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\markboth{\hfil Uniform stability  \hfil EJDE--2001/25}
{EJDE--2001/25\hfil A. Soufyane \hfil}
\begin{document}

\title{\vspace{-1in}
\parbox{\linewidth}{\footnotesize\noindent
{\sc  Electronic Journal of Differential Equations},
Vol. {\bf 2001}(2001), No. 25, pp. 1--10. \newline
ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu
\newline ftp  ejde.math.swt.edu  (login: ftp)} \vspace{\bigskipamount} 
\\
Uniform stability of displacement coupled second-order equations
% 
\thanks{\emph{Mathematics Subject Classifications:} 
34K35, 35B37, 37N35, 93B52, 93B05, 93D15. \hfil\break\indent 
\emph{Key words:} 
uniform stability, exact controllability, velocity coupled dissipator, 
\hfil\break\indent  displacement coupled dissipator. 
\hfil\break\indent \copyright 2001
Southwest Texas State University. \hfil\break\indent 
Submitted November 28, 2000. Published April 17, 2001.} }
\date{}
\author{ A. Soufyane }
\maketitle

\begin{abstract}
We prove that the uniform stability of semigroups associated to displacement
coupled dissipator systems is equivalent to the uniform stability of
velocity coupled dissipator systems. Using this equivalence, we give
sufficient conditions for obtaining uniform stability and exact
controllability of displacement coupled dissipator systems.
\end{abstract}

\newtheorem{theorem}{Theorem} 
\newtheorem{proposition}[theorem]{Proposition} 
\newtheorem{remark}[theorem]{Remark}

\section{Introduction}

We consider a linear oscillator in a Hilbert space $H$ represented by 
\begin{equation}
\partial _{tt}u(t)+\mathcal{A}u(t)=h,  \label{1}
\end{equation}
where $\mathcal{A}$ is a (generally unbounded) positive self-adjoint
operator on $H$. Russell \cite[p. 340]{Ru} proposed to introduce

\begin{quote}
certain \textit{indirect damping} mechanisms which arise, not from insertion
of damping terms into the original equations describing the mechanical
motion, but by coupling those equations to further equations describing
other processes in the structure \dots
\end{quote}

He described, in the same work, two types of \textit{indirect damping}: the
velocity coupled dissipator and the displacement coupled dissipator. Works
on \textit{indirect damping} mechanisms of the first type leading to
exponential decay of the total energy may be found in \cite{A-B,A-B-B,
A-B-T,H-L-P,Ru}. In this paper, our attention will be focused on the second
type of indirect damping mechanisms. The description given in \cite{Ru} is
the following. Consider a system with displacement vector $(w,z)$, velocity 
$(\partial _tw,\partial _tz)$ and energy form 
\begin{equation}
E(w,z)(t)=\frac{1}{2}\left( \langle\left(\begin{array}{c}
w \\ 
z
\end{array}
\right) ,S\left(
\begin{array}{c}
w \\ 
z
\end{array}
\right) \rangle_{H\times G} +\| \partial _tw\| _{H}^2+\| \partial _tz\|
_{G}^2\right) .  \label{2}
\end{equation}
where $G$ is a second Hilbert space and $S$ is a positive self-adjoint
operator on $H\times G$ representable in operator matrix form as 
\begin{equation}
S=\left(
\begin{array}{cc}
A & B \\ 
B^{\ast } & C
\end{array}
\right)  \label{3}
\end{equation}
The energy $E(w,z)(t)$ is conserved for the second order system 
\begin{equation}
\left(
\begin{array}{c}
\partial _{tt}w \\ 
\partial _{tt}z
\end{array}
\right) +S\left(
\begin{array}{c}
w \\ 
z
\end{array}
\right) =0.  \label{sd}
\end{equation}
Damping is then introduced in the second equation of the system: 
\[
\left(
\begin{array}{c}
\partial _{tt}w \\ 
\partial _{tt}z
\end{array}
\right) +S\left(
\begin{array}{c}
w \\ 
z
\end{array}
\right) +\gamma \left(
\begin{array}{c}
0 \\ 
\partial _tz
\end{array}
\right) =0.
\]
At this level, Russell \cite{Ru} assumes the inertial forces in the $z$
system are small in comparison with the damping and, then, $\partial _{tt}z$
is discarded. In our work, we do not adopt this last assumption. Moreover,
we replace the constant $\gamma $ by an (eventually unbounded) positive
self-adjoint operator $D\ $acting on $G$ 
\begin{equation}
\left(
\begin{array}{c}
\partial _{tt}w \\ 
\partial _{tt}z
\end{array}
\right) +S\left(
\begin{array}{c}
w \\ 
z
\end{array}
\right) +\left(
\begin{array}{c}
0 \\ 
D\partial _tz
\end{array}
\right) =0.  \label{depl}
\end{equation}
Our problem is then to find, among all these damping mechanisms, those for
which the energy of the resulting system has an exponential decay to zero
(Russell was interested in the analyticity of the associated semigroup).

