\documentclass[reqno]{amsart} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2003(2003), No. 29, pp. 1--14. \newline ISSN: 1072-6691. URL: http://ejde.math.swt.edu or http://ejde.math.unt.edu \newline ftp ejde.math.swt.edu (login: ftp)} \thanks{\copyright 2003 Southwest Texas State University.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE--2003/29\hfil Uniform stabilization for the Timoshenko beam] {Uniform stabilization for the Timoshenko beam by a locally distributed damping} \author[A. Soufyane \& A. Wehbe \hfil EJDE--2003/29\hfilneg] {Abdelaziz Soufyane \& Ali Wehbe} \address{Abdelaziz Soufyane\hfill\break Math \& Comp Sciences department, UAE University, PO Box 17551 Al Ain, UAE} \email{Soufyane.Abdelaziz@uaeu.ac.ae} \address{Ali Wehbe \hfill \break Universite Libanaise Faculte des Sciences, Section 1\\ Hadath, Beyrouth, Lebanon} \email{ ali\_wehbe@yahoo.fr} \date{} \thanks{Submitted September 2, 2002. Published March 16, 2003.} \subjclass[2000]{35L55, 93C05, 93C20, 93D15, 93D20, 93D30} \keywords{Timohsenko beam, strong stability, non-uniform stability, \hfill\break\indent uniform stability, Lyapunov function} \begin{abstract} We study the uniform stabilization of a Timoshenko beam by one control force. We prove that under, one locally distributed damping, the exponential stability for this system is assured if and only if the wave speeds are the same. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{proposition}[theorem]{Proposition} \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} A basic linear model, developed in \cite{Tim}, for describing the transverse vibration of beams is given by two coupled partial differential equations \begin{equation} \begin{gathered} \rho w_{tt}=(K(w_{x}-\varphi ))_{x}, \\ I_{\rho }\varphi _{tt}(x,t)=(EI\varphi _{x})_{x}+K(w_{x}-\varphi ). \end{gathered} \quad\text{% on } (0,l)\times \mathbb{R}^{+} \label{1} \end{equation} Here, $t$ is the time variable and $x$ the space coordinate along the beam, the length of which is $l$, in its equilibrium position. The function $w$ is the transverse displacement of the beam and $\varphi $ is the rotation angle of a filament of the beam. The coefficients $\rho,I_{\rho },E,I$ and $K$ are the mass per unit length, the polar moment of inertia of a cross section, Young's modulus of elasticity, the moment of inertia of a cross section and the shear modulus respectively. The natural energy of the beam is \begin{equation} \mathcal{E}(t)=\frac{1}{2}\int_{0}^{l}\rho \left| \partial _{t}w\right| ^{2}+I_{\rho }\left| \partial _{t}\varphi \right| ^{2}+EI\left| \partial _{x}\varphi \right| ^{2}+K\left| \partial _{x}w-\varphi \right| ^{2})dx. \label{2} \end{equation} The aim of this paper is to study two types of stabilization of this system, the internal and the boundary stabilization. Let us mention some results about the stabilizability of the Timoshenko beam. The case of \ two boundary control force has already been considered by Kim and Renardy \cite{K-R} for the Timoshenko beam. They proved the exponential decay of the energy $% \mathcal{E}(t)$ by using a multiplier technique and provided numerical estimates of the eigenvalues of the operator associated to this system, and by Lagnese \& Lions \cite{lag} for the study of the exact controllability, Taylor \cite{Tayl} studied the boundary control of system (\ref{1}) with variable physical characteristics. Recently Shi \& Feng \cite{shi} established the exponential decay of the energy $\mathcal{E}(t)$ with locally distributed feedback (two feedback).We will first prove that it is possible to stabilize uniformly (with respect to the initial data) this beam, which is assumed to be clamped at the two ends of $(0,l)$, by using a unique locally distributed feedback \begin{equation} \begin{gathered} \rho w_{tt}=(K(w_{x}-\varphi ))_{x}, \\ I_{\rho }\varphi _{tt}=(EI\varphi _{x})_{x}+K(w_{x}-\varphi )-b(x)\varphi _{t}, \\ w(0,t)=w(l,t)=0;\quad \varphi (0,t)=\varphi (l,t)=0. \end{gathered} \quad \text{on }(0,l)\times \mathbb{R}^{+} \label{1'} \end{equation} where $b$ is a positive continuous function of the space variable. Indeed, we prove the uniform stability holds for system (\ref{1'}) if and only if \ the wave speeds $\frac{K}{\rho }\;$and $\frac{EI}{I_{\rho }}\,\,\,\,\,$are the same. If not, the asymptotic stability for this system is proved. The techniques we use consist in the computation of the essential type (see the definition in the next section) of the associated semigroups, thanks to the result of Neves et al. \cite{N-R-L}. Our paper is organized as follows. In