x_{1}, \\ \bar{C_{1}} & = - \frac{1 - (q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{1}, t) d y d t, \bar{C_{2}} = - \frac{(q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{2}, t) d y d t \end{aligned}

(5.6)

where

\bar{v} \in W

satisfies

\begin{aligned} {\begin{cases} ({\bar{v}}^{+})^{'} - \frac{\partial^{2} {\bar{v}}^{+}}{\partial x_{2}^{2}} + {\bar{v}}^{+} = {\bar{u}}^{+} - u_{d} in Ω^{+} \times (0, T), \\ ({\bar{v}}^{-})^{'} - Δ {\bar{v}}^{-} + {\bar{v}}^{-} = {\bar{u}}^{-} - u_{d} in Ω^{-} \times (0, T), \\ \frac{\partial {\bar{v}}^{+}}{\partial x_{2}} = 0 on Γ_{u} \times (0, T), \\ {\bar{v}}^{+} = {\bar{v}}^{-}, \frac{\partial {\bar{v}}^{-}}{\partial x_{2}} - (q - p) \frac{\partial {\bar{v}}^{+}}{\partial x_{2}} = 0 on Γ_{c} \times (0, T), \\ {\bar{v}}^{-} = 0 on Γ_{b} \times (0, T), \\ \bar{v} (x, T) = 0, in Ω^{+}, \\ \bar{v} is Γ_{s^{'}} periodic . \end{cases} \end{aligned}

(5.7)

Conversely, let a pair

(\bar{u}, \bar{v}) \in W \times W

solves the optimality system:

\begin{aligned} {\begin{cases} ({\bar{u}}^{+})^{'} - \frac{\partial^{2} {\bar{u}}^{+}}{\partial x_{2}^{2}} + {\bar{u}}^{+} = f - \bar{θ} in Ω^{+} \times (0, T), \\ ({\bar{v}}^{+})^{'} - \frac{\partial^{2} {\bar{v}}^{+}}{\partial x_{2}^{2}} + {\bar{v}}^{+} = {\bar{u}}^{+} - u_{d} in Ω^{+} \times (0, T), \\ ({\bar{u}}^{-})^{'} - Δ {\bar{u}}^{-} + {\bar{u}}^{-} = f in Ω^{-} \times (0, T), \\ ({\bar{v}}^{-})^{'} - Δ {\bar{v}}^{-} + {\bar{v}}^{-} = {\bar{u}}^{-} - u_{d} in Ω^{-} \times (0, T), \\ \frac{\partial {\bar{u}}^{+}}{\partial x_{2}} = \bar{C_{2}}, \frac{\partial {\bar{v}}^{+}}{\partial x_{2}} = 0 on Γ_{u} \times (0, T), \\ {\bar{u}}^{+} = {\bar{u}}^{-}, \frac{\partial {\bar{u}}^{-}}{\partial x_{2}} - (q - p) \frac{\partial {\bar{u}}^{+}}{\partial x_{2}} = \bar{C_{1}} on Γ_{c} \times (0, T), \\ {\bar{v}}^{+} = {\bar{v}}^{-}, \frac{\partial {\bar{v}}^{-}}{\partial x_{2}} - (q - p) \frac{\partial {\bar{v}}^{+}}{\partial x_{2}} = 0 on Γ_{c} \times (0, T), \\ {\bar{u}}^{-} = 0, {\bar{v}}^{-} = 0 on Γ_{b} \times (0, T), \\ \bar{u} (x, 0) = 0, \bar{v} (x, T) = 0, in Ω^{+}, \\ \bar{u}, \bar{v} is Γ_{s^{'}} periodic, \end{cases} \end{aligned}

(5.8)

where

\begin{aligned} \bar{θ} (x_{2}, t) & = \frac{2}{(q - p)} \int_{0}^{1} \bar{v} (x_{1}, x_{2}, t) d x_{1}, \\ \bar{C_{1}} & = - \frac{1 - (q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{1}, t) d y d t, \bar{C_{2}} = - \frac{(q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{2}, t) d y d t \end{aligned}

Then

(\bar{u}, \bar{θ}, \bar{C_{1}}, \bar{C_{2}})

is the optimal solution of (5.4).

Similarly, we have the following theorem corresponding to (5.5) (case $α > 1$ ).

Theorem 5.3

Let $(\bar{u}, \bar{C_{1}}, \bar{C_{2}})$ be the optimal solution of (5.5), then

\begin{aligned} \bar{C_{1}} = - \frac{1 - (q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{1}, t) d y d t, \bar{C_{2}} = - \frac{(q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{2}, t) d y d t \end{aligned}

(5.9)

where

\bar{v} \in W

satisfies (5.7). Conversely, let a pair

(\bar{u}, \bar{v}) \in W \times W

solves the optimality system:

\begin{aligned} {\begin{cases} ({\bar{u}}^{+})^{'} - \frac{\partial^{2} {\bar{u}}^{+}}{\partial x_{2}^{2}} + {\bar{u}}^{+} = f in Ω^{+} \times (0, T), \\ ({\bar{v}}^{+})^{'} - \frac{\partial^{2} {\bar{v}}^{+}}{\partial x_{2}^{2}} + {\bar{v}}^{+} = {\bar{u}}^{+} - u_{d} in Ω^{+} \times (0, T), \\ ({\bar{u}}^{-})^{'} - Δ {\bar{u}}^{-} + {\bar{u}}^{-} = f in Ω^{-} \times (0, T), \\ ({\bar{v}}^{-})^{'} - Δ {\bar{v}}^{-} + {\bar{v}}^{-} = {\bar{u}}^{-} - u_{d} in Ω^{-} \times (0, T), \\ \frac{\partial {\bar{u}}^{+}}{\partial x_{2}} = \bar{C_{2}}, \frac{\partial {\bar{v}}^{+}}{\partial x_{2}} = 0 on Γ_{u} \times (0, T), \\ {\bar{u}}^{+} = {\bar{u}}^{-}, \frac{\partial {\bar{u}}^{-}}{\partial x_{2}} - (q - p) \frac{\partial {\bar{u}}^{+}}{\partial x_{2}} = \bar{C_{1}} on Γ_{c} \times (0, T), \\ {\bar{v}}^{+} = {\bar{v}}^{-}, \frac{\partial {\bar{v}}^{-}}{\partial x_{2}} - (q - p) \frac{\partial {\bar{v}}^{+}}{\partial x_{2}} = 0 on Γ_{c} \times (0, T), \\ {\bar{u}}^{-} = 0, {\bar{v}}^{-} = 0 on Γ_{b} \times (0, T), \\ \bar{u} (x, 0) = 0, \bar{v} (x, T) = 0, in Ω^{+}, \\ \bar{u}, \bar{v} is Γ_{s^{'}} periodic, \end{cases} \end{aligned}

(5.10)

where

\begin{aligned} \bar{C_{1}} = - \frac{1 - (q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{1}, t) d y d t, \bar{C_{2}} = - \frac{(q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{2}, t) d y d t \end{aligned}

Then

(\bar{u}, \bar{C_{1}}, \bar{C_{2}})

is the optimal solution of (5.5).