This paper is organized as follows. In the second section, we give the
relation between the velocity coupled dissipator and the displacement
coupled dissipator. The third section is devoted to uniform stability of
displacement coupled dissipator. In the fourth section we give some
applications (two displacement coupled wave equations and Timoshenko beam).
And in the last section, we give a result of exact controllability of the
displacement coupled dissipator.

\section{Relation between displacement coupling and velocity coupling}

Under suitable assumptions, we will show the equivalence between coupling
through displacements and coupling through velocities of two elastic systems.

Let $H$ and $G$ be Hilbert spaces. Let $A$ and $C$ be positive self-adjoint
unbounded operators acting on $H$ and $G$ respectively, with compact
resolvents. Let $B$ be an unbounded operator from $G$ to $H$ such that $
D(A)\subset D(B^{\ast })$ and $D(C)\subset D(B)$. Let $D$ be a positive
self-adjoint operator on $G$.

We start by giving conditions for the well-posedeness of system (\ref{sd}).
We denote by $(.,.)_{H}$ and $(.,.)_{G}$ the scalar products on $H$ and $G$
respectively and we set $X=H\times G$.

\begin{proposition}
If there exists $c\in \lbrack 0,\frac{1}{2}[$ such that, for all $(u,v)\in
D(A)\times D(C)$
\begin{equation} \label{e6}
\left| (u,Bv)\right| <c\left( \| A^{1/2}u\| _{H}^2+\|
C^{1/2}v\| _{G}^2\right)
\end{equation}
 then $S=\left(\begin{array}{cc}
A & B \\ 
B^{\ast } & C
\end{array}
\right)$  is positive self-adjoint on $X$ with domain $D(S)=D(A)\times D(C)$.
\end{proposition}

