\documentclass[twoside]{article} \usepackage{amsfonts, amsmath} % \pagestyle{myheadings} \markboth{\hfil Convergence of the boundary control \hfil EJDE--2002/35} {EJDE--2002/35\hfil A. K. Nandakumaran \hfil} \begin{document} \title{\vspace{-1in}\parbox{\linewidth}{\footnotesize\noindent {\sc Electronic Journal of Differential Equations}, Vol. {\bf 2002}(2002), No. 35, pp. 1--10. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \vspace{\bigskipamount} \\ % Convergence of the boundary control for the wave equation in domains with holes of critical size % \thanks{ {\em Mathematics Subject Classifications:} 35L05, 35B37, 93D05. \hfil\break\indent {\em Key words:} perforated domain, exact controllability, Hilbert uniqueness method, \hfil\break\indent homogenization. \hfil\break\indent \copyright 2002 Southwest Texas State University. \hfil\break\indent Submitted September 2, 2001. Published April 23, 2002.} } \date{} % \author{A. K. Nandakumaran} \maketitle \begin{abstract} In this paper, we consider the homogenization of the exact controllability problem for the wave equation in periodically perforated domain with holes of critical size. We show that the boundary control converges to the boundary control of the homogenized system under the assumption that the perforations are uniformly away from the boundary. \end{abstract} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \catcode`\@=11 \def\theequation{\@arabic{\c@section}.\@arabic{\c@equation}} \catcode`\@=12 \section{Introduction} In this article, we consider the following exact boundary controllability problem for the wave equation in the perforated domain $\Omega_{\varepsilon T}$: \begin{equation} \label{e1.1} \begin{gathered} y_{\varepsilon}'' - \Delta y_{\varepsilon} = 0 \quad\mbox{in } \Omega_{\varepsilon T}\\ y_{\varepsilon} = v_{\varepsilon}\quad\mbox{on } \Sigma_{\varepsilon T}\\ y_{\varepsilon}(0) =y_{\varepsilon}^0, \; y_{\varepsilon}' = y_{\varepsilon}^1 \quad\mbox{in } \Omega_{\varepsilon}, \end{gathered} \end{equation} where $\Omega_{\varepsilon T} =\Omega_{\varepsilon} \times (0, T)$, $\Sigma_{\varepsilon T} =\Sigma_{\varepsilon} \times (0, T)$, $\Sigma_{\varepsilon}=\partial\Omega_{\varepsilon}$ and $\Omega_{\varepsilon}$ is a perforated domain obtained from $\Omega$ by removing small holes periodically distributed (with period $\varepsilon > 0$, a small parameter) of size $a_{\varepsilon}$ which is of critical size. We make this precise later. The controllability problem consists in finding a control $v_{\varepsilon}$ so that the corresponding solution $y_{\varepsilon}$ satisfies $y_{\varepsilon}(T) = y_{\varepsilon}'(T) =0$. The controllability problem (i.e., the existence of a control $v_{\varepsilon}$ and the corresponding solution $y_{\varepsilon}$) and homogenization (limit analysis as $\varepsilon \to 0$) for wave equation have been extensively studied by various authors \cite{cdmz,cdz1,cdz2,l1,l2,z}. Our aim in this article is to study the convergence of the outer boundary control $v_{\varepsilon}|_{\Gamma_{T}}$ when the holes are of critical size which seems to be open in the literature quoted above. The (strong) convergence of this control when the size is smaller than the critical one has been studied in \cite{cdz1}. Of course even this article does not yield the convergence of the controls on the boundary of the holes, but our expectation is that it should converge to an internal control in some sense. In the next section, we make the problem precise and state our result, while it will be proved in Section 3. \section{Preliminaries and Main Result} \paragraph{Notation} Let $\Omega \subset \mathbb{R}^N$ be a bounded smooth domain with boundary $\Gamma$. Let $Y = (-1, 1)^N$ and $S\subset\!