section 2, we give a result of a well-posedness of the solution of the system and we study the strong asymptotic stability and the nonuniform stability of (\ref{1'}) under the assumption $\frac{K}{\rho }\neq \frac{EI}{I_{\rho }}$. In section 3, we study the uniform stability of (\ref{1'}) under the assumption $\frac{K}{% \rho }= \frac{EI}{I_{\rho }}$. \section{Well-posedness, asymptotic and nonuniform stability} \subsection*{Well-posedness} Here we outline the existence theory of the solution of the Timoshenko beam. To establish the existence and uniqueness of the solution of the studied model we use the\ semigroup theory, we suppose that $b\in C([0,l])$ . The energy space associated to system (\ref{1'}) is \begin{equation*} H:=(H_{0}^{1}(]0,l[)\times L^{2}(]0,l[))^{2} \end{equation*} The inner product in the energy space is defined as follows: \begin{equation*} \left( Y_{1},Y_{2}\right) :=\int_{0}^{l}(K(\partial _{x}u_{1}-w_{1})(\partial _{x}u_{2}-w_{2})+\rho v_{1}v_{2} +I_{\rho }f_{1}f_{2}+EI(\partial _{x}w_{1}\partial _{x}w_{2}))dx, \end{equation*} where $Y_{k}=% \begin{pmatrix} u_{k} \\ v_{k} \\ w_{k} \\ f_{k}% \end{pmatrix} \in H$, $k=1,2$. In the sequel we will denote by $\left\| Y\right\| ^{2}:=\left( Y,Y\right)$, the norm in the energy space. The system \ref{1'} can be written as \begin{equation*} \partial _{t}Y(t)=LY(t), \end{equation*} where \begin{equation*} Y(t):=% \begin{pmatrix} w(t) \\ \partial _{t}w(t) \\ \varphi (t) \\ \partial _{t}\varphi (t)% \end{pmatrix}% ,\quad L:= \begin{pmatrix} 0 & I & 0 & 0 \\ \frac{K}{\rho }\partial _{xx} & 0 & -\frac{K}{\rho }\partial _{x} & 0 \\ 0 & 0 & 0 & I \\ \frac{K}{I_{\rho }}\partial _{x} & 0 & \frac{EI}{I_{\rho }}\partial _{xx}- \frac{K}{I_{\rho }} & -b(x)% \end{pmatrix}% \end{equation*} with $D(L):=(( H^{2}(]0,l[)\cap H_{0}^{1}(]0,l[)) \times H_{0}^{1}(]0,l[)) ^{2}$. \begin{proposition}[\cite{Souf},\cite{shi}] \label{prop1} The operator $(L$, $D(L))$ generates a $C_0$-semigroup of contractions $(e^{Lt})_{t\geq 0}$ on $H$. \end{proposition} \subsection*{Asymptotic strong Stability and nonuniform stability} Before giving our first result, we need to recall some results and the following definitions: $e^{Lt}$ is asymptotically stable if, for any $Y_{0}\in H$ $% \lim_{t\rightarrow \infty }e^{Lt}Y_{0}=0$. $e^{Lt}$ is uniformly (or exponentially) stable if there exist $\omega <0$ and $M>0$ such that \begin{equation*} \| e^{Lt}\| \leq Me^{\omega t}\quad t\in \mathbb{R}^{+} \end{equation*} For a continuous linear operator from a Banach space into itself, we define its essential spectral radius $r_{e}(L)$ as \begin{equation} \begin{aligned} r_{e}(L)=\inf \Big\{& R>0:\mu \in \sigma (L),| \mu| >R \text{ implies }\mu \text{ is an isolated } \\ &\text{eigenvalue of finite multiplicity} \end{aligned} \Big\} \label{ress} \end{equation} It is well-known (see Gohberg-Krein \cite{G-K}) that, if $r(L)$ is the spectral radius of $L$ \begin{gather*} r_{e}(L) \leq r(L);\,\, \\ r_{e}(L+K) =r_{e}(L)\quad \forall K\in L(X),\,K\text{ compact} \end{gather*} If $e^{Lt}$ is a $C_{0}$-semigroup generated by $L$, then (see for instance \cite{vN}) there exist two real numbers $\omega =\omega (L)$ and $\omega _{e}=\omega _{e}(L)$ such that \begin{gather} r(e^{Lt}) =e^{\omega t}, \label{type} \\ r_{e}(e^{Lt}) =e^{\omega _{e}t}\quad \forall t\in \mathbb{R}^{+}. \label{typess} \end{gather} $\omega $ is often called the\textit{\ type} and $\omega _{e}$ the \textit{% essential type} of the semigroup. A third real number which plays an essential role in stability theory is the \textit{spectral abscissa} $s(L)$ of $L$ defined by \begin{equation*} s(L)=\sup \left\{ \mathop{\rm Re}\lambda : \lambda \in \sigma (L)\right\} . \end{equation*} These three real numbers are related as follows \cite[Theorem 3.6.1., p. 107]% {vN}: \begin{equation*} \omega (L)=\max \left( \omega _{e}(L),s(L)\right). \end{equation*} Clearly, the uniform stability of $e^{Lt}$ is equivalent to $\omega (L)<0$. We are now ready to state our first result. \begin{theorem} \label{thm2} Assume that $b\in C([0,l])$ and \begin{equation} b(x)\geq \overline{b}>0\quad \mbox{on }[b_{0},b_{1}]\subset \;[0,l]. \label{ahme2} \end{equation} Then: \begin{itemize} \item $e^{Lt}$ is asymptotically stable. \item If \ $\frac{K}{\rho }\neq \frac{EI}{I_{\rho }},\;$then $e^{Lt}$ is non uniformly stable. \end{itemize} \end{theorem} Before giving the proof of this theorem we need to recall the following result. \begin{theorem}[Benchimol \cite{Ben}] \label{thmben} Let $L$ be a maximal linear operator in a complex Hilbert space $H$ and assume that: \begin{itemize} \item[(a)] $L$ has a compact resolvent. \item[(b)] $L$ does not have purely imaginary eingenvalues. \end{itemize} Then $e^{Lt}$ is strongly stable. \end{theorem} \begin{proof}[Theorem \protect\ref{thm2}] To prove strong stability of $e^{Lt}$, it remains to verify the properties (a) and (b) of theorem \ref{thmben}. Property (a) follows at once from the compactness of the imbedding $D(L)$ in $H,$ a consequence of Rellich's theorem. Suppose the conclusion of (b) is false, then $L$ have a purely imaginary eingenvalues $i\omega $. Let $U_{1}$ the eigenvectors associated to $i\omega$% . Using the definition of $L$ it follows that $LU_{1} =i\omega U_{1}$ if and only if \begin{equation} \begin{gathered} K\partial _{xx}u-K\partial _{x}v=-\rho \omega ^{2}u, \\ EI.\partial _{xx}v+K\partial _{x}u-Kv-ib(x)\omega v=-I_{\rho }\omega ^{2}v, \\ u\big| _{0,l} =v\big| _{0,l} =0. \end{gathered} \label{ahm} \end{equation} We multiply the second equation of this system by $v$ to obtain \begin{equation*} \int_{0}^{l}(EI| \partial _{x}v| ^{2}-K\partial _{x}u.v+(K-I_{\rho }\omega ^{2})| v| ^{2})dx=0\;\text{\ and\ \ }\int_{0}^{l}\omega b(x)|v|^{2}dx=0. \end{equation*} Then $v=0$, on $[b_{0},b_{1}]$, and \begin{equation*} EI.\partial _{xx}v+K\partial _{x}u-Kv =-I_{\rho }\omega ^{2}v, \end{equation*} which implies $\partial _{x}u =0$ on $]b_{0},b_{1}[$. From \begin{equation*} K\partial _{xx}u-K\partial _{x}v =-\rho \omega ^{2}u, \end{equation*} it follows that $u =0$ on $]b_{0},b_{1}[$. Now, it is trivial to see that $u=v=0$ is the solution of the system \begin{equation} \begin{gathered} K\partial _{xx}u-K\partial _{x}v=-\rho \omega ^{2}u, \\ EI.\partial _{xx}v+K\partial _{x}u-Kv=-I_{\rho }\omega ^{2}v, \\ v=u=0 \quad \text{on }]b_{0},b_1[,\\ u\big|_{0,l} =v\big| _{0,l} =0. \end{gathered} \label{ahme1} \end{equation} This complete the proof of the strong stability of $(e^{Lt})_{t\geq 0}$. The proof of the nonuniform stability for $e^{Lt}$ when $\frac{K}{\rho }\neq \frac{EI}{I\rho }$ is based on the computation of the essential type of $% (e^{Lt})_{t\geq 0}$. Let \begin{equation*} D=% \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & -\frac{K}{I_{\rho }} & 0% \end{pmatrix}% \end{equation*} This operator is compact in the energy space $H$. Then $r_{e}(L)=r_{e}(L-D)$% . Now we calculate the essential spectral radius of $L_{1}:=L-D$ , for this reason we transform the equations by introducing the following variables: \begin{gather*} p=-\sqrt{\frac{K}{\rho }}\partial _{x}u+\partial _{t}u, \quad q=\sqrt{\frac{K% }{\rho }}\partial _{x}u+\partial _{t}u, \\ \varphi =-\sqrt{\frac{EI}{I_{\rho }}}\partial _{x}v+\partial _{t}v, \quad \psi =\sqrt{\frac{EI}{I_{\rho }}}\partial _{x}v+\partial _{t}v. \end{gather*} which are solutions of the system \begin{equation} \partial _{t}% \begin{pmatrix} p \\ \varphi \\ q \\ \psi% \end{pmatrix} +K\partial _{x}% \begin{pmatrix} p \\ \varphi \\ q \\ \psi% \end{pmatrix} +C% \begin{pmatrix} p \\ \varphi \\ q \\ \psi% \end{pmatrix} =0, \end{equation} with the boundary conditions \begin{equation*} (p+q)(0,t)=(\varphi +\psi )(0,t)=(p+q)(l,t)=(\varphi +\psi )(l,t)=0, \end{equation*} where \begin{equation*} K=% \begin{pmatrix} \sqrt{K/\rho } & 0 & 0 & 0 \\ 0 & \sqrt{EI/I_{\rho}} & 0 & 0 \\ 0 & 0 & -\sqrt{K/\rho} & 0 \\ 0 & 0 & 0 & -\sqrt{EI/I_{\rho}}% \end{pmatrix}% , \end{equation*} \begin{equation*} C=% \begin{pmatrix} 0 & -\frac{K}{2\rho }\sqrt{\frac{I_{\rho }}{EI}} & 0 & \frac{K}{2\rho } \sqrt{\frac{I_{\rho }}{EI}} \\ \frac{\sqrt{K\rho }}{2I_{\rho }} & -\frac{b(x)}{2I_{\rho }} & -\frac{\sqrt{% K\rho }}{2I_{\rho }} & -\frac{b(x)}{2I_{\rho }} \\ 0 & -\frac{K}{2\rho }\sqrt{\frac{I_{\rho }}{EI}} & 0 & \frac{K}{2\rho } \sqrt{\frac{I_{\rho }}{EI}} \\ \frac{\sqrt{K\rho }}{2I_{\rho }} & -\frac{b(x)}{2I_{\rho }} & -\frac{\sqrt{% K\rho }}{2I_{\rho }} & -\frac{b(x)}{2I_{\rho }}% \end{pmatrix}% \end{equation*} When $\frac{K}{\rho }\neq \frac{EI}{I_{\rho }}$, we can apply a result in \cite[p 324]{N-R-L} which implies that $r_{e}(e^{L_{1}t})=r_{e}(e^{L_{2}t})=% \exp (\alpha t)$ where \begin{equation*} L_{2}=% \begin{pmatrix} \sqrt{K/\rho} & 0 & 0 & 0 \\ 0 & \sqrt{EI/I_{\rho}} & 0 & 0 \\ 0 & 0 & -\sqrt{K/\rho} & 0 \\ 0 & 0 & 0 & -\sqrt{EI/I_{\rho}}% \end{pmatrix} \partial _{x} +% \begin{pmatrix} 0 & 0 & 0 & 0 \\ 0 & -\frac{b(x)}{2I_{\rho }} & 0 & 0 \\ 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & -\frac{b(x)}{2I_{\rho }}% \end{pmatrix}% \end{equation*} and $\alpha =\sup \{\mathop{\rm Re}(\lambda ):\lambda \in \sigma(L_{2})\}$. To compute $\alpha $ we solve the system \begin{equation*} \begin{gathered} \sqrt{\frac{K}{\rho }}\partial _{x}p=\lambda p, \quad \sqrt{\frac{K}{\rho }}\partial _{x}q=-\lambda q, \\ \sqrt{\frac{EI}{I_{\rho }}}\partial _{x}\varphi =(\lambda +\frac{b(x)}{2I_{\rho }})\varphi , \quad \sqrt{\frac{EI}{I_{\rho }}}\partial _{x}\psi =-(\lambda +\frac{b(x)}{2I_{\rho }})\psi , \end{gathered} \end{equation*} with the boundary conditions \begin{equation*} (p+q)(0,t)=(\varphi +\psi )(0,t)=(p+q)(l,t)=(\varphi +\psi )(l,t)=0. \end{equation*} Then we have \begin{gather*} p(x) = c_{1}\exp (\lambda \sqrt{\frac{\rho }{K}}x), \quad q(x) = -c_{1}\exp (-\lambda \sqrt{\frac{\rho }{K}}x), \\ \varphi (x) =c_{2}\exp (\sqrt{\frac{I_{\rho }}{EI}}(\lambda x +\int_{0}^{x}% \frac{b(y)}{2I_{\rho }}dy)), \\ \psi (x) =-c_{2}\exp (-\sqrt{\frac{I_{\rho }}{EI}}(\lambda x +\int_{0}^{x}% \frac{b(y)}{2I_{\rho }}dy)), \end{gather*} where $c_{1}$ and $c_{2}$ are in $\mathbb{R}\backslash \{0\}$. Using the boundary conditions, we see that one of the following two equations must hold for non-trivial solutions, \begin{gather*} \exp (\lambda \sqrt{\frac{\rho }{K}}l) -\exp (-\lambda \sqrt{\frac{\rho }{K}}% l) =0, \\ \exp (\sqrt{\frac{I_{\rho }}{EI}}(\lambda l+\int_{0}^{l}\frac{b(y)}{2I_{\rho }}dy))-\exp (-\sqrt{\frac{I_{\rho }}{EI}}(\lambda l +\int_{0}^{l}\frac{b(y)}{% 2I_{\rho }}dy) =0, \end{gather*} these imply that \begin{equation*} \mathop{\rm Re}(\lambda )=0\ \ or\ \ \mathop{\rm Re}(\lambda )=-\frac{1}{l}% \int_{0}^{l}(\frac{b(x)}{2I_{\rho }})dx, \end{equation*} and thus \begin{equation*} \alpha =\sup \{\mathop{\rm Re}(\lambda ),\lambda \in \sigma(L_{2})\}=0. \end{equation*} Then $r_{e}(e^{L_{2}t})=1$. Using the property $r(e^{Lt})\geq r_{e}(e^{Lt})$ we deduce, the non uniform stability of the semigroup associated to the Timoshenko beam if $\frac{K}{\rho }\neq \frac{EI}{I\rho }$. \end{proof} \section{Uniform stability} In this section our main result is the following theorem. \begin{theorem} \label{thm3} Assuming that $b(x)$ satisfies (\ref{ahme2}), $e^{Lt}$ is exponentially stable if $\frac{K}{\rho }=\frac{EI}{I_{\rho }}$. \end{theorem} The proof of the uniform stability is based on the construction of a Lyapunov function $\phi (Y(t))$ satisfying the following inequalities: \begin{itemize} \item[i)] There exist two positives constants, $d_{0},d_{1}$ such that \begin{equation} d_{0}\left\| Y\right\| ^{2}\leq \phi (Y)\leq d_{1}\left\| Y\right\| ^{2}\quad \forall \;Y\;\epsilon \;D(L) \label{ahm3} \end{equation} \item[ii)] There exist a positive constant $d_{2}$ such that \begin{equation} \partial _{t}\phi (Y(t))\leq -d_{2}\left\| Y(t)\right\| ^{2} \label{ahm4} \end{equation} \end{itemize} To construct this function we will use the multiplicative technique. \begin{lemma} \label{lm4} If \ $\phi (Y(t))$ satisfies (\ref{ahm3}), (\ref{ahm4}), then there exist $m$ and $e$ a positive constant such that \[ \| Y(t)\| ^{2}\leq m \exp (-et)\| Y(0)\| ^{2} \] \end{lemma} \begin{proof} It follows from (\ref{ahm3}) that $-\left\| Y(t)\right\| ^{2}\leq -\frac{1}{% d_{1}}\phi (Y(t))$. Using (\ref{ahm4}) then we have \begin{equation*} \partial _{t}\phi (Y(t))\leq -\frac{d_{2}}{d_{1}}\phi (Y(t)) \end{equation*} thus \begin{equation*} \phi (Y(t))\leq \exp (-\frac{d_{2}}{d_{1}}t)\phi (Y)\,. \end{equation*} Using (\ref{ahm3}), we have $\| Y(t)\| ^{2}\leq \frac{d_{1}}{d_{0}}\exp (-% \frac{d_{2}}{d_{1}}t)\| Y\| ^{2}$ \end{proof} \begin{lemma} \label{lm5} For any positive constant $\varepsilon _{1}$ and scalar functions $h,c$ such that: $h(0) >0$, $h(l)<0$, $h_{x}<0$ on $[b_{0},b_{1}]$, $h_{x}>0$ on $[0,l]\backslash \lbrack b_{0},b_{1}]$, and $c_{xx} <0$ on $[0,l]$, we have the following statements: \\ i) \begin{align*} \partial _{t}\int_{0}^{l}I_{\rho }\partial _{t}\varphi .h(x)\varphi _{x}dx =&\frac{-EI}{2}\int_{0}^{l}h_{x}(\varphi _{x})^{2}dx-\frac{I_{\rho }% }{2}\int_{0}^{l}h_{x}(\partial _{t}\varphi )^{2}dx+\frac{EI}{2}[h.(\varphi _{x})^{2}]_{0}^{l} \\ &-\int_{0}^{l}(b(x)\partial _{t}\varphi ).h(x)\varphi _{x}dx+K\int_{0}^{l}(w_{x}-\varphi ).h\varphi _{x}dx, \end{align*} ii) \begin{align*} \partial _{t}\int_{0}^{l}I_{\rho }\partial _{t}\varphi .c(x)\varphi dx =&-EI\int_{0}^{l}c(\varphi _{x})^{2}dx+\frac{EI}{2}\int_{0}^{l}c_{xx}(% \varphi )^{2}dx+I_{\rho }\int_{0}^{l}c(\partial _{t}\varphi )^{2}dx. \\ &+K\int_{0}^{l}(w_{x}-\varphi ).c\varphi dx-\int_{0}^{l}(b(x)\partial _{t}\varphi ).c(x)\varphi dx. \end{align*} iii) \begin{align*} &\partial _{t}\int_{0}^{l}\rho \partial _{t}w.\varepsilon _{1}(-\partial _{xx})^{-1}\partial _{x}(h(x)\varphi _{x}+c(x)\varphi )dx \\ &=K\int_{0}^{l}\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi dx.