\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2010(2010), No. 24, pp. 1--13.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2010 Texas State University - San Marcos.}
\vspace{8mm}}

\begin{document}
\title[\hfilneg EJDE-2010/24\hfil Simultaneous exact controllability]
{Simultaneous exact controllability for Maxwell equations
and for a second-order hyperbolic system}

\author[B. V. Kapitonov, G. Perla Menzala \hfil EJDE-2010/24\hfilneg]
{Boris V. Kapitonov, Gustavo Perla Menzala}  % in alphabetical order

\address{Boris V. Kapitonov \newline
Sobolev Institute of Mathematics,
Siberian Branch of the Russian Academy of Science,
 Novosibirsk, Russia}

\address{Gustavo Perla Menzala \newline
National Laboratory of Scientific Computation (LNCC/MCT),
Av. Getulio Vargas 333, Quitandinha, Petr\'opolis,
RJ, 25651-070, Brazil\newline
Institute of Mathematics, Federal University of Rio de Janeiro \\
RJ, P.O. Box 68530, Rio de Janeiro, Brazil}
\email{perla@lncc.br}

\thanks{Submitted November 10, 2009. Published February 10, 2010.}
\subjclass[2000]{35Q99, 74F99, 35B40}
\keywords{Distributed systems; simultaneous exact controllability;
 \hfill\break\indent  boundary observability}

\begin{abstract}
 We present a result on ``simultaneous'' exact controllability
 for two models that describe two hyperbolic dynamics.
 One is the system of Maxwell equations and the other a
 vector-wave equation with a pressure term.
 We obtain the main result using modified multipliers in order
 to generate a necessary observability estimate which allow
 us to use the Hilbert Uniqueness Method (HUM) introduced by Lions.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}[theorem]{Corollary}
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

We consider two models which describe two distinct hyperbolic dynamics:
One of them is the system of Maxwell equations and the other is a vector
wave equation with a pressure term.
In the solution (and vector-valued) variables $\{E,H,u\}$ satisfy
\begin{gather}
\left. \begin{gathered}
\mathcal{E} E_t-\mathop{\rm curl} H=0\\
\mu H_t+\mathop{\rm curl} E=0\\
\mathop{\rm div} E=0,\quad \mathop{\rm div} H=0
\end{gathered} \right\}
\text{ in } \Omega\times(0,T)\label{e1.1}
\\
E(x,0)=E_0(x),\quad H(x,0)=H_0(x) \quad \text{in }\Omega\label{e1.2}
\\
\eta \times E=Q(x,t)\quad \text{on } \partial\Omega\times(0,T)\label{e1.3}
\end{gather}
and
\begin{gather}
\left. \begin{gathered}
\rho u_{tt}-\alpha\Delta u+\text{grad }p=0\\
\mathop{\rm div} u=0
\end{gathered} \right\}\text{ in } \Omega\times(0,T) \label{e1.4}
\\
u(x,0)=g_1(x),\quad u_t(x,0)=g_2(x)\text{ in } \Omega\label{e1.5}
\\
u=P(x,t)\quad\text{ on }\partial\Omega\times(0,T).\label{e1.6}
\end{gather}
In \eqref{e1.1}, $E=(E_1,E_2,E_3)$ and $H=(H_1,H_2,H_3)$ denote
the electric and magnetic field res\-pectively and $\mathcal{E}$ and $\mu$
are positive constants representing the permittivity and magnetic
permeability respectively.

In \eqref{e1.4}, $\rho$ denotes the scalar density which we  will
assume to be a positive constant and $p=p(x,t)$ is the pressure
term (an scalar function). Also, $\alpha$ is a positive constant
depending on the elastic properties of the material. In
\eqref{e1.3}, $\eta=\eta(x)$ denotes the unit normal vector at
$x\in\partial\Omega$ pointing the exterior of $\Omega$. Finally, grad, curl,
$\Delta,\mathop{\rm div}$ and $\times$ denote the usual gradient, rotational,
Laplace operator, divergent and vector product in $\mathbb{R}^3$.

Generally speaking the problem of exact controllability  (for
either problem \eqref{e1.1}--\eqref{e1.3} or
\eqref{e1.4}--\eqref{e1.6}) can be state as follows: Given a time
$T>0$ and any initial data and desired terminal data, to find a
corresponding control $F$ driving the system to the terminal data
at time $T$. One of the most usefull methods to solve such
problems of controllability is the Hilbert Uniqueness Method (HUM)
introduced by J.L. Lions in the middle 80's and is based on the
construction of appropriate Hilbert space structures on the space
of initial data. These Hilbert structures are connected with
uniqueness properties.

Several authors considered the problem of exact controllability
either for problem \eqref{e1.1}--\eqref{e1.3} (see for instance
Russell \cite{r2},  Lagnese \cite{l1},  Eller and  Masters \cite{e1},
Eller \cite{e2}, Kapitonov \cite{k2},  Weck \cite{w1},  Phung \cite{p1}) or problem
\eqref{e1.4}--\eqref{e1.6} (see for instance  Lions \cite{l4} and Rocha
dos Santos \cite{r1}).

Clearly, the above results provide the existence of controls
$Q(x,t)$ and $P(x,t)$ which are not necessarily related one to the
other. In the middle 80's,  Russell \cite{r2} and  Lions \cite{l2}
raised the question if it is possible to solve the exact
controllability problem for two evolution models using only one
control. They named this problem ``simultaneous'' exact
controllability. In the absence of dissipative effects, due to
technical difficulties ``simultaneous'' exact controllability were
only treated for one system with two different boundary conditions
(see for instance,  Lions \cite{l2},  Kapitonov \cite{k3},
Kapitonov, Raupp \cite{k4} and  Kapitonov,  Perla Menzala \cite{k5}).

Now, let us formulate  the ``simultaneous'' exact controllability
for systems \eqref{e1.1}--\eqref{e1.3} and
\eqref{e1.4}--\eqref{e1.6}: Given initial the states
$(f_1,f_2,g_1,g_2)$ and the terminal data
$(\varphi_1,\varphi_2,\psi_1,\psi_2)$ in suitable function spaces we ask
if it is possible to find one vector-valued function $P=P(x,t)$
such that the solution $\{E,H,u\}$ de \eqref{e1.1}--\eqref{e1.6}
with $Q$ in terms of $P$ satisfies at terminal time $T$
$$
(E(T),H(T),u(T),u_t(T))=(\varphi_1,\varphi_2,\psi_1,\psi_2).
$$
The main purpose of this work is to prove that this is indeed  the
case, $P$ serving as a control function for
\eqref{e1.4}--\eqref{e1.6} while the vector-valued function
$$
Q=\mu \eta \times (\eta \times  P_t)
$$
is a control function for \eqref{e1.1}--\eqref{e1.3}.

