\documentclass[reqno]{amsart}
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\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2021 (2021), No. 49, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2021 Texas State University.}
\vspace{8mm}}
\begin{document}
\title[\hfilneg EJDE-2021/49\hfil Exact boundary controllability for the wave equation]
{Exact boundary controllability for the wave equation with moving boundary domains in
a star-shaped hole}
\author[R. S. O. Nunes \hfil EJDE-2021/49\hfilneg]
{Ruikson S. O. Nunes}
\address{Ruikson S. O. Nunes \newline
UFMT- Federal University of Mato Grosso, ICET,
Department of Mathematics, 78060-900,
Cuiab\'a, MT, Brazil}
\email{ruiksonsillas@hotmail.com}
\thanks{Submitted August 16, 2020. Published June 1, 2021.}
\subjclass[2010]{35L05, 35L20, 35L53, 35B40, 93B05, 49J20}
\keywords{Wave equation; energy decay; exact boundary controllability;
\hfill\break\indent non-cylindrical domains; holed domains}
\begin{abstract}
We consider an exact boundary control problem for the wave equation in
a moving bounded domain which has a star-shaped hole.
The boundary domain is composed by two disjoint parts, one is the boundary
of the hole, which is fixed, and the other one is the external boundary
which is moving. The initial data has finite energy and the control obtained
is square integrable and is obtained by means of the conormal derivative.
We use the method of controllability presented by Russell in \cite{rus2},
and assume that the control acts only in the moving part of the boundary.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\allowdisplaybreaks
\section{Introduction}
There is a large number of works available for the exact boundary controllability
problems of wave equations, because such problems have a great importance both from a practical
and theoretical point of view.
Since the 1970s, many works have appeared giving seminal contributions to the advancement
of this branch of mathematics, among them we have \cite{BLR,Lions,rus} which are
canonical references for the subject. Many works have appeared
dealing with control problems for wave equations on Euclidean domains with
diverse types of geometry, for example, domains with fixed and moving boundary and
domains with perforated interior; see
\cite{Chen,basfe,zuazua,dona,lag,MilM,Nu}.
In this work we study an exact boundary control problem for the standard wave equation
on a domain with moving boundary which has a single fixed hole.
The boundary of such domains is composed by two disjoint parts:
one it is the boundary on hole which is fixed, and the other one is the external
boundary which is moving. We shall consider the control acting only on the moving
boundary part. In practical situations, many processes involve domains with a geometry
as described above. For example, a flexible body that is crossed by a cylindrical pillar
and is fixed to it. Without any variation in the temperature of the environment the
body has no dilation and thus its external boundary remains static.
However, if there is a variation in the temperature, the body would have a dilation
or a contraction, causing the mobility of the its external boundary.
In this work when we deal with a domain with a hole, and we refer by external boundary
as being the part of the boundary of the domain that does not coincide with that
one of the hole.
To establish these concepts in more detail, we consider $B\subset \mathbb{R}^{n}$,
$n\geq 2$, a convex compact set with the origin in its interior with smooth boundary
$\Gamma_0$.
We set $\Omega_{\infty}=\mathbb{R}^{n}-B$. Let $\Xi\subset \mathbb{R}^{n}$ be a simply
connected bounded domain with piecewise smooth boundary $\Gamma_1$, with no cusps,
such that $B\subset \Xi$. We assume that $\operatorname{dist}(\Gamma_0, \Gamma_1)\geq \epsilon> 0$
and set $\Omega=\Xi-B$. Hence, the boundary of $\Omega$ is
$\partial \Omega=\Gamma_0\cup \Gamma_1$. Note that $\Omega$ is a holed domain whose
hole has the shape of $B$.
We also consider the moving boundary domain $\Xi_t\subset \mathbb{R}^{n}$ where
$$
\Xi_t=\{x\in \mathbb{R}^{n}: x=\alpha(t)y,\, y\in \Xi\},\quad t\in [0,+\infty)
$$
whose boundary is denoted by $\Gamma_t$ and $\Xi_0=\Xi$.
Here $\alpha:\mathbb{R}^{+}\cup\{0\}\to \mathbb{R}$ is a piecewise bounded smooth
function, where
\begin{equation}\label{pri}
\Xi_t\times \mathbb{R}\subset \cup_{\overline{x}\in \Xi}\{(x,t) \in \mathbb{R}^{N}
\times \mathbb{R}: |x-\overline{x}|^2\leq t^2 \},
\end{equation}
with $B\subset \Xi_t$ and $\operatorname{dist}(\Gamma_0\cap \Gamma_t)\geq \epsilon>0$,
for all $t>0$. The boundedness of $\alpha$ implies in the existence of $r>0$ such that
$\Xi_t\subset B(0,r)$ for all $t\in [0,+\infty)$.
Defining $\widetilde{\Omega}=B(0,r)-B$ and $\Omega_t=\Xi_t-B$ we can see that
$\Omega_t \subset \widetilde{\Omega}$ for all $t\in [0,+\infty)$.
The boundary of $\Omega_t$ is $\partial \Omega_t=\Gamma_0\cup \Gamma_t$.