6. Convergence Results

Now, we will state the main convergence theorems depending on the scaling parameter $α$ . Firstly, we state and prove the convergence theorem for critical case $α = 1$ .

Theorem 6.1
Let $({\bar{u}}_{ϵ}, {\bar{θ}}_{ϵ})$ be the optimal solution of (2.2) with $α = 1$ and let $(\bar{u}, \bar{θ}, \bar{C_{1}}, \bar{C_{2}})$ be the optimal solution of (5.4). Let ${\bar{v}}_{ϵ}, \bar{v}$ be the corresponding adjoint state satisfying (4.2) and (5.7) respectively. Then
$\begin{aligned} \tilde{{\bar{u}}_{ϵ} |_{Ω_{ϵ}^{+}}} ⇀ (q - p) \bar{u} |_{Ω^{+}} weakly in L^{2} ((0, 1) \times (0, T); H^{1} (h_{1}, h_{2})), \\ \tilde{{\bar{v}}_{ϵ} |_{Ω_{ϵ}^{+}}} ⇀ (q - p) \bar{v} |_{Ω^{+}} weakly in L^{2} ((0, 1) \times (0, T); H^{1} (h_{1}, h_{2})), \\ {\bar{u}}_{ϵ} |_{Ω^{-}} ⇀ \bar{u} |_{Ω^{-}} weakly in L^{2} (0, T; H^{1} (Ω^{-})), \\ {\bar{v}}_{ϵ} |_{Ω^{-}} ⇀ \bar{v} |_{Ω^{-}} weakly in L^{2} (0, T; H^{1} (Ω^{-})), \\ ({\hat{\bar{θ}}}_{ϵ}, ϕ) \to (Φ, ϕ), \end{aligned}$
(6.1)
for all $ϕ \in L^{2} (0, T; H^{1} (Ω^{+})),$ $Φ = Φ (\bar{θ}, \bar{C_{1}}, \bar{C_{2}})$ where
$\begin{aligned} (Φ, ϕ) = \int_{0}^{1} \int_{0}^{T} \bar{C_{1}} ϕ (x_{1}, h_{1}, t) d x_{1} d t + \int_{0}^{1} \int_{0}^{T} \bar{C_{2}} ϕ (x_{1}, h_{2}, t) d x_{1} d t + \int_{Ω^{+} \times (0, T)} \bar{θ} ϕ (x_{1}, x_{2}, t) d x_{1} d x_{2} d t, \end{aligned}$
and
$\begin{aligned} \bar{θ} & = \frac{2}{(q - p)} \int_{0}^{1} \bar{v} (x_{1}, x_{2}, t) d x_{1}, \bar{C_{1}} = - \frac{1 - (q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{1}, t) d y d t and \\ \bar{C_{2}} & = - \frac{(q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{2}, t) d y d t . \end{aligned}$

Proof.
The restriction of ${\bar{u}}_{ϵ}$ in $Ω_{ϵ}^{+}$ is denoted by ${\bar{u}}_{ϵ}^{+} = {\bar{u}}_{ϵ} |_{Ω_{ϵ}^{+}} .$ Similarly ${\bar{u}}_{ϵ}^{-} = {\bar{u}}_{ϵ} |_{Ω^{-}} .$ Using the Propositions 4.3 and 3.2 $(P 3),$ we conclude that $T^{ϵ} ({\bar{u}}_{ϵ}^{+})$ is uniformly bounded in $G$ (independent of $ϵ$ ). Thus, using weak compactness argument, there exist a subsequence (still denoted by $ϵ$ ) and a $u^{+} \in G$ , such that
$\begin{aligned} T_{+}^{ϵ} ({\bar{u}}_{ϵ}^{+}) ⇀ u^{+} weakly in G . \end{aligned}$
(6.2)
From the Proposition 3.2 $(P 3)$ and (6.2), it follows that
$\begin{aligned} T^{ϵ} ({\bar{u}}_{ϵ}^{+}) & ⇀ u^{+} weakly in L^{2} (Ω^{+} \times (p, q) \times (0, T)), \end{aligned}$
(6.3a)

$\begin{aligned} T^{ϵ} (\frac{\partial {\bar{u}}_{ϵ}^{+}}{\partial x_{2}}) & ⇀ \frac{\partial u^{+}}{\partial x_{2}} weakly in L^{2} (Ω^{+} \times (p, q) \times (0, T)), \end{aligned}$
(6.3b)

$\begin{aligned} ϵ T^{ϵ} (\frac{\partial {\bar{u}}_{ϵ}^{+}}{\partial x_{1}}) & ⇀ \frac{\partial u^{+}}{\partial x_{3}} weakly in L^{2} (Ω^{+} \times (p, q) \times (0, T)) . \end{aligned}$
(6.3c)