Let us briefly describe the sections in this paper:  In Section 2
we briefly indicate the function space where the solutions of
systems \eqref{e1.1}--\eqref{e1.3} and \eqref{e1.4}--\eqref{e1.6}
will be considered. Then, we use modified multipliers in order to
obtain a boundary observability inequality valid for system
\eqref{e1.1}--\eqref{e1.6} with $P$ and $Q$ identically equal to
zero (see inequalities \eqref{e2.17}, \eqref{e2.18}). However, the
right hand side of this inequality is not suitable to apply the
steps of the Hilbert Uniqueness Method. That is why we assume a
suitable (numerically) relation between important parameters of
\eqref{e1.1}--\eqref{e1.6}. The observability inequality is obtain
assuming that the region is substar-shaped. In Section 3, the
``simultaneous'' exact controllability is studied by means of the
Hilbert Uniqueness Method (HUM) introduced by  Lions (see
\cite{l2,l3}).

Systems \eqref{e1.1}--\eqref{e1.3} and \eqref{e1.4}--\eqref{e1.6}
are not directly coupled to each other. A more interesting problem
would be to study the case when they are coupled say with coupling
terms -- $\gamma \mathop{\rm curl} E$ and $\gamma\mathop{\rm curl} u_t$ (with $\gamma>0$) for
system \eqref{e1.1}--\eqref{e1.3} and \eqref{e1.4}--\eqref{e1.6}
respectively. As far as we know this remains an open problem. Our
techniques seem to work when the coefficients $\mathcal{E}, \mu,\rho$ and
$\alpha$ are functions of $x$ which are smooth and they and their
partial derivatives of first order are bounded below by strictly
positive constants.

We will use standard notations which can be found in  Duvaut and
Lions' book \cite{d1}. For example, $H^m(\Omega)$ and $H^r(\partial\Omega)$ will
denote the Sobolev spaces of order $m$ and $r$ on $\Omega$ and
$\partial\Omega$ respectively. Given a real-valued function $g$ the
notation $\int_{\partial\Omega} g d\Gamma$ means the surface integral of $g$
over the surface $\partial\Omega$. If $X$ is a vector space, then $[X]^m$
means $X\times X\times X \times\cdots\times X$ $m$-times. For any
vector $v\in\mathbb{R}^3$, $|v|$ denotes the usual norm of $v$ in
$\mathbb{R}^3$.

\section{A Boundary Observability Inequality}

We consider suitable function spaces where the solutions
$\{E,H,u,u_t\}$ of problem \eqref{e1.1}--\eqref{e1.6} (with $P$
and $Q$ identically equal to zero) will be considered. Let $\Omega$
be a bounded region of $\mathbb{R}^3$ with smooth boundary $\partial\Omega$ and
$\mathcal{E}>0$, $\mu >0$ as indicated in Section 1. We consider the
Hilbert space $\mathcal{H}$ defined as follows
$$
\mathcal{H}=[L^2(\Omega)]^3\times [L^2(\Omega)]^3
$$
with inner product given by
$$
\langle v,w\rangle_{\mathcal{H}}=\int_\Omega [\mathcal{E} v_1\cdot w_1+\mu v_2\cdot w_2]dx
$$
for any $v=(v_1,v_2)$, $w=(w_1,w_2)$ in $\mathcal{H}$. Here the central dot
$\cdot$ means the usual inner product in $\mathbb{R}^3$. We also
consider the Hilbert space
$$
H(\rm curl,\Omega)=\{w\in[L^2(\Omega)]^3;\mathop{\rm curl} w\in[L^2(\Omega)]^3\}
$$
with inner product
$$
\langle v_1,v_2\rangle_{H(\rm curl,\Omega)}=\int_\Omega[v_1\cdot v_2 +\mathop{\rm curl}
v_1\cdot\mathop{\rm curl} v_2] dx
$$
for any $v_1,v_2\in H(\rm curl,\Omega)$. It is well known (see \cite{d1})
that the map $Z\mapsto\eta \times  Z|_{\partial\Omega}$ from
$[C_0^1(\overline\Omega)]^3$ into $[C^1(\partial\Omega)]^3$ extends by continuity
to a continuous linear map from $H(\rm curl,\Omega)$ into
$[H^{-1/2}(\partial\Omega)]^3$. This result allow us to consider the space
$$
H_0(\rm curl,\Omega)=\{w\in H(\rm curl,\Omega),\quad \eta \times  w=0
\text{ on } \partial\Omega\}.
$$
Here $\eta=\eta(x)$ denotes the unit normal vector at $x\in\partial\Omega$
pointing the exterior of $\Omega$. We define the  operator
$A\colon\mathcal{D}(A)\subseteq\mathcal{H}\mapsto\mathcal{H}$ with domain $\mathcal{D}(A)$ given
by
$$
\mathcal{D}(A)=H_0(\rm curl,\Omega)\times H(\rm curl,\Omega)
$$
and $A$ is defined as follows
$$
A(v_1,v_2)=(\mathcal{E}^{-1}\mathop{\rm curl} v_2,-\mu^{-1}\mathop{\rm curl} v_1)
$$
The skew-selfadjointness of $A$ can be easily verified.  By
Stone's Theorem, the operator $A$ generates a one-parameter group
of unitary operators $\{U(t)\}_{t\in\mathbb{R}}$ on $\mathcal{H}$. Observe that
in order $U(t)f$ to solve problem \eqref{e1.1}--\eqref{e1.3} with
$Q\equiv 0$ and given $f\in\mathcal{D}(A)$ remains to prove that the
components of $U(t)f$ are divergent free. In order to overcome
this issue we consider $M=\{(\text{grad }\varphi_1,\text{grad }\varphi_2)$
with $\varphi_1,\varphi_2\in C_0^\infty(\Omega)\}$ and  $M_1=M^\perp$. We
observe that $M$ is not closed in $\mathcal{H}$ but $M_1$ is closed in
$\mathcal{H}$. In the distributional sense it is easy to prove that
whenever $w=(w_1,w_2)\in M_1$ then $\mathop{\rm div} w_1=0$ and $\mathop{\rm div} w_2=0$.
Furthermore, we claim that $U(t)$ takes $M_1\cap\mathcal{D}(A)$ into
itself. Indeed, for any $w$ in $M$ and $v$ in $M_1\cap\mathcal{D}(A)$ we
have
\[
\frac d{dt} \langle U(t)v,w\rangle_{\mathcal{H}}
=\langle AU(t)v,w\rangle_{\mathcal{H}}
=\langle U(t)v,A^*w\rangle_{\mathcal{H}}
=0 \quad \text{for any } t\in\mathbb{R}.
\]
Thus
$$
\langle U(t)v,w\rangle_{\mathcal{H}}=\text{constant} \quad \text{for any } t\in\mathbb{R}.
$$
Therefore, taking $t=0$ we have $\langle v,w\rangle=0$. Consequently
$U(t)v\in M_1\cap\mathcal{D}(A)$ for any $t\in\mathbb{R}$  which proves our
claim. Observe that for any element $v=(v_1,v_2)$ belonging to
$M_1\cap \mathcal{D}(A)$ and $w=(0,\mathop{\rm grad}\varphi_2)$ with $\varphi_2\in H^2(\Omega)$
we have
$$
0=\langle v,w\rangle_{\mathcal{H}}=\int_\Omega \mu v_2\cdot\mathop{\rm grad}\varphi_2 dx =\mu
\int_{\partial\Omega}\varphi_2 v_2\cdot\eta d\Gamma.
$$
Since $\varphi_2\in H^2(\Omega)$ is arbitrary, the above identify say that
\begin{equation}
\eta\cdot v_2=0\quad \text{on }\partial\Omega.\label{e2.1}
\end{equation}
We conclude that problem \eqref{e1.1}--\eqref{e1.3} with
$Q\equiv0$  has a generator $A_1$ which applies $M_1\cap\mathcal{D}(A)$
into $M_1\cap\mathcal{D}(A)$ and for any $w=(w_1,w_2)\in M_1\cap\mathcal{D}(A)$
the relation $\eta\cdot w_2=0$ on $\partial\Omega$ holds.