\subset Y$ be an open set containing the origin. Let $\varepsilon > 0$ be a small parameter and $0 0$ be any positive real number and define $$ D_{\alpha} = \{ x \in \Omega : d(x, \partial\Omega) \geq\alpha\}. $$ Our assumption is that, we make perforations uniformly away from the boundary, i.e., let $I_{\varepsilon} = \{ k \in Z^{N}: Y_{\varepsilon}^{k} \subset \Omega \setminus D_{\alpha}\}$ and define $$ \Omega_{\varepsilon} = \Omega \setminus(\cup_{k\in I_{\varepsilon}} \overline{S_{\varepsilon}^{k}}). $$ The boundary of $\Omega _{\varepsilon}$ is $\Sigma_{\varepsilon} = \partial\Omega_{\varepsilon} = \Gamma \bigcup (\cup_{k\in I_{\varepsilon}} \partial S_{\varepsilon}^{k})$. Let $T > 0 $ and $\Omega_{T} = \Omega \times (0, T) $ and define $\Omega_{\varepsilon T}, \Sigma_{\varepsilon T}, \Gamma_{T}$ etc. analogously. The critical size $a_\varepsilon$ is given by \begin{equation} \label{2.1} a_{\varepsilon} = \begin{cases} C_{0} \varepsilon ^{N / (N-2)} &\mbox{if } N \geq 3 \\ \exp(-C_{0}/\varepsilon^2) &\mbox{if } = 2, \end{cases}\end{equation} where $C_{0}$ is a constant. We now introduce the construction of the controls $v_{\varepsilon}$ given in \cite{cdz1,cdz2} using the Hilbert Uniqueness Method (HUM) introduced by J. L. Lions \cite{l1,l2}. Let $m(x) = x - x^0$, $x^0 \in \mathbb{R}^{N}$ fixed and $T_{0} =2\|m\|_{L^{\infty}(\Omega)}$. Let $T > T_{0}$ and consider a real function $\psi \in C^1[0, T]$ such that $\psi' \leq 0$ for all $t\in [0, T]$, $\psi(t) = 1$ for all $t \in [0, \frac{T_{0}+T}{2}]$ and $\psi(T) = 0 $. Let $\{\phi_{\varepsilon}^0, \phi_{\varepsilon}^1 \} \in {H_{0}^1(\Omega_{\varepsilon})} \times L^2(\Omega_{\varepsilon})$ and solve the system \begin{equation}\label{2.2} \begin{gathered} \phi_{\varepsilon}'' -\Delta\phi_{\varepsilon} = 0 \quad\mbox{in } \Omega_{\varepsilon T}\\ \phi_{\varepsilon} = 0 \quad \mbox{on } \Sigma_{\varepsilon T}\\ \phi_{\varepsilon}(0)=\phi_{\varepsilon}^0, \; \phi_{\varepsilon}'(0)= \phi_{\varepsilon}^1\quad\mbox{in } \Omega_{\varepsilon} \end{gathered} \end{equation} Then $y_{\varepsilon}$ is obtained by solving \begin{equation} \label{2.3} \begin{gathered} y_{\varepsilon}''-\Delta y_{\varepsilon} = \quad\mbox{in } \Omega_{\varepsilon T}\\ y_{\varepsilon}= \psi(t)(m. \nu_{\varepsilon}) \frac{\partial\phi_{\varepsilon}}{\partial\nu_{\varepsilon}} \quad \mbox{on } \Sigma_{\varepsilon T}\\ y_{\varepsilon}(T) =y_{\varepsilon}'(T) = 0. \end{gathered} \end{equation} In the above equation $\nu_{\varepsilon}$ denotes the exterior unit normal to the boundary. The system (\ref{2.3}) has a unique solution given by the transposition method and $$ y_{\varepsilon} \in C^0([0, T]; L^2(\Omega_{\varepsilon})) \bigcap C^1([0, T]; H^{-1}(\Omega_{\varepsilon})). $$ Define $\Lambda_{\varepsilon}:H_{0}^1(\Omega_{\varepsilon}) \times L^2(\Omega_{\varepsilon})\to H^{-1}(\Omega_{\varepsilon}) \times L^2(\Omega_{\varepsilon})$ by $$\Lambda_{\varepsilon} \{\phi_{\varepsilon}^0, \phi_{\varepsilon}^1\} =\{y_{\varepsilon}'(0), -y_{\varepsilon}(0)\}. $$ It is proved in \cite{cdz1, cdz2} that $\Lambda_{\varepsilon}$ is an isomorphism. Thus for a given $\{y_{\varepsilon}^0,y_{\varepsilon}^1\} \in