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

\AtBeginDocument{{\noindent\small
{\em Electronic Journal of Differential Equations},
Vol. 2007(2007), No. 139, pp. 1--11.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu  (login: ftp)}
\thanks{\copyright 2007 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2007/139\hfil Stabilization and exact controllability]
{Uniform stabilization and exact controllability for hyperbolic
systems with discontinuous coefficients}

\author[F. P. Q. G\'omez, B. V. Kapitonov \hfil EJDE-2007/139\hfilneg]
{F\'elix P. Quispe G\'omez, Boris V.  Kapitonov}  % in alphabetical order

\address{F\'elix P. Quispe G\'omez\newline
 Department of Mathematics, Federal University of Santa Catarina\\
 Rua Defino Conti, s/n, Trindade, 88040-900, Santa Catarina, Brazil}
\email{quispe@mtm.ufsc.br}

\address{Boris V.  Kapitonov \newline
Sobolev Institute of Mathematics\\
Russian Academy of Sciences, Russia}
\curraddr{National Laboratory for Scientific Computation - LNCC/MCT\\
Rua Getulio Vargas, 333, 25651-70, Rio de Janeiro, Brazil}
\email{boris@lncc.br}

\thanks{Submitted December 21, 2006. Published October 19, 2007.}
\subjclass[2000]{05C38, 15A15, 05A15, 15A18}
\keywords{Uniform stabilization; exact control; hyperbolic systems}

\begin{abstract}
 This paper considers a hyperbolic system with discontinuous coefficients
 in a bounded, open, connected set with smooth boundary and controlled
 through the Robin boundary condition.
 Uniform stabilization of the solutions are established.
 Exact boundary controllability is obtained through the Russell's
``Controllability via Stabilizability'' principle.
\end{abstract}

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

\section{Introduction}\label{sec:pre}

Let $\Omega$ be a bounded domain in $\mathbb{R}^{n}$
with a smooth boundary $S$ which consists of the disjoint closed
surfaces $S_0$  and $S_1$ (the case $S_1=\emptyset$ is not
excluded).
In the cylinder $\Omega\times ]0,T[$ we consider the mixed problem
\begin{gather}
\partial^2_t\mathbf{u}(\mathbf{x},t)-
\sum_{i=1}^{n}\partial_{x_{i}}\left[P(\mathbf{x})\partial_{x_{i}}
\mathbf{u}(\mathbf{x},t)\right]=0 \quad
(\mathbf{x},t)\in \Omega\times ]0,T[\label{elenise}\\
\mathbf{u}(\mathbf{x},0)=f_1(\mathbf{x}),\quad \partial_t
\mathbf{u}(\mathbf{x},0)=f_2(\mathbf{x})\quad \mathbf{x}\in
\Omega\label{elen-1} \\
P\partial_{\mathbf{\nu}}
\mathbf{u}(\mathbf{x},t)+a \mathbf{u}(\mathbf{x},t)
+b\partial_t\mathbf{u}(\mathbf{x},t)=0
\quad (\mathbf{x},t)\in \Sigma_0=S_0\times ]0,T[,\label{elen-2}\\
\mathbf{u}(\mathbf{x},t)=0 \quad (\mathbf{x},t) \in \Sigma_1
=S_1\times ]0,T[\label{elen-3}
\end{gather}
Here $\mathbf{u}=(u^{1}(\mathbf{x},t),\dots,u^{m}(\mathbf{x},t))$,
$\mathbf{x}=(x_1,\dots,x_n)$, $P(\mathbf{x})=P^\ast(\mathbf{x})$ are square
matrices of order $m$, $\mathbb{\nu} = (\nu_1,\dots,\nu_n)$ is the
unit outward normal to the  boundary $S$, and $a, b$ are positive
constants.

Assume that
$$
P(\mathbf{x})\mathbb{\xi}\cdot\mathbb{\xi}
 \geq c_{_{0}}|\mathbf{\xi}|^2,\quad c_{_{0}}>0
$$
where $\mathbb{\xi}=(\xi^1,\dots,\xi^m)$ is an arbitrary vector.

Assume that $\Omega_0\subset\Omega$ is a bounded domain with sufficiently
smooth boundary $\Gamma$. We set $\Omega_1=\Omega\setminus\overline{\Omega}_0$
and assume that the entries $a_{pq}(\mathbf{x})$ of the matrix
$P(\mathbf{x})$ lose continuity on the surface $\Gamma$.

We shall use the notation
\begin{alignat*}{1}
 P(\mathbf{x})=
   \begin{cases}
    A(\mathbf{x})& \text{if } \mathbf{x}\in \Omega_0,\\
    B(\mathbf{x})& \text{if }  \mathbf{x}\in \Omega_1.
   \end{cases}
&\quad \mathbf{u}(\mathbf{x},t)=
\begin{cases}
w(\mathbf{x},t)& \text{if } \mathbf{x}\in \Omega_0,\\
v(\mathbf{x},t)& \text{if } \mathbf{x}\in \Omega_1.
\end{cases}
\end{alignat*}

For simplicity we assume that $A$ and $B$ are constants matrices.
We add to \eqref{elenise}
the following interface conditions on $\Gamma$:
\begin{equation}
\label{bibiana}
w\big\vert_{\Sigma}=v\big\vert_{\Sigma},\quad A \partial_{\nu} w
\big\vert_{\Sigma}=B\partial_{\nu}v\big\vert_{\Sigma}\quad
\text{in } \Sigma=\Gamma\times ]0,T[
\end{equation}
where $\mathbf{\nu}$ is the unit outward (with respect to
$\Omega_0$) normal to the surface $\Gamma$.

In one space dimension it is well known that stabilization holds for
wave operators with piecewise smooth but possibly discontinuous
coefficients (BV is the right class) regardless of the sign of the
jump. Thus, one dimension is much better than several dimensions.
The proof of this is based on simple sidewise energy estimates.
See \cite{coxzua} and \cite{zua1} and the references therein.

Our purpose is to prove the uniform stabilization of solutions to
the problem \eqref{elenise}-\eqref{elen-3} and \eqref{bibiana}.
Using this result we obtain exact boundary controllability for the
corresponding evolution system. Several approaches are known to
solve the problem of exact boundary controllability. A systematic
method (named HUM) was proposed by Lions \cite{leo} and
\cite{leon1}.