\int_{0}^{l}\varphi dx+\int_{0}^{l}K(w_{x}-\varphi ).(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx \\ &+\int_{0}^{l}\rho \partial _{t}w.((-\partial _{xx})^{-1}\partial _{x}(\varepsilon _{1}h(x)\partial _{t}\varphi _{x}+\varepsilon _{1}c(x)\partial _{t}\varphi ))dx, \end{align*} where $(-\partial _{xx})^{-1}$ is taken in the sense of Dirichlet boundary conditions. \end{lemma} \begin{proof} i) Multiplying the second equation in \ref{1'}) by $h(x)\varphi _{x}$, we get \begin{align*} &\partial _{t}\int_{0}^{l}I_{\rho }\partial _{t}\varphi .h(x)\varphi _{x}dx=\int_{0}^{l}I_{\rho }\partial _{tt}\varphi .h(x)\varphi _{x}dx+\int_{0}^{l}I_{\rho }\partial _{t}\varphi .h(x)\partial _{t}\varphi _{x}dx \\ &=\int_{0}^{l}(EI\varphi _{xx}+K(w_{x}-\varphi )-b(x)\partial _{t}\varphi ).h(x)\varphi _{x}dx+\int_{0}^{l}I_{\rho }\partial _{t}\varphi .h(x)\partial _{t}\varphi _{x}dx \\ &=\int_{0}^{l}(-\frac{EI}{2}(\varphi _{x})^{2}h_{x}-\frac{I_{\rho }}{2} (\partial _{t}\varphi )^{2}h_{x})dx \\ &\quad+\int_{0}^{l}(K(w_{x}-\varphi)-b(x)\partial _{t}\varphi ).h(x) \varphi _{x}dx+\frac{EI}{2}[h.(\varphi_{x})^{2}]_{0}^{l} \end{align*} ii) Multiplying the second equation of (\ref{1'}) by $c(x)\varphi $, we get \begin{align*} &\partial _{t}\int_{0}^{l}I_{\rho }\partial _{t}\varphi .c(x)\varphi dx \\ &=\int_{0}^{l}I_{\rho }\partial _{tt}\varphi .c(x)\varphi dx+\int_{0}^{l}I_{\rho }\partial _{t}\varphi .c(x)\partial _{t}\varphi dx \\ &=\int_{0}^{l}(EI\varphi _{xx}+K(w_{x}-\varphi )-b(x)\partial _{t}\varphi ).c(x)\varphi dx+\int_{0}^{l}I_{\rho }\partial _{t}\varphi .c(x)\partial _{t}\varphi dx \\ &=-EI\int_{0}^{l}c(\varphi _{x})^{2}dx+\frac{EI}{2}\int_{0}^{l}c_{xx}(% \varphi )^{2}dx+K\int_{0}^{l}(w_{x}-\varphi ).c\varphi dx \\ &\quad -\int_{0}^{l}(b(x)\partial _{t}\varphi ).c(x)\varphi dx+I_{\rho }\int_{0}^{l}c(\partial _{t}\varphi )^{2}dx \end{align*} iii) Multiplying the first equation of (\ref{1'}) by \begin{equation*} f=(-\partial _{xx})^{-1}\partial _{x}(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi ),\ \ f(0)=f(l)=0, \end{equation*} we obtain \begin{align*} &\partial _{t}\int_{0}^{l}(\rho \partial _{t}w.f)dx \\ &=\int_{0}^{l}K\partial_{x}(w_{x}-\varphi ).f\,dx +\int_{0}^{l}\rho \partial _{t}w.(\partial _{t}f)dx \\ &=-\int_{0}^{l}K(w_{x}-\varphi ).f_{x}\;dx+\int_{0}^{l}\rho \partial _{t}w.(\partial _{t}f)dx \\ &=-\int_{0}^{l}K(w_{x}-\varphi ).\partial _{x}(-\partial _{xx})^{-1}\partial _{x}(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx+\int_{0}^{l}\rho \partial _{t}w.(\partial _{t}f)dx \end{align*} Note that $-\partial _{x}(-\partial _{xx})^{-1}\partial _{x}v=v-\frac{1}{l} \int_{0}^{l}v\;dx$. Then \begin{align*} &\partial _{t}\int_{0}^{l}(\rho \partial _{t}w.f)dx \\ &=\int_{0}^{l}K(w_{x}-\varphi ).(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx \\ &\quad-\frac{1}{l}\int_{0}^{l}\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi dx.\int_{0}^{l}K(w_{x}-\varphi )dx+\int_{0}^{l}\rho \partial _{t}w.(\partial _{t}f)dx \\ &=\int_{0}^{l}K(w_{x}-\varphi ).(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx+\frac{K}{l}\int_{0}^{l}\varepsilon _{1}h(x)\varphi _{x} \\ &\quad+\varepsilon _{1}c(x)\varphi dx.\int_{0}^{l}\varphi dx -\int_{0}^{l}\rho \partial _{t}w.((-\partial _{xx})^{-1}\partial _{x}(\varepsilon _{1}h(x)\partial _{t}\varphi _{x}+\varepsilon _{1}c(x)\partial _{t}\varphi ))dx. \end{align*} \end{proof} Now we introduce the function \begin{equation*} \phi _{1}(Y(t)):=\mathcal{E}(t)+\varepsilon _{1}\int_{0}^{l}I_{\rho }\partial _{t}\varphi .h(x)\varphi _{x}dx+\varepsilon _{1}\int_{0}^{l}I_{\rho }\partial _{t}\varphi .c(x)\varphi dx-\int_{0}^{l}(\rho \partial _{t}w.f)dx \end{equation*} \begin{lemma} \label{lm6} For positive constants $a_{i}$ ($i=1,4$) and \[ \alpha =| (h_{x}\partial _{x}(-\partial _{xx})^{-1}-h)| _{L^{2}(0,l)}\,,\quad \beta =| \partial _{x}(-\partial _{xx})^{-1}| _{L^{2}(0,l)} \] we have \begin{align*} &| \partial _{t}\phi _{1}(Y(t))| \\ &\leq \int_{0}^{l}(-b(x)-\varepsilon _{1}\frac{I_{\rho }}{2}h_{x}+\varepsilon _{1}I_{\rho }c+\varepsilon _{1}\frac{a_{1}b^{2}}{2}+\varepsilon _{1}\frac{% a_{2}b^{2}}{2}+\frac{\varepsilon _{1}\alpha }{2a_{3}}+\frac{\varepsilon _{1}\beta c^{2}}{2a_{4}})(\partial _{t}\varphi )^{2}dx\\ &\quad+\int_{0}^{l}(-\varepsilon _{1}\frac{EI}{2}h_{x}-\varepsilon _{1}EIc+\varepsilon _{1}\frac{h^{2}}{2a_{1}})(\varphi _{x})^{2}dx +\varepsilon _{1}\frac{EI}{2}[h.(\varphi _{x})^{2}]_{0}^{l}\\ &\quad +\int_{0}^{l}(\varepsilon _{1}\frac{EI}{2} c_{xx}+\varepsilon _{1}\frac{c^{2}}{2a_{2}})(\varphi )^{2}dx \\ &\quad+\big| K\int_{0}^{l}(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx.\int_{0}^{l}\varphi dx\big| \\ &\quad+(\frac{\varepsilon _{1}a_{3}\alpha }{2}+\frac{\varepsilon _{1}a_{4}\beta }{2})\int_{0}^{l}(\rho \partial _{t}w)^{2}dx. \end{align*} \end{lemma} \begin{proof} Note that \begin{align*} & \partial _{t}\phi _{1}(Y(t)) \\ =& \partial _{t}\mathcal{E}(t)+\varepsilon _{1}\partial _{t}\int_{0}^{l}I_{\rho }\partial _{t}\varphi .h(x)\varphi _{x}dx+\varepsilon _{1}\partial _{t}\int_{0}^{l}I_{\rho }\partial _{t}\varphi .c(x)\varphi dx-\partial _{t}\int_{0}^{l}(\rho \partial _{t}w.f)dx \\ =& -\int_{0}^{l}b(x)(\partial _{t}\varphi )^{2}dx+\varepsilon _{1}(\int_{0}^{l}(-\frac{EI}{2}(\varphi _{x})^{2}h_{x}-\frac{I_{\rho }}{2}% (\partial _{t}\varphi )^{2}h_{x})dx \\ & +\int_{0}^{l}(K(w_{x}-\varphi )-b(x)\partial _{t}\varphi ).h(x)\varphi _{x}dx+\frac{EI}{2}[h.(\varphi _{x})^{2}]_{0}^{l}) \\ & +\varepsilon _{1}(-EI\int_{0}^{l}c(\varphi _{x})^{2}dx+\frac{EI}{2}% \int_{0}^{l}c_{xx}(\varphi )^{2}dx+K\int_{0}^{l}(w_{x}-\varphi ).c\varphi dx \\ & -\int_{0}^{l}(b(x)\partial _{t}\varphi ).c(x)\varphi dx+I_{\rho }\int_{0}^{l}c(\partial _{t}\varphi )^{2}dx) \\ & -\int_{0}^{l}K(w_{x}-\varphi ).(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx \\ & -\frac{K}{l}\int_{0}^{l}(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx.\int_{0}^{l}\varphi dx \\ & -\int_{0}^{l}\rho \partial _{t}w.((-\partial _{xx})^{-1}\partial _{x}(\varepsilon _{1}h(x)\partial _{t}\varphi _{x}+\varepsilon _{1}c(x)\partial _{t}\varphi ))dx \\ =& \int_{0}^{l}(-b(x)-\varepsilon _{1}\frac{I_{\rho }}{2}h_{x}+\varepsilon _{1}I_{\rho }c)(\partial _{t}\varphi )^{2}dx+\int_{0}^{l}(-\varepsilon _{1}% \frac{EI}{2}h_{x}-\varepsilon _{1}EIc)(\varphi _{x})^{2}dx \\ & -\varepsilon _{1}\int_{0}^{l}(b(x)\partial _{t}\varphi ).h(x)\varphi _{x}dx-K\int_{0}^{l}(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx.\int_{0}^{l}\varphi dx \\ & +\varepsilon _{1}\frac{EI}{2}[h.(\varphi _{x})^{2}]_{0}^{l}+\varepsilon _{1}\frac{EI}{2}\int_{0}^{l}c_{xx}(\varphi )^{2}dx-\varepsilon _{1}\int_{0}^{l}(b(x)\partial _{t}\varphi ).c(x)\varphi dx \\ & -\int_{0}^{l}\rho \partial _{t}w.((-\partial _{xx})^{-1}\partial _{x}(\varepsilon _{1}h(x)\partial _{t}\varphi _{x}+\varepsilon _{1}c(x)\partial _{t}\varphi ))dx. \\ =& \int_{0}^{l}(-b(x)-\varepsilon _{1}\frac{I_{\rho }}{2}h_{x}+\varepsilon _{2}I_{\rho }c)(\partial _{t}\varphi )^{2}dx+\int_{0}^{l}(-\varepsilon _{1}% \frac{EI}{2}h_{x}-\varepsilon _{1}EIc)(\varphi _{x})^{2}dx \\ & -\varepsilon _{1}\int_{0}^{l}(b(x)\partial _{t}\varphi ).h(x)\varphi _{x}dx-\frac{K}{l}\int_{0}^{l}(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx.\int_{0}^{l}\varphi dx \\ & +\varepsilon _{1}\frac{EI}{2}[h.(\varphi _{x})^{2}]_{0}^{l}+\varepsilon _{1}\frac{EI}{2}\int_{0}^{l}c_{xx}(\varphi )^{2}dx-\varepsilon _{1}\int_{0}^{l}(b(x)\partial _{t}\varphi ).c(x)\varphi dx \\ & -\int_{0}^{l}\rho \varepsilon _{1}(h_{x}\partial _{x}(-\partial _{xx})^{-1}-h)\partial _{t}w.\partial _{t}\varphi dx+\int_{0}^{l}\rho \varepsilon _{1}\partial _{x}(-\partial _{xx})^{-1}\partial _{t}w.c(x)\partial _{t}\varphi dx. \end{align*}% Using Young's inequality we obtain \begin{gather*} \big|\int_{0}^{l}(b(x)\partial _{t}\varphi ).h(x)\varphi _{x}dx\big|\leq \frac{a_{1}}{2}\int_{0}^{l}(b(x)\partial _{t}\varphi )^{2}dx+\frac{1}{2a_{1}}% \int_{0}^{l}(h(x)\varphi _{x})^{2}dx\text{,} \\ \big|\int_{0}^{l}(b(x)\partial _{t}\varphi ).c(x)\varphi dx\big|\leq \frac{% a_{2}}{2}\int_{0}^{l}(b(x)\partial _{t}\varphi )^{2}dx+\frac{1}{2a_{2}}% \int_{0}^{l}(c(x)\varphi )^{2}dx, \\ \big|\int_{0}^{l}\rho \varepsilon _{1}(h_{x}\partial _{x}(-\partial _{xx})^{-1}-h)\partial _{t}w.\partial _{t}\varphi dx\big|\leq \frac{% \varepsilon _{1}a_{3}\alpha }{2}\int_{0}^{l}(\rho \partial _{t}w)^{2}dx+% \frac{\varepsilon _{1}\alpha }{2a_{3}}\int_{0}^{l}(\partial _{t}\varphi )^{2}dx\,, \\ \big|\int_{0}^{l}\rho \varepsilon _{1}\partial _{x}(-\partial _{xx})^{-1}\partial _{t}w.c(x)\partial _{t}\varphi dx\big|\leq \frac{% \varepsilon _{1}a_{4}\beta }{2}\int_{0}^{l}(\rho \partial _{t}w)^{2}dx+\frac{% \varepsilon _{1}\beta }{2a_{4}}\int_{0}^{l}(c(x)\partial _{t}\varphi )^{2}dx. \end{gather*}% Then \begin{align*} & |\partial _{t}\phi _{1}(Y(t))| \\ & \leq \int_{0}^{l}(-b(x)-\varepsilon _{1}\frac{I_{\rho }}{2}% h_{x}+\varepsilon _{1}I_{\rho }c+\varepsilon _{1}\frac{a_{1}b^{2}}{2}% +\varepsilon _{1}\frac{a_{2}b^{2}}{2}+\frac{\varepsilon _{1}\alpha }{2a_{3}}+% \frac{\varepsilon _{1}\beta c^{2}}{2a_{4}})(\partial _{t}\varphi )^{2}dx \\ & \quad +\int_{0}^{l}(-\varepsilon _{1}\frac{EI}{2}h_{x}-\varepsilon _{1}EIc+\varepsilon _{1}\frac{h^{2}}{2a_{1}})(\varphi _{x})^{2}dx \\ & +\varepsilon _{1}\frac{EI}{2}[h.