L^2(\Omega_{\varepsilon}) \times H^{-1}(\Omega_{\varepsilon})$, define $\{\phi_{\varepsilon}^0, \phi_{\varepsilon}^1\} = \Lambda_{\varepsilon}^{-1} \{y_{\varepsilon}^1, -y_{\varepsilon}^0\}$. Then the solution $y_{\varepsilon}$ of (\ref{2.3}) associated to $\phi_{\varepsilon}$ given by (\ref{2.2}) satisfies $y_{\varepsilon}(0) =y_{\varepsilon}^0, \; y_{\varepsilon}'(0) = y_{\varepsilon}^1$. Thus the controllability problem is solved. Regarding the convergence of (\ref{2.2}) and (\ref{2.3}), the following result can be found in \cite{cdz1,cdz2}. Throughout the paper, $\tilde{g}$ denotes the extension of $g$ by zero in the holes. \begin{theorem} \label{2t1} Assume the notations as in Section 2.1 and Let $T>T_0$. Let $\{y_{\varepsilon}^0, \; y_{\varepsilon}^1\} = \{y ^0, \; y^1\}\chi_{\Omega_{\epsilon}}$ with $\{y ^0, \; y^1\} \in L^2(\Omega)\times L^2(\Omega)$, where $\chi_{\Omega_{\epsilon}}$ is the characteristic function of $\Omega_{\epsilon}$ . Let $y_{\varepsilon}$ be the solution of (\ref{2.3}). Then as $\varepsilon\to 0$, one has $$ \tilde{y_{\varepsilon}}\rightharpoonup y \quad\text{in } L^{\infty}(0, T;L^2(\Omega)) \quad\mbox{weak }\star, $$ where $y$ is the solution of \begin{equation} \label{2.4} \begin{gathered} y''-\Delta y + \mu y = 2\mu\psi\phi \quad\mbox{in }\Omega_{T}\\ y = \psi(t)(m. \nu) \frac{\partial\phi}{\partial\nu}\quad\mbox{in }\Gamma_{T}\\ y(0) = y^0, \quad y'(0) =y^1 \quad\mbox{in } \Omega, \end{gathered} \end{equation} and $\phi$ is the solution of \begin{equation} \label{2.5} \begin{gathered} \phi''-\Delta \phi + \mu \phi = 0 \quad\mbox{in }\Omega_T\\ \phi = 0 \quad\mbox{on }\Gamma_{T}\\ \phi (0) = \phi^0, \quad \phi'(0) =\phi^1, \end{gathered} \end{equation} and such that \begin{equation} \label{2.6} \begin{gathered} \{\tilde{\phi_{\varepsilon}^0}, \tilde{\phi_{\varepsilon}^1} \} \rightharpoonup \{ \phi^0, \phi^1 \}\quad\mbox{in } H_{0}^1(\Omega) \times L^2(\Omega) \quad\mbox{weakly}, \\ \tilde{\phi_{\varepsilon}}\rightharpoonup\phi \quad\mbox{in } L^{\infty}(0, T; H_{0}^1 (\Omega) )\quad\mbox{weak } \star . \end{gathered} \end{equation} Moreover $ y(T) = y '(T) = 0$. \end{theorem} The non-negative constant $\mu$ is called the {\bf strange term} in Cioranescu and Murat \cite{cm} in the study of elliptic equation in perforated domains. We recall this result in the following lemma. Similar type of test functions are also studied in \cite{n} for other systems. \begin{lemma}{\label{2l2}} Let $\Omega_{\varepsilon}$ be as in Section 2. Then there exists a sequence $w_{\varepsilon} \in H^1(\Omega)$ and a non negative constant $\mu \in \mathbb{R}_{+}$ such that \begin{description} \item{i.} $0 \leq w_{\varepsilon} \leq 1, w_{\varepsilon} = 0$ on $S_{\varepsilon}$ for every $\varepsilon > 0$ \item{ii.} $w_{\varepsilon}\rightharpoonup 1, $ weakly in $H^1(\Omega)$ as $\varepsilon \to 0 $ \item{iii.} $\langle -\Delta w_{\varepsilon}, \zeta u_{\varepsilon} \rangle_{H^{-1}, H_{0}^1} \to \mu \int\zeta u$, \end{description} for every $ \zeta \in {\mathcal{D}}(\Omega), $ every sequence $u_{\varepsilon}$ such that $u_{\varepsilon} = 0 $ on $S_{\varepsilon}$ and $u_{\varepsilon} \rightharpoonup u$ weakly in $H^1(\Omega)$ as $\varepsilon \to 0$. Further if $a_{\varepsilon}$ is sub critical (holes are much smaller), i.e., \begin{equation} \label{2. 