In \cite{quiz1} we obtained exact controllability for the system
\eqref{elenise}-\eqref{elen-3} using HUM. The exact
controllability for a system in elasticity theory is established
by Lagnese, with method HUM in \cite{lag3}. We
obtain the same result for the class of systems $\partial^2_t u
-\partial_{ x_{i}}\left(A_{ij}\partial_{x_{i}} u  \right)=0$ which includes
the system in elasticy theory.

Here we use another approach which is based on D.~Russell's
``controllability via stabilizability'' principle \cite{ruse}, which
is  different from of  Lagnese's in \cite{lag3}.
Both techniques are well known.

There is an extensive number of publications on these topics. Exact
controllability and uniform energy decay (boundary damping) are obtained
for various equations and systems: the wave equation, the
Schr\"odinger equation, Euler-Bernoulli
beam equation, the system of elasticity, Maxwell's equation and
others \cite{che}, \cite{kapi}-\cite{ruse1}, \cite{zua}.
Although for equations with discontinuous coefficients
very few results are known: Maxwell's equations in multilayered
media \cite{kapi1}, Euler-Bernoulli
beam equation in the one-dimensional case \cite{che1}.

\section{Well-Posedness} \label{sec:well-pos}

Denote by $\mathcal{H}$ the Hilbert space of pairs
$\{\mathbf{u},\mathbf{u}_1\}$ of $m$-component vector-functions such that
$$
\mathbf{u}\in H^1(\Omega_k),\quad \mathbf{u}_1\in L^2(\Omega_k),\quad
 k=0,1,\quad \mathbf{u}\Big\vert_{S_1}=0.$$
The scalar product in $\mathcal{H}$  is defined by the formula
$$
\langle \{\mathbf{u},\mathbf{u}_1\}, \{f,f_1\}\rangle
=\int_{S_0}a\,\mathbf{u}\cdot f\,dS+\int_{\Omega}\Big(P\,
\partial_{x_i}\mathbf{u}\cdot\partial_{x_{i}}f+\mathbf{u}_1\cdot f_1\Big)dx.
$$
Define an unbounded operator $\mathcal{A}$ in $\mathcal{H}$ whose
domain $D(\mathcal{A})$ consists of the elements
$\{ \mathbf{u},\mathbf{u}_1\} \in ~\mathcal{H}$ such that
$\mathbf{u}\in H^2(\Omega_k)$, $\mathbf{u}_1\in H^1(\Omega_k)$, $ k=0,1$,
\begin{gather}
P\partial_{\mathbf{\nu}} \mathbf{u}(\mathbf{x},t)+
a\mathbf{u}+b\mathbf{u}_1\Big\vert_{S_0}=0,\quad
\mathbf{u}_1\Big\vert_{S_1}=0,\quad
\mathbf{u}\Big\vert_{S_1}=0\label{interf-1}\\
\mathbf{u}^0\Big\vert_{\Gamma}=\mathbf{u}^1\Big\vert_{\Gamma},\quad
\mathbf{u}_1^0\Big\vert_{\Gamma}=\mathbf{u}_1^1\Big\vert_{\Gamma},\quad
A\partial_{\mathbf{\nu}} \mathbf{u}^0\Big\vert_{\Gamma}= B\partial_{\nu}
\mathbf{u}^1\Big\vert_{\Gamma},\label{interf-2}
\end{gather}
where $\mathbf{u}^k,\;\mathbf{u}^k_1$ are the restrictions of the
functions $\mathbf{u}, \;\mathbf{u}_1$ on $\Omega_k$.
For $\{\mathbf{u},\mathbf{u}_1\}\in D(\mathcal{A})$ we set
$$
\mathcal{A}\{\mathbf{u},\mathbf{u}_1\}
=\{\mathbf{u}_1,\;\partial_{x_i}(P \partial_{x_i}\mathbf{u})\}.
$$
In a standard way we  construct the adjoint operator
$\mathcal{A}^{\ast}$. The domain of the operator
$\mathcal{A}^{\ast}$ consists of
elements
$\{ \mathbf{v},\mathbf{v}_1\} \in \mathcal{H}$ such that
$\mathbf{v}\in H^2(\Omega_k)$, $\mathbf{v}_1\in H^1(\Omega_k)$, $k=0,1$,
\begin{gather*}
P\partial_{\nu} \mathbf{v}(\mathbf{x},t)+a\mathbf{v}
-b\mathbf{v}_1\Big\vert_{S_0}=0,\quad
\mathbf{v}_1\Big\vert_{S_1}=0,\quad \mathbf{v}\Big\vert_{S_1}=0
\\
\mathbf{v}^0\Big\vert_{\Gamma}=\mathbf{v}^1\Big\vert_{\Gamma},\quad
\mathbf{v}_1^0\Big\vert_{\Gamma}=\mathbf{v}_1^1\Big\vert_{\Gamma}, \quad A\,
\partial_{\nu}\mathbf{v}^0\Big\vert_{\Gamma}= B\, \partial_{\nu}\mathbf{v}^1\Big\vert_{\Gamma},
\end{gather*}
where $\mathbf{v}^k,\;\mathbf{v}^k_1$ are the restrictions of the
functions $\mathbf{v}, \;\mathbf{v}_1$ on $\Omega_k$.