Now, for $T>0$, let us consider the non-cylindrical domain of $\mathbb{R}^{n+1}$,
$$
Q_T=\cup_{00$ sufficiently large and a control function $h(\cdot,t)\in L^2(\Sigma_T)$
such that the solution $u \in H^1(Q_T)$ of the problem
\begin{equation}\label{02}
\begin{gathered}
u_{tt}-\Delta u=0 \quad \text{in } Q_T \\
u(\cdot, 0)=f,\quad u_t(\cdot,0)=g, \quad \text{in } \Omega \\
u(\cdot, t)=0, \quad \text{on } \Gamma_0 \times [0,T] \\
\nu_tu_t-\nabla u\cdot \nu_x=h(\cdot, t), \quad \text{on } \Sigma_T.
\end{gathered}
\end{equation}
satisfy the final condition
\begin{equation}\label{03}
u(\cdot, T)=0=u_t(\cdot,T)\quad\text{in } \Omega_T.
\end{equation}
\end{theorem}
In the literature some papers deal with exact boundary controllability problems
for wave equation on domains with holes. However, as far as we know, there are no papers
dealing with such problems in holed moving domains, as it is proposed in the present paper.
In this paper \cite{Lions} the author seeking to show the wide applicability of the HUM method,
considers control problems on domains with different types of geometry.
In particular, he has shown how the HUM can be applied to solve some exact boundary
control problems on a fixed domain which has a hole.
In \cite{zuazua} also using the HUM method the authors have considered exact boundary
control problems for wave equation in domains which have one or a family of small holes.
In the literature there are also many works dealing with exact boundary control problems
on domains that has a mobile boundary but with no hole; see
\cite{Chen,basfe,MilM} and citing references.
At this point we highlight the contribution of the present paper by
making available in the literature a work that deals with exact control problems
on non-cylindrical holed domains.
Russell \cite{rus2} developed a technique, based in \cite{rus1},
for studding an exact boundary control problem for wave equation with control
acting only on a part of boundary of the domain. Here we shall utilize such technique
to obtain the desirable exact boundary control problem proposed in Theorem \ref{01}.
The applicability of Russell's method requires some properties of the system to be considered,
the principals are: linearity, time reversibility, local energy decay in a exterior domain
(obtained via \cite{Ikehat,maraw}) and suitable traces theorems that are obtained in \cite{Ta}.
As seen above the Russell's method requires many properties of the system to be controlled
but it has the advantage of requiring very little on the geometry of the domain.
From this fact we can consider the limiting function $\alpha$, defined above,
to be only piecewise smooth.
Here, it is proposed that the boundary of the domain be comprise two disjoint parts:
the internal part (the boundary of the hole) fixed, and the other part which is moving.
An interesting point is to consider an exact boundary control problem for wave equation
on a holed domain where both external and internal boundary moving.
In the literature there are papers \cite{Coop,CoopSt} that obtain local energy decay estimates
for the wave equation in the exterior of a domain with moving boundary.
Another interesting point it is to study on exact boundary control problems,
in holed domains, for systems of coupled waves equations as proposed in
\cite{wBJ,Rui}. We intend to return on this questions in posterior works.
The rest of this article is organized as follows.
In Section 2 we presents a brief summary with respect to trace and extension properties.
In Section 3 we obtain local energy decay estimates for the wave equation in exterior domain.
In Section 4 we explore exact boundary controllability results in a holed domain with
fixed boundary.
In Section 5 we prove Theorem \ref{01}.
\section{Extension and traces}
In this section, we make a brief presentation about the trace and extension theorems,
such properties are essential in the proof of Theorem \ref{01}.
Remembering that we are considering $\Omega$ as a holed domain whose its boundary is
performed by two disjoint parts which is denoted by $\Gamma_0$ and $\Gamma_1$ respectively,
with $\operatorname{dist}(\Gamma_0,\Gamma_1)>0$ and that $\Gamma_0$ is smooth surface
and $\Gamma_1$ a surface smooth by parts with no cusps.
In this work we shall use the following extension lemma.
\begin{lemma}\label{le1}
Let $\Omega \subset \Omega_{\infty}$ as defined above and $V \subset \Omega_{\infty}$
an open set such that $\overline{\Omega} \subset V$. So, there exist a bounded linear
operator
$E: H^1(\Omega)\times L^2(\Omega)\to H^1(\Omega_{\infty})\times L^2(\Omega_{\infty})$
such that for $ u \in H^1(\Omega)\times L^2(\Omega)$:
%%
\marginpar{Please define $A\Subset B$}
%%
\begin{enumerate}
\item $Eu=u$ in $\Omega$;
\item $\operatorname{supp} Eu\Subset V$;
\item $\| Eu\|_{H^1(\Omega_{\infty})\times L^2(\Omega_{\infty})}
\leq C \| u\|_{H^1(\Omega)\times L^2(\Omega)}$,
where $C$ is a positive real constant independent on $u$.
\end{enumerate}
\end{lemma}
\begin{proof}
To sketch the proof let us firstly to define the projection operators
$$
P_{i-1}:H^1(\Omega)\times L^2(\Omega)\to H^{i-1}(\Omega_{\infty}),\quad i=1,2.
$$
As we are considering $\operatorname{dist}(\Gamma_0, \Gamma_1)>0$ and $\Gamma_1$
is a piecewise smooth surface, we obtain a finite cover $(U_{i})_{i=1}^{k}$ of
$\Gamma_1$ where $U_{i}\subset \Omega_{\infty}$, $i=1,\cdots,k$.
So, we can adapt the proof of the \cite[Theorems 3.9 and 3.10]{Necas}
to obtain the extension operators $E_1:H^1(\Omega)\to H^1(\Omega_{\infty})$,
where $E_1u=u$ in $\Omega$ with $\|E_1u\|_{H^1(\Omega_{\infty})}\leq C\|u\|_{H^1(\Omega)}$.