7} a_{\varepsilon} = \begin{cases} C_{o}\varepsilon^{\alpha} \quad\mbox{with } \alpha >\frac{N}{N-2} &\mbox{if }N \geq 3\\ \exp(-C_{0}/\varepsilon^{\alpha}) \quad\mbox{with } \alpha > 2 &\mbox{if }N=2, \end{cases}\end{equation} then one can take $\mu = 0$ and the convergence in (ii) is strong in $H^1(\Omega)$. \end{lemma} As mentioned earlier the solution $y_{\varepsilon}$ is obtained via transposition method. i.e., $y_{\varepsilon}$ satisfies: $$ \int\int_{\Omega_{\varepsilon T}} y_{\varepsilon} f_{\varepsilon} = \int_{0}^{T}\int_{\partial\Omega_{\varepsilon} } \psi(t)(m. \nu_{\varepsilon}) \frac{\partial\phi_{\varepsilon}}{\partial \nu_{\varepsilon}}\frac{\partial \theta_{\varepsilon}}{\partial \nu_{\varepsilon}}, $$ for all $f_\varepsilon =f. \chi_{\Omega_{\varepsilon}} \in L^1(0, T; L^2(\Omega_{\varepsilon}))$, where $\theta_{\varepsilon}$ is the unique solution of \begin{equation} \label{2.8} \begin{gathered} \theta ''_{\varepsilon}-\Delta \theta_{\varepsilon} = f_{\varepsilon}\quad\mbox{in }\Omega_{\varepsilon T}\\ \theta_{\varepsilon} = 0 \quad\mbox{on }\Sigma_{\varepsilon T}\\ \theta_{\varepsilon}(0) = 0 = \theta_{\varepsilon}'(0). \end{gathered} \end{equation} The following convergence is also true: \begin{equation} \label{2.9} \int\int_{\Gamma_{T}\bigcup S_{\varepsilon T}}\psi(m. \nu_{\varepsilon})\frac{\partial \phi_{\varepsilon}}{\partial \nu_{\varepsilon}} \frac{\partial\theta_{\varepsilon}}{\partial \nu_{\varepsilon}}\to \int\int_{\Gamma_{T}}\psi(m. \nu) \frac{\partial \phi}{\partial \nu}\frac{\partial \theta}{\partial \nu} + \int\int_{\Omega_{T}}2 \mu \varphi \phi \theta. \end{equation} Here $\theta$ is the unique solution of \begin{equation} \label{2.10} \begin{gathered} \theta '' -\Delta\theta +\mu \theta = f\\ \theta = 0 \\ \theta(0) = 0 = \theta'(0). \end{gathered} \end{equation} The natural questions which would arise, at this stage are the convergence of the controls $$ \psi(m. \nu_{\varepsilon}) \frac{\partial\phi_{\varepsilon}}{\partial \nu} \big|_{\Gamma_{T}}\quad \mbox{and}\quad \psi (m. \nu_{\varepsilon}) \frac{\partial \phi_{\varepsilon}}{\partial \nu_{\varepsilon}}\Big|_{\partial S_{\varepsilon T}}. $$ We have the following theorem which will be proved in the next section. \begin{theorem}{\label{2t3}} Let $a_{\varepsilon}$ be of critical size as in (\ref{2.1}) and $\Omega_{\varepsilon}$ be given as in Section 2.1. Then the outer boundary controls $$ \psi (t)(m. \nu)\frac{\partial\phi_{\varepsilon}}{\partial\nu}\rightharpoonup \psi (t)(m. \nu)\frac{\partial\phi}{\partial\nu} \quad\mbox{in } L^2(\Gamma_{T})\;\; weak. $$ where $\phi_{\varepsilon}, \phi $ respectively are the solutions of (\ref{2.2}) and (\ref{2.5}). \end{theorem} \begin{remark} \label{2r4} \rm We do not have the convergence of the internal boundary controls. Also without the assumption that the perforations are located uniformly away from the boundary, the problem still remains open. \end{remark} \begin{remark}\label{2r5} \rm Comparing the convergence in (\ref{2.9}), it is not yet clear that whether the following convergence are true: \begin{gather}\label{2.10a} \iint_{\Gamma_T}\psi(m. \nu)\frac{\partial \phi_{\varepsilon}}{\partial \nu} \frac{\partial\theta_{\varepsilon}}{\partial \nu}\to \iint_{\Gamma_{T}}\psi(m. \nu) \frac{\partial \phi}{\partial \nu}\frac{\partial \theta}{\partial \nu} \\ \label{2.10b} \iint_{S_{\varepsilon T}}\psi(m. \nu_{\varepsilon})\frac{\partial \phi_{\varepsilon}}{\partial \nu_{\varepsilon}} \frac{\partial\theta_{\varepsilon}}{\partial \nu_{\varepsilon}}\to \iint_{\Omega_{T}}2 \mu \varphi \phi \theta. \end{gather} Of course the proof of