For $\{\mathbf{v},\mathbf{v}_1\}\in D(\mathcal{A}^{\ast})$ we set
$$
\mathcal{A}^{\ast}\{\mathbf{v},\mathbf{v}_1\}=
-\left\{\mathbf{v}_1,\;\partial_{x_i}\left(P\partial_{x_i} \mathbf{v}
\right)\right\}.
$$
It can be shown that $\mathcal{A}$ and $\mathcal{A}^{\ast}$ are
dissipative operators in $\mathcal{H}$; i.e.,
\begin{gather*}
\langle  \mathcal{A}\{\mathbf{u}, \mathbf{u}_1\},\{\mathbf{u}, \mathbf{u}_1\}
\rangle \leq 0 \quad \{\mathbf{u},
\mathbf{u}_1\} \in D(\mathcal{A})
\\
\langle  \mathcal{A}^{\ast}\{\mathbf{v}, \mathbf{v}_1\},\{\mathbf{v}, \mathbf{v}_1\} \rangle \leq 0 \quad \{\mathbf{v},
\mathbf{v}_1\} \in D(\mathcal{A}^{\ast}).
\end{gather*}
Assume that $\{\mathbf{u},\mathbf{u}_1\}\in D(\mathcal{A})$. Then
$$
\frac{d}{dt}\langle \mathcal{A}\{\mathbf{u},\mathbf{u}_1\},
\{\mathbf{u},\mathbf{u}_1\}\rangle
=-\int_{S_0}b|\mathbf{u}_1|^2d\,S\leq 0.
$$
Similarly,
$$
\frac{d}{dt}\langle \mathcal{A}^{\ast}\{\mathbf{v},\mathbf{v}_1\},
\{\mathbf{v},\mathbf{v}_1\}\rangle
=-\int_{S_0}b|\mathbf{v}_1|^2d\,S\leq 0,\quad \{\mathbf{v},
\mathbf{v}_1\}\in D(\mathcal{A}^{\ast}).
$$
Indeed, if $\{\mathbf{u},\mathbf{u}_1\}\in D(\mathcal{A})$, then
\begin{align*}
&\langle  \mathcal{A}\{\mathbf{u}, \mathbf{u}_1\},\{\mathbf{u},
\mathbf{u}_1\} \rangle\\
& = \int_{S_{0}}a\mathbf{u}_1\cdot\mathbf{u}\;dS+\int_{\Omega_{0}}
\left(A\frac{\partial \mathbf{u}_1^0}{\partial x_i}\cdot
\frac{\partial \mathbf{u}^0}{\partial x_i}
+  \frac{\partial }{\partial x_i}
\Big( A \frac{\partial \mathbf{u}^0}{\partial x_i}  \Big)\cdot
\mathbf{u}_1^0 \right)dx \\
&\quad +\int_{\Omega_{1}}\left(B\frac{\partial \mathbf{u}_1^1}{\partial x_i}
\cdot\frac{\partial \mathbf{u}^1}{\partial x_i}
+  \frac{\partial }{\partial x_i}\Big( B
\frac{\partial \mathbf{u}^1}{\partial x_i}  \Big)\cdot \mathbf{u}_1^1 \right)dx
\\
&= \int_{S_{0}}a\mathbf{u}_1\cdot\mathbf{u}\;dS+\int_{\Omega_{0}}
\left(A\frac{\partial \mathbf{u}_1^0}{\partial x_i}\cdot
\frac{\partial \mathbf{u}^0}{\partial x_i}
-\frac{\partial  \mathbf{u}^0}{\partial x_i}A
 \frac{\partial \mathbf{u}^0_1}{\partial x_i}  \right)dx
+\int_{\Gamma}A\frac{\partial \mathbf{u}^0}{\partial \nu}\mathbf{u}^0_1 dS
\\
&\quad +\int_{\Omega_{1}}\left(B\frac{\partial \mathbf{u}_1^1}{\partial x_i}
\cdot\frac{\partial \mathbf{u}^1}{\partial x_i}
- \frac{\partial \mathbf{u}^1}{\partial x_i}B
\frac{\partial \mathbf{u}^1_1}{\partial x_i}   \right)dx
-\int_{\Gamma}B\frac{\partial \mathbf{u}^1}{\partial \nu}\mathbf{u}^1_1 dS
+ \int_{S}B\frac{\partial \mathbf{u}^1}{\partial \nu}\mathbf{u}^1_1 dS
\\
&=\int_{\Gamma}\left(A\frac{\partial \mathbf{u}^0}{\partial \nu}
\mathbf{u}^0_1-B\frac{\partial \mathbf{u}^1}{\partial \nu}\mathbf{u}^1_1\right)
 dS + \int_{S_{0}}a\mathbf{u}_1\cdot\mathbf{u}\,dS
+\int_{S_{0}}B\frac{\partial \mathbf{u}^1}{\partial \nu}\mathbf{u}^1_1 dS
\\
&=\int_{S_{0}}a\mathbf{u}_1\cdot\mathbf{u}\;dS+\int_{S_{0}}P
 \frac{\partial \mathbf{u}}{\partial \nu}\cdot\mathbf{u}_1\;dS\\
& =\int_{S_{0}}[a\mathbf{u}_1\cdot\mathbf{u}+(-a\mathbf{u}
 -b\mathbf{u}_1)\mathbf{u}_1]\;dS=\int_{S_{0}}b|\mathbf{u}_1|^2\;dS\leq 0.
\end{align*}
It can be shown in a similar way that $\mathcal{A}^{\ast}$ is dissipative.

Thus, $\mathcal{A}$ generates a $C_0$-semigroup of contractions
$U(t)\colon \mathcal{H} \to \mathcal{H},\;t>0$ where
\begin{equation*}
U(t)\{f_1,f_2\}\in
C([0,\infty);D(\mathcal{A}))\cup~C^1([0,\infty);\mathcal{H})
\end{equation*}
when $\{f_1,f_2\}\in D(\mathcal{A})$
and  $U(t)\{f_1,f_2\}$ is strongly
differentiable with respect to $t$ for $\{f_1,f_2\}\in
D(\mathcal{A})$. Moreover,
$$
\frac{d}{dt}U(t)\{f_1,f_2\}=\mathcal{A} U(t)\{f_1,f_2\}
$$
and $U(t)$ carries $D(\mathcal{A})$ onto $D(\mathcal{A})$ and
commutes with $\mathcal{A}$.

Let  $\{f_1, f_2\} \in D(\mathcal{A})$ and
$\{\mathbf{u},\mathbf{u}_1\}=U(t)\{f_1,f_2\}$.
Then $\mathbf{u}=\mathbf{u}_1$ and
$\mathbf{u}_{1t}=\sum\partial_{x_i}(P\partial_{x_{i}}\mathbf{u})$; i.e,
the first component of $U(t)\{f_1,f_2\}$ is a solution to the
problem \eqref{elenise}, \eqref{bibiana}.

 Observe that, for $F=\{f_1,f_2\}\in \mathcal{H},\; U(t)F$ is a weak solution
in $\mathcal{H}$ to the abstract Cauchy problem
$$
\frac{d}{dt}\{\mathbf{u},\mathbf{u}_1\}=\left\{\mathbf{u}_1,\; \partial_{x_i}\left(P\partial_{x_i} \mathbf{u}\right)\right\}=
\left\{\mathbf{u}_1, \mathcal{P}\mathbf{u} \right\}
$$
in the following sense
$$
\int_0^T\Big(\langle U(t)F, \frac{d \phi}{dt} \rangle
+\langle U(t)F,\mathcal{A}^{\ast}\phi\rangle \Big)dt
=-\langle F,\phi(0)\rangle
$$
for every $\phi\in L^2(0,T;D(\mathcal{A}^{\ast}))$,
$\phi_{{}_t} \in  L^2(0,T;\mathcal{H})$, $\phi(T)=0$.