Next, we consider problem \eqref{e1.3}--\eqref{e1.6} with
$P\equiv0$.  We can use Galerkin's method to find $u$ and $p$
(defined up to a constant). We choose the spaces
$$
V=\{\varphi\in[C_0^\infty(\Omega)]^3, \mathop{\rm div}\varphi=0\text{ in } \Omega\}.
$$
Let $Y$ be the closure of $V$ with respect to the norm of
$[H_0^1(\Omega)]^3$ and
$$
W=Y\cap[H^2(\Omega)]^3.
$$
Considering $u_0\in W$, $u_1\in Y$ we obtain by using the Galerkin
method a unique solution of problem \eqref{e1.3}--\eqref{e1.6}
such that $u\in C([0,+\infty);V)\cap C^1([0,+\infty);H)$ where $H$
denotes the closure of $V$ with respect to the norm of
$[L^2(\Omega)]^3$. We can also obtain more regularity. For example,
if $u_0\in V\cap [H^4(\Omega)]^3$ and $u_1\in W$ then the solution
$u\in C([0,\infty);W)\cap C^1([0,+\infty);V)$ with $p\in
H^2(\Omega)$.

Let us now concern ourselves with the simultaneous boundary
observability problem. We use the multiplier method. They are
conveniently modified in such a way that we can handle the extra
boundary terms appearing in the identities. Let $h(x)$ be an
auxiliary scalar smooth function on $\overline\Omega$, which we will choose
later on. For problem \eqref{e1.1} we consider the multipliers
\begin{gather*}
M_1=tE+\mu\mathop{\rm grad} h \times  H, \\
M_2=tH-\mathcal{E}\mathop{\rm grad} h \times  E
\end{gather*}
Since $\{E,H\}$ solves \eqref{e1.1}--\eqref{e1.2} then we have the
identity
\begin{equation}
\begin{aligned}
0 &= 2M_1\cdot(\mathcal{E} E_t-\mathop{\rm curl} H)+2M_2\cdot(\mu H_t+\mathop{\rm curl} E)\\
&\quad + 2\mathcal{E}(\mathop{\rm grad} h\cdot E)\mathop{\rm div} E+2\mu(\mathop{\rm grad} h\cdot H)\mathop{\rm div} H.
\end{aligned}\label{e2.2}
\end{equation}
We can rearrange the terms on the right hand of \eqref{e2.2} to obtain
\begin{equation}
\frac{\partial A}{\partial t}=\mathop{\rm div}(\vec B)+D\label{e2.3}
\end{equation}
where
\begin{gather}
A=A(x,t)=t(\mathcal{E}|E|^2+\mu|H|^2)+2\mathcal{E}\mu\mathop{\rm grad} h\cdot(H  \times E)\label{e2.4}
\\
\begin{aligned}
\vec B =\vec B(x,t)
&= 2t H \times  E+\mathop{\rm grad} h(\mathcal{E}|E|^2+\mu|H|^2)\\
&\quad - 2\mathcal{E} E(E\cdot\mathop{\rm grad} h)-2\mu H(H\cdot\mathop{\rm grad} h),
\end{aligned}\label{e2.5}
\\
D=2\sum_{i,j=1}^3\frac{\partial^2h}{\partial x_i\partial x_j}(\mathcal{E} E_iE_j+\mu
H_iH_j) -(\Delta h-1)(\mathcal{E}|E|^2+\mu|H|^2), \label{e2.6}
\end{gather}
where
$$
|E|^2=\sum_{j=1}^3E_j^2,\quad |H|^2=\sum_{j=1}^3H_j^2.
$$
Similarly, for problem \eqref{e1.4} we consider the multipliers
\begin{gather*}
M_3=tu_t+(\mathop{\rm grad} h \cdot \mathop{\rm grad})u+u, \\
M_4=tp\frac\partial{\partial t}+p(\mathop{\rm grad} h\cdot \mathop{\rm grad})+p.
\end{gather*}
Here $\mathop{\rm grad} h \cdot\mathop{\rm grad}=\sum_{j=1}^3\frac{\partial h}{\partial x_j}
\frac{\partial}{\partial x_j}$. Observe that $M_4$ is actually an operator.
Since $\{u,p\}$ is a solution of \eqref{e1.4} then we have the
identity
\begin{equation}
0=2M_3\cdot(\rho u_{tt}-\alpha\Delta u+\mathop{\rm grad} p)+2M_4 \mathop{\rm div} u.\label{e2.7}
\end{equation}
We can rearrange terms in identity \eqref{e2.7} to obtain
\begin{equation}
\frac{\partial A_1}{\partial t}=\mathop{\rm div}\vec G+\mathop{\rm div}\vec F+D_1\label{e2.8}
\end{equation}
where
\begin{gather*}
A_1=t \Big(\rho|u_t|^2+\alpha\sum_{i=1}^3\big|\frac{\partial u}{\partial x_i}
\big|^2\Big)+2\rho[u_t\cdot(\mathop{\rm grad} h\cdot\mathop{\rm grad})u+u],
\\
\vec G=(G_1,G_2,G_3)
\end{gather*}
with
\begin{gather}
G_i = 2[tu_t+(\mathop{\rm grad} h\cdot\mathop{\rm grad})u+u]\cdot
 \alpha\frac{\partial u}{\partial x_i}
+\frac{\partial h}{\partial x_i}\Big(\rho|u_t|^2-\alpha\sum_{j=1}^3
\big|\frac{\partial u}{\partial
x_j}\big|^2\Big),
\label{e2.9}
\\
\vec F=-2p[tu_t+(\mathop{\rm grad} h\cdot \mathop{\rm grad}) u+u] \nonumber
\end{gather}
and
\begin{align*}
D_1 &=(3-\Delta h)\rho|u_t|^2+(\Delta h-1)\alpha\sum_{j=1}^3
 \big|\frac{\partial u}{\partial x_j}\big|^2 \\
&\quad - 2\sum_{i,q=1}^3\frac{\partial^2h}{\partial x_i\partial x_q}\alpha
 \Big( \frac{\partial u}{\partial x_i}\cdot \frac{\partial u}{\partial x_q}\Big)
+2p\sum_{i,j=1}^3\frac{\partial^2 h}{\partial x_i\partial x_j}\frac{\partial u_j}{\partial x_i}.
\end{align*}


\begin{remark} \label{rmk1} \rm
 If we choose $h(x)=\frac12|x-x_0|^2$ for
some  $x_0\in\mathbb{R}^3$ then we can verify that $D$ and $D_1$ in
\eqref{e2.3} and \eqref{e2.8} are identically zero. Therefore in
the case \eqref{e2.3} and \eqref{e2.8} represent a conservation
law.
\end{remark}