\paragraph{Proof}
$S$ is clearly symmetric and condition (\ref{e6}) implies its invertibility.
In fact, for $(h,g)\in $ $H\times G$, we look for a solution $(u,v)\in D(S)$
of the system 
\begin{equation}
\begin{gathered} Au+Bv=h \\ B^{\ast }u+Cv=g. \end{gathered}  \label{9}
\end{equation}
Solving for $u$ in the first equation of (\ref{9}) and replacing in the
second, we obtain 
\[
\begin{gathered}
u=-A^{-1}Bv+A^{-1}h, \\ 
(C-B^{\ast }A^{-1}B)v=g-B^{\ast }A^{-1}h.
\end{gathered}
\]
Thus, (\ref{9}) has a solution in $D(S)$ if and only if $ C-B^{\ast }A^{-1}B$
is boundedly invertible. But, actually, one has 
\[
C-B^{\ast }A^{-1}B=C^{1/2}(I-C^{-1/2}B^{\ast }A^{-1}BC^{-1/2})C^{1/2}.
\]
Now, condition (\ref{e6}) may be rewritten as 
\[
\left| (A^{1/2}u,(A^{-1/2}BC^{-1/2})C^{1/2}v)\right| <c\left( \| A^{1/2}u\|
_{H}^2+\| C^{1/2}v\| _{G}^2\right)
\]
for all $(u,v)\in D(S)$. Set $x=A^{1/2}u\in D(A^{1/2}),\,y=C^{1/2}v\in
D(C^{1/2})$ and $T=A^{-1/2}BC^{-1/2}$. Then 
\[
\left| (x,Ty)\right| <c\left( \| x\| _{H}^2+\| y\| _{G}^2\right) .
\]
Let $x=Ty$ in this last inequality. Then 
\[
\| Ty\| _{H}^2\leq \frac{c}{1-c}\| y\| _{G}^2\quad y\in D(C^{1/2}).
\]
Then, since $C^{-1/2}B^{\ast }A^{-1}BC^{-1/2}=T^{\ast }T$ and $\dfrac{c}{1-c}
<1$, it follows that $I-C^{-1/2}B^{\ast }A^{-1}BC^{-1/2}$ is boundedly
invertible and, moreover maps $D(C^{1/2})$ into itself. So, the conclusion
follows since 
\[
v=C^{-1/2}(I-C^{-1/2}B^{\ast }A^{-1}BC^{-1/2})^{-1}C^{-1/2}(g-B^{\ast
}A^{-1}h)\in D(C).
\]
This implies that $S$ is self-adjoint. The positivity follows from (\ref{e6}
). \hfill$\diamondsuit$\medskip

As a consequence of this proposition, it follows that system (\ref{sd}) is
well-posed in the energy space $V=D(S^{\frac{1}{2} })\times X$.

\begin{remark} \rm
Using (\ref{e6}) we have 
\begin{eqnarray*}
\lefteqn{(1-2c)\left( \langle \left(\begin{array}{c}
w \\ z \end{array}
\right) ,\left(\begin{array}{cc}
A & 0 \\ 
0 & C
\end{array}
\right) \left(\begin{array}{c}
w \\ z \end{array}
\right) \rangle_{H\times G}+\| \partial _tw\| _{H}^2+\|
\partial _tz\| _{G}^2\right) }\\
&\leq &\left( \langle\left(\begin{array}{c}
w \\ z \end{array}
\right) ,\left(\begin{array}{cc}
A & B \\ 
B^{*} & C
\end{array}
\right) \left(\begin{array}{c}
w \\ z \end{array}
\right) \rangle_{H\times G}+\| \partial _tw\| _{H}^2+\|
\partial _tz\| _{G}^2\right) \\
&\leq &(1+2c)\left( \langle \left(\begin{array}{c}
w \\ z \end{array}
\right) ,\left(\begin{array}{cc}
A & 0 \\ 
0 & C
\end{array}
\right) \left(\begin{array}{c}
w \\ z \end{array}
\right) \rangle_{H\times G}+\| \partial _tw\| _{H}^2+\|
\partial _tz\| _{G}^2\right)
\end{eqnarray*}
\end{remark}

\begin{theorem}
If $(w,z)$ is a solution of system (\ref{depl}), then $(u,v)$ defined by 
\begin{equation} \label{e8}
\begin{gathered}
u=-A^{-1/2}\partial _t w  \\ 
v=z \end{gathered}
\end{equation}
is a solution of the system 
\begin{equation} \label{e9}
\begin{gathered}
\partial _{tt}u=-Au+A^{-1/2}B\partial _tv \\ 
\partial _{tt}v=-B^{\ast }A^{-1/2}\partial _tu-(C-B^{\ast
}A^{-1}B)v-D\partial _tv
\end{gathered}
\end{equation}
Moreover, under the assumptions of Proposition 1, $C-B^{\ast
}A^{-1}B$ is self-adjoint positive.  Conversely, if $(u,v)$ is a
solution of (\ref{e9}), then $(w,z)$ defined by 
\begin{equation} \label{e10}
\begin{gathered}
w=A^{-1/2}u_t-A^{-1}Bv \\ 
z=v
\end{gathered}
\end{equation}
 is a solution of (\ref{depl}).
\end{theorem}