\section{Stabilization} \label{sec:stab}

We start from geometrical conditions on $\Omega$. We consider the problem:
\begin{equation} \label{andes}
\Delta \Psi=\frac{a_{_{0}}}{c_{_{0}}},\quad x\in \Omega, \quad
\partial_{\nu} \Psi\big\vert_{S_0}=\frac{a_{_{0}}\mathop{\rm meas}
(\Omega)}{c_{_{0}}
\mathop{\rm meas}(S_0)},
\quad \partial_ {\nu}\Psi\big\vert_{S_1}=0,
\end{equation}
where $\Psi(x)\in C^2(\Omega)\cup C^1(\overline{\Omega})$ be a solution to
the problem, $a_{_{0}}=\max |a_{pq}|$, $a_{pq}$
are the entries of the matrix $P$, and  $c_{_{0}}$ is a constant defined
as above (observe that for the wave operator $P=I$ and $c_{_{0}}=a_{_{0}}=1$).

For an arbitrary bounded domain $\Omega$ with smooth boundary $S$ we define
the  quantity
$$
\kappa=\max_{i,j}\sup_{x\in \overline{\Omega}}|\partial_{x_{i}x_{j}}^2\Psi(x)|.
$$
Suppose that $\Omega$ satisfies the conditions: There is a point
$x^{0}\in \mathbb{R}^n$ such that
\begin{itemize}
\item[(a)] $S_1$ is star-like with respect to $x^{0}$: $(x-x^{0},\nu)\leq 0$ for $x\in S_1$;
\item[(b)] for some $0<\epsilon\leq 1$
$$
(x-x^{0},\nu)>-\frac{1}{\epsilon+n{\kappa}}
\frac{\mathop{\rm meas}(\Omega)}{\mathop{\rm meas}(S)},\quad x\in S_0.
$$
\end{itemize}
Clearly, (b) holds if $S_0$ is star-like with respect to the
point $x^0$ when it be taken sufficiently close to the domain, see
Figure \ref{unis}.

\begin{figure}[htb]
\begin{center}
\includegraphics[width=0.7\textwidth]{fig1}
\end{center}
  \caption{Surface $\mathbf{S}_0$ is starlike with respect to the point $\mathbf{x}^o$}\label{unis}
\end{figure}

\begin{theorem}\label{vic}
Let a domain $\Omega$ and surface $\Gamma$ satisfy the above-listed
conditions with a parameter $0<\epsilon\leq 1$
and let the coefficient  $a$  in the boundary conditions satisfy
$0<a<\delta c_0 n^2 \kappa/(3r)$,
where $r={\sup_{x\in\, \overline{\Omega}}|\nabla \varphi|}$.
Suppose that $AB=BA$ and matrix $A-B$ is nonnegative.
Then there are $T^{\ast}>0$ and $C^{\ast}>0$ such that for $t>T^{\ast}$
$$
\| U(t)\{f_1,f_2\}\|^2\leq C^{\ast}(T^{\ast})^{\epsilon-1}
\frac{1}{t^{\epsilon}}\|\{f_1,f_2\}\|^2
$$
for every $\{f_1,f_2\}\in \mathcal{H}$.
\end{theorem}