Let $(E_0,H_0)\in M_1\cap\mathcal{D}(A)$ and $(E,H)$ be the corresponding
solution of problem \eqref{e1.1}--\eqref{e1.3} with $Q\equiv0$.
Integration in $\Omega\times(0,T)$ of identity \eqref{e2.3} give us
\begin{equation}
\begin{aligned}
& T\int_\Omega\{\mathcal{E}|E|^2+\mu|H|^2\} dx+2\mathcal{E}\mu\int_\Omega\mathop{\rm grad} h\cdot
 (H \times  E) dx \Big|_{t=0}^{t=T} \\
&= \int_0^T \int_{\partial \Omega}\beta(E,H,h)d\Gamma dt+\int_0^T\int_\Omega D \, dxds
\end{aligned} \label{e2.10}
\end{equation}
where
\begin{equation}
\begin{aligned}
\beta(E,H,h) &= 2t\eta\cdot(H \times  E)+ \frac{\partial h}{\partial\eta} (\mathcal{E}|E|^2+\mu|H|^2)\\
&\quad - 2\mathcal{E}(E\cdot\eta)(E\cdot\mathop{\rm grad} h)-2\mu(H\cdot\eta)(H\cdot\mathop{\rm grad} h).\end{aligned} \label{e2.11}
\end{equation}
and $D$ is given as in \eqref{e2.6}. Using the boundary
conditions,  \eqref{e1.3} (with $Q\equiv 0$) and the vector
identities on $\partial\Omega$ we get
\begin{gather*}
\eta\cdot(H \times  E)=-(\eta \times  E)\cdot H=0,\quad
|E|^2=(E\cdot\eta)^2+|E \times \eta|^2=(E\cdot\eta)^2 \\
E=\eta  \times (\eta \times  E)+\eta(E\cdot\eta)=\eta(E\cdot\eta),\quad
|H|^2=(H\cdot\eta)^2+|H \times  \eta|^2=|H \times \eta|^2
\end{gather*}
because $|\eta|=1$ and \eqref{e2.1} holds for $H$. Thus we obtain
from \eqref{e2.11}
\begin{equation}
\beta(E,H,h)=\frac{\partial h}{\partial\eta} \{ \mu|H \times \eta|^2
-\mathcal{E}(E\cdot\eta)^2\}.\label{e2.12}
\end{equation}
Now, we want to get appropriate estimates for the term
$\int_0^T\int_\Omega D\, dxdt$ in \eqref{e2.10}. Let us choose a
convenient function $h(x)$. We consider the elliptic problem
\begin{gather*}
\Delta \Phi = 1 \quad \text{in } \Omega \\
\frac{\partial\Phi}{\partial\eta} = \frac{{\rm vol}(\Omega)}{{\rm area}(\partial \Omega)}
 \quad \text{on } \partial \Omega
\end{gather*}
which admits a solution $\Phi\in C^2(\Omega)\cap C^1(\overline \Omega)$.  Here
${\rm area}(\partial\Omega)$ means the surface area of $S=\partial\Omega$. Let
$0<\delta<1$ and $x_0\in\mathbb{R}^3$ we define
\begin{equation}
h(x)=\delta\Phi(x)+\frac12|x-x_0|^2.\label{e2.13}
\end{equation}
substitution of such $h(x)$ into \eqref{e2.6} give us
$$
D=2\delta\sum_{i,j=1}^3\frac{\partial^2\Phi}{\partial x_i\partial x_j}(\mathcal{E} E_iE_j+\mu H_iH_j)-\delta(\mathcal{E}|E|^2+\mu|H|^2).
$$
Let $C_1=C_1(\Phi)$ be the constant given by
$$
C_1=\max_{x\in\overline\Omega,\, i,j=1,2,3}
\big| \frac{\partial^2\Phi(x)}{\partial x_i\partial x_j}\big|.
$$
We can easily verified that $C_1\ge1/3$ and
\begin{equation}
|D|\le\delta C_2(\mathcal{E}|E|^2+\mu|H|^2)\label{e2.14}
\end{equation}
where $C_2=6C_1-1>0$ and $0<\delta<1$. Since the quantity
$\frac12\int_\Omega\{\mathcal{E}|E|^2+\mu|H|^2\} dx$ is constant for any
$t\in\mathbb{R}$ for the solution $(E,H)$ of \eqref{e1.1}--\eqref{e1.3}
(with $Q\equiv0$) then, from \eqref{e2.14} it follows that
\begin{equation}
\int_0^T\int_\Omega D\, dxdt\le \delta C_2T\int_\Omega\{\mathcal{E}|E|^2+\mu|H|^2\}
 dx.\label{e2.15}
\end{equation}
Now, we want to estimate the term $2\mathcal{E}\mu\int_\Omega\mathop{\rm grad} h\cdot(H
\times  E) dx\Big|_{t=0}^{t=T}$  in \eqref{e2.10}. Let $C_3>0$
given by
$$
C_3=\max_{x\in\overline\Omega}\{|\mathop{\rm grad}\Phi(x)|+|x-x_0|\}
$$
Since $|H \times E|\le 2|E| |H|$ and $h$ as in \eqref{e2.13} we deduce
\begin{equation}
\begin{aligned}
2\int_\Omega\mathcal{E}\mu\mathop{\rm grad} h\cdot(H \times  E) dx
&\le 4(1+\delta)C_3\sqrt{\mathcal{E}\mu} \int_\Omega\sqrt{\mathcal{E}\mu} |E| |H| dx\\
&\le 4(1+\delta)C_3\sqrt{\mathcal{E}\mu} \int_\Omega(\mathcal{E}|E|^2+\mu|H|^2)dx
\end{aligned}\label{e2.16}
\end{equation}
for any $0\le t\le T$. Thus, from identity \eqref{e2.10} and
inequalities \eqref{e2.12}, \eqref{e2.15} and \eqref{e2.16} we get
\begin{equation}
\begin{aligned}
& (1-\delta C_2)(T-T_0)\int_\Omega \{\mathcal{E}|E|^2+\mu|H|^2\} dx \\
& \le\int_0^T\int_{\partial\Omega} \frac{\partial h}{\partial\eta}  \{\mu|H
\times \eta|^2-\mathcal{E}(E\cdot\eta)^2\} d\Gamma \end{aligned}\label{e2.17}
\end{equation}
where $T_0=4(1+\delta) C_3\sqrt{\mathcal{E}\mu} \Big/ 1-\delta C_2$.
Similarly,  using identity \eqref{e2.8} and assuming some
geometric condition on the region $\Omega$ (say for instance,
``substar'' like \cite{k1,k5}) we can prove that the solution of
problem \eqref{e1.4}--\eqref{e1.6} (with $P\equiv 0$) satisfies
the inequality
\begin{equation}
\begin{aligned}
& (1-\delta\tilde C_2)(T-\tilde T_0) \int_\Omega
\Big\{ \rho|u_t|^2+\alpha\sum_{i=1}^3 \big| \frac{\partial u}{\partial x_i}\Big|^2
\Big\} dx \\
& \le \int_0^T \int_{\partial\Omega} \frac{\partial h}{\partial\eta} \alpha
\big| \eta \times  \frac{\partial u}{\partial\eta} \big|^2 d\Gamma dt
\end{aligned}\label{e2.18}
\end{equation}
for some $\tilde C_2>0$ and $\tilde T_0>0$.