\paragraph{Proof}
We use direct computations. From (\ref{e8}), one has 
\begin{equation}
\partial _tu =-A^{-1/2}\partial _{tt}w =-A^{-1/2}(-Aw-Bz) =
A^{1/2}w+A^{-1/2}Bz  \label{8}
\end{equation}
and thus 
\[
\partial _{tt}u =A^{1/2}\partial _tw+A^{-1/2}B\partial _tz
=-Au+A^{-1/2}B\partial _tv\,. 
\]
For the second equation in $v$, using (\ref{8}) one has 
\begin{eqnarray*}
\partial _{tt}v &=&-B^{\ast }w-Cv-D\partial _tv \\
&=&-B^{\ast }A^{-1/2}\partial _tu-(C-B^{\ast }A^{-1}B)v-D\partial _tv.
\end{eqnarray*}
and this gives system (\ref{e9}). The converse is obtained by the same
computations.

\section{Uniform stability}

Our main result is the uniform stability of (\ref{depl}). Before giving the
assumptions on the operators $A,B,C$ and $D$ to obtain the uniform stability
we define the operators 
\[
\begin{gathered}
\mathcal{A}_1 =\left(\begin{array}{cccc}
0 & I & 0 & 0 \\ 
-A & 0 & -B & 0 \\ 
0 & 0 & 0 & I \\ 
-B^{\ast } & 0 & -C & -D
\end{array}
\right) \,, \\
\widetilde{\mathcal{A}} =\left(\begin{array}{cccc}
0 & I & 0 & 0 \\ 
-A\;\;\; & 0 & 0 & A^{-1/2}B \\ 
0 & 0 & 0 & I \\ 
0 & -B^{\ast }A^{-1/2}\; & -C+B^{\ast }A^{-1}B\; & -D\;
\end{array}
\right) \,.
\end{gathered}
\]
The operator $\mathcal{A}_1$ has dense domain, 
\[
D(\mathcal{A}_1\text{)}=D(A)\times D(A^{1/2})\times D(C)\times
(D(C^{1/2})\cap D(D)).
\]

The system (\ref{depl}) can be written as 
\begin{equation}
\partial _tY=\mathcal{A}_1Y\,.  \label{kari1}
\end{equation}
and (\ref{e9}) as 
\begin{equation}
\partial _tZ=\widetilde{\mathcal{A}}Z\,.  \label{kari2}
\end{equation}

\begin{theorem}
The energy of system (\ref{kari1}) decays exponentially if and only
if the energy of system  (\ref{kari2}) decays exponentially in the same
energy space.
\end{theorem}