By the Proposition 3.2 (P4), we get
$\begin{aligned} | | T^{ϵ} (\frac{\partial {\bar{u}}_{ϵ}^{+}}{\partial x_{1}}) | |_{L^{2} (Ω^{+} \times (p, q) \times (0, T))}^{2} = | | \frac{\partial {\bar{u}}_{ϵ}^{+}}{\partial x_{1}} | |_{L^{2} (Ω_{ϵ}^{+} \times (0, T))}^{2} \leq | | {\bar{u}}_{ϵ} | |_{L^{2} (0, T; H^{1} (Ω_{ϵ}^{+}))}^{2} . \end{aligned}$
(6.4)
With the aid of inequality (6.4) and Proposition 4.3, we obtain the sequence $T^{ϵ} (\frac{\partial {\bar{u}}_{ϵ}^{+}}{\partial x_{1}})$ which is uniformly bounded in $L^{2} (Ω^{+} \times (p, q) \times (0, T))$ . Hence, using weak compactness argument, there exist a subsequence (still denoted by $ϵ$ ) and a $ξ_{1} \in L^{2} (Ω^{+} \times (p, q) \times (0, T))$ such that
$\begin{aligned} T^{ϵ} (\frac{\partial {\bar{u}}_{ϵ}^{+}}{\partial x_{1}}) ⇀ ξ_{1} weakly in L^{2} (Ω^{+} \times (p, q) \times (0, T)) . \end{aligned}$
(6.5)
Further from (6.3c), it follows that $\frac{\partial u^{+}}{\partial x_{3}} = 0$ , i.e., $u^{+}$ is independent of $x_{3}$ variable. Also, using Proposition 3.2 $(P 7)$
$\begin{aligned} \tilde{{\bar{u}}_{ϵ}^{+}} ⇀ (q - p) u^{+} weakly in L^{2} ((0, 1) \times (0, T); H^{1} (h_{1}, h_{2})) . \end{aligned}$
(6.6)
Since $| | {\bar{u}}_{ϵ}^{-} | |_{L^{2} (0, T; H^{1} (Ω^{-}))}$ is uniformly bounded, therefore, there exists a subsequence (still denoted by $ϵ$ ) and $u^{-} \in L^{2} (0, T; H^{1} (Ω^{-}))$ such that
$\begin{aligned} {\bar{u}}_{ϵ}^{-} ⇀ u^{-} weakly in L^{2} (0, T; H^{1} (Ω^{-})) . \end{aligned}$
(6.7)
Define $u$ as
$\begin{aligned} u (x, t) = {\begin{cases} u^{+} & if (x, t) \in Ω^{+} \times (0, T), \\ u^{-} & if (x, t) \in Ω^{-} \times (0, T) . \end{cases} \end{aligned}$
(6.8)
It can be proved that $u \in W$ defined in (5.2)(see, Nandakumaran et al., 2015).

Now, we claim that $ξ_{1} = 0.$ To show this, let’s define
$\begin{aligned} ϕ_{ϵ} = ϵ ϕ (x_{1}, x_{2}, t) ψ ({\frac{x_{1}}{ϵ}}) . \end{aligned}$
where $ϕ \in C_{c}^{\infty} (Ω^{+} \times (0, T)), ψ \in C_{p e r}^{\infty} (0, 1) .$ Observe that
$\begin{aligned} T^{ϵ} (ϕ_{ϵ}) & = ϵ ϕ (ϵ [\frac{x_{1}}{ϵ}] + ϵ x_{3}, x_{2}, t) ψ (x_{3}), \\ T^{ϵ} (\frac{\partial ϕ_{ϵ}}{\partial x_{1}}) & = ϵ \frac{\partial ϕ}{\partial x_{1}} (ϵ [\frac{x_{1}}{ϵ}] + ϵ x_{3}, x_{2}, t) ψ (y_{1}) + ϕ (ϵ [\frac{x_{1}}{ϵ}] + ϵ x_{3}, x_{2}, t) \frac{\partial ψ}{\partial x_{3}}, \\ T^{ϵ} (\frac{\partial ϕ_{ϵ}}{\partial x_{2}}) & = ϵ \frac{\partial ϕ}{\partial x_{2}} (ϵ [\frac{x_{1}}{ϵ}] + ϵ x_{3}, x_{2}, t) ψ (x_{3}) \\ T^{ϵ} (ϕ_{ϵ}^{'}) & = ϵ ϕ^{'} (ϵ [\frac{x_{1}}{ϵ}] + ϵ x_{3}, x_{2}, t) ψ (x_{3}) . \end{aligned}$