On the other hand, using the classical extension by zero out $\Omega$, we obtain
the extension operator $E_0:L^2(\Omega)\to L^2(\Omega_{\infty})$ where
$\|E_0u\|_{L^2(\Omega_{\infty})}\leq \|u\|_{L^2(\Omega)}$.
Thus, we obtain the desirable extension operator by defines $E=(E_1P_1, E_0P_0)$.
\end{proof}
Next we mention a result on the regularity of the traces of the solution of the wave
equation which it is essential in the proof of Theorem \ref{01}.
Let us begin with some notation and definitions.
Let $P(\xi, D)$ be a linear second order hyperbolic partial differential equation with
$C^{\infty}$ coefficients depending on $\xi$ in some open bounded domain
$\Xi \subset \mathbb{R}^{N}$. Being $\Sigma \subset \Xi$ an oriented
smooth hypersurface which is time-like and non-characteristic with respect to $P(\xi, D)$.
Let $\eta=(\eta_1,\cdots, \eta_{N})$ be a unit normal to $\Sigma$.
If $\sum a^{ij}\frac{\partial^2}{\partial\xi_{i} \partial\xi_{j}}$ is the principal part
of $P(\xi, D)$, then the expression
$\frac{\partial u}{\partial \eta}=\sum a^{ij}\frac{\partial u}{\partial \xi_{i}}\eta_{j}$
defines the conormal derivative of $u$ relative to the $P(\xi, D)$ along $\Sigma$.
An important fact it is to know what the regularity of the traces of the conormal derivative
on surfaces, for this purpose we turn to \cite{Ta}.
Considering $\Xi \subset \mathbb{R}^{N}$, with $N\geq 2$, \cite[Theorem 2]{Ta} proves that
if $u \in H^1_{\rm loc}(\Xi)$ is such that $P(\xi, D)u \in L_{\rm loc}^2(\Xi)$ then
$\frac{\partial u}{\partial \eta} \in L_{\rm loc}^2(\Sigma)$.
Particularly, if we consider $P(\xi, D)$ as being the standard wave operator,
its principal part will be
$\frac{\partial^2}{\partial t^2}-\sum_{i=1}^{N} \frac{\partial^2}{\partial x_{i}^2}$.
Now, if $\gamma$ is a smooth hypersurface in $\mathbb{R}^{N}$ we consider the surface
$\gamma \times \mathbb{R}$ whose unit normal vector is $\nu=(\nu_x,\nu_t)$, where
$\nu_x=(\nu_1,\cdots,\nu_{N})$. In this case the conormal derivative of $u$ along
$\gamma \times \mathbb{R}$ is $\frac{\partial u}{\partial \nu}=\nu_tu_t-\nabla u\cdot \nu_x$.
Particularly, if we apply the trace result mentioned in the previous paragraph for the
wave operator we obtain the following result.
\begin{lemma}\label{L3}
Let $u$ be the solution of the initial-boundary value problem
\begin{equation}\label{02.}
\begin{gathered}
u_{tt}-\Delta u=0 \quad \text{in } \Omega_{\infty}\times \mathbb{R} \\
u(\cdot, 0)=u_0,\quad u_t(\cdot,0)=u_1, \quad\text{in } \Omega_{\infty} \\
u(\cdot, t)=0, \quad\text{on } \Gamma_0 \times\mathbb{R}
\end{gathered}
\end{equation}
with initial data $(u_0, u_1) \in H^1(\Omega_{\infty})\times L^2(\Omega_{\infty})$,
where $\operatorname{supp}(u_0),\operatorname{supp}(u_1) \Subset \Omega_{\infty}$.
Let $\gamma$ be a smooth hypersurface in $\Omega_{\infty} $, with no self intersection
and considers the surface $\gamma \times \mathbb{R}$ which the unit normal vector is
$\nu=(\nu_x,\nu_t)$. Then the conormal derivative of $u$ along $\gamma \times \mathbb{R}$
has trace $\nu_tu_t-\nabla u\cdot \nu_x \in L^2_{\rm loc}(\gamma \times \mathbb{R})$.
\end{lemma}
\section{Local energy decay}
There are many publications dedicated to obtain local energy decay estimates for
hyperbolic equations in exterior domains; see
\cite{Ikehat,maraw, Zac} and references there in.
In this paper such estimates play a fundamental role in the proof of Theorem \ref{01}.
So, we dedicate this section to such subject.
Let us consider the initial initial-boundary value problem
\begin{equation}\label{04}
\begin{gathered}
u_{tt}-\Delta u=0 \quad \text{in } \Omega_{\infty} \times (0,+\infty) \\
u(\cdot, 0)=f,\quad u_t(\cdot,0)=g, \quad\text{in } \Omega_{\infty} \\
u(\cdot, t)=0,\quad\text{in } \Gamma_0 \times (0,+\infty).
\end{gathered}
\end{equation}
Let $\mathcal{O}\subset \Omega_{\infty}$ a bounded domain, the energy of the solution
$u$ of the \eqref{04} confined in $\mathcal{O}$ is defined by
\begin{equation}\label{05}
E(t,\mathcal{O},u)=\frac{1}{2} \int_{\mathcal{O}}[|\nabla u|^2 +u_t^2 ](x,t)dx.