\paragraph{Proof}
Using the result of the Theorem 2, we have a bounded invertible
transformation $P\in \mathcal{L}(D(A^{1/2})\times H\times D(C^{1/2})\times G)
$, such that 
\begin{equation}
\mathcal{A}_1P=P\widetilde{\mathcal{A}}  \label{(9)}
\end{equation}
where 
\[
P=\left(
\begin{array}{cccc}
0 & A^{-1/2} & -A^{-1}B & 0 \\ 
-A^{1/2}\; & 0 & 0 & 0 \\ 
0 & 0 & I & 0 \\ 
0 & 0 & 0 & I
\end{array}
\right)
\]
and 
\[
P^{-1}=\left(
\begin{array}{cccc}
0 & -A^{-1/2} & 0 & 0 \\ 
A^{1/2}\; & 0 & A^{-1/2}B & 0 \\ 
0 & 0 & I & 0 \\ 
0 & 0 & 0 & I
\end{array}
\right) \,.
\]
Applying $P$ to $(\ref{kari2})$ leads to 
\begin{eqnarray*}
\partial _tPZ &=&P\widetilde{\mathcal{A}}Z\; \\
&=&\mathcal{A}_1PZ,
\end{eqnarray*}
Then $PZ:=W\,$\ is a solution of (\ref{kari1}). If we suppose the uniform
stability of (\ref{kari2}), that is: there exist positive constants $
M,\omega $ such that 
\[
\left\| Z(t)\right\| _{D(A^{1/2})\times H\times D(C^{\frac{1}{2} })\times
G}^2\leq M\left\| Z(0)\right\| _{D(A^{1/2})\times H\times D(C^{1/2})\times
G}^2\exp (-\omega t)\;,\;t\geq 0 \,,
\]
then for $t\geq 0$, we have 
\begin{eqnarray*}
\lefteqn{ \left\| P^{-1}W(t)\right\| _{D(A^{1/2})\times H\times D(C^{\frac{1
}{2}})\times G}^2 } \\
&\leq& M\left\| P^{-1}W(0)\right\| _{D(A^{1/2})\times H\times
D(C^{1/2})\times G}^2\exp (-\omega t)\,.
\end{eqnarray*}
Using the boundedness of $P^{-1}$, we obtain the uniform stability of (\ref
{kari1}). \hfill$\diamondsuit$

\begin{remark} \rm
1)\ $\mathcal{A}_1$ (respectively $\widetilde{\mathcal{A}}$ ) generates a $
C_{0}-$semigroup on $D(A^{1/2})\times H\times D(C^{1/2})\times G$, 
denoted by $S_{\mathcal{A}_1}(t)$ (respectively 
$S_{\widetilde{\mathcal{A}}}(t)$). And 
\begin{equation*}
S_{\mathcal{A}_1}(t)=PS_{\widetilde{\mathcal{A}}}(t)P^{-1}\;\;,t\geq 0 
\end{equation*}
 2)\; The semigroups $S_{\mathcal{A}_1}(t)$ and $S_{\widetilde{\mathcal{A
}}}(t)$ have the same type.
\end{remark}

Denote by $B_1:=A^{-1/2}B$ and $C_1:=C-B^{\ast }A^{-1}B$. To give a
sufficient conditions for the uniform stabilization of the system (\ref{depl}
) we will need the following assumption: (H) $B_1$ is boundedly invertible
and the operators $D^{-\frac{1}{2} }B_1{}^{-1}A^{1/2}$, $
D^{-1/2}C_1B_1^{-1}A^{-1/2}$, $D^{1/2}B_1^{-1}$ and $A^{-1/2}B_1^{-1}D^{-1/2}
$ all extend to bounded operators on $X$.

\begin{proposition}
Under condition (\ref{e6}) and assumption $(H)$ we have the uniform stability 
of $(\ref{depl})$.
\end{proposition}