We obtain the following convergences as $ϵ \to 0$
$\begin{aligned} T^{ϵ} (ϕ_{ϵ}) & \to 0 in L^{2} (Ω^{+} \times (p, q) \times (0, T)), \\ T^{ϵ} (\frac{\partial ϕ_{ϵ}}{\partial x_{1}}) & \to ϕ (x_{1}, x_{2}, t) \frac{\partial ψ}{\partial x_{3}} in L^{2} (Ω^{+} \times (p, q) \times (0, T)), \\ T^{ϵ} (\frac{\partial ϕ_{ϵ}}{\partial x_{2}}) & \to 0 in L^{2} (Ω^{+} \times (p, q) \times (0, T)), \\ T^{ϵ} (ϕ_{ϵ}^{'}) & \to 0 in L^{2} (Ω^{+} \times (p, q) \times (0, T)) . \end{aligned}$
(6.9)
Using the weak formulation (2.4) with $ϕ_{ϵ}$ as a test function, we obtain the weak solution ${\bar{u}}_{ϵ}$ for ${\hat{θ}}_{ϵ} = {\hat{\bar{θ}}}_{ϵ}$ . Using the properties of the unfolding operator $T^{ϵ}$ , we obtain
$\begin{aligned} - \int_{Ω^{+} \times (p, q) \times (0, T)} T^{ϵ} ({\bar{u}}_{ϵ}^{+}) T^{ϵ} (ϕ_{ϵ}^{'}) + \int_{Ω^{+} \times (p, q) \times (0, T)} T^{ϵ} (\nabla {\bar{u}}_{ϵ}^{+}) \cdot T^{ϵ} (\nabla ϕ_{ϵ}) \\ + \int_{Ω^{+} \times (p, q) \times (0, T)} T^{ϵ} ({\bar{u}}_{ϵ}^{+}) T^{ϵ} (ϕ_{ϵ}) = \int_{Ω^{+} \times (p, q) \times (0, T)} T^{ϵ} (f) T^{ϵ} (ϕ_{ϵ}) \\ + ϵ^{α + 1} \sum_{j = 0}^{N - 1} [\int_{h_{1}}^{h_{2}} \int_{0}^{T} {\bar{θ}}_{ϵ} (p, y_{2}, t) ϕ (j ϵ + ϵ p, y_{2}, t) ψ (p) + \int_{h_{1}}^{h_{2}} \int_{0}^{T} {\bar{θ}}_{ϵ} (q, y_{2}, t) ϕ (j ϵ + ϵ q, y_{2}, t) ψ (q)] . \end{aligned}$
(6.10)
Using the convergence results obtained in (6.3), (6.5) and (6.9), we pass the limit $ϵ \to 0$ in equation (6.10), as a result, we get
$\begin{aligned} \int_{Ω^{+} \times (p, q) \times (0, T)} ξ_{1} ϕ (x_{1}, x_{2}, t) \frac{\partial ψ}{\partial x_{3}} = 0. \end{aligned}$
Since $ϕ$ and $ψ$ are arbitrary, we get $ξ_{1} = 0 a.e. (x_{1}, x_{2}, x_{3}, t) \in Ω^{+} \times (p, q) \times (0, T) .$ This proves the claim. In the similar way, we obtain the following convergence for the adjoint state ${\bar{v}}_{ϵ}$ described in (4.2):
$\begin{aligned} T^{ϵ} ({\bar{v}}_{ϵ} |_{Ω_{ϵ}^{+}}) ⇀ v |_{Ω^{+}} weakly in G, \\ \tilde{{\bar{v}}_{ϵ}^{+}} ⇀ (q - p) v |_{Ω^{+}} weakly in L^{2} ((0, 1) \times (0, T); H^{1} (h_{1}, h_{2})), \\ {\bar{v}}_{ϵ} |_{Ω^{-}} ⇀ v |_{Ω^{-}} weakly in L^{2} (0, T; H^{1} (Ω^{-})), \end{aligned}$
(6.11)
where $v \in W$ satisfies (5.7) for $\bar{u} = u$ defined in (6.8). Now, we take test function $ϕ \in C^{\infty} (\bar{Ω} \times (0, T))$ with $ϕ |_{Γ_{b}} = 0$ in the weak formulation (2.4) for ${\hat{θ}}_{ϵ} = {\hat{\bar{θ}}}_{ϵ}$ and passing the limit $ϵ \to 0,$ the left -hand side of (2.4) becomes
$\begin{aligned} \int_{Ω_{ϵ} \times (0, T)} {\bar{u}}_{ϵ}^{'} ϕ + \int_{Ω_{ϵ} \times (0, T)} \nabla {\bar{u}}_{ϵ} \cdot \nabla ϕ + \int_{Ω_{ϵ} \times (0, T)} {\bar{u}}_{ϵ} ϕ \\ = - \int_{Ω^{+} \times (p, q) \times (0, T)} T^{ϵ} ({\bar{u}}_{ϵ}^{+}) T^{ϵ} (ϕ^{'}) + T^{ϵ} (\frac{\partial {\bar{u}}_{ϵ}^{+}}{\partial x_{1}}) T^{ϵ} (\frac{\partial ϕ}{\partial x_{1}}) + T^{ϵ} (\frac{\partial {\bar{u}}_{ϵ}^{+}}{\partial x_{2}}) T^{ϵ} (\frac{\partial ϕ}{\partial x_{2}}) + T^{ϵ} ({\bar{u}}_{ϵ}^{+}) T^{ϵ} (ϕ) \\ - \int_{Ω^{-} \times (0, T)} ({\bar{u}}_{ϵ}^{-}) ϕ^{'} + \int_{Ω^{-} \times (0, T)} \nabla {\bar{u}}_{ϵ}^{-} \cdot \nabla ϕ + \int_{Ω^{-} \times (0, T)} {\bar{u}}_{ϵ}^{-} ϕ \\ \to \int_{Ω^{+} \times (p, q) \times (0, T)} ((u^{+})^{'} ϕ + \frac{\partial u^{+}}{\partial x_{2}} \frac{\partial ϕ}{\partial x_{2}} + u^{+} ϕ) + \int_{Ω^{-} \times (0, T)} ((u^{-})^{'} ϕ + \nabla u^{-} \cdot \nabla ϕ + u^{-} ϕ), \end{aligned}$
(6.12)
and the right-hand side of (2.4) becomes
$\begin{aligned} \int_{Ω_{ϵ} \times (0, T)} f ϕ + \int_{Γ_{ϵ} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ = \int_{Ω_{ϵ}^{+} \times (0, T)} f ϕ + \int_{Ω^{-} \times (0, T)} f ϕ + \int_{Γ_{ϵ} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ . \end{aligned}$
(6.13)
Using Proposition 3.2 (P5), we get
$\begin{aligned} lim_{ϵ \to 0} \int_{Ω_{ϵ}^{+} \times (0, T)} f ϕ = lim_{ϵ \to 0} \int_{Ω^{+} \times (p, q) \times (0, T)} T^{ϵ} (f) T^{ϵ} (ϕ) = (q - p) \int_{Ω^{+} \times (0, T)} f ϕ . \end{aligned}$
(6.14)
Further,
$\begin{aligned} \int_{Γ_{ϵ} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ = \int_{Γ_{1}^{ϵ} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ + \int_{Γ_{2}^{ϵ} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ + \int_{Γ_{ϵ} / (Γ_{1}^{ϵ} \cup Γ_{2}^{ϵ}) \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ \end{aligned}$
(6.15)