\begin{proof}
The following identity is proved in the Appendix:
\begin{equation} \label{genal}
\begin{aligned}
&2\Big[t\partial_tu+(\nabla\varphi, \nabla)u+\frac{n-1}{2}u\Big]\cdot
\Bigl[\partial^2_{t}u-\partial_{x_i}\left(P\partial_{x_i} u \right)\Bigr]\\
&=\partial_ t\Big[t\Big(|\partial_tu|^2
+\sum_{i=1}^{n}P\partial_{x_i} u\cdot\partial_{x_i} u  \Big)
+2(\nabla\varphi, \nabla)u\cdot \partial_t u+(n-1)u\cdot \partial_t u\Big]\\
&\quad -\partial_{x_i}\Big[P\partial_{x_i} u\cdot\Bigl(2t
 \partial_tu+2(\nabla\varphi, \nabla)u+(n-1)u\Bigr)\\
&\quad + \partial_{x_i} \varphi\Big(|\partial_tu|^2
  -\sum_{i=1}^{n}P\partial_{x_i} u\cdot\partial_{x_i} u\Big)\Big]\\
&\quad - \left[(\Delta \varphi-n+2)P\partial_{x_i}u\cdot\partial_{x_i} u
 -(\Delta\varphi-n)|\partial_tu|^2
-2\partial_{x_px_i}^2\varphi\, \partial_{x_p} u\cdot P\partial_{x_{i}} u\right].
\end{aligned}
\end{equation}
For $\varphi=2^{-1}|x-x^{o}|$, it represents a conservation law, a
consequence of invariance of the system relative to the one-parameter
group of dilations in all variables with the infinitesimal operator
$$
t\partial_{t}+(x_i-x^o_i)\partial_{x_i}-\frac{n-1}{2}u^j\partial_{ u^j}.
$$
Let $\{f_1,f_2\}\in D(\mathcal{A})$ and $\mathbf{u}(\mathbf{x},t)$
be a solution of
\eqref{elenise}, \eqref{bibiana}. After integration by parts over
$\Omega_0\times ]0,T[$ and $\Omega_1\times ]0,T[$ using \eqref{bibiana},
we obtain the formula
\begin{equation}\label{lucimar}
\begin{aligned}
&-\int_{\Omega}u\partial_t u\,dx\Big\vert_{t=T_{0}}^{t=T}
-\int_{S_{0}}\frac{1}{2}b|u|^2\,dS\Big\vert_{t=T_{0}}^{t=T}\\
&=\int_{T_{0}}^{T}\int_{\Omega}\Big(\sum_{i=1}^{n}P\partial_{x_i}u\cdot
\partial_{ x_i}u-|\partial_tu|^2\Big)dx\,dt
+\int_{T_{0}}^{T}\int_{S_0}a|u|^2\,dS dt.
\end{aligned}
\end{equation}
An application of \eqref{genal}, together with \eqref{lucimar} multiplied
by the constant $\gamma$, leads to the formula
\begin{equation}\label{cordi}
\begin{aligned}
&\Big\{\int_{\Omega}\left[ t\mathcal{I}(u)+2(\nabla\varphi,
 \nabla)u\cdot \partial_tu+(n-1-\gamma)u\cdot \partial_tu\right]dx\\
&+ \int_{S_0}\Big[ta|u|^2+\frac{n-1-\gamma}{2}b|u|^2\Big]dS\Big\}
\Big\vert_{t=T_{0}}^{t=T}\\
&=\int_{T_{0}}^{T}\int_{\Omega}\Big[(\Delta\varphi-n+2+\gamma)\Phi(u)
 -(\Delta\varphi-n+\gamma)|\partial_tu|^2\\
&\quad -2\partial^2_{x_px_i}\varphi P \partial_{x_i}u \partial_{x_p} u\Big]dxdt
 + \int_{T_{0}}^{T}\int_{S_1}\partial_{\nu}\varphi\;\Phi(u)\,dS\,dt
\\
&\quad +\int_{T_{0}}^{T}\int_{S_0}\Big\{\partial_{\nu}\varphi
 \left(|\partial_tu|^2-\Phi(u)\right)
 -2t b|\partial_tu|^2-(n-2-\gamma)a|u|^2\\
&\quad -2b(\nabla\varphi, \nabla)u\cdot \partial_tu-2a(\nabla\varphi,
  \nabla)u\cdot u\Big\}dS\, dt\\
&\quad +\int_{T_{0}}^{T}\int_{\Gamma}\Big\{-\partial_{\nu}\varphi
 \left(A-B\right)\partial_{x_i} w\cdot\partial_{x_i}w
-\partial_{\nu} \varphi(AB^{-1}A\\
&\quad +B-2A)\partial_{\nu} w\cdot \partial_{\nu} w
\Big\}d\Gamma\,dt,
\end{aligned}
\end{equation}
here we use the notation:
$$
\Phi(u)=\sum_{i=1}^{n}P\partial_{x_i}u \cdot\partial_{x_i}u,\quad
\mathcal{I}(u)=|\partial_t u|^2+\Phi(u).
$$
Choose the function $\varphi(x)$ in \eqref{cordi} as follows
$$
\varphi(x)=\frac{c_0}{a_0}\Psi(x)+\frac{1}{2\theta}|x-x^0|^2,\quad
\theta>0, \quad x^{0} \in \mathbb{R}^n
$$
We obtain
\begin{align*}
\mathcal{K}&\equiv (\Delta\varphi-n+2+\gamma)\Phi(u)
 -(\Delta\varphi-n+\gamma)|\partial_t u|^2-2\partial^2_{x_{p}x_{i}}\varphi
P\partial_{x_{i}} u\cdot \partial_{x_{p}} u\\
&\leq \left(n-1-\frac{n}{\theta}-\gamma\right)|\partial_tu|^2
+\Big(3+\frac{n-2}{\theta}+2\kappa n+\gamma-n\Big)\Phi(u).
\end{align*}
Set $\theta=(\epsilon+n\kappa)^{-1}$ and
$\gamma=n-2+\epsilon-n(\epsilon+\kappa n)$. Then
$\mathcal{K}\leq (1-\epsilon)\mathcal{I}(u)$,
and for $x\in S_{0}$ we have
$$
\partial_{\nu}\,\varphi=(\epsilon+n\kappa)\left[(x-x^0,\nu)+
\frac{1}{(\epsilon+n\kappa)}\,\frac{\mathop{\rm meas}(\Omega)}
{\mathop{\rm meas}(S_0)}\right]>0,
$$
which by compactness of $S_0$ leads to the inequality
$$
\partial_{\nu}\,\varphi\geq |\nabla \varphi|\delta,\quad \delta>0.
$$
We now assume that the surface $\Gamma$ satisfies the condition
$$
\partial_{\nu}\;\varphi\big\vert_{\Gamma}\geq 0.
$$
Note that  if $S_0$ is strictly star-shaped with respect to $x^0\in
\mathbb{R}^n$; i.e,
$$
(x-x^0,\, \nu)>0
$$
we can choose
${\varphi(\mathbf{x})=\frac{1}{2}|\mathbf{x}-\mathbf{x}^0|^2}$.
In this case,  $\Gamma$ is
an arbitrary star-shaped surface with respect to $\mathbf{x}^0$.

Moreover, we assume that matrices $A$ and $B$ are constant and
\[
  AB=BA,\quad (A-B)\mathbb{\xi}\cdot\mathbb{\xi}\geq 0, \quad
  \forall \mathbb{\xi} \in \mathbb{R}^m,
\]
for examples on the last condition, see \cite{lag3} and
\cite{leon1}. For examples where monotonicity fails, so that the
uniform decay does not hold, see \cite{tay1}.

Then we obtain that integral over $\Gamma\times ]T_0,T[$ in \eqref{cordi}
is non positive. Denote by $\mathcal{G}$ the integrand of the integral over
$S_0\times ]T_0,T[$ on the right side of the formula \eqref{cordi}.
We have the following estimate for $\mathcal{G}$:
\begin{equation}
\label{vera} \mathcal{G} \leq |u|^2\left[n^2\kappa a-|\nabla
\varphi|\frac{3a^2}{\delta c_0}\right]-|u_t|^2\left[2tb-|\nabla
\varphi| -|\nabla \varphi|\frac{3b^2}{\delta c_0}\right].
\end{equation}
By hypotheses we have
\begin{equation} \label{marilda}
0<a<\frac{\delta c_{_{0}} n^2 \kappa}{3r},
\end{equation}
where ${r=\sup_{x\in\, \overline{\Omega}}|\nabla \varphi|}$.