\paragraph{Proof}
The result of this proposition follows from the relation between
displacement coupling and velocity coupling, using Theorem 3, then the
uniform stability of the system $(5)$ is equivalent to the uniform stability
of the system 
\[
\begin{gathered} \partial _{tt}u=-Au+A^{-1/2}B\partial _tv \\ \partial
_{tt}v==-B^{\ast }A^{-1/2}\partial _tu-(C-B^{\ast }A^{-1}B)v-D\partial
_tv\,. \end{gathered}
\]
Using a result in \cite{A.K} we construct a Lyapunov function associated
with the system above. Let $U=\left( 
\begin{array}{c}
u \\ 
v
\end{array}
\right)$ and 
\begin{eqnarray*}
\chi _{\varepsilon }(\left(
\begin{array}{c}
U(t) \\ 
\partial _tU(t)
\end{array}
\right) )&:=&\frac{1}{2}(\left\| U(t)\right\| _{D(S^{1/2})}^2+\left\|
\partial _tU(t)\right\| _{X}^2) \\
&&+\varepsilon \left( \frac{1}{2} (u,\partial _tu)+\frac{1}{4}\left\|
D^{1/2}v\right\| +\frac{1}{2} (v,\partial _tv)\right) \\
&&+\varepsilon \left( \frac{1}{2}(u,B_1v)+(\partial _tv,B_1^{-1}\partial
_tu)+(C_1v,B_1^{-1}u)\right) \,.
\end{eqnarray*}
Next, we prove (see \cite{A.K}) that for $\varepsilon >0$ sufficiently small
there exist $a_{\varepsilon },b_{\varepsilon }$ and $c_{\varepsilon }$
positive constants such that 
\begin{eqnarray}
a_{\varepsilon }(\left\| U(t)\right\| _{D(S^{1/2})}^2+\left\| \partial
_tU(t)\right\| _{X}^2) &\leq& \chi _{\varepsilon }(\left(
\begin{array}{c}
U(t) \\ 
\partial _tU(t)
\end{array}
\right) )  \label{(10)} \\
&\leq& b_{\varepsilon }(\left\| U(t)\right\| _{D(S^{1/2})}^2+\left\|
\partial _tU(t)\right\| _{X}^2)  \nonumber
\end{eqnarray}
and 
\begin{equation}
\frac{d}{dt}\chi _{\varepsilon }(\left(
\begin{array}{c}
U(t) \\ 
\partial _tU(t)
\end{array}
\right) )\leq -c_{\varepsilon }.\chi _{\varepsilon }(\left(
\begin{array}{c}
U(t) \\ 
\partial _tU(t)
\end{array}
\right) )\,.  \label{(11)}
\end{equation}
Then the proof of this proposition is derived from  (\ref{(10)}) and (\ref
{(11)}). In fact, from (\ref{(11)}) it follows that 
\[
\chi _{\varepsilon }(\left(
\begin{array}{c}
U(t) \\ 
\partial _tU(t)
\end{array}
\right) \leq \exp (-c_{\varepsilon }t)\chi _{\varepsilon }(\left(
\begin{array}{c}
U(0) \\ 
\partial _tU(0)
\end{array}
\right)
\]
and from (\ref{(10)}) we have 
\[
\left\| U(t)\right\| _{D(S^{1/2})}^2+\left\| \partial _tU(t)\right\|
_{X}^2\leq \frac{b_{\varepsilon }}{a_{\varepsilon }}(\left\| U(0)\right\|
_{D(S^{1/2})}^2+\left\| \partial _tU(0)\right\| _{X}^2)\exp (-c_{\varepsilon
}t)\,.
\]

\section{Applications}

\subsection*{Particular cases}

In this subsection we set $H=G$ and assume that $B,C$ and $D$ are powers of
the positive self-adjoint operator $A$, in this case we consider the system 
\begin{equation}
\begin{gathered} \partial _{tt}u+Au+a.A^{\alpha }v=0 \\ \partial
_{tt}v+A^{\beta }v+a.A^{\alpha }u+A^{\gamma }\partial _tv=0\,, \end{gathered}
\label{st}
\end{equation}
where $a\neq 0$ is a real constant such that 
\[
\left| a\right| \left\| A^{\alpha -(\frac{\beta +1}{2})}\right\| _{H} < 1\,.
\]
and $\alpha ,\beta ,\gamma $ are real constants. Our objective is to find
conditions on $\alpha ,\beta ,\gamma $ in order to obtain the uniform
stability of the above system. By using our result (Theorem 3), the uniform
stability of system (\ref{st}) is equivalent to the uniform stability of the
system 
\begin{equation}
\begin{gathered} \partial _{tt}u+Au-a.A^{\alpha -\frac{1}{2}}\partial _tv=0
\\ \partial _{tt}v+A^{\beta }(I-a^2A^{2\alpha -\beta -1})v+a.A^{\alpha
-\frac{ 1}{2}}\partial _tu+A^{\gamma }\partial _tv=0\,. \end{gathered}
\label{st1}
\end{equation}