\end{equation}
If there exist a positive constant $C$ and a function $p(t)$ such that
\begin{equation}\label{06}
E(t,\mathcal{O},u)\leq Cp(t)E(0,\mathcal{O},u),
\end{equation}
with $p(t)\rightarrow 0$, as $t\rightarrow + \infty$, we say the energy of \eqref{04}
decays locally.
Adapting process in \cite{maraw,Ikehat} we show the validity of the following
local energy decay estimate.
\begin{lemma}\label{07}
Let $(f,g)\in H^1(\Omega_{\infty})\times L^2(\Omega_{\infty})$ with
$\operatorname{supp}(f), \operatorname{supp}(g)\subset \mathcal{O} \subset\Omega_{\infty}$,
then there exist a positive real constant $K$, independent of $f$ and $g$, such that the
solution $u$ of \eqref{04} satisfies
\begin{equation}\label{08}
\|u(\cdot,t) \|^2_{H^1(\mathcal{O})}+\|u_t(\cdot,t) \|^2_{L^2(\mathcal{O})}
\leq\frac{K}{t-R}\big\{\|u(\cdot,0) \|^2_{H^1(\mathcal{O})}
+\|u_t(\cdot,0) \|^2_{L^2(\mathcal{O})}\big\},
\end{equation}
for $t>R$ sufficiently large, where $R$ is such that $\mathcal{O} \subset \Omega_{R}$
being $B(0,R)$ the ball of center 0 and radius $R$ and
$\Omega_{R}=B(0,R)\cap\Omega_{\infty}$.
\end{lemma}
The proof of Lemma \ref{07} follows the ideas presented by Morawetz \cite{maraw},
so it is essential to prove some preliminaries results.
The compactness of the initial data $f$ and $g$ implies finite propagation speed property
for solution of the system \eqref{04}, that is the solution $u$ of \eqref{04},
in the instant $t$, has support contained in some bounded region of the space
$\Omega_{\infty}$. So if we take $R>0$ and the ball $B(0,R)$ of center 0 and radius $R$,
such that
$\operatorname{supp}(f) \cup \operatorname{supp}(g)\subset \Omega_{R}=B(0,R)\cap\Omega_{\infty}$,
then from some $t>R$ the function $u(x,t)$ as well as its derivatives are null for
$x \in \Omega_{\infty}$ with $|x|\geq t$.
Other important and classical property of the solution $u$ of the system \eqref{04} is
that his total energy
\begin{equation}\label{09}
E(t,\Omega_{\infty},u)=\frac{1}{2} \int_{\Omega_{\infty}}[|\nabla u|^2+u_t^2](x,t)dx,
\end{equation}
is conserved with respect to $t$, that is,
\begin{equation}\label{10}
E(t,\Omega_{\infty},u)=E(0,\Omega_{\infty},u),\quad \text{for all $t>0$}.
\end{equation}
In the proof of estimate \eqref{08} we will used the following result.
\begin{lemma}\label{prop1}
Let $u$ be the solution of the boundary-value problem \eqref{04}, if $w$ is the solution
of the problem
\begin{equation}\label{11}
\begin{gathered}
w_{tt}-\Delta w=0 \quad \text{in } \Omega_{\infty} \times (0,+\infty) \\
w(\cdot, 0)=h,\quad w_t(\cdot,0)=f, \quad \text{in } \Omega_{\infty} \\
w(\cdot, t)=0, \quad \text{in } \Gamma_0 \times (0,+\infty)
\end{gathered}
\end{equation}
with $\Delta h=g$ and $h=0$ in $B$. Then $w_t=u$.
\end{lemma}
\begin{proof}
Note that $v=w_t$ satisfies
\begin{equation}\label{12}
\begin{gathered}
v_{tt}-\Delta v=0 \quad \text{in } \Omega_{\infty} \times (0,+\infty) \\
v(\cdot, 0)=f,\quad v_t(\cdot,0)=\Delta h, \quad \text{in } \Omega_{\infty} \\
v(\cdot, t)=0, \quad \text{in } \Gamma_0 \times (0,+\infty)
\end{gathered}
\end{equation}
with $\Delta h=g$ and $h=0$ on $B$. Then $v=w_t$ satisfies \eqref{04} and by the
uniqueness of solution we conclude that $v=w_t=u$
\end{proof}
From Lemma \ref{prop1} we obtain the estimate
\begin{equation}\label{13}
\begin{aligned}
\int_{\Omega_{\infty}}|u(x,t)|^2dx
&=\int_{\Omega_{\infty}}|w_t(x,t)|^2dx \\
&\leq E(t,\Omega_{\infty}, w_t)= E(0,\Omega_{\infty}, w_t)\\
&\leq C E(0,\Omega_{\infty},u).