Theorem \ref{2t3} can be applied to see that $\frac{\partial\theta_{\varepsilon}}{\partial \nu_{\varepsilon}}|_{\Gamma_{T}}$ also converges weakly to $\frac{\partial \theta}{\partial \nu}|_{\Gamma_{T}}$, but this does not directly yield (\ref{2.10a}) (and hence (\ref{2.10b})). But in the sub critical case, one gets the strong convergence of $\frac{\partial\theta_{\varepsilon}}{\partial \nu_{\varepsilon}}$ and hence (\ref{2.10a}) and (\ref{2.10b}) (see \cite{cdz1}) with $\mu=0$. \end{remark} \section{Proof of Theorem \ref{2t3}} The proof consists of the following two steps. \noindent{\bf Claim 1:} There exists a constant $C > 0$ such that \begin{equation} \label{3. 1} \|\frac{\partial\rho_{\varepsilon}}{\partial\nu} \| \leq C, \end{equation} where $\rho_{\varepsilon}$ is the solution of \begin{equation} \label{3.2} \begin{gathered} \rho''_{\varepsilon} -\Delta\rho_{\varepsilon} = f_{\varepsilon} \quad\text{in }\Omega_{\varepsilon T}\\ \rho_{\varepsilon} = 0 \quad\mbox{on } \Gamma_{T} \bigcup S_{\varepsilon T}\\ \rho_{\varepsilon}(0) =\rho^0_{\varepsilon}, \quad \rho'_{\varepsilon}(0) =\rho^1_{\varepsilon}, \end{gathered} \end{equation} where \begin{equation} \label{3.3} \begin{gathered} f_{\varepsilon}\to f \quad\mbox{in } L^1(0, T; L^2(\Omega)) \quad\mbox{strong}\\ \rho_{\varepsilon}^0 \rightharpoonup \rho^0 \quad\mbox{in }H_{0}^1(\Omega) \quad\mbox{weak}\\ \rho_{\varepsilon}^1 \rightharpoonup \rho^1 \quad\mbox{in } L^2(\Omega) \quad\mbox{weak.} \end{gathered} \end{equation} From the homogenization results of \cite{cdmz}, one has \begin{equation} \label{3.4} \begin{gathered} \rho_{\varepsilon}\rightharpoonup \rho \quad\mbox{in } L^{\infty} (0, T, H_{0}^1)\quad\mbox{weak} \\ \rho'_{\varepsilon} \rightharpoonup \rho' \quad\mbox{in } L^{\infty}(0, T; L^2) \quad\mbox{weak}, \end{gathered} \end{equation} where $\rho$ satisfies \begin{equation} \label{3.5} \begin{gathered} \rho'' - \Delta\rho + \mu \rho= f\\ \rho = 0 \\ \rho(0) = \rho^0, \quad \rho'(0) = \rho^1. \end{gathered} \end{equation} \\ {\bf Claim 2: } $\frac{\partial\rho_{\varepsilon}}{\partial\nu} \rightharpoonup \frac{\partial\rho}{\partial\nu}$ in $L^2(\Gamma_{T})$ weak. \paragraph{Proof Claim 1:} Recall $D_\alpha = \{ x \in \Omega : d(x, \partial\Omega)\leq \alpha \}$ and choose $q \in C^1(\bar{\Omega})^N$ such that \begin{equation} q = \begin{cases} \nu &\mbox{on } \partial \Omega \\ 0 &\mbox{in }\Omega\backslash D_{\alpha}, \end{cases} \end{equation} where $\nu$ is the unit normal to $\Gamma$. Since $D_{\alpha}$ does not contains any holes, we have $q=0$ on $S_{\varepsilon}$, that is on the boundary of the holes. Now, multiplying (\ref{3.2}) by $q_k \frac{\partial \rho_{\varepsilon}}{\partial x_{k}}$ (where we use the repeated indices convention) and integrating by parts, we get \begin{equation} \label{3.6} \begin{aligned} \int_{\Omega_{\varepsilon}}[\rho_{\varepsilon}'q_k {\frac{\partial \rho_{\varepsilon}}{\partial x_{k}}}]_{0}^{T} - \int_{\Omega_{\varepsilon T}} \rho_{\varepsilon}'q_k {\frac{\partial \rho_{\varepsilon}'} {\partial x_{k}}} + \int_{\Omega_{\varepsilon T}} \nabla \rho_{\varepsilon} \nabla q_{k}{\frac{\partial \rho_{\varepsilon}}{\partial x_{k}}}&\\ + \int_{\Omega_{\varepsilon T}} \nabla \rho_{\varepsilon}. q_{k} \nabla {\frac{\partial \rho_{\varepsilon}}{\partial x_{k}}} -\int_{\Gamma_{\varepsilon T}\bigcup S_{\varepsilon T}} {\frac{\partial\rho_{\varepsilon}}{\partial\nu}} q_k {\frac{\partial \rho_{\varepsilon}}{\partial