Using Proposition 3.4 (iii) and the characterization of the optimal control ${\bar{θ}}_{ϵ},$ we get
$\begin{aligned} \int_{Γ_{1}^{ϵ} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ & = \int_{(0, 1) \times A_{1} \times (0, T)} T_{1}^{ϵ} ({\hat{\bar{θ}}}_{ϵ}) (x_{1}, h_{1}, x_{3}, t) T_{1}^{ϵ} (ϕ) (x_{1}, h_{1}, x_{3}, t) d x_{1} d x_{3} d t \\ = \int_{(0, 1) \times A_{1} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} (ϵ [\frac{x_{1}}{ϵ}] + ϵ x_{3}, h_{1}, t) T_{1}^{ϵ} (ϕ) (x_{1}, h_{1}, x_{3}, t) d x_{1} d x_{3} d t \\ = \int_{(0, 1) \times A_{1} \times (0, T)} {\bar{θ}}_{ϵ} (x_{3}, h_{1}, t) T_{1}^{ϵ} (ϕ) (x_{1}, h_{1}, x_{3}, t) d x_{1} d x_{3} d t \\ = - \frac{1}{β} \int_{(0, 1) \times A_{1} \times (0, T)} (\int_{0}^{1} T_{1}^{ϵ} ({\bar{v}}_{ϵ}) (y, h_{1}, x_{3}, t) d y) T_{1}^{ϵ} (ϕ) (x_{1}, h_{1}, x_{3}, t) d x_{1} d x_{3} d t . \end{aligned}$
Also, from the convergence results obtained in (6.11) and by The Trace theorem ${\bar{v}}_{ϵ}^{-} \to v^{-}$ in $L^{2} (Γ_{c} \times (0, T))$ , and by using Proposition 3.4 (iii, iv), we get
$\begin{aligned} lim_{ϵ \to 0} \int_{Γ_{1}^{ϵ} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ & = - \frac{1}{β} \int_{(0, 1) \times A_{1} \times (0, T)} (\int_{0}^{1} v^{-} (y, h_{1}, t) d y) ϕ (x_{1}, h_{1}, t) d x_{1} d x_{3} d t \\ = - \frac{1 - (q - p)}{β} \int_{0}^{1} \int_{0}^{T} (\int_{0}^{1} v^{-} (y, h_{1}, t) d y) ϕ (x_{1}, h_{1}, t) d x_{1} d t . \end{aligned}$
(6.16)
Similarly
$\begin{aligned} \int_{Γ_{2}^{ϵ} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ & = \int_{(0, 1) \times A_{2} \times (0, T)} T_{2}^{ϵ} ({\hat{\bar{θ}}}_{ϵ}) (x_{1}, h_{2}, x_{3}, t) T_{2}^{ϵ} (ϕ) (x_{1}, h_{2}, x_{3}, t) d x_{1} d x_{3} d t \\ = \int_{(0, 1) \times A_{2} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} (ϵ [\frac{x_{1}}{ϵ}] + ϵ x_{3}, h_{2}, t) T_{2}^{ϵ} (ϕ) (x_{1}, h_{2}, x_{3}, t) d x_{1} d x_{3} d t \\ = \int_{(0, 1) \times A_{2} \times (0, T)} {\bar{θ}}_{ϵ} (x_{3}, h_{2}, t) T_{2}^{ϵ} (ϕ) (x_{1}, h_{2}, x_{3}, t) d x_{1} d x_{3} d t \\ = - \frac{1}{β} \int_{(0, 1) \times A_{2} \times (0, T)} (\int_{0}^{1} T_{2}^{ϵ} ({\bar{v}}_{ϵ}) (y, h_{2}, x_{3}, t) d y) T_{2}^{ϵ} (ϕ) (x_{1}, h_{2}, x_{3}, t) d x_{1} d x_{3} d t . \end{aligned}$
Because $T^{ϵ} ({\bar{v}}_{ϵ}) ⇀ v^{+}$ in $G$ and $T_{2}^{ϵ} ({\bar{v}}_{ϵ}) = T^{ϵ} ({\bar{v}}_{ϵ}) |_{Γ_{u}}$ , we get
$\begin{aligned} lim_{ϵ \to 0} \int_{Γ_{2}^{ϵ} \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ & = - \frac{1}{β} \int_{(0, 1) \times A_{2} \times (0, T)} (\int_{0}^{1} v^{+} (y, h_{2}, t) d y) ϕ (x_{1}, h_{2}, t) d x_{1} d x_{3} d t, \\ = - \frac{(q - p)}{β} \int_{0}^{1} \int_{0}^{T} (\int_{0}^{1} v^{+} (y, h_{2}, t) d y) ϕ (x_{1}, h_{2}, t) d x_{1} d t . \end{aligned}$
and
$\begin{aligned} \int_{Γ_{ϵ} / (Γ_{1}^{ϵ} \cup Γ_{2}^{ϵ}) \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ & = \sum_{j = 0}^{N - 1} [\int_{h_{1}}^{h_{2}} \int_{0}^{T} {\hat{\bar{θ}}}_{ϵ} (j ϵ + ϵ p, x_{2}, t) ϕ (j ϵ + ϵ p, x_{2}, t) d x_{2} d t \\ + \int_{h_{1}}^{h_{2}} \int_{0}^{T} {\hat{\bar{θ}}}_{ϵ} (j ϵ + ϵ q, x_{2}, t) ϕ (j ϵ + ϵ q, x_{2}, t) d x_{2} d t] \\ = \sum_{j = 0}^{N - 1} ϵ^{α} [\int_{h_{1}}^{h_{2}} \int_{0}^{T} {\bar{θ}}_{ϵ} (p, x_{2}, t) ϕ (j ϵ + ϵ p, x_{2}, t) d x_{2} d t \\ + \int_{h_{1}}^{h_{2}} \int_{0}^{T} {\bar{θ}}_{ϵ} (q, x_{2}, t) ϕ (j ϵ + ϵ q, x_{2}, t) d x_{2} d t] . \end{aligned}$

Utilizing the characterization of optimal control in terms of unfolding operator, we get
$\begin{aligned} \int_{Γ_{ϵ} / (Γ_{1}^{ϵ} \cup Γ_{2}^{ϵ}) \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ & = \sum_{j = 0}^{N - 1} ϵ^{2 α - 1} [\int_{h_{1}}^{h_{2}} \int_{0}^{T} (\int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, x_{2}, p, t) d x_{1}) ϕ (j ϵ + ϵ p, x_{2}, t) d x_{2} d t \\ + \int_{h_{1}}^{h_{2}} \int_{0}^{T} (\int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, x_{2}, q, t) d x_{1}) ϕ (j ϵ + ϵ q, x_{2}, t) d x_{2} d t] \\ = \frac{ϵ^{2 α - 1}}{ϵ} \sum_{j = 0}^{N - 1} \int_{h_{1}}^{h_{2}} \int_{0}^{T} (\int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, x_{2}, p, t) d x_{1}) \\ \times (\int_{j ϵ}^{(j + 1) ϵ} \int_{0}^{T} T^{ϵ} (ϕ) (x_{1}, x_{2}, p, t) d x_{1}) d x_{2} d t \\ + \frac{ϵ^{2 α - 1}}{ϵ} \sum_{j = 0}^{N - 1} \int_{h_{1}}^{h_{2}} \int_{0}^{T} (\int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, x_{2}, q, t) d x_{1}) \\ \times (\int_{j ϵ}^{(j + 1) ϵ} \int_{0}^{T} T^{ϵ} (ϕ) (x_{1}, x_{2}, q, t) d x_{1}) d x_{2} d t \end{aligned}$
(6.17)