\end{aligned}
\end{equation}
Now, for $T>0$, let $D=\Omega_{\infty}\times [0,T]$ be an exterior region which boundary
is $\partial D= \partial D_1\cup \partial D_2 \cup \partial D_3$, where
$$
\partial D_1=\Omega_{\infty}\times \{0\},\quad
\partial D_2=\Omega_{\infty}\times \{T\},\quad
\partial D_3=\Gamma_0\times [0,T].
$$
Multiplying the equality $u_{tt}-\Delta u=0$ by $(x\cdot \nabla u)+t u_t+\frac{N-1}{2}u$
and integrating it on $D$, from the Gauss' divergence formula, as in \cite{Ikehat},
we obtain
\begin{equation}\label{13s}
\begin{aligned}
tE(t,\Omega_{\infty},u)
&=\int_{\Omega_{R}}\left(x\cdot \nabla u(\cdot,0)\right)u_t(\cdot,0) dx
+\frac{(N-1)}{2}\int_{\Omega_{R}}u(\cdot,0) u_t(\cdot,0) dx\\
&\quad - \int_{\Omega_{\infty}}\left(x\cdot \nabla u(\cdot,t)\right)u_t(\cdot,t) dx
-\frac{(N-1)}{2}\int_{\Omega_{\infty}}u(\cdot,t) u_t(\cdot,t)dx\\
&\quad + \frac{1}{2}\int_{ D_3}\{x\cdot \nu(\cdot)\}
\big|\frac{\partial u}{\partial \nu}(.,s)\big|^2\,dx\,ds.
\end{aligned}
\end{equation}
Since $B=\Omega_{\infty}^{c}$ is star-shaped with respect to origin, that is
$\{x\cdot \nu(\cdot)\}\leq 0$, it follow that
$$
\frac{1}{2}\int_{ D_3}\{x\cdot \nu(\cdot)\}\big|\frac{\partial u}{\partial \nu}(.,s)\big|^2
\,dx\,ds\leq 0.
$$
So, from \eqref{13s} we obtain the estimate
\begin{equation}\label{14}
\begin{aligned}
&tE(t,\Omega_{\infty},u) \\
&\leq\int_{\Omega_{R}}\left(x\cdot \nabla u(\cdot,0)\right)u_t(\cdot,0) dx
+\frac{(N-1)}{2}\int_{B(0,R)}u(\cdot,0) u_t(\cdot,0) dx \\
&\quad -\int_{\Omega_{\infty}}\left(x\cdot \nabla u(\cdot,t)\right)u_t(\cdot,t) dx
-\frac{(N-1)}{2}\int_{\Omega_{\infty}}u(\cdot,t) u_t(\cdot,t)dx.
\end{aligned}
\end{equation}
Now observe that
\begin{align*}
&\big|\int_{\Omega_{\infty}}\left(x\cdot \nabla u(\cdot,t)\right)u_t(\cdot,t) dx\big|\\
&\leq R\int_{\Omega_{R}}\left|\nabla u(\cdot,t)\right | |u_t(\cdot,t)| dx
+\int_{|x|\geq R}|x||\nabla u(\cdot,t)||u_t(\cdot,t)| dx.
\end{align*}
Because of the finite propagation speed property, we can take $t> R$ such that $u(x,t)$
and its derivatives are null, so
\begin{align*}
\int_{|x|\geq R}|x||\nabla u(\cdot,t)||u_t(\cdot,t)| dx
& =\int_{t\geq|x|\geq R}|x||\nabla u(\cdot,t)||u_t(\cdot,t)| dx \\
&\leq\frac{t}{2}\int_{|x|\geq R}\{|\nabla u(\cdot,t)|^2 +|u_t(\cdot,t)|^2\} dx.
\end{align*}
Therefore,
\begin{equation}\label{15}
\begin{aligned}
&\big|\int_{\Omega_{\infty}}\left(x\cdot \nabla u(\cdot,t)\right)u_t(\cdot,t) dx\big| \\
&\leq \frac{R}{2}\int_{\Omega_{R}}\{|\nabla u(\cdot,t)|^2 +|u_t(\cdot,t)|^2\} dx
+\frac{t}{2}\int_{|x|\geq R}\{|\nabla u(\cdot,t)|^2 +|u_t(\cdot,t)|^2\} dx.
\end{aligned}
\end{equation}
On the other hand,
\begin{equation}\label{16}
\big|\int_{\Omega_{\infty}} u(\cdot,t)u_t(\cdot,t) dx\big|
\leq\frac{1}{2}\int_{\Omega_{\infty}}\left[|u(\cdot,t)|^2+|u_t(\cdot,t)|^2 \right]dx.
\end{equation}
Joining \eqref{13} and \eqref{16} we obtain
\begin{equation}\label{17}
\big|\int_{\Omega_{\infty}} u(\cdot,t)u_t(\cdot,t) dx\big|\leq C E(0,\Omega_{\infty},u),
\end{equation}
where $C$ is a positive constant independent of the initial data $f$ and $g$.
The following inequalities are also valid
\begin{gather}\label{18}
\big|\int_{\Omega_{R}}\left(x\cdot \nabla u(\cdot,0)\right)u_t(\cdot,0) dx\big|
\leq C(R) E(0,\Omega_{R},u), \\
\label{19}
\big|\int_{\Omega_{R}}u(\cdot,0) u_t(\cdot,0) dx\big|
\leq C(R) E(0,\Omega_{R},u),
\end{gather}
where $C(R)$ is a positive constant which vary from line to line and depend on $R$
but not on of the initial data $f$ and $g$.
Now, joining and manipulating the the inequalities \eqref{14}-\eqref{19} we obtain
\begin{align*}
tE(t,\Omega_{R},u) + tE(t,|x|
&\geq R,u) \\
&\leq C E(0,\Omega_{R},u)+ RE(t, \Omega_{R},u)
&\quad +tE(t,|x|\geq R,u),
\end{align*}
which implies
\begin{equation}\label{20}
E(t,\Omega_{R},u) \leq \frac{K}{t-R} E(0,\Omega_{R},u),
\end{equation}
where $K$ is a positive constant independent of the initial data.