Choose $T_1$ so large that for $t\geq T_1$ the last term in \eqref{vera}
is non positive. Since $(\nabla \varphi, \nu)\leq 0$ for $x\in S_1$,
the surface integrals on the right-hand side of  \eqref{cordi} are
nonnegative as $T_0\geq T_1$. Thus, we obtain that for $T_0\geq T_1$:
\begin{equation}\label{dani}
\begin{aligned}
&\Big\{\int_{\Omega}\Bigl[t\mathcal{I}(u)+2(\nabla\varphi,
 \nabla)u\cdot\partial_t u+ (n-1-\gamma)u\cdot \partial_tu\Bigr]dx\\
&+\int_{S_0}\left[ta|u|^2+\frac{n-1-\gamma}{2}b|u|^2\right]dS\,
\Big\}\Big\vert_{t=T_{0}}^{t=T}\\
&\leq (1-\epsilon)\int_{T_{0}}^{T}\int_{\Omega}\mathcal{I}(u)\,dxdt.
\end{aligned}
\end{equation}
here $\gamma=n+\epsilon-2-n(\epsilon+\kappa n)$. Denote by
$\tau_{{_{0}}}$ the smallest constant for which the following
inequality holds
$$
\int_{\Omega}\left(|u|^2+|\nabla u|^2\right)dx
\leq \tau_{0}\Big(\int_{\Omega}P\partial_{x_{i}} u\cdot \partial_{x_{i}}u\,dx+
\int_{S_0}a|u|^2\,dS \Big),\quad u\in H^1(\Omega).
$$
We have
$$
\int_{\Omega}\bigl[2(\nabla\varphi, \nabla)u\cdot \partial_t u+(n-1-\gamma)u\cdot \partial_t u\bigr]dx\leq C_0
\| U(T_0)\{f_1,f_2\}\|^2,\quad t\geq T_0
$$
Combining this estimate with \eqref{dani}, we arrive at the inequality
\begin{equation} \label{claudia}
T\| U(T)\{f_1,f_2\}\|^2-(T_0+C_1)\| U(T_0)\{f_1,f_2\}\|^2\leq
(1-\epsilon)\int_{T_0}^{T}\| U(t)\{f_1,f_2\}\|^2dt
\end{equation}
in which $T_0\geq T_1$.
With the help of Gronwall's inequality, and \eqref{claudia} we obtain
$$
t\| U(t)\{f_1,f_2\}\|^2\leq C_2\big(\frac{t}{T_2}\big)^{1-\epsilon}
\| U(T_2)\{f_1,f_2\}\|^2
$$
for $t>T_2$.

Given an arbitrary element $\{f_1,f_2\}\in \mathcal{H}$, approximate
it by smooth data for which the inequality of the
theorem was established above. Taking  the limit finishes the proof.
\end{proof}

\begin{corollary}
The operator $U(t)$ takes $\mathcal{H}$ into itself and
$$
\| U(t)\| <1  \quad \text{for } t>t^{\ast}
=(C^{\ast}(T^{\ast})^{\epsilon-1})^{1/\epsilon}.
$$
\end{corollary}

By applying semigroup properties, we obtain the following result.

\begin{corollary} \label{coro3.3}
Assume $\{f_1,f_2\}\in \mathcal{H}$. There are $C,\;\beta >0$ such that
$$\| U(t)\{f_1,f_2\}\|^2 \leq C\exp(-\beta t)\| \{f_1,f_2\}\|^2.$$
\end{corollary}

\section{Exact Controllability}

In this section, we shall use the estimate of the Theorem \ref{vic} to
prove exact controllability of the
evolution system studied in the previous sections.
In $\Omega\times ]0,T[$ we consider the  problem
\begin{gather}\label{sandra}
\partial^2_{t} \mathbf{u}(\mathbf{x},t)-\sum_{i=1}^{n}\partial_{
x_{i}}\left[P(\mathbf{x})\partial_{x_{i}} \mathbf{u}(\mathbf{x},t)\right]=0
 \quad (\mathbf{x},t) \in \Omega\times ]0,T[
\\
\mathbf{u}(\mathbf{x},0)=f_1(\mathbf{x}),\quad
\partial_{t} \mathbf{u}(\mathbf{x},0)=f_2(\mathbf{x})\quad
 \mathbf{x} \in \Omega \\
P\partial_{\mathbf{\nu}} \mathbf{u}(\mathbf{x},t)
+a\mathbf{u}(\mathbf{x},t)=\mathbb{q}(\mathbf{x},t) \quad
(\mathbf{x},t)\in \Sigma_0=S_0\times ]0,T[,\\
\mathbf{u}(\mathbf{x},t)=0 \quad (\mathbf{x},t)\in \Sigma_1=S_1\times ]0,T[\\
\mathbf{w}=\mathbf{v},\quad A\partial_{\mathbf{\nu}} \mathbf{w}
=B\partial_{\mathbf{\nu}}\mathbf{v},\quad
(\mathbf{x},t)\in \Sigma=\Gamma\times ]0,T[
\end{gather}
where $A (B)$ and $\mathbf{w} (\mathbf{v})$ are the restrictions of
matrix $P$ and vector-function $\mathbf{u}$ on
$\Omega_0\; (\Omega_1), \mathbf{f}=\{f_1,f_2\}$ is an arbitrary element
of the space $\mathcal{H}$.

For every $\mathbf{g}=\{g_1,g_2\}\in \mathcal{H}$, we have to find
a vector-function $\mathbf{q}(x,t)$ such that the
solution of \eqref{sandra} satisfies the conditions
$$
\mathbf{u}\Big\vert_{t=T}=g_1(x),\quad
\partial_t \mathbf{u} \Big\vert_{t=T}=g_2(x), \quad\text{for }T>t^{\ast}.
$$

\begin{theorem} \label{thm4.1}
Let the coefficient $a$ in the boundary conditions of problem \eqref{sandra}
satisfies \eqref{marilda}. There is a
$t^{\ast}>0$ such that, for $T>t^{\ast}$, arbitrary initial data
$\mathbf{f}=\{f_1,f_2\}\in \mathcal{H}$, and any
element $\mathbf{g}=\{g_1,g_2\}\in \mathcal{H}$, there exists a
boundary control $\mathbf{q}(x,t)\in L^2(S_0\times]0,T[)$
transferring a solution of \eqref{sandra} to the state
$ \mathbf{g}=\{g_1,g_2\}$ at time $T$. Moreover,
$$
\| \mathbb{q} \|^2_{L^2(\Gamma_0\times]0,T[)}\leq C(\| \mathbb{f}\|^2
+\| \mathbb{g}\|^2).
$$
\end{theorem}