\begin{remark} \label{rmk2} \rm
By choosing $\delta>0$ sufficiently
small and adding  inequalities \eqref{e2.17} and \eqref{e2.18} we
would obtain a boundary observability provided $\frac{\partial
h}{\partial\eta}\ge0$ on $\partial\Omega$ . However, in order to apply the
techniques to use the HUM would not help that much.
\end{remark}

We want to prove the following observability inequality.


\begin{theorem} \label{thm1}
Let $\{E,H,u,u_t\}$ be the
solution of  \eqref{e1.1}--\eqref{e1.6} with $P=Q=0$ on $\partial\Omega$.
Suppose there exist $\delta_1>0$ and $x_0\in\mathbb{R}^3$
($\delta_1<\min\{C_2^{-1},\tilde C_2^{-1}\}$) where $C_2$ and
$\tilde C_2$ are as in \eqref{e2.17} and \eqref{e2.18} such that
\begin{equation}
\delta_1\frac{{\rm vol}(\Omega)}{{\rm area}(\partial\Omega)}+(x-x_0)\cdot\eta>0\quad
\text{for all } x\in\partial\Omega\label{e2.19}
\end{equation}
and the parameters in \eqref{e1.1} and \eqref{e1.4} satisfy
(numerically) the relation $\rho=\mathcal{E}\mu\alpha$. Then, there exist
constants $C_5,C_6$ and a $T_1>0$ such that
\begin{align*}
&(2-\delta_1C_6)(T-T_1)  \int_\Omega
\Big\{\mathcal{E}|E|^3+\mu|H|^2+\rho|u_t|^2+\alpha\sum_{i=1}^3
\big|\frac{\partial u}{\partial x_i}\big|^2\Big\} \\
& \le \int_0^T \int_{\partial\Omega} \Big\{ \frac{C_5}2
\big| \mu H+\alpha\frac{\partial u}{\partial\eta}\big|^2-\frac{\partial h}{\partial\eta}
\mathcal{E}(E\cdot\eta)^2 \Big\} d\Gamma dt
\end{align*}
\end{theorem}

\begin{proof}
In \eqref{e2.13} we choose $\delta=\delta_1$. Observe
that \eqref{e2.19} tell us that $\frac{\partial h}{\partial\eta}>0$ for all
$x\in\partial\Omega$. we can easily verify the identity
\begin{equation}
\begin{aligned}
& \mu H\cdot(\rho u_{tt}-\alpha\Delta u+\mathop{\rm grad} p)+\rho\mathcal{E}^{-1}\mathop{\rm curl} u \cdot(\mathcal{E} E_t-\mathop{\rm curl} H) \\
& + \rho u_t\cdot(\mu H_t+\mathop{\rm curl} E)+(\mu p-\alpha\mu \mathop{\rm div} u)\mathop{\rm div} H \\
& + (\rho\mathcal{E}^{-1}-\alpha\mu)\mathop{\rm curl} u \cdot\mathop{\rm curl} H \\
&= \frac\partial{\partial t}A_2-\mathop{\rm div}\vec B_2.
\end{aligned}\label{e2.20}
\end{equation}
where
\begin{gather*}
A_2=\rho u_t\cdot\mathcal{E} H+\rho\mathop{\rm curl} u\cdot E, \\
\vec B_2=\rho u_t \times  E+\alpha\mu(\mathop{\rm div} u)H+\alpha\mu H \times
\mathop{\rm curl} u-\mu p H.
\end{gather*}
 From \eqref{e2.20} it follows that $\frac\partial{\partial t}A_2=\mathop{\rm div} B_2$.
Integration over $\Omega\times(0,T)$ give us
\begin{equation}
 \int_\Omega \{\rho u_t\cdot\mathcal{E} H+\rho\mathop{\rm curl} u\cdot E\} dx
 \Big|_{t=0}^{t=T}
 = -\alpha\mu\int_0^T\int_{\partial\Omega}(H \times \eta)\cdot
\mathop{\rm curl} u \, d\Gamma dt.
\label{e2.21}
\end{equation}
We use the identity
$$
|\mu(H \times \eta)-\alpha\mathop{\rm curl} u|^2=\mu^2|H
\times \eta|^2-2\alpha\mu(H \times  \eta)\cdot \mathop{\rm curl} u+\alpha^2|\mathop{\rm curl} u|^2.
$$
Substitution into \eqref{e2.21} give us
\begin{equation}
\begin{aligned}
& \int_\Omega\{\rho u_t\cdot\mathcal{E} H+\rho\mathop{\rm curl} u\cdot E\} dx\Big|_{t=0}^{t=T} \\
&= \int_0^T\int_{\partial\Omega} \Big\{ \frac12 |\mu(H \times \eta)
 -\alpha\mathop{\rm curl} u|^2-\frac12\mu^2|H \times \eta|^2
 -\frac{\alpha^2}2|\mathop{\rm curl} u|^2 \Big\} d\Gamma dt\\
&= \int_0^T \int_{\partial\Omega} \Big\{ \frac12|\mu (H \times \eta)
 -\alpha\mathop{\rm curl} u|^2-\frac12\mu^2|H \times  \eta|^2-\frac{\alpha^2}2
 \big|\frac{\partial u}{\partial \eta} \times \eta\big|^2\Big\} d\Gamma dt
\end{aligned} \label{e2.22}
\end{equation}
because $u=0$ on $\partial\Omega\times(0,T)$
(and $u\in[H^2(\Omega)\cap H_0^1(\Omega)]^3$ then
$\frac{\partial u_i}{\partial x_j}= \eta_j \frac{\partial u_i}{\partial \eta}$;
 therefore $\mathop{\rm curl} u=\eta \times \frac{\partial u}{\partial\eta}$ on
$\partial\Omega\times(0;T)$. Observe that
\begin{equation}
|\mathop{\rm curl} u|^2\le 2\sum_{i,j=1}^3
 \big(\frac{\partial u_i}{\partial x_j}\big)^2
=2\sum_{j=1}^3 \big|\frac{\partial u}{\partial x_j}\big|^2.\label{e2.23}
\end{equation}
Using \eqref{e2.23} we can also obtain the inequality
\begin{equation}
\begin{aligned}
& \big| \int_\Omega \{\rho u_t\cdot\mathcal{E} H+\rho\mathop{\rm curl} u\cdot E\} dx \big| \\
& \le C_4 \int_\Omega \Big\{ \rho|u_t|^2+\alpha \sum_{j=1}^3
\big| \frac{\partial u}{\partial x_j} \big|^2+\mathcal{E}|E|^2+\mu |H|^2 \Big\} dx
\end{aligned} \label{e2.24}
\end{equation}
where
$C_4=\max\{(\rho/\mu)^{1/2}\mathcal{E}, 2\rho/\sqrt{\mathcal{E}}\}$.
Consider
$$
C_5=2\max \big\{ \big\|\frac{\partial h}{\partial\eta}\big\|_{L^\infty(\partial\Omega)}
\alpha^{-1}, \big\|\frac{\partial h}{\partial\eta}\big\|_{L^\infty(\partial\Omega)} \mu^{-1}
\big\}.
$$
We multiply \eqref{e2.22} by $C_5$ and obtain from
\eqref{e2.17}, \eqref{e2.18}, \eqref{e2.22} and \eqref{e2.24}
\begin{align*}
& (1-\delta_1C_6)(T-T_1)\int_\Omega
\Big\{\mathcal{E}|E|^2+\mu|H|^2+\rho|u_t|^2+\alpha\sum_{i=1}^3
\big| \frac{\partial u}{\partial x_i}\big|^2 \Big\} dx \\
&\le \int_0^T \int_{\partial\Omega}\frac{\partial h}{\partial\eta}
 \Big\{\mu|H \times \eta|^2-\mathcal{E}(E\cdot\eta)^2+\alpha
 \big| \eta \times \frac{\partial u}{\partial \eta}\big|^2 \Big\} \\
&\quad + \frac{C_5}{2} |\mu(H \times \eta)-\alpha\mathop{\rm curl} u|^2
 -\frac{C_5}2\mu^2|H \times \eta|^2-\frac{\alpha^2}2 C_5
 \big| \frac{\partial u}{\partial \eta} \times  \eta\big|^2 d\Gamma dt \\
&\le \int_0^T \int_{\partial\Omega}
 \Big\{ \frac{C_5}2 |\mu(H \times  \eta)-\alpha\mathop{\rm curl} u|^2
 -\frac{\partial h}{\partial\eta}\mathcal{E}(E\cdot\eta)^2 \Big\} d\Gamma dt \label{e2.25}
\end{align*}
where
$C_6=C_2+\tilde C_2$ and
$$
T_1=\frac{(T_0+\tilde T_0) -\delta_1(C_2T_0+\tilde C_2\tilde T_0)}{2-\delta_1 C_6} >0
$$
We claim that the term $|\mu(H \times \eta)-\alpha\mathop{\rm curl} u|$ on the
right hand side of \eqref{e2.19} equals to $|\alpha\frac{\partial u}{\partial
\eta}+\mu H|$ for any $(x,t)\in\partial\Omega\times(0,T)$ q.t.p. In fact,
using the boundary conditions we have
$$
|\mu(H \times \eta)-\alpha\mathop{\rm curl} u|
= \big|\mu(H \times \eta)+\alpha\big(\frac{\partial u}{\partial\eta} \times \eta\big)
\big|.
$$
Using the identity $|v\times\eta|^2+(v\cdot\eta)^2=|v|^2$ valid
for any vector of $v\in\mathbb{R}^3$ we obtain
$$
\big| \big(\mu H+\alpha\frac{\partial u}{\partial\eta}\big) \times  \eta \big|^2
+\Big[\big(\mu H+\alpha\frac{\partial u}{\partial\eta}\big)\cdot\eta\Big]^2
= \big|\alpha\frac{\partial u}{\partial\eta}+\mu H\big|^2
$$
because $H\cdot \eta=0$ and $\frac{\partial u}{\partial\eta} \cdot \eta=0$
on $\partial\Omega\times (0,T)$. This proves  our claim and the conclusion
of Theorem \ref{thm1}.
\end{proof}