Applying \eqref{20} to the solution $w$ of \eqref{11}, and by Lemma \ref{prop1} we obtain
\begin{equation}\label{21}
\begin{aligned}
\int_{\Omega_{R}}|u|^2dx
&=\int_{B(0,R)}|w_t|^2dx \\
&\leq E(t,\Omega_{R},w_t)\leq E(t,\Omega_{R},u)\\
&\leq \frac{K}{t-R} E(0,\Omega_{R},u).
\end{aligned}
\end{equation}
Joining \eqref{20} and \eqref{21} and considering $\mathcal{O}\subset \Omega_{R}$,
we obtain \eqref{08}, completes the proof. Lemma \ref{07}.
\section{Control in a holed domain with fixed boundary}
In this section we prove an exact boundary control problem for the wave equation
on a holed domain with fixed boundary. This section plays a important rule
in the proof Theorem \ref{01}.
Let $\Omega$ be as define in the first section and $T>0$.
The boundary of $\Omega$ is $\partial\Omega=\Gamma_0\cup \Gamma_1$.
\begin{lemma}\label{22}
Given $(v_0,v_1) \in \mathcal{H}^1(\Omega)\times L^2(\Omega)$ and $T>0$,
there exists a control function $h(\cdot,t)\in L^2(\Gamma_1\times [0,T])$
such that the solution $v \in H^1(\Omega\times [0,T])$ of the problem
\begin{equation}\label{23}
\begin{gathered}
v_{tt}-\Delta v=0 \quad \text{in } \Omega\times [0,T] \\
v(\cdot, T)=v_0, \quad v_t(\cdot, T)=v_1, \quad \text{in }\Omega \\
v(\cdot, t)=0, \quad \text{on } \Gamma_0 \times [0,T] \\
\nu_tv_t-\nabla v\cdot \nu_x=h(\cdot, t), \quad \text{on } \Gamma_1\times [0,T].
\end{gathered}
\end{equation}
satisfies the condition
\begin{equation}\label{24}
v(\cdot, 0)=0=v_t(\cdot,0)\quad\text{in } \Omega.
\end{equation}
\end{lemma}
\begin{proof}
Let $\delta$ be a positive number, and
$\Omega_{\delta}=\{y\in \Omega_{\infty}: \exists x\in \Omega; |x-y|<\delta \}$ be
an open neighborhood of $\Omega$.
Given an arbitrary $ (w_0, w_1) \in \mathcal{H}^1(\Omega)\times L^2(\Omega)$,
according Lemma \ref{le1} we can extend $ (w_0, w_1)$, for all $\Omega_{\infty}$.
Let $ (\widetilde{w}_0, \widetilde{w}_1)$ be the extension of $ (w_0, w_1)$,
that is, $ (\widetilde{w}_0, \widetilde{w}_1)=E(w_0, w_1)$.
Let $w$ the solution of the backward initial boundary value problem
\begin{equation}\label{25}
\begin{gathered}
w_{tt}-\Delta w=0 \quad \text{in } \Omega_{\infty} \times (0,+\infty) \\
w(\cdot, T)=\widetilde{w}_0,\quad w_t(\cdot,T)=\widetilde{w}_1, \quad \text{in } \Omega_{\infty} \\
w(\cdot, t)=0, \quad \text{in } \Gamma_0 \times (0,+\infty).
\end{gathered}
\end{equation}
Now, for $T>0$ we define the bounded linear operator
$$
\overline{S}_T: \mathcal{H}^1_0(\Omega_{\delta})\times L^2(\Omega_{\delta})
\to \mathcal{H}^1(\Omega_{\infty})\times L^2(\Omega_{\infty})
$$
such that $\overline{S}_T(w(\cdot, T), w_t(\cdot,T))=(w(\cdot, 0), w_t(\cdot, 0))$,
where $w$ is the solution of \eqref{25}. From the decay estimate \eqref{08}, with
$\mathcal{O}=\Omega_{\delta}$, applied to $w$ we obtain the estimate
\begin{equation}\label{26}
\|(w(\cdot, 0), w_t(\cdot,0)) \|^2_{ \mathcal{H}^1(\Omega_{\delta})
\times L^2(\Omega_{\delta})}
\leq \frac{K}{T-R}\|(\widetilde{w}_0, \widetilde{w}_1) \|^2_{ \mathcal{H}^1(\Omega_{\infty})
\times L^2(\Omega_{\infty})}\,,
\end{equation}
for $T>R$ sufficiently large and $K$ is a constant independent on data
$(\widetilde{w}_0, \widetilde{w}_1)$.
Now we consider the cut off function $\phi \in C^{\infty}_0(\Omega_{\infty})$ such that
$\phi\equiv 1$ in $\Omega_{\delta/2}$, and $\phi\equiv 0$ out side of $\Omega_{\delta}$.
Then we solve the forward initial boundary value problem
\begin{equation}\label{27}
\begin{gathered}
z_{tt}-\Delta z=0 \quad \text{in } \Omega_{\infty} \times (0,+\infty) \\
z(\cdot, 0)=\phi w(\cdot,0),\quad z_t(\cdot, 0)=\phi w_t(\cdot,0), \quad \text{in } \Omega_{\infty} \\
z(\cdot, t)=0, \quad \text{in } \Gamma_0 \times (0,+\infty).