\begin{proof}
Let $U(t)$ be the semigroup defined above and let $U^{\ast}(t)$ be
semigroup constructed from the operator $\mathcal{A}^{\ast}$.
Consider the following equation in $\mathcal{H}$:
$$
\{h,h_1\}-U^{\ast}(T)U(T)\{h,h_1\}=f-U^{\ast}(T)g.
$$
The operator $G(T)=U^{\ast}(T)U(T)$ takes $\mathcal{H}$ into
itself and $\| G(T)\| <1$ for $T>t^{\ast}$. Thus we can
solve this equation for any $\mathbf{f},\ \mathbf{g}\in \mathcal{H}$ and
$$
\| \mathbf{h}\| = \| \{h,h_1\}\| \leq C(\| \mathbf{f}\| +\| \mathbf{g}\|).
$$
Consequently, if we choose
$\mathbf{h}=(I-G(T))^{-1}(\mathbf{f}-U^{\ast}(T)\mathbf{g})$, then
\begin{equation*}
\{\alpha,\alpha_1\}=U(t)\mathbf{h}\quad{and}\quad
\{\beta,\beta_1\}=U^{\ast}(T-t)(U(T)\mathbf{h}-\mathbf{g})
\end{equation*}
are weak solutions to the problems
\begin{gather*}
  {\frac{d}{dt}}\{\alpha, \alpha_1\}=\{\alpha_1, \mathcal{P} \alpha\}\\
  P\partial_{\nu}\alpha+a \alpha +b \alpha_1\Big\vert_{S_0}=0, \quad
 \alpha\Big\vert_{S_1}=0,
\end{gather*}
and
\begin{gather*}
 {\frac{d}{dt}}\{\beta, \beta_1\}=\{\beta_1, \mathcal{ P}\beta\} \\
P\partial_{\nu} \beta+a\;\beta -b\;\beta_1\Big\vert_{S_0}=0,
\quad \beta\Big\vert_{S_1}=0\,.
\end{gather*}
By the energy identity, the following estimates hold
\[
\int_{0}^{T}\int_{S_0}b|\alpha_1|^2\,dS\, dt\leq C\| \mathbf{h}\|^2,
\quad \int_{0}^{T}\int_{S_0}b|\beta_1|^2dS\, dt\leq C(\| \mathbf{
h}\|^2+\| \mathbf{g}\|^2).
\]
Clearly,  $\{u,u_1\}=\{\alpha,\alpha_1\}-\{\beta,\beta_1\}$ is a
solution to problem \eqref{sandra} with boundary data on $S_0$:
$$
\mathbf{q}(x,t)=-b(\alpha_1+\beta_1),
$$
which belongs to $L^2(S_0\times]0,T[)$ and
$$
\| \mathbf{q} \|^2_{L^2(S_0\times(0,T))}
 \leq C(\| \mathbf{f}\|^2+\| \mathbf{g}\|^2).
$$
\end{proof}

\begin{remark} \label{rmk4.1}
We can study in the same way the more general case. Assume that
$B_k\subset \Omega$ is a bounded domain with boundary
$\Gamma_k, \overline{B}_k\subset B_{k+1}$ for $k=1,\dots,n$.

Assume that $\Gamma_1,\dots,\Gamma_n$ and $ S_0, S_1$ are star-shaped
with respect to the point $x^{0}\in \mathbb{R}^n$. Suppose that
matrix $P(x)$ lose the continuity on $\Gamma_1,\dots,\Gamma_n$. We set
$$
\Omega_0=B_1,\quad \Omega_k=B_{k+1}\setminus \overline{B}_k,\quad
k=1,\dots,n-1,\quad \Omega_n=\Omega\setminus\overline{B}_n
$$
The interface conditions are
\begin{gather*}
\mathbf{u}^{k-1}\Big\vert_{\Gamma_k \times]0,T[}
=\mathbf{u}^{k}\Big\vert_{\Gamma_k\times]0,T[}\\
P^{k-1}\partial_{\nu}\mathbf{u}^{k-1}\Big\vert_{\Gamma_k
\times]0,T[}=P^k\partial_{\nu}\mathbf{u}^{k}\Big\vert_{\Gamma_k
\times]0,T[},\quad k=1,\dots,n
\end{gather*}
where $\mathbb{\nu}=\mathbb{\nu}(\mathbf{x})$ (for $x\in \Gamma_k$)
is the unit normal vector pointing into the exterior of
$B_k$; $P^k,\;\mathbf{u}^k$ are the restrictions of  $P$ and
$\mathbf{u}$ on $\Omega_k$.
\end{remark}