$\begin{aligned} = ϵ^{2 α - 2} \sum_{j = 0}^{N - 1} \int_{h_{1}}^{h_{2}} \int_{0}^{T} [\int_{j ϵ}^{(j + 1) ϵ} [\int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, x_{2}, p, t) d x_{1}] \\ T^{ϵ} (ϕ) (x_{1}, x_{2}, p, t) d x_{1}] d x_{2} d t + ϵ^{2 α - 2} \sum_{j = 0}^{N - 1} \int_{h_{1}}^{h_{2}} \int_{0}^{T} [\int_{j ϵ}^{(j + 1) ϵ} [\int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, x_{2}, q, t) d x_{1}] \\ T^{ϵ} (ϕ) (x_{1}, x_{2}, q, t) d x_{1}] d x_{2} d t \end{aligned}$

$\begin{aligned} = ϵ^{2 α - 2} [\int_{Ω^{+} \times (0, T)} [\int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, x_{2}, p, t) d x_{1}] \\ T^{ϵ} (ϕ) (x_{1}, x_{2}, p, t) d x_{1} d x_{2} d t + \int_{Ω^{+} \times (0, T)} [\int_{0}^{1} T^{ϵ} ({\bar{v}}_{ϵ}) (x_{1}, x_{2}, q, t) d x_{1}] \\ T^{ϵ} (ϕ) (x_{1}, x_{2}, q, t) d x_{1} d x_{2} d t] . \end{aligned}$
Considering the case $α = 1$ in (6.17), and passing to the limit $ϵ \to 0$ , we obtain
$\begin{aligned} \int_{Γ_{ϵ} / (Γ_{1}^{ϵ} \cup Γ_{2}^{ϵ}) \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ \to 2 \int_{Ω^{+} \times (0, T)} (\int_{0}^{1} v (x_{1}, x_{2}, t) d x_{1}) ϕ (x_{1}, x_{2}, t) d x_{1} d x_{2} d t . \end{aligned}$
(6.18)

Therefore, we obtain the following limit equation:
$\begin{aligned} {\begin{cases} (q - p) \int_{Ω^{+} \times (0, T)} ((u^{+})^{'} ϕ + \frac{\partial u^{+}}{\partial x_{2}} \frac{\partial ϕ}{\partial x_{2}} + u^{+} ϕ) + \int_{Ω^{-} \times (0, T)} ((u^{-})^{'} ϕ + \nabla u^{-} \cdot \nabla ϕ + u^{-} ϕ) \\ = (q - p) \int_{Ω^{+} \times (0, T)} f ϕ + \int_{Ω^{-} \times (0, T)} f ϕ \\ - \int_{0}^{1} \int_{0}^{T} (\frac{1 - (q - p)}{β} \int_{0}^{1} v^{-} (y, h_{1}, t) d y) ϕ (x_{1}, h_{1}, t) d x_{1} d t \\ - \int_{0}^{1} \int_{0}^{T} (\frac{q - p}{β} \int_{0}^{1} v^{+} (y, h_{2}, t) d y) ϕ (x_{1}, h_{2}, t) d x_{1} d t \\ + (q - p) \int_{Ω^{+} \times (0, T)} [\frac{2}{(q - p)} \int_{0}^{1} v (x_{1}, x_{2}, t) d x_{1}] ϕ (x_{1}, x_{2}, t) d x_{1} d x_{2} d t . \end{cases} \end{aligned}$
(6.19)
For all $ϕ \in C^{\infty} (\bar{Ω} \times (0, T))$ with $ϕ |_{Γ_{b}} = 0$ , and hence it’s true for all $ϕ$ in $W$ by density. Therefore, $u$ satisfies the equation (5.3) for $j = 1$ with $θ = θ^{0}$ , $C_{1} = C_{1}^{0}$ , $C_{2} = C_{2}^{0}$ , where
$\begin{aligned} θ^{0} & = \frac{2}{(q - p)} \int_{0}^{1} v (x_{1}, x_{2}, t) d x_{1}, \end{aligned}$
(6.20)

$\begin{aligned} C_{1}^{0} & = - \frac{1 - (q - p)}{β} \int_{0}^{1} \int_{0}^{T} v^{-} (y, h_{1}, t) d y d t, C_{2}^{0} = - \frac{(q - p)}{β} \int_{0}^{1} \int_{0}^{T} v^{+} (y, h_{2}, t) d y d t . \end{aligned}$
(6.21)

Therefore, we obtain the optimality system corresponding to the minimization problem (5.4). Using Theorem 5.2, the optimal solution is given by $(u, θ^{0}, C_{1}^{0}, C_{2}^{0})$ . Thus, by uniqueness, we have
$\begin{aligned} \bar{u} = u, \bar{v} = v, \bar{θ} = θ^{0}, {\bar{C}}_{1} = C_{1}^{0}, {\bar{C}}_{2} = C_{2}^{0} . \end{aligned}$
This completes the proof of Theorem 6.1.