We remark that the operator $A^{2\alpha -\beta -1}$ is a compact
perturbation from $D(A^{1/2})$ to $H$, then the uniform stability of system (
\ref{st1}) is equivalent to the uniform stability of the system 
\begin{equation}
\begin{gathered} \partial _{tt}u+Au-a.A^{\alpha -\frac{1}{2}}\partial _tv=0
\\ \partial _{tt}v+A^{\beta }v+a.A^{\alpha -\frac{1}{2}}\partial
_tu+A^{\gamma }\partial _tv=0\,. \end{gathered}  \label{st2}
\end{equation}

\begin{proposition}[{\cite{A.K}}]
 When $\beta \neq 1$, system (\ref{st2}) is uniformly stable
if 
\begin{equation*}
\gamma \in [\max (2-2\alpha ,2\alpha -2,2\beta -2\alpha ),2\alpha -1]\,. 
\end{equation*}
When $\beta =1$, system (\ref{st2}) is uniformly stable if and only if 
\begin{equation*}
\gamma \in \lbrack \max (0,2\alpha -2),2\alpha -1]\,. 
\end{equation*}
\end{proposition}

\subsection*{Timoshenko beam}

We consider in this example a model of Timoshenko beam \cite{Sou,Sou1}. The
equations of motion for this system are given by 
\begin{equation}
\begin{gathered} \rho \partial _{tt}u=K\partial _{xx}u-K\partial _{x}v
\quad\mbox{in } ]0,l[\times \mathbb{R}^{+} \\ I_{\rho }\partial
_{tt}v=EI\partial _{xx}v+K(\partial _{x}u-v)-b(x)\partial _tv\quad \mbox{in
} ]0,l[\times \mathbb{R}^{+} \\ u(0,t)=u(l,t)=v(0,t)=v(l,t)=0\,.
\end{gathered}  \label{a1}
\end{equation}
This system is coupled with the initial conditions 
\begin{equation}
\begin{gathered} u(x;0) =u_{0}(x) \quad \partial _tu(x;0)=u_1(x) \\ v(x;0)
=v_{0}(x) \quad \partial _tv(x;0)=v_1(x) \end{gathered}  \label{a2}
\end{equation}
Here, $t$ is the time variable and $x$ is the space coordinate along the
beam in its equilibrium position. The functions$ \;u(x,t) $ is the
transverse displacement of the beam and $v(x,t)$ is the rotation angle of a
filament of the beam. The coefficients $\rho ,I_{\rho },E $ and $I$ are the
mass per unit length, the mass moment of inertia of the cross section,
Young's modulus and the moment of inertia of the cross section,
respectively. The coefficient $K$ is the shear modulus and $b(x)$ is a
positive function on $[0,l]$ . The energy of the beam is given by 
\[
E(t)=\frac{1}{2} \int_{0}^{l}(\rho (\partial _tu)^2+I_{\rho }(\partial
_tv)^2+EI(\partial _{x}v)^2+K(\partial _{x}u-v)^2)dx
\]

\begin{remark} \rm 
In this example  $B=-\frac{K}{\rho }\partial _{x}$ is not a
power of the operator $A=\frac{K}{\rho }\partial _{xx}$.
\end{remark}

\begin{theorem}[{\cite{Sou,Sou1}}] 
If $b(x)>0$ on $[0,l]$, then
\begin{equation*}
E(t)\leq M.e^{-at}E(0)\quad\mbox{if and only if} \quad
\frac{K}{\rho }=\frac{EI}{I\rho }\,. 
\end{equation*}
\end{theorem}

\section{Exact Controllability}

In this section we obtain an exact controllability result for displacement
coupled dissipator systems. We suppose that $D$ is \textbf{bounded} on $G$.
Our result is as follows.