\end{gathered}
\end{equation}
We define the linear operator
$S_T: \mathcal{H}^1_0(\Omega_{\delta})\times L^2(\Omega_{\delta})
\to \mathcal{H}^1(\Omega_{\infty})\times L^2(\Omega_{\infty})$ by
$S_T(z(\cdot,0), z_t(\cdot,0))=(z(\cdot, T), z_t(\cdot, T))$.
Applying again the decay estimate \eqref{08}, with $\mathcal{O}=\Omega_{\delta}$,
for function $z$, we obtain
\begin{equation}\label{28}
\|(z(\cdot,T), z_t(\cdot,T)) \|^2_{ \mathcal{H}^1(\Omega_{\delta})
\times L^2(\Omega_{\delta})}
\leq \frac{K}{T-R}\|(z(\cdot,0), z_t(\cdot,0))) \|^2_{ \mathcal{H}^1(\Omega_{\delta})
\times L^2(\Omega_{\delta})}\,,
\end{equation}
for $T>R$ sufficiently large and $K$ is a constant independent on data $(z_0, z_1)$.
We define $\widetilde{v}(\cdot,t)=w(\cdot,t)-z(\cdot,t)$ and see that
$\widetilde{v}$ satisfies
\begin{equation}\label{29}
\begin{gathered}
\widetilde{v}_{tt}-\Delta\widetilde{v}=0 \quad\text{in } \Omega_{\infty} \times (0,+\infty) \\
\widetilde{v}(\cdot, T)=w(\cdot,T)-z(\cdot, T),\quad
\widetilde{v}_t(\cdot, T)=w_t(\cdot, T)-z_t(\cdot, T) \quad \text{in } \Omega_{\infty} \\
\widetilde{v}(\cdot, t)=0, \quad \text{in } \Gamma_0 \times (0,+\infty)
\end{gathered}
\end{equation}
and
\begin{gather*}
\widetilde{v}(\cdot, 0)=w(\cdot, 0)-\phi w(\cdot, 0)=0\quad\text{in } \Omega,\\
\widetilde{v}_t(\cdot, 0)=w_t(\cdot, 0)-\phi w_t(\cdot, 0)=0\quad\text{in } \Omega,
\end{gather*}
since $\phi=1$ in $\Omega$.
Note that the function $\widetilde{v}$ solves the homogeneous wave equation and has
the desirable final state $(\widetilde{v}(\cdot, 0), \widetilde{v}_t(\cdot, 0))=(0,0)$
in $\Omega$.
Now an important step it is to know if we may obtain $T>0$ such that
$(\widetilde{v}(\cdot, T), \widetilde{v}_t(\cdot, T))$ extend the initial data
$(v_0, v_1)$. That is, we wish establish solution for the equations
$$
w(\cdot, T)-z(\cdot, T)=v_0,\quad
w_t(\cdot, T)-z_t(\cdot, T)=v_1\quad\text{in } \Omega.
$$
The last equations can be rewriting as
\begin{equation}\label{30}
E(w_0, w_1)-(z(\cdot, T), z_t(\cdot, T))=(v_0,v_1)\quad\text{in } \Omega.
\end{equation}
We want to solve \eqref{30} for unknown $(w_0, w_1) \in \mathcal{H}^1(\Omega)\times L^2(\Omega)$.
For this purpose we rewrite equation \eqref{30} in terms of the operators $S_T$ and
$\overline{S}_T$. Note that
\begin{align*}
(z(\cdot, T), z_t(\cdot, T))
&=S_T(\phi w(\cdot,0), \phi w_t(\cdot,0))\\
&=S_TM_{\phi}( w(\cdot,0), w_t(\cdot,0))\\
&=S_TM_{\phi}\overline{S}_T(w(\cdot, T), w_t(\cdot,T))\\
&=[S_TM_{\phi}\overline{S}_TE](w_0, w_1),
\end{align*}
where $M_{\phi}$ is the operator multiplication by $\phi$.
Thus, \eqref{30} becomes
\begin{equation}\label{31}
(w_0, w_1)-\mathcal{R}S_TM_{\phi} \overline{S}_TE(w_0, w_1)=(v_0, v_1)
\quad\text{in } \Omega,
\end{equation}
where $\mathcal{R}$ denotes the restriction to $\Omega$.
Denoting $\mathcal{R}S_TM_{\phi} \overline{S}_TE$ by $K_T$, equation \eqref{31} can be
rewritten as
\begin{equation}\label{32}
\left(I-K_T\right)(w_0, w_1)=(v_0, v_1)\quad\text{in }\Omega,
\end{equation}
where $I$ is the identity operator in $ \mathcal{H}^1(\Omega)\times L^2(\Omega)$.
Now, for solving equation \eqref{31} it is sufficient to show that $K_T$ is a contraction
in $ \mathcal{H}^1(\Omega)\times L^2(\Omega)$.