Similarly, we present following theorem for $α > 1.$
Theorem 6.2
Let $({\bar{u}}_{ϵ}, {\bar{θ}}_{ϵ})$ be the optimal solution of (2.2) with $α > 1$ and let $(\bar{u}, \bar{C_{1}}, \bar{C_{2}})$ be the optimal solution of (5.5). Let ${\bar{v}}_{ϵ}, \bar{v}$ be the corresponding adjoint state satisfying (4.2) and (5.7) respectively. Then
$\begin{aligned} \tilde{{\bar{u}}_{ϵ} |_{Ω_{ϵ}^{+}}} ⇀ (q - p) \bar{u} |_{Ω^{+}} weakly in L^{2} ((0, 1) \times (0, T); H^{1} (h_{1}, h_{2})), \\ \tilde{{\bar{v}}_{ϵ} |_{Ω_{ϵ}^{+}}} ⇀ (q - p) \bar{v} |_{Ω^{+}} weakly in L^{2} ((0, 1) \times (0, T); H^{1} (h_{1}, h_{2})), \\ {\bar{u}}_{ϵ} |_{Ω^{-}} ⇀ \bar{u} |_{Ω^{-}} weakly in L^{2} (0, T; H^{1} (Ω^{-})), \\ {\bar{v}}_{ϵ} |_{Ω^{-}} ⇀ \bar{v} |_{Ω^{-}} weakly in L^{2} (0, T; H^{1} (Ω^{-})), \\ {\hat{\bar{θ}}}_{ϵ} ⇀ Φ = Φ (\bar{C_{1}}, \bar{C_{2}}) weakly in L^{2} (0, T; {(H^{1} (Ω^{+}))}^{*}), \end{aligned}$
(6.22)
where
$\begin{aligned} (Φ, ϕ) = \int_{0}^{1} \int_{0}^{T} \bar{C_{1}} ϕ (x_{1}, h_{1}, t) d x_{1} d t + \int_{0}^{1} \int_{0}^{T} \bar{C_{2}} ϕ (x_{1}, h_{2}, t) d x_{1} d t, \end{aligned}$
and
$\begin{aligned} \bar{C_{1}} = - \frac{1 - (q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{1}, t) d y d t, \bar{C_{2}} = - \frac{(q - p)}{β} \int_{0}^{1} \int_{0}^{T} \bar{v} (y, h_{2}, t) d y d t . \end{aligned}$

Proof.
The proof up to equation (6.17) is similar. Now, take $α > 1$ in (6.17) and pass to the limit $ϵ \to 0$ , to obtain
$\begin{aligned} \int_{Γ_{ϵ} / (Γ_{1}^{ϵ} \cup Γ_{2}^{ϵ}) \times (0, T)} {\hat{\bar{θ}}}_{ϵ} ϕ \to 0. \end{aligned}$
(6.23)

Thus, we have the following limit problem:
$\begin{aligned} {\begin{cases} (q - p) \int_{Ω^{+} \times (0, T)} ((u^{+})^{'} ϕ + \frac{\partial u^{+}}{\partial x_{2}} \frac{\partial ϕ}{\partial x_{2}} + u^{+} ϕ) + \int_{Ω^{-} \times (0, T)} ((u^{-})^{'} ϕ + \nabla u^{-} \cdot \nabla ϕ + u^{-} ϕ) \\ = (q - p) \int_{Ω^{+} \times (0, T)} f ϕ + \int_{Ω^{-} \times (0, T)} f ϕ \\ - \int_{0}^{1} \int_{0}^{T} (\frac{1 - (q - p)}{β} \int_{0}^{1} v^{-} (y, h_{1}, t) d y) ϕ (x_{1}, h_{1}, t) d x_{1} d t \\ - \int_{0}^{1} \int_{0}^{T} (\frac{q - p}{β} \int_{0}^{1} v^{+} (y, h_{2}, t) d y) ϕ (x_{1}, h_{2}, t) d x_{1} d t . \end{cases} \end{aligned}$
(6.24)
which is true for all $ϕ \in C^{\infty} (\bar{Ω} \times (0, T))$ . Therefore, $u$ satisfies the equation (5.3) for $j = 0$ with $C_{1} = C_{1}^{0}$ , $C_{2} = C_{2}^{0}$ , where
$\begin{aligned} C_{1}^{0} = - \frac{1 - (q - p)}{β} \int_{0}^{1} \int_{0}^{T} v^{-} (y, h_{1}, t) d y d t, C_{2}^{0} = - \frac{(q - p)}{β} \int_{0}^{1} \int_{0}^{T} v^{+} (y, h_{2}, t) d y d t . \end{aligned}$
(6.25)
Using Theorem 5.3, the optimal solution is given by $(u, C_{1}^{0}, C_{2}^{0})$ . Thus, by uniqueness, we have
$\begin{aligned} \bar{u} = u, \bar{v} = v, {\bar{C}}_{1} = C_{1}^{0}, {\bar{C}}_{2} = C_{2}^{0} . \end{aligned}$
This completes the proof of Theorem 6.2.
7. Conclusions

The characterization of Neumann boundary optimal control is obtained in this article. Associated with heat equation via the unfolding operator which is further utilized to do limiting analysis and obtain the homogenized optimal control problems. We obtain two homogenized optimal control problems corresponding to $α = 1$ and $α > 1$ . In the homogenized optimal control problem, we obtain three controls, acting on $Γ_{ϵ} \cap Ω^{+}$ (called interior control), $Γ_{ϵ} \cap Γ_{u}$ (called boundary control) and $Γ_{ϵ} \cap Γ_{c}$ (called interfacial control). Here, we would like to remark that problem considered in this article can be generalized for heat equation involving oscillating coefficients or incorporating the Dirichlet type cost functional.

Footnotes

Acknowledgments

The author wish to thank the Editor-in-Chief, Dr. Alain Miranville, and the anonymous reviewers for their valuable comments and helpful suggestions that have strengthened this work. The author also acknowledge Dr. Bidhan Chandra Sardar, Assistant Professor, Department of Mathematics, IIT Madras for his guidance and suggestions in this work. Ritu Raj acknowledges the Council of Scientific & Industrial Research (CSIR), for providing the Research Fellowship (09/1005(0035)/2020-EMR-I). This work was partially supported by the FIST program of the Department of Science and Technology, Government of India, Reference No. SR/FST/MS-I/2018/22(C).

ORCID iDs

Ritu Raj

Funding

The author received no financial support for the research, authorship and/or publication of this article.

Conflicting Interests

The author declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Data Availability Statement

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

References

Aiyappan

Nandakumaran

A. K.

(2017). Optimal control problem in a domain with branched structure and homogenization. Mathematical Methods in the Applied Sciences, 40(8), 3173–3189.

Aiyappan

Nandakumaran

A. K.

Prakash

(2018). Generalization of unfolding operator for highly oscillating smooth boundary domains and homogenization. Calculus of Variations and Partial Differential Equations, 57, 1–30.

Aiyappan

Nandakumaran

A. K.

Sufian

(2019). Asymptotic analysis of a boundary optimal control problem on a general branched structure. Mathematical Methods in the Applied Sciences, 42, 6407–6434.

Aiyappan

Sardar

B. C.

(2019). Biharmonic equation in a highly oscillating domain and homogenization of an associated control problem. Applicable Analysis, 98(16), 2783–2801.