\begin{corollary} \label{coro1}
 Let $\{E,H,u,u_t\}$ be the solution
of \eqref{e1.1}-\eqref{e1.6} with zero boundary conditions and
assume the conditions of Theorem \ref{thm1}. If the condition
$$
\mu H+\alpha\frac{\partial u}{\partial\eta}=0\quad \text{on } \partial\Omega\times(0,T)
$$
holds, then for any $T>T_1$ we will have
$$
E(x,t)\equiv H(x,t)\equiv u(x,t)\equiv 0\quad \text{in }\Omega\times(0,T).
$$
\end{corollary}

\section{Simultaneous exact controllability}

Let $\{E,H,u,u_t\}$ be the solution of
\eqref{e1.1}--\eqref{e1.6}  with zero boundary conditions. In the
function space of initial data (for strong solutions) we consider
the Hilbert space $\mathcal{F}$ obtained by completing such space with
respect to the norm
$$
\|(f,g)\|_{\mathcal{F}}= \Big( \int_0^T\int_{\partial\Omega}
\big|\mu H+\alpha\frac{\partial u}{\partial\eta} \big|^2 d\Gamma dt \Big)^{1/2}
$$
for $T>T_1$ where $f=(f_1,f_2)$ and $g=(g_1,g_2)$ are the initial
data  of problems \eqref{e1.1}--\eqref{e1.3} and
\eqref{e1.4}--\eqref{e1.6} respectively. From Corollary \ref{coro1} it
follows that $\|\cdot\|_{\mathcal{F}}$ is indeed a norm. Let us denote by
$\|\cdot\|_K$ the energy norm
$$
\|(f,g)\|_K^2=\int_\Omega\Big\{ \mathcal{E}|E|^2+\mu|H|^2+\rho|u_t|^2
+\alpha\sum_{j=1}^3 \big| \frac{\partial u}{\partial x_j} \big|^2 \Big\} dx
$$
then, clearly we have
$\mathcal{F}\subseteq K$  and
$\|(f,g)\|_K\le C\|(f,g)\|_{\mathcal{F}}$
for some positive constant $C$. Let us consider the dual space of
$\mathcal{F}$ with respect to $K$. We will denoted by $\mathcal{F}'$.



\noindent\textbf{Definition.}
 Given $R=R(x,t)\in[L^2(\partial\Omega\times(0,T))]^3$  and
$(f_1,f_2,g_1,g_2)\in\mathcal{F}'$. We say that $\{E,H,u,u_t\}$ is a
solution of the Maxwell/elasticity system if $\{E,H\}$ solves
\eqref{e1.1}--\eqref{e1.2} with boundary condition.
\begin{equation}
\eta \times  E=0=\mu\eta \times (\eta \times  R)\quad\text{ on }
\partial\Omega\times(0,T)\label{e2.26}
\end{equation}
and $(u,u_t)$ solves \eqref{e1.4}--\eqref{e1.5} with boundary
conditions $u_t=R$ on $\partial\Omega\times(0,T)$.
Furthermore,
\begin{itemize}