(\varphi _{x})^{2}]_{0}^{l}+\int_{0}^{l}(\varepsilon _{1}\frac{EI}{2}% c_{xx}+\varepsilon _{1}\frac{c^{2}}{2a_{2}})(\varphi )^{2}dx \\ & \quad +\big|\frac{K}{l}\int_{0}^{l}(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx.\int_{0}^{l}\varphi dx\big|+(\frac{% \varepsilon _{1}a_{3}\alpha }{2}+\frac{\varepsilon _{1}a_{4}\beta }{2}% )\int_{0}^{l}(\rho \partial _{t}w)^{2}dx. \end{align*} \end{proof} \begin{lemma} \label{lm7} For scalar positive constants $a_{i}$ ($i=5,6$), and scalar functions $k$, $d$ such that $k(0) >0$, $k(l)<0$, $k_{x}<0$ on $[b_{0},b_{1}]$, $k_{x}>0$ on $[0,l]\backslash [b_{0},b_{1}]$, and $d >0$ on $[0,l]$, we have: i) \begin{align*} \big| \partial _{t}\int_{0}^{l}\rho \partial _{t}w.k(x)w_{x}dx\big| &\leq \int_{0}^{l}(-\frac{K}{2}k_{x}+K\frac{a_{5}}{2})(w_{x})^{2}dx +K\frac{1}{2a_{5}}\int_{0}^{l}(k(x)\varphi _{x})^{2}dx \\ &\quad -\int_{0}^{l}\frac{\rho }{2}k_{x}(\partial _{t}w)^{2}dx +\frac{K}{2}[k.(w_{x})^{2}]_{0}^{l}, \end{align*} ii) \[-\partial _{t}\int_{0}^{l}\rho \partial _{t}w.d(x)wdx=\int_{0}^{l}K(w_{x}-\varphi ).\partial _{x}(d(x)w)dx-\int_{0}^{l}\rho d(x)(\partial _{t}w)^{2}dx, \] and iii) \begin{align*} &\big| \partial _{t}\int_{0}^{l}(\partial _{t}\varphi .(\varphi -w_{x})-\partial _{t}w.\varphi _{x})dx\big| \\ &\leq \int_{0}^{l}(-\frac{K}{I_{\rho }} +\frac{a_{6}}{2I_{\rho }})(\varphi -w_{x})^{2}dx + \int_{0}^{l}(1+\frac{I_{\rho }b^{2}}{2a_{6}})(\partial _{t}\varphi )^{2}dx+\big| \frac{K}{\rho }[\varphi _{x}.w_{x}]_{0}^{l}\big| . \end{align*} \end{lemma} \begin{proof} i) Multiplying the first equation in (\ref{1'}) by $k(x)w_{x}$, we obtain \begin{align*} &\partial _{t}\int_{0}^{l}\rho \partial _{t}w.k(x)w_{x}dx \\ &=\int_{0}^{l}\rho\partial _{tt}w.k(x)w_{x}dx+\int_{0}^{l}\rho \partial _{t}w.k(x)\partial_{t}w_{x}dx \\ &=\int_{0}^{l}K\partial _{x}(w_{x}-\varphi ).k(x)w_{x}dx+\int_{0}^{l}\rho \partial _{t}w.k(x)\partial _{t}w_{x}dx \\ &=-\int_{0}^{l}\frac{K}{2}k_{x}(w_{x})^{2}dx-\int_{0}^{l}K\varphi _{x}.k(x)w_{x}dx -\int_{0}^{l}\frac{\rho }{2}k_{x}(\partial _{t}w)^{2}dx+ \frac{K}{2}[k.(w_{x})^{2}]_{0}^{l} \end{align*} Using Young's inequality we obtain \begin{equation*} \int_{0}^{l}\varphi _{x}.k(x)w_{x}dx\leq \frac{a_{5}}{2}% \int_{0}^{l}(w_{x})^{2}dx+\frac{1}{2a_{5}}\int_{0}^{l}(k(x)\varphi _{x})^{2}dx. \end{equation*} Then \begin{align*} \partial _{t}\int_{0}^{l}I_{\rho }\partial _{t}\varphi .h(x)\varphi _{x}dx \leq &\int_{0}^{l}(-\frac{K}{2}k_{x}+K\frac{a_{5}}{2})(w_{x})^{2}dx +K\frac{1% }{2a_{5}}\int_{0}^{l}(k(x)\varphi _{x})^{2}dx \\ &-\int_{0}^{l}\frac{\rho }{2}k_{x}(\partial _{t}w)^{2}dx+\frac{K}{2}% [k.(w_{x})^{2}]_{0}^{l}\,. \end{align*} ii) Multiplying the first equation in (\ref{1'}) by $d(x)w$, we obtain \begin{align*} -\partial _{t}\int_{0}^{l}\rho \partial _{t}w.d(x)wdx &=-\int_{0}^{l}\rho \partial _{tt}w.d(x)wdx-\int_{0}^{l}\rho \partial _{t}w.d(x)\partial _{t}wdx \\ &=-\int_{0}^{l}K\partial _{x}(w_{x}-\varphi ).d(x)wdx-\int_{0}^{l}\rho d(x)(\partial _{t}w)^{2}dx \\ &=\int_{0}^{l}K(w_{x}-\varphi ).\partial _{x}(d(x)w)dx-\int_{0}^{l}\rho d(x)(\partial _{t}w)^{2}dx \end{align*} iii) Multiplying the first equation in (\ref{1'}) by $\varphi _{x}$ and the second equation of system (\ref{1'}) by $\varphi -w_{x}$, we obtain \begin{align*} &\partial _{t}\int_{0}^{l}(\partial _{t}\varphi .(\varphi -w_{x})-\partial _{t}w.\varphi _{x})dx \\ &=\int_{0}^{l}\partial _{tt}\varphi .(\varphi-w_{x})dx+\int_{0}^{l} \partial _{t}\varphi .(\partial _{t}\varphi -\partial_{t}w_{x})dx \\ &\quad -\int_{0}^{l}\partial _{tt}w.\varphi _{x}dx-\int_{0}^{l}\partial _{t}w\partial _{t}\varphi _{x}dx \\ &=\int_{0}^{l}\frac{EI}{I_{\rho }}\varphi _{xx}.(\varphi -w_{x})dx-\int_{0}^{l}\frac{K}{I_{\rho }}(\varphi -w_{x})^{2}dx-\int_{0}^{l}% \frac{b(x)}{I_{\rho }}\partial _{t}\varphi .(\varphi -w_{x})dx \\ &\quad +\int_{0}^{l}\partial _{t}\varphi .(\partial _{t}\varphi -\partial _{t}w_{x})dx-\int_{0}^{l}\frac{K}{\rho }\partial _{x}(w_{x}-\varphi ).\varphi _{x}dx-\int_{0}^{l}\partial _{t}w\partial _{t}\varphi _{x}dx. \end{align*} Using the fact that $\frac{K}{\rho }=\frac{EI}{I_{\rho }}$, we have \begin{align*} &\partial _{t}\int_{0}^{l}(\partial _{t}\varphi .(\varphi -w_{x})-\partial _{t}w.\varphi _{x})dx \\ &=-\int_{0}^{l}\frac{K}{I_{\rho }}(\varphi-w_{x})^{2}dx -\int_{0}^{l}\frac{% b(x)}{I_{\rho }}\partial _{t}\varphi.(\varphi -w_{x})dx +\int_{0}^{l}(\partial _{t}\varphi )^{2}dx-\frac{K}{\rho }[\varphi _{x}.w_{x}]_{0}^{l}. \end{align*} Using Young's inequality we have \begin{equation*} \big| \int_{0}^{l}\frac{b(x)}{I_{\rho }}\partial _{t}\varphi .