Amirat

Simon

(1997). Riblets and drag minimization. In Optimization methods in partial differential equations. Contemporary Mathematics (Vol. 209, pp. 9–17). American Mathematical Society.

Birnir

Hou

Wellander

(2007). Derivation of the viscous moore–greitzer equation for aeroengine flow. Journal of Mathematical Physics, 48(6), 065209.

Brizzi

Chalot

(1997). Boundary homogenization and neumann boundary value problem. Ricerche di Matematica: A Journal of Pure and Applied Mathematics, 46, 341–388.

Casado-Díaz

Fernández-Cara

Simon

(2003). Why viscous fluids adhere to rugose walls: a mathematical explanation. Journal of Differential Equations, 189(2), 526–537.

Cioranescu

Damlamian

Griso

(2002). Periodic unfolding and homogenization. Comptes Rendus. Mathématique, 335(1), 99–104.

10.

Cioranescu

Damlamian

Griso

(2008). The periodic unfolding method in homogenization. SIAM Journal on Mathematical Analysis, 40(4), 1585–1620.

11.

Damlamian

Pettersson

(2009). Homogenization of oscillating boundaries. Discrete and Continuous Dynamical Systems (DCDS), 23(1–2), 197–219.

12.

D’Angelo

Panasenko

Quarteroni

(2013). Asymptotic-numerical derivation of robin-type coupling conditions for the macroscopic pressure at a reservoir–capillaries interface. Applicable Analysis, 92(1), 158–171.

13.

D’Apice

De Maio

Kogut

P. I.

(2008). Gap phenomenon in the homogenization of parabolic optimal control problems. IMA Journal of Mathematical Control and Information, 25(4), 461–489.

14.

De Maio

Faella

Perugia

(2015). Optimal control for a second-order linear evolution problem in a domain with oscillating boundary. Complex Variables and Elliptic Equations, 60(10), 1392–1410.

15.

De Maio

Gaudiello

Lefter

(2004). Optimal control for a parabolic problem in a domain with highly oscillating boundary. Applicable Analysis, 83(12), 1245–1264.

16.

De Maio

Gaudiello

Lefter

(2024). Null internal controllability for a kirchhoff–love plate with a comb-like shaped structure. SIAM Journal on Control and Optimization, 62(5), 2456–2474.

17.

Durante

Faella

Perugia

(2007). Homogenization and behaviour of optimal controls for the wave equation in domains with oscillating boundary. NoDEA Nonlinear Differential Equations Applications (NoDEA), 14(5–6), 455–489.

18.

Durante

Mel’nyk

(2010). Asymptotic analysis of an optimal control problem involving a thick two-level junction with alternate type of controls. Journal of Optimization Theory and Applications, 144(2), 205–225.

19.

Faella

Raj

Sardar

B. C.

(2024). Optimal control problem governed by wave equation in an oscillating domain and homogenization. Zeitschrift für Angewandte Mathematik und Physik, 75(2), 1–22.

20.

Gajewski

Gröger

Zacharias

(1974). Nichtlineare operatorgleichungen Und operatordifferentialgleichungen. Mathematische Lehrbücher und Monographien, II. Abteilung, Band 38, Akademie-Verlag, Berlin.

21.

Gaudiello

Guibé

Murat

(2017). Homogenization of the brush problem with a source term in

L^{1}

. Archive for Rational Mechanics and Analysis, 225, 1–64.

22.

Gaudiello

Lenczner

(2020). A two-dimensional electrostatic model of interdigitated comb drive in longitudinal mode. SIAM Journal on Applied Mathematics, 80, 792–813.

23.

Gaudiello

Mel’nyk

(2018). Homogenization of a nonlinear monotone problem with nonlinear signorini boundary conditions in a domain with highly rough boundary. Journal of Differential Equations, 265(10), 5419–5454.

24.

Gaudiello

Mel’nyk

(2019). Homogenization of a nonlinear monotone problem with a big nonlinear signorini boundary interaction in a domain with highly rough boundary. Nonlinearity, 32(12), 5150.

25.

Jäger

Mikelić

(2003). Couette flows over a rough boundary and drag reduction. Communications in Mathematical Physics, 232(3), 429–455.

26.

Lenczner

(2007). Multiscale model for atomic force microscope array mechanical behavior. Applied Physics Letters, 90(9), 1–3.

27.

Lions

J. L.

(1971). Optimal control of systems governed by partial differential equations (Vol. 170). Die Grundlehren der Mathematischen Wissenschaften, Springer-Verlag, New York–Berlin. (Translated from the French by S. K. Mitter).

28.

Mel’nik

T. A.

Sivak

O. A.

(2010). Asymptotic analysis of a parabolic semilinear problem with nonlinear boundary multiphase interactions in a perforated domain. Applicable Analysis, 164, 427–453.

29.

Mel’nyk

Sadovyi

(2019). Multiple-scale analysis of boundary-value problems in thick multi-level junctions of type 3:2:2. SpringerBriefs in Mathematics, Springer, Cham.

30.

Nandakumaran

A. K.

Prakash

Sardar

B. C.

(2013). Homogenization of an optimal control problem in a domain with highly oscillating boundary using periodic unfolding method. Mathematics in Engineering, Science and Aerospace, 4(3), 281–303.

31.

Nandakumaran

A. K.

Prakash

Sardar

B. C.

(2015). Periodic controls in an oscillating domain: Controls via unfolding and homogenization. SIAM Journal on Control and Optimization, 53(5), 3245–3269.

32.

Nandakumaran

A. K.

Prakash

Sardar

B. C.

(2016). Asymptotic analysis of neumann periodic optimal boundary control problem. Mathematical Methods in the Applied Sciences, 39(15), 4354–4374.

33.

Nandakumaran

A. K.

Sufian

(2021). Oscillating PDE in a rough domain with a curved interface: Homogenization of an optimal control problem. ESAIM: Control, Optimisation and Calculus of Variations, 27(4), 37.

34.

Sardar

B. C.

Sufian

(2022). Homogenization of a boundary optimal control problem governed by Stokes equations. Complex Variables and Elliptic Equations, 67(12), 2944–2974.

35.

Showalter

R. E.

(1997). Monotone operators in Banach space and nonlinear partial differential equations (Vol. 49). American Mathematical Society, Mathematical Surveys and Monographs.