\item[(a)]
$(E(\cdot,t),H(\cdot,t),u(\cdot,t),u_t(\cdot,t))\in L^\infty(0,T;\mathcal{F}')$
and
\item[(b)]
\begin{equation}
\begin{aligned}
& \langle (E(\cdot,t),H(\cdot,t),u(\cdot,t),u_t(\cdot,t)),
(\tilde E(\cdot,t),\tilde H(\cdot,t),\tilde u(\cdot,t),
 \tilde u_t(\cdot,t))\rangle_K \\
&= \langle(f_1,f_2,g_1,g_2), (\tilde f_1,\tilde f_2,
 \tilde g_1,\tilde g_2)\rangle_K \\
&\quad + \int_0^t \int_{\partial\Omega} R\cdot
\Big( \mu\tilde H_0+\alpha\frac{\partial\tilde u}{\partial\eta}-\tilde p\eta \Big)
d\Gamma d\tau
\end{aligned} \label{e2.27}
\end{equation}
holds for any $(\tilde f_1,\tilde f_2,\tilde g_1,\tilde
g_2)\in\mathcal{F}$  and $t\in(0,T)$ where $(\tilde E,\tilde H,\tilde
u,\tilde u_t)$ is a solution of \eqref{e1.1}--\eqref{e1.6} with
zero boundary conditions. In \eqref{e2.26}
\begin{align*}
& \langle(f_1,f_2,g_1,g_2), (\tilde f_1,\tilde f_2,\tilde g_1,\tilde g_2)\rangle_K \\
&= \int_\Omega \Big\{ \mathcal{E} f_1\cdot\tilde f_1+\mu f_2\cdot\tilde f_2
+\alpha\sum_{i=1}^3 \frac{\partial g_1}{\partial x_i}\cdot
\frac{\partial\tilde g_1}{\partial x_i}+\rho g_2\cdot \tilde g_2 \Big\} dx.
\end{align*}
\end{itemize}
Here $\tilde p$ denotes the pressure term for the solution
$(\tilde u,\tilde u_t)$ of \eqref{e1.4}--\eqref{e1.6} with zero
boundary conditions.


\noindent\textbf{Definition.}
 Given $R=R(x,t)\in[L^2(\partial\Omega\times(0,T)]^3$ we  say that
$\{E,H,u,u_t\}$ is a solution of the Maxwell/elasticity system
with zero initial data at time $t=T$ if $\{E,H\}$ solves
\eqref{e1.1}--\eqref{e1.2} with boundary condition
$$
\eta \times  E=\mu\eta \times (\eta \times  R)\quad \text{on }
\partial\Omega\times(0,T)
$$
and $\{u,u_t\}$ solves \eqref{e1.4}--\eqref{e1.5} with boundary
condition
\begin{equation}
u_t=R\quad\text{on } \partial\Omega\times(0,T)\label{e2.28}
\end{equation}
Furthermore,
\begin{itemize}

\item[(a)]
$(E(\cdot,t),H(\cdot,t),u(\cdot,t),u_t(\cdot,t))\in L^\infty(0,T;\mathcal{F}')$
and

\item[(b)]
\begin{equation}
\begin{aligned}
& \langle (E(\cdot,t),H(\cdot,t),u(\cdot,t),u_t(\cdot,t)),
(\tilde E(\cdot,t),\tilde H(\cdot,t),\tilde u(\cdot,t),\tilde u_t(\cdot,t))\rangle_K \\
&= -\int_t^T \int_{\partial\Omega} R\cdot
\Big( \alpha \frac{\partial \tilde u}{\partial\eta}+\mu\tilde H
-\tilde p\eta\Big) d\Gamma d\tau
\end{aligned}\label{e2.29}
\end{equation}
holds for any $(\tilde f_1,\tilde f_2,\tilde g_1,\tilde g_2)\in\mathcal{F}$
and $t\in(0,T)$.
\end{itemize}

Due to the linearity and reversibility of system \eqref{e1.1}-\eqref{e1.6}
it is clear that in order to solve the problem of exact controllability
it is sufficient to prove that for any initial data
$(f_1,f_2,g_1,g_2)\in \mathcal{F}'$ then the corresponding solution can be
driven to the equilibrium state at time $T$.



 \begin{theorem} \label{thm2}
 Under the assumptions of Theorem \ref{thm1}. If
$T>T_1$, then for any initial data $(f_1,f_2,g_1,g_2)\in\mathcal{F}'$ of
problems \eqref{e1.1}--\eqref{e1.2} and \eqref{e1.4}--\eqref{e1.5}
there exist a control  $P\in H^1(0,T;[L^2(\partial\Omega)]^3)$
such that $u=P$ on $\partial\Omega\times(0,T)$ and the corresponding
solution satisfies
$$
(u,u_t)\big|_{t=T}=(0,0)
$$
while the vector-valued function $Q=\mu\eta \times (\eta \times
P_t)$  drives system \eqref{e1.1}--\eqref{e1.2} such that $\eta
\times E=Q$ on $\partial\Omega\times(0,T)$ and the corresponding solution
satisfies
$$
(E,H)\big|_{t=T}=(0,0)
$$
\end{theorem}