x_{k}}} &= \int_{\Omega_{\varepsilon T}} f_{\varepsilon} q_k {\frac{\partial \rho_{\varepsilon}}{\partial x_{k}}}, \end{aligned} \end{equation} which can be written under the form $$ I_1 + I_2 + I_3 + I_4- I = I_5. $$ Since $\rho_\varepsilon = 0 $ on $\Gamma_T$, we have $\frac{\partial \rho_{\varepsilon}}{\partial x_{k}} = \nu_k \frac{\partial \rho_{\varepsilon}}{\partial \nu} $. By the choice of $q$, it follows that $$ I = \int_{\Gamma_{T}} | \frac{\partial \rho_{\varepsilon}}{\partial \nu} |^2 d\sigma dt. $$ We estimate the other terms as follows: \begin{align*} I_{4} &= \frac{1}{2} \int_{\Omega_{\varepsilon T}} q_k {\frac{\partial}{\partial x_{k}}} (\nabla \rho_{\varepsilon}. \nabla\rho_{\varepsilon})\\ &=-\frac{1}{2}\int_{\Omega_{\varepsilon T}} \mathop{\rm div}q. |\nabla\rho_{\varepsilon}|^2 + \frac{1}{2}\int_{\Gamma_{T}} \nabla\rho_{\varepsilon} \nabla\rho_{\varepsilon} \\ &= - \frac{1}{2}\int_{\Omega_{\varepsilon T}} \mathop{\rm div}q. |\nabla\rho_{\varepsilon}|^2 + \frac{1}{2} \int_{\Gamma_{T}} |{\frac{\partial\rho_{\varepsilon}}{\partial\nu}}|^2. \end{align*} \begin{align*} I_2 &= -\frac{1}{2} \int_{\Omega_{\varepsilon T}} q_k {\frac{\partial}{\partial x_{k}}} (\rho'^2_{\varepsilon})\\ &= \frac{1}{2}\int_{\Omega_{\varepsilon T}} \mathop{\rm div}q. |\rho'_{\varepsilon}|^2 - \frac{1}{2} \int_{\Gamma_{T}\bigcup S_{\varepsilon T}}q_k \rho'^2_{\varepsilon} \nu_k, \end{align*} The second term vanishes as $\rho'_{\varepsilon} = 0 $ on $\Gamma_{T} \bigcup S_{\varepsilon T}$, so we get from (\ref{3.6}): \begin{equation}\label{3.7} \begin{aligned} \frac{1}{2}\int_{\Gamma_{T}}|{\frac{\partial\rho_{\varepsilon}} {\partial\nu}}|^2 =& \int_{\Omega_{\varepsilon}} [\rho'_{\varepsilon}(T) q_k {\frac{\partial\rho_{\varepsilon}}{\partial x_k}}(T) - \rho'_{\varepsilon}(0) q_k {\frac{\partial\rho_{\varepsilon}}{\partial x_k}}(0)] \\ &+ \frac{1}{2} \int_{\Omega_{\varepsilon T}} \mathop{\rm div}q. (\rho'^2_{\varepsilon} - |\nabla\rho_{\varepsilon}|^2 ) + \int_{\Omega_{\varepsilon T}} \nabla\rho_{\varepsilon} \nabla q_k {\frac{\partial\rho_{\varepsilon}}{\partial x_k}}\\ &- \int_{\Omega_{\varepsilon T}} f_{\varepsilon} q_k {\frac{\partial\rho_{\varepsilon}}{\partial x_k}}\\ \leq &C [ ( \| \rho'_{\varepsilon}(T)\|_{L^2(\Omega_{\varepsilon})}^2 + \| \nabla \rho_{\varepsilon}(T)\|_{L^2(\Omega_{\varepsilon})}^2)\\ &+( \|\rho'_{\varepsilon}(0) \| _{L^2(\Omega_{\varepsilon})}^2 + \| \nabla \rho_{\varepsilon}(0)\|_{L^2(\Omega_{\varepsilon})}^2)] \\ &+ C [\int_{\Omega_{\varepsilon T}}(|\rho'_{\varepsilon}|^2 + |\nabla \rho_{\varepsilon}|^2 )] +C [\int_{\Omega_{\varepsilon T}} |f_{\varepsilon}| |\nabla \rho_{\varepsilon} |]. \end{aligned}\end{equation} To estimate the right-hand side of the above inequality, we multiply (\ref{3.2}) by $\rho'_{\varepsilon}$ and integrate from $0$ to $ t$ to get $$ \frac{1}{2}\int_{0}^{t} \int_{\Omega_{\varepsilon}} {\frac{\partial}{\partial t}} (\rho'^2_{\varepsilon} + \nabla\rho_{\varepsilon} \nabla\rho_{\varepsilon} ) = \int_{0}^{t} \int_{\Omega_{\varepsilon}} f_\varepsilon \rho'_{\varepsilon}, $$ i.e., $$E_{\varepsilon}(t) - E_{\varepsilon}(0) = \int_0^t\int_{\Omega_{\varepsilon}} f_{\varepsilon}\rho_{\varepsilon}', $$ where $$ E_\varepsilon(t) = {\frac{1}{2}}\int_{\Omega_{\varepsilon}} (|\rho'_{\varepsilon}(t)|^2 + |\nabla\rho_{\varepsilon}(t)|^2)\,dx. $$ Now \begin{equation}\label{3.8} \begin{aligned} |\int_{0}^{t}\int_{\Omega_{\varepsilon}} f_{\varepsilon} \rho'_{\varepsilon}| \leq &\int_{0}^{t} \|f_\varepsilon\|_{L^2(\Omega_{\varepsilon})} \|\rho'_{\varepsilon} \|_{L^2(\Omega_{\varepsilon})}\\ \leq &\|\rho'_{\varepsilon} \|_{L^{\infty}(0, T; L^2(\Omega_\varepsilon))}\|f_{\varepsilon}\|_{L^1(0, T;L^2(\Omega_{\varepsilon}))}. \end{aligned} \end{equation} Therefore, \begin{align*} E_\varepsilon(t) \leq & E_\varepsilon(0) + \|\rho'_{\varepsilon}\|_{L^{\infty}(0, T;L^2(\Omega_\varepsilon))} \|f_\varepsilon\|_{L^1(0, T;L^2(\Omega_{\varepsilon}))}\\ \leq& C[|\nabla\rho_{\varepsilon}^0(t)\|_{L^2(\Omega_{\varepsilon})}^2 + \|\rho^1_{\varepsilon}(t) \|_{L^2(\Omega_{\varepsilon})}^2 + \|f_\varepsilon\|_{L^1(0, T;L^2(\Omega_{\varepsilon}))}]. \end{align*} Obviously, we also have \begin{align*} \int_{\Omega_{\varepsilon T}}|f_\varepsilon\|\nabla\rho_{\varepsilon}| \leq& \|\nabla\rho_{\varepsilon}\|_{L^{\infty}(0, T;L^2 (\Omega_{\varepsilon}))} \|f_\varepsilon\|_{L^1(0, T;L^2(\Omega_{\varepsilon}))} \\ \leq& C \|f_\varepsilon \|_{L^1(0, T; L^2(\Omega_{\varepsilon}))}. \end{align*} So, using these inequalities in (\ref{3.7}), it follows that \begin{equation} \label{3.9} \begin{aligned} \int_{\Gamma_{T}} |{\frac{\partial \rho_{\varepsilon}}{\partial \nu}}|^2 \leq & C[\|\nabla \rho_{\varepsilon}^0\|^2_{L^2(\Omega_{\varepsilon})} + \|\rho^1_{\varepsilon}\|^2_{L^2(\Omega_\varepsilon)} + \|f_\varepsilon\|_{L^1(0, T: L^2(\Omega_\varepsilon))}]\\ \leq & \mbox{constant}, \end{aligned} \end{equation} for all the choices of $\rho_{\varepsilon}^0$, $\rho'_{\varepsilon}$ and $f_\varepsilon$ as in (\ref{3.3}). This completes the proof of Claim~1. \paragraph{Proof of Claim 2:} From claim 1, it follows that $$ {\frac{\partial\rho_{\varepsilon}}{\partial\nu}} \rightharpoonup \eta \quad\mbox{in } L^2(\Gamma_{T}) \quad \mbox{weak}. $$ We have now to identify $\eta$. Let $g \in {\mathcal D}(0, T), v \in C^{\infty}( \bar{\Omega}). $ Multiplying (\ref{3.2}) by $g v \omega_{\varepsilon}, $ where $\omega_{\varepsilon} $ are the test functions given by Lemma \ref{2l2}, we get \begin{equation} \label{3.10} \begin{aligned} -\int_{\Omega_{\varepsilon T}} \rho'_{\varepsilon} g' v \omega_{\varepsilon} + \int_{\Omega_{\varepsilon T}}\nabla\rho_{\varepsilon} g \nabla v \omega_{\varepsilon} &\\ + \int_{\Omega_{\varepsilon T}} \nabla\rho_{\varepsilon} g v \nabla \omega_{\varepsilon} - \int_{\Gamma_{T}\bigcup S_{\varepsilon T}}{\frac{\partial \rho_{\varepsilon}}{\partial \nu}} g v \omega_{\varepsilon} &=\int_{\Omega_{\varepsilon T}} f_{\varepsilon} g v\omega_{\varepsilon} \end{aligned} \end{equation} which can be written in the form $$ I_1 + I_2 + I_3 - I = I_4. $$ Note that since the holes are away from the boundary, we have $\omega_{\varepsilon} = 1$ in a neighborhood of $\Gamma $ (this is from the construction of $\omega_{\varepsilon})$. Since $\omega_\varepsilon = 0 $ on $S_\varepsilon$, from (\ref{3.4}) and Lemma \ref{2l2} we can easily get the convergence of $I, I_1, I_2$ and $I_4$ as follows: \begin{gather*} I \to \int_{\Gamma_{T}}\eta g v , \quad I_1 \to \int_{\Omega_{T}} \rho' g' v ,\\ I_2 \to \int_{\Omega_{T}} \nabla\rho g \nabla v, \\ I_4 \to \int_{\Omega_{T}} f gv. \end{gather*} Then, it remains to pass to the limit in $I_3$. Of course, we can write formally, $$ I_3 = \int_{0}^{T} g \langle - \Delta \omega_{\varepsilon}, \rho_{\varepsilon} v \rangle $$ At this point, we cannot apply Lemma \ref{2l2}, because $v \not\in {\mathcal D}(\Omega)$, but we can overcome this using the fact that $\rho_{\varepsilon} = 0 $ on