\section{Appendix}

We shall show here the details in the proof of the identity used in
Theorem \ref{vic}.
We use the following notation
\begin{gather*}
\mathbf{u} =\left( \mathbf{u}^1,\dots,\mathbf{u}^m\right),\quad
\partial_t\mathbf{u}=\left(\partial_t \mathbf{u}^1,\dots,
\partial_t\mathbf{u}^m\right),\quad
\mathbf{\nabla}=\left(\partial_{x_i},\dots,\partial_{x_i}\right),\\
\partial^2_t\mathbf{u}=\left(\partial^2_t\mathbf{u}^1,\dots,
\partial^2_t\mathbf{u}^m\right),\quad
\partial_{x_{i}}\mathbf{u} =\left(\partial_{x_i}\mathbf{u}^1,\dots,
\partial_{x_i}\mathbf{u}^m\right),
\end{gather*}
the matrix $P(\mathbf{x})=\big(a_{_{pq}}(\mathbf{x})\big)_{_{m \times m}}$
and
$$
\left(\nabla \varphi,\nabla\right)\mathbf{u}
=\Big(\sum^n_{i=1}\partial_{x_i} \varphi\,
\partial_{x_i} \mathbf{u}^q\Big)_{1\leq q\leq m}
$$
The identity \eqref{genal} can be verified by direct computations as follows
\begin{align*}
&2\left(tu^q_t+\frac{\partial \varphi}{\partial x_i}\frac{\partial u^q}{\partial
x_i}+\frac{n-1}{2}u^q\right)\left(u^q_{tt}-\frac{\partial}{\partial
x_i}\Big(a_{pq}\frac{\partial u^q}{\partial x_i}\Big)\right)
\\
&=2tu^q_{tt}u^q_t-2tu^p_t\frac{\partial}{\partial
x_i}\Big(a_{pq}\frac{\partial u^q}{\partial x_i}\Big)+
2\frac{\partial \varphi}{\partial x_i}\frac{\partial u^q}{\partial
x_i}u^q_{tt}-2\frac{\partial \varphi}{\partial x_i}\frac{\partial
u^p}{\partial x_i}\frac{\partial}{\partial
x_i}\Big(a_{pq}\frac{\partial u^q}{\partial x_i}\Big)
\\
&\quad +(n-1) u^qu^q_{tt}-(n-1)u^p\frac{\partial}{\partial x_i}
\Big(a_{pq}\frac{\partial u^q} {\partial x_i}\Big)
\\
&=t\frac{\partial |u^q_t|^2}{\partial t}-2t\frac{\partial}{\partial x_i}
 \left(a_{pq}u^p_t \frac{\partial u^q}{\partial x_i}\right)
 +2ta_{pq}\frac{\partial u^p_t}{\partial x_i}
 \frac{\partial u^q}{\partial x_i}
 +2\frac{\partial \varphi}{\partial x_i}\frac{\partial u^q}
 {\partial x_i}u^q_{tt}\\
&\quad -\frac{\partial}{\partial x_i}\left(2\frac{\partial \varphi}{\partial x_i}a_{pq}
 \frac{\partial u^p}{\partial x_i}\frac{\partial u^q}{\partial x_i}\right)
 +\frac{\partial}{\partial x_i}\left(2 \frac{\partial \varphi}
 {\partial x_i}\frac{\partial u^p}{\partial x_i}\right)a_{pq}
 \frac{\partial u^q}{\partial x_i}\\
&\quad +(n-1)\frac{\partial }{\partial t}\left(u^qu^q_t \right)
 -(n-1)|u^q_t|^2  -(n-1)\frac{\partial}{\partial x_i}
 \left(a_{pq}u^p\frac{\partial u^q}{\partial x_i}\right) \\
&\quad +(n-1)a_{pq}\frac{\partial u^p}{\partial x_i}
  \frac{\partial u^q}{\partial x_i}
\\
&=t\frac{\partial |u^q_t|^2}{\partial t}-2t\frac{\partial}{\partial x_i}
 \left(a_{pq}u^p_t \frac{\partial u^q}{\partial x_i}\right)
 + 2ta_{pq}\frac{\partial u^p_t}{\partial x_i}\frac{\partial
 u^q}{\partial x_i} +\frac{\partial}{\partial t}
 \left(2\frac{\partial \varphi}{\partial x_i}\frac{\partial
 u^q}{\partial x_i}u^q_{t}\right)\\
&\quad -\frac{\partial}{\partial t}\left(2\frac{\partial
 \varphi}{\partial x_i}\frac{\partial u^q}{\partial x_i}\right) u^q_{t}
 -\frac{\partial}{\partial x_i}\left( 2a_{pq}\frac{\partial\varphi}{\partial x_i}
\frac{\partial u^p}{\partial x_i}\frac{\partial u^q}{\partial x_i}\right)+\frac{\partial}{\partial
x_i}\left(2\frac{\partial \varphi}{\partial x_i}\frac{\partial
u^p}{\partial x_i}\right)a_{pq}\frac{\partial u^q}{\partial
x_i}
\\
&\quad +(n-1)\frac{\partial }{\partial t}\Big(u^qu^q_t\Big)
 -(n-1)|u^q_t|^2-(n-1)\frac{\partial }{\partial
 x_i}\left(a_{pq}u^p\frac{\partial u^q}{\partial x_i}\right)\\
&\quad +(n-1)a_{pq}\frac{\partial u^p}{\partial
  x_i}\frac{\partial u^q}{\partial x_i}
\\
&=t\frac{\partial |u^q_t|^2}{\partial t}+t\frac{\partial}{\partial t}
 \left(a_{pq}\frac{\partial u^p}{\partial x_i}
 \frac{\partial u^q}{\partial x_i}\right)+|u_t^q|^2+
a_{pq}\frac{\partial u^p}{\partial x_i}\frac{\partial u^q}{\partial
x_i}+\frac{\partial}{\partial t}\left(2\frac{\partial
\varphi}{\partial x_i}
\frac{\partial u^q}{\partial x_i}u_t^q  \right)\\
&\quad +\frac{\partial}{\partial t} \Big((n-1)u^qu^q_t \Big)
-2t\frac{\partial}{\partial x_i}\left(a_{pq}\frac{\partial u^q}{\partial x_i}
 u^p_t \right)
-\frac{\partial}{\partial x_i}\left(2a_{pq}\frac{\partial \varphi}{\partial x_i}
\frac{\partial u^p}{\partial x_i} \frac{\partial u^q}{\partial x_i}  \right)
\\
&\quad +\frac{\partial}{\partial  x_i}\left(2 \frac{\partial
\varphi}{\partial x_i}\frac{\partial u^p}{\partial x_i}\right) a_{pq}
\frac{\partial u^q}{\partial x_i}
-2\frac{\partial}{\partial t}\left( \frac{\partial \varphi}{\partial x_i}
\frac{\partial u^q}{\partial x_i} \right)u_t^q
-(n-1)\frac{\partial}
{\partial x_i}\left(a_{pq}u^p\frac{\partial u^q}{\partial x_i}\right)
\\
&\quad +(n-2)a_{pq}\frac{\partial u^p}{\partial
x_i}\frac{\partial u^q}{\partial x_i}-n|u^q_t|^2
\\
&=\frac{\partial}{\partial t}\left[t\Big(|u^q_t|^2+a_{pq}
\frac{\partial u^p}{\partial x_i}\frac{\partial u^q}{\partial x_i}\Big)+
2\frac{\partial \varphi}{\partial x_i}\frac{\partial u^q}{\partial
x_i}u^q_t+(n-1)u^qu^q_t \right]
\\
&\quad -\frac{\partial}{\partial
x_i}\Big[a_{pq}\frac{\partial u^p}{\partial
x_i}\left(2tu^q_t+2\frac{\partial \varphi}{\partial
x_i}\frac{\partial u^q}{\partial x_i}+(n-1)u^q\right)
\\
&\quad +\frac{\partial \varphi}{\partial
x_i}\Big(|u^q|^2-a_{pq}\frac{\partial u^p}{\partial
x_i}\frac{\partial u^q}{\partial x_i}\Big)\Big]
+\frac{\partial^2 \varphi}{\partial x_i}a_{pq}\frac{\partial
u^p}{\partial x_i}\frac{\partial u^q}{\partial x_i}
\\
&\quad +(n-2)a_{pq}\frac{\partial u^p}{\partial x_i}\frac{\partial
u^q}{\partial x_i}-n|u^q_t|^2+\frac{\partial^2 \varphi}{\partial
x_i^2}+2\frac{\partial \varphi}{\partial x_i}\frac{\partial^2
u^p}{\partial x^2_i}a_{pq}\frac{\partial u^q}{\partial x_i}.
\end{align*}
The computation is now complete.

To obtain \eqref{cordi} it's enough to add to the above identity for equation
\eqref{lucimar} after multiplication by a parameter fixed $\gamma$, and finally
apply Green's formula  and interface conditions
\eqref{interf-1} and \eqref{interf-2}.

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\end{document}