It is in this point where the energy decay takes place, by considering inequalities \eqref{26}
and \eqref{28} note that
\begin{align*}
\|K_T(w_0, w_1) \|_{ \mathcal{H}^1(\Omega)\times L^2(\Omega)}
&=\|(z(\cdot, T), z_t(\cdot, T))\|_{ \mathcal{H}^1(\Omega)\times L^2(\Omega)} \\
&\leq \|(z(\cdot,T), z_t(\cdot,T)) \|^2_{ \mathcal{H}^1(\Omega_{\delta})
\times L^2(\Omega_{\delta})}\\
&\leq \frac{K}{T-R}\|(z(\cdot,0), z_t(\cdot,0)) \|^2_{ \mathcal{H}^1(\Omega_{\delta})
\times L^2(\Omega_{\delta})}\\
&= \frac{K}{T-R}\|(\phi w(\cdot,0), \phi w_t(\cdot,0)) \|^2_{ \mathcal{H}^1(\Omega_{\delta})
\times L^2(\Omega_{\delta})} \\
&\leq \frac{\widetilde{K}}{T-R}\|( w(\cdot,0), w_t(\cdot,0)) \|^2_{ \mathcal{H}^1
(\Omega_{\delta})\times L^2(\Omega_{\delta})}\\
&\leq \frac{K\widetilde{K}}{(T-R)^2}\|E(w_0, w_1) \|^2_{ \mathcal{H}^1(\Omega)
\times L^2(\Omega)} \\
&\leq \frac{K\widetilde{K}}{(T-R)^2}\|(w_0, w_1) \|^2_{ \mathcal{H}^1(\Omega)
\times L^2(\Omega)}.
\end{align*}
From the inequalities above we obtain
\begin{equation}\label{33}
\|K_T(w_0, w_1) \|_{ \mathcal{H}^1(\Omega)\times L^2(\Omega)}
\leq \frac{\sqrt{K\widetilde{K}}}{T-R}\|(w_0, w_1) \|_{ \mathcal{H}^1(\Omega)\times L^2(\Omega)},
\end{equation}
for $T>R$ sufficiently large, where $\widetilde{K}$ is a positive constant depending only $K$
and $\Omega$.
So, we choose a $T>R$ such that $\frac{\sqrt{K\widetilde{K}}}{T-R}\leq c<1$
and for such $T$, $K_T$ is a contraction. Thus, we take the solution $(w_0, w_1)$
for \eqref{32} and take it to the begin of the proof in order to obtain $w$, $z$
and $\widetilde{v}=w-z$, where $\widetilde{v}$ solves \eqref{29} and has
the desirable final condition $(\widetilde{v}(\cdot,0), \widetilde{v}(\cdot,0))=(0,0)$.
To complete the proof we define $v=\widetilde{v}_{|\Omega}$ and apply Lemma \ref{L3}
from the previous section to read off the trace of $\nu_tu_t-\nabla u\cdot \nu_x$
as a function of $ \in L^2(\Gamma_1 \times [0,T])$ completing the proof.
\end{proof}
\section{Proof of Theorem \ref{01}}
Let $\Omega$, $\Omega_{\infty}$ and $\widetilde{\Omega}$ as defined in the initial section.
See that the $\widetilde{\Omega}$ is a holed domain with fixed boundary
$\partial \widetilde{\Omega}=\widetilde{\Gamma} \cup \Gamma_0$, where $\Gamma_0$
is the boundary of the hole and $\widetilde{\Gamma}$ is the external boundary.
Let $(\widetilde{f},\widetilde{g}) \in \mathcal{H}^1(\Omega_{\infty})\times L^2(\Omega_{\infty})$
such that $\widetilde{f}$ and $\widetilde{g}$ are extensions of $f$ and $g$ respectively,
with $\operatorname{supp}(\widetilde{f})\subset B(0,r)\cap \Omega_{\infty}$ and
$\operatorname{supp}( \widetilde{g})\subset B(0,r)\cap \Omega_{\infty}$.
Here the number $r$ is that one defined in the Section 1.
Let $\widetilde{u}$ be the solution of the initial-boundary value problem
\begin{equation}\label{33b}
\begin{gathered}
\widetilde{u}_{tt}-\Delta \widetilde{u}=0 \quad \text{in } \Omega_{\infty} \times (0,+\infty) \\
\widetilde{u}(\cdot, 0)=\widetilde{f},\quad
\widetilde{u}_t(\cdot,0)=\widetilde{g}, \quad \text{in } \Omega_{\infty} \\
\widetilde{u}(\cdot, t)=0, \quad \text{in } \Gamma_0 \times (0,+\infty).
\end{gathered}
\end{equation}
Now, for a $T>r$, we take the state
$(\widetilde{u}(\cdot, T), \widetilde{u}_t(\cdot,T)) \in \mathcal{H}^1(\widetilde{\Omega})
\times L^2(\widetilde{\Omega}) $ and according to Lemma \ref{22}, changing $\Omega$ by
$\widetilde{\Omega}$, we solve the exact boundary control problem
\begin{equation}\label{34}
\begin{gathered}
v_{tt}-\Delta v=0 \quad \text{in } \widetilde{\Omega}\times [0,T] \\
v(\cdot, T)=\widetilde{u}(\cdot, T),\quad
v_t(\cdot, T)=\widetilde{u}_t(\cdot,T), \quad \text{in } \widetilde{\Omega} \\
v(\cdot, t)=0, \quad \text{on } \Gamma_0 \times [0,T] \\
\nu_tv_t-\nabla v\cdot \nu_x=h(\cdot, t), \quad \text{on } \widetilde{\Gamma}\times [0,T],
\end{gathered}
\end{equation}
which satisfies, at the instant $t=0$, the condition
\begin{equation}\label{35}
v(\cdot, 0)=0=v_t(\cdot,0)\quad \text{in } \widetilde{\Omega}.
\end{equation}
Considering $\Omega_t$ as defined in the Section 1, note that
$\Omega_t\subset \widetilde{\Omega}$ for all $t>0$, so follows that for each $T>0$
we have $Q_T=\cup_{0