the boundary $\Gamma_{T}$. We proceed as follows: Let $\delta > 0 $, put $$ A_\delta = \{ x \in \mathbb{R}^{N}\backslash\Omega:d(x, \partial\Omega) < \delta \} $$ and let $\Omega_{1} = \bar{\Omega}\bigcup A_{\delta} $ be the $\delta$ neighborhood of $\Omega$. Extend $\rho_{\varepsilon}$ to $\tilde{\rho_{\varepsilon}}$ by zero in $A_{\delta}$ and since $\rho_{\varepsilon} = 0$ on $\Gamma_{T}$, we have $\tilde{\rho_{\varepsilon}}\in L^{\infty}(0, T;H^1_0(\Omega_1))$ and $$ \tilde{\rho_{\varepsilon}} \rightharpoonup \tilde{\rho} \quad \mbox{in } L^{\infty}(0, T; H_{0}^1(\Omega_{1}) \quad \mbox{weak} *. $$ Moreover, $$ \tilde{\rho} = \begin{cases} \rho & \mbox{in }\Omega\\ 0 &\mbox{in } A_{\delta}. \end{cases} $$ Let $ \tilde{v}$ be any extension of $ v$ such that $\tilde{v} \in {\mathcal D}(\Omega_{1})$ and extend $\omega_{\varepsilon}$ by $1$ outside $\Omega$ and again denote the extension by $\omega_{\varepsilon}$. We can apply Lemma \ref{2.2} and since $\tilde{\rho}\tilde{v}\big|_{\Omega} =\rho v \in H^1_0 (\Omega)$, we get $$ \langle- \Delta \omega_{\varepsilon}, \tilde{\rho_{\varepsilon}} \tilde{v} \rangle_{H_{0}^{-1}(\Omega_{1}), H_{0}^1(\Omega_{1})} \to \mu \int_{\Omega_{1}} \tilde{\rho}\tilde{v} = \mu \int_{\Omega} \rho v. $$ Now, \begin{align*} I_3 =& \int_{\Omega_{\varepsilon T}} \nabla \rho_{\varepsilon} g v \nabla \omega_{\varepsilon} = \int_{\Omega_{1T}} \nabla \tilde{\rho_{\varepsilon}} g \tilde{v} \nabla\omega_{\varepsilon}\\ =& \int_{0}^{T} g \langle - \Delta \omega_{\varepsilon}, \tilde{\rho_{\varepsilon}} \tilde{v} \rangle_{H^{-1} (\Omega_{1}), H_{0}^1(\Omega_{1})} \\ \to& \int_{\Omega_{1T}} \mu g \tilde{v}\tilde{\rho} = \int_{\Omega_{T}} \mu g v\rho. \end{align*} So, passing to the limit in (\ref{3.10}), we get \begin{equation} \label{3.11} \int_{\Gamma_{T}} \eta g v = - \int_{\Omega_{T}} \rho' g' v + \int_{\Omega_{T}} \nabla\rho g \nabla v + \int \mu \rho g v - \int f g v. \end{equation} On the other hand, multiplying (\ref{3.5}) by $g v$, we get \begin{equation}\label{3.12} -\int_{\Omega_{T}} \rho' g' v + \int_{\Omega_{T}} \nabla\rho g \nabla v -\int_{\Gamma_{T}} {\frac{\partial\rho} {\partial\nu}} g v + \mu \int_{\Omega_{T}} \rho g v = \int_{\Omega T} f g v. \end{equation} From (\ref{3.11}) and (\ref{3.12}), it follows that $$ \int_{\Gamma_{T}} \eta g v = \int_{\Gamma_{T}} {\frac{\partial\rho} {\partial\nu}} g v, \;\; \forall\;\;g \in {\mathcal D}(0, T), v \in C^{\infty}( \bar{\Omega}), $$ which implies that $$ \eta = {\frac{\partial\rho}{\partial\nu}}. $$ Hence, Claim 2 is proved, which ends the proof of the Theorem \ref{2t3}. \begin{thebibliography}{99} \frenchspacing \bibitem{cdmz} D. 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Lions, {\it Exact controllability, stabilization and perturbations for distributed systems}, SIAM Review 30(1988), 1-68. \bibitem{l2} J. L. Lions, {\it Contr\^{o}labilit\'{e} exacte, perturbations et stabilisation de syst\`{e}mes distribu\'{e}s. } Tome 1 and 2, RMA 8 and 9, 1988. \bibitem{n} A. K. Nandakumar, {\it Homogenization of eigenvalue problems of elasticity in perforated domain}, Asymptotic Analysis 9 (1994) 337-358. \bibitem{z} E. Zuazua, {\it Exact controllability for the semi linear Wave Equation}, J. Math. Pures et Appl. 69 (1) 1990, 1-32. \end{thebibliography} \noindent\textsc{A. K. Nandakumaran}\\ Department of Mathematics \\ Indian Institute of Science\\ Bangalore - 560 012, India. \\ e-mail: nands@math.iisc.ernet.in \end{document}