\begin{theorem}
 There exists $T>0$ and $c_1\geq 0$ such that the solution of the
 system 
\begin{equation}
\begin{gathered}
\left(\begin{array}{c}
\partial _{tt}\varphi (t) \\ 
\partial _{tt}\phi (t)
\end{array}
\right) =\left(\begin{array}{cc}
-A & -B \\ 
-B^{\ast } & -C
\end{array}
\right) \left(\begin{array}{c}
\varphi (t) \\ 
\phi (t)
\end{array}
\right) \\ 
\left(\begin{array}{c}
\varphi (0) \\ 
\phi (0)
\end{array}
\right) =\left(\begin{array}{c}
\varphi _{0} \\ 
\phi _{0}
\end{array}
\right) ,\quad \left(\begin{array}{c}
\partial _t\varphi (0) \\ 
\partial _t\phi (0)
\end{array}
\right) =\left(\begin{array}{c}
\varphi _1 \\ 
\phi _1
\end{array}
\right)
\end{gathered}
\end{equation}
 satisfies
\begin{eqnarray*}
\lefteqn{\left| \left(\begin{array}{c}
\partial _t\varphi (0) \\ 
\partial _t\phi (0)
\end{array}
\right) \right| _{H\times G}^2+\Big\langle\left(\begin{array}{cc}
A & B \\ 
B^{\ast } & C
\end{array}
\right) \left(\begin{array}{c}
\varphi (0) \\ 
\phi (0)
\end{array}
\right) ,  \left(\begin{array}{c}
\varphi (0) \\ 
\phi (0)
\end{array}
\right) \Big\rangle_{H\times G} }\\
&\leq& c_1\int_{0}^{T}\left| D^{1/2}\partial _t\phi
(t)\right| _{G}^2\,dt \hspace{5cm} 
\end{eqnarray*}
 if and only if, the system 
\begin{equation} \label{e24}
\begin{gathered}
\partial _{tt}u=-Au+A^{-1/2}B\partial _tv \\ 
\partial _{tt}v==-B^{\ast }A^{-1/2}\partial _tu-(C-B^{\ast
}A^{-1}B)v-D\partial _tv \\ 
(u(0),v(0)) =  (u_{0},v_{0})  \quad
(\partial _tu(0),\partial _tv(0)) =  (u_1,v_1) 
\end{gathered}
\end{equation}
 is exponentially stable.
\end{theorem}

\paragraph{Proof}
By Theorem 3, the uniform stability of (\ref{e24}) is equivalent to the 
uniform stability of (\ref{depl}). We write the system (\ref{depl}) as 
\[
\begin{gathered} \partial _{tt}\left(\begin{array}{c} u \\ v \end{array}
\right) +\left(\begin{array}{cc} A & B \\ B^{\ast } & C \end{array} \right)
\left(\begin{array}{c} u \\ v \end{array} \right) +\left(\begin{array}{cc} 0
& 0 \\ 0 & D \end{array} \right) \partial _t\left(\begin{array}{c} u \\ v
\end{array} \right) =0 \\ \left(\begin{array}{c} u(0) \\ v(0) \end{array}
\right) =\left(\begin{array}{c} u_{0} \\ v_{0} \end{array} \right) ,\quad
\left(\begin{array}{c} \partial _tu(0) \\ \partial _tv(0) \end{array}
\right) =\left(\begin{array}{c} u_1 \\ v_1 \end{array} \right)\,.
\end{gathered}
\]
Using Proposition 1 we deduce that $S$ is a self-adjoint, coercive operator
on $X$, and $\left(
\begin{array}{cc}
0 & 0 \\ 
0 & D
\end{array}
\right) $is a bounded positive operator on $X$. Applying a result of Haraux 
\cite{H1}, we conclude the present proof.

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\noindent \textsc{Abdelaziz Soufyane }\newline
University of Franche Comt\'{e}\newline
Laboratoire de Physique et M\'{e}trologie des Oscillateurs\newline
32, Avenue de l'observatoire, \newline
25044 Besan\c{c}on Cedex. France. \newline
e-mail: asoufyane@lpmo.edu

\end{document}