 \begin{proof} We use our previous discussion to apply the
Hilbert  Uniqueness Method (HUM). Let $(h_1,h_2,q_1,q_2)$ be an
(arbitrary) element of $\mathcal{F}$ and $(\varphi,\psi,v,v_t)$ the solution
of \eqref{e1.1}--\eqref{e1.6} with zero boundary conditions and
initial data
\begin{equation}
(\varphi,\psi,v,v_t)\big|_{t=0}=(h_1,h_2,q_1,q_2).\label{e2.30}
\end{equation}
Finally, let $(E,H,u,u_t)$ be the solution of  \eqref{e1.1},
\eqref{e2.26}, \eqref{e1.4}, \eqref{e2.28} with zero initial data
at $t=T>T_1$ where $R$ is chosen to be
\begin{equation}
-R(x,t)=\mu\psi+\alpha\frac{\partial v}{\partial \eta}\quad \text{on } \partial\Omega\times(0,T).\label{e2.31}
\end{equation}
We consider the map $M\colon \mathcal{F}\mapsto\mathcal{F}'$ given by
$$
M(h_1,h_2,q_1,q_2)=(E,H,u,u_t)\big|_{t=0}.
$$
Our objective is to show that $M$ is an isomorphism from $\mathcal{F}$
onto $\mathcal{F}'$. From to \eqref{e2.29} (with $t=0$) and \eqref{e2.31}
it follows
\begin{equation}
\begin{aligned}
& \langle M(h_1,h_2,q_1,q_2), (\tilde f_1,\tilde f_2,
 \tilde q_1,\tilde q_2)\rangle_K \\
&= \int_0^T \int_{\partial\Omega} \Big(\mu\psi+\alpha\frac{\partial v}{\partial\eta}\Big)
\cdot \Big( \alpha\frac{\partial\tilde u}{\partial\eta}+\mu\tilde H-\tilde p\eta\Big)
d\Gamma d\tau
\end{aligned} \label{e2.32}
\end{equation}
where $(\tilde E,\tilde H,\tilde u,\tilde u_t)$ is a solution  of
\eqref{e1.1}--\eqref{e1.6} with zero boundary conditions. Since we
know that $\psi\cdot\eta=0$ and $\frac{\partial v}{\partial\eta}\cdot\eta=0$
on $\partial\Omega\times(0,T) $ because \eqref{e2.1} and $v=0$ on
$\partial\Omega\times(0,T)$, then it follows that
$\left(\mu\psi+\alpha\frac{\partial v}{\partial\eta} \right) \cdot\tilde p\eta=0$ on
$\partial\Omega\times(0,T)$. Therefore from \eqref{e2.32} we deduce
\begin{equation}
\begin{aligned}
& \langle M(h_1,h_2,q_1,q_2), (\tilde f_1,\tilde f_2,\tilde q_1,
 \tilde q_2)\rangle_K \\
&= \int_0^T \int_{\partial\Omega} \Big(\mu\psi+\alpha\frac{\partial v}{\partial\eta}\Big)
 \cdot \Big( \alpha\frac{\partial\tilde u}{\partial\eta}+\mu\tilde H\Big) d\Gamma d\tau \\
&= \langle (h_1,h_2,q_1,q_2), (\tilde f_1,\tilde f_2,\tilde q_1,\tilde q_2)\rangle_{\mathcal{F}}\end{aligned} \label{e2.33}
\end{equation}
for any $(\tilde f_1,\tilde f_2,\tilde q_1,\tilde q_2)\in\mathcal{F}$.
Clearly \eqref{e2.33} implies that $M$ is an isomorphism from
$\mathcal{F}$ onto $\mathcal{F}'$. Now, we return to problems \eqref{e1.1},
\eqref{e1.2}, \eqref{e2.26} and \eqref{e1.4}, \eqref{e1.5},
\eqref{e2.28}. Let $(f_1,f_2,g_1,g_2)\in\mathcal{F}'$. We set
$$
(h_1,h_2,q_1,q_2)=M^{-1}(f_1,f_2,g_1,g_2), \quad
R=-\Big(\mu\psi+\alpha\frac{\partial v}{\partial \eta} \Big)
$$
where $(\varphi,\psi,v,v_t)$ is the solution of \eqref{e1.1},
\eqref{e1.6}  with zero boundary conditions and initial data at
$t=0$ as in \eqref{e2.30}. From \eqref{e2.29} with $t=T>T_1$ we
deduce
\begin{equation}
\begin{aligned}
& \langle(E(T),H(T),u(T),u_t(T)),
(\tilde E(T),\tilde H(T),\tilde u(T),\tilde u_t(T))\rangle_K  \\
& = \langle M(h_1,h_2,q_1,q_2),
(\tilde f_1,\tilde f_2,\tilde q_1,\tilde q_2)\rangle_K \\
&\quad - \langle(h_1,h_2,q_1,q_2),
(\tilde f_1,\tilde f_2,\tilde q_1,\tilde q_2)\rangle_{\mathcal{F}}.
\end{aligned}
\label{e2.34}
\end{equation}
Using \eqref{e2.33}, we conclude that the right hand side  of
\eqref{e2.34} is equal to zero. This means that $(E(T),H(T),u(T),
u_t(T))$ generates the zero functional on $\mathcal{F}$. Now, the
conclusion of Theorem \ref{thm2} is a consequence of the above discussion:
Construct $R(x,t)$ as in \eqref{e2.31} and let
$$
P(x,t)=\int_0^tR(x,s)ds+g_1(x)
$$
Obviously $P=u$ and $\eta\times E=\mu\eta\times(\eta\times P_t)$
by construction.
\end{proof}



\subsection*{Acknowledgements}
B. V. Kapitonov is partially supported by Research Grant
303981-03-2 from CNPq, Brazil.
G. Perla M. is partially supported by Research Grant 306282-2003-8
from CNPq, Brazil.


\begin{thebibliography}{00}

\bibitem{d1} G. Duvaut, J. L. Lions;
 Les Inequations en Mecanique et en Physique, Dunod, Paris, (1972).

\bibitem{e1} M. M. Eller, J. Masters;
Exact boundary controlability of electromagnetic fields
in a general region, Appl. Math. Optim. \textbf{45} (2002), 99--123.

\bibitem{e2} M. M. Eller;
Continuous Observability for the Anisotropic Maxwell System, Appl.
Math. Optim. \textbf{55} (2007), 185--201.

\bibitem{k1} B. V. Kapitonov;
 Simultaneous precise control of systems of linear elasticity,
Russian Acad. Sci. Dokl. Math. \textbf{48(2)} (1994),  332--337.

\bibitem{k2} B. V. Kapitonov;
Stabilization and exact boundary controllability for Maxwell's equations,
SIAM, J. Control Optim. \textbf{28} (1990), 2, 294--319.

\bibitem{k3} B. V. Kapitonov;
Stabilization and simultaneous boundary controllability for a pair
of Maxwell's equations, Computational and Applied Mathematics,
\textbf{(15(3)} (1996), 213--225.

\bibitem{k4} B. V. Kapitonov, M. A. Raupp;
 Simultaneous exact control of piezoelectric  systems in
multiplayered media, Comp. Appl. Math., Vol. \textbf{22(2)} (2003), 249--277.

\bibitem{k5} B. V. Kapitonov, G. Perla Menzala;
 Simultaneous exact control of quasi-electrostatic piezoelectric
systems in multiplayered media, Acta. Appl. Math.,
\textbf{94} (2006), 49--66.

\bibitem{l1} J. Lagnese;
Exact controllability of Maxwell's equations
in a general region, SIAM, J. Control and Optimization,
 Vol. \textbf{27(2)} (1989), 374--388.

\bibitem{l2} J. L. Lions;
Exact controllability, stabilization and
perturbations for distributed systems, SIAM Review,
 Vol. \textbf{30(1)} (1988), 1--68.

\bibitem{l3} J. L. Lions;
Controlabilite exacte des systems distribues, C. R. Acad. Sci. Paris,
Serie I, Mathematics, \textbf{302} (1986), 471--475.

\bibitem{l4} J. L. Lions;
On some hyperbolic equations with a pressure term,
Partial Differential Equations and related subjects:
Proceedings of the Conference dedicated do L. Nirenberg, (1992)
Pitman Res. Notes Math. Ser. \textbf{269}, Longman House Tech.,
 Harlow (1992).

\bibitem{n1} S. Nicaise, C. Pignotti; Internal stabilization of
Maxwell equations in heterogeneous media, Abstract and Applied
Analysis 2005:7 (2005), 791--811.

\bibitem{p1} K. Phung;
Controle et stabilisation d'ondes electromagnetiques,
ESAIM, Control, Optimization and Calculus of Variations,
\textbf{5} (200), 87--137.

\bibitem{r1} A. Rocha dos Santos;
Controlabilidade exacta para a equa\c cao de ondas com coeficientes
 variaveis e um termo de pressao (in Portuguese). Atas XVII CNMAC,
Vitoria, UFES (Brasil) (1994), 1--5.

\bibitem{r2} D. L. Russell;
Dirichlet-Neumann boundary control problem  associated with Maxwell
equations in a cylindrical region, SIAM, J. Math. Anal. \textbf{24} (1986),
199-229.

\bibitem{w1} N. Weck;
Exact boundary controllability of a Maxwell
problem, SIAM, J. Control Optim. \textbf{38(2)} (2000), 736--750.

\bibitem{z1} E. Zuazua;
Controllability and Observability of PDEs: some results and open
problems. Hanbook of Differential Equations:
Evolutionary Differential Equations, Vol. \textbf{3}, C. M. Dafermos
and E. Feireisl Eds (2006), elsevier Science, 52--621.

\end{thebibliography}

\end{document}