(\varphi -w_{x})dx\big| \leq \frac{I_{\rho }}{2a_{6}}\int_{0}^{l}(b(x)\partial _{t}\varphi )^{2}dx+\frac{a_{6}}{2I_{\rho }}\int_{0}^{l}(\varphi -w_{x})^{2}dx \end{equation*} Therefore, \begin{align*} &\big| \partial _{t}\int_{0}^{l}(\partial _{t}\varphi .(\varphi -w_{x})-\partial _{t}w.\varphi _{x})dx\big| \\ &\leq \int_{0}^{l}(-\frac{K}{I_{\rho }}+\frac{a_{6}}{2I_{\rho }}) (\varphi -w_{x})^{2}dx +\int_{0}^{l}(1+\frac{I_{\rho }b^{2}}{2a_{6}})(\partial _{t}\varphi )^{2}dx+\big| \frac{K}{\rho }[\varphi _{x}.w_{x}]_{0}^{l}\big| . \end{align*} \end{proof} Now, we define the Lyapunov function associated to this problem. \begin{equation} \begin{aligned} \phi (Y(t)) =&\phi _{1}(Y(t))+\varepsilon _{2}\int_{0}^{l}(\partial _{t}\varphi .(\varphi -w_{x})-\partial _{t}w.\varphi _{x})dx \\ &+\varepsilon _{3}\int_{0}^{l}\rho \partial _{t}w.(k(x)w_{x}-d(x)w)dx. \end{aligned} \label{e13} \end{equation} \begin{lemma} \label{lm8} For $\varepsilon _{i}$ ($i=1,2,3$) sufficiently small and $c,h,k,d$ satisfying \begin{gather*} -\frac{h_{x}(x)}{2}+c(x) <0\quad\forall \; x \in ]0,l[\backslash]b_{0},b_{1}[ \\ \frac{h_{x}(x)}{2}+c(x) >0\quad \forall \; x \in ]0,l[ \\ \frac{k_{x}(x)}{2}+d(x) > 0\quad \forall \;x \in ]0,l[, \end{gather*} the function $\phi (Y(t))$ satisfies (\ref{ahm3}) and (\ref{ahm4}). \end{lemma} \begin{proof} Note that \begin{align*} \partial _{t}\phi (Y(t))=& \partial _{t}\phi _{1}(Y(t))+\varepsilon _{2}\partial _{t}\int_{0}^{l}(\partial _{t}\varphi .(\varphi -w_{x})-\partial _{t}w.\varphi _{x})dx \\ & +\varepsilon _{3}\partial _{t}\int_{0}^{l}\rho \partial _{t}w.(k(x)w_{x}-d(x)w)dx. \end{align*}% Using the Lemmas \ref{lm5} and \ref{lm6}, we obtain \begin{align*} & |\partial _{t}\phi (Y(t))| \\ & \leq \int_{0}^{l}(-b(x)-\varepsilon _{1}\frac{I_{\rho }}{2}% h_{x}+\varepsilon _{1}I_{\rho }c(x)+\varepsilon _{1}\frac{a_{1}b^{2}}{2}% +\varepsilon _{1}\frac{a_{2}b^{2}}{2}+\frac{\varepsilon _{1}\alpha }{2a_{3}} \\ & \quad +\frac{\varepsilon _{1}\beta c^{2}}{2a_{4}}+\varepsilon _{2}(1+\frac{% I_{\rho }b^{2}}{2a_{6}}))(\partial _{t}\varphi )^{2}dx \\ & \quad +\int_{0}^{l}(-\varepsilon _{1}\frac{EI}{2}h_{x}-\varepsilon _{1}EIc+\varepsilon _{1}\frac{(h(x))^{2}}{2a_{1}}+K\frac{\varepsilon _{3}}{% 2a_{5}}(k(x))^{2})(\varphi _{x})^{2}dx \\ & \quad +\int_{0}^{l}(\varepsilon _{1}\frac{EI}{2}c_{xx}+\varepsilon _{1}% \frac{(c(x))^{2}}{2a_{2}})(\varphi )^{2}dx+\varepsilon _{2}\int_{0}^{l}(-% \frac{K}{I_{\rho }}+\frac{a_{6}}{2I_{\rho }})(\varphi -w_{x})^{2}dx \\ & \quad +\varepsilon _{3}\big|\int_{0}^{l}K(w_{x}-\varphi ).\partial _{x}(d(x)w)dx\big|+\big|\frac{K}{l}\int_{0}^{l}(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx.\int_{0}^{l}\varphi dx\big| \\ & \quad +\varepsilon _{2}|\frac{K}{\rho }[\varphi _{x}.w_{x}]_{0}^{l}|+\varepsilon _{1}\frac{EI}{2}[h.(\varphi _{x})^{2}]_{0}^{l}+\frac{K\varepsilon _{3}}{2}[k.(w_{x})^{2}]_{0}^{l} \\ & \quad +\varepsilon _{3}\int_{0}^{l}(-\frac{K}{2}k_{x}+K\frac{a_{5}}{2}% )(w_{x})^{2}dx \\ & \quad +\int_{0}^{l}(-\frac{\varepsilon _{3}\rho }{2}k_{x}-\varepsilon _{3}\rho d(x)+(\frac{\varepsilon _{1}a_{3}\alpha }{2}+\frac{\varepsilon _{1}a_{4}\beta }{2})\rho ^{2})(\partial _{t}w)^{2}dx\,. \end{align*}% Using Cauchy Shwartz's inequality and Young's inequality, we obtain \begin{equation*} \big|\int_{0}^{l}K(w_{x}-\varphi ).(d(x)w_{x})dx\big|\leq \frac{Ka_{7}}{2}% \int_{0}^{l}(w_{x}-\varphi )^{2}dx+\frac{K}{2a_{7}}% \int_{0}^{l}(d(x)w_{x})^{2}dx \end{equation*}% where $a_{7}>0$, \begin{equation*} \big|\int_{0}^{l}K(w_{x}-\varphi ).d_{x}wdx\big|\leq \frac{Ka_{7}}{2}% \int_{0}^{l}(w_{x}-\varphi )^{2}dx+\frac{K\overline{d}^{2}c_{0}}{2a_{7}}% \int_{0}^{l}(w_{x})^{2}dx \end{equation*}% where $\overline{d}=\max (d_{x})$, \begin{align*} & \big|\frac{K}{l}\int_{0}^{l}(\varepsilon _{1}h(x)\varphi _{x}+\varepsilon _{1}c(x)\varphi )dx.\int_{0}^{l}\varphi dx\big| \\ & \leq \frac{K}{l}\frac{l^{2}\varepsilon _{1}}{2}\int_{0}^{l}(h(x)\varphi _{x})^{2}dx+\int_{0}^{l}(\frac{Kl^{2}\varepsilon _{1}(2+c^{2}(x))}{2}% )\varphi ^{2}dx. \end{align*}% Then \begin{align*} |\partial _{t}\phi (Y(t))|& \leq \int_{0}^{l}(-b(x)-\varepsilon _{1}\frac{% I_{\rho }}{2}h_{x}+\varepsilon _{1}I_{\rho }c(x)+\varepsilon _{1}\frac{% a_{1}b^{2}}{2}+\varepsilon _{1}\frac{a_{2}b^{2}}{2}+\frac{\varepsilon _{1}\alpha }{2a_{3}} \\ & \quad +\frac{\varepsilon _{1}\beta c^{2}}{2a_{4}}+\varepsilon _{2}(1+\frac{% I_{\rho }b^{2}}{2a_{6}}))(\partial _{t}\varphi )^{2}dx \\ & \quad +\int_{0}^{l}(-\varepsilon _{1}\frac{EI}{2}h_{x}-\varepsilon _{1}EIc+\varepsilon _{1}\frac{(h(x))^{2}}{2a_{1}}+K\frac{\varepsilon _{3}}{% 2a_{5}}(k(x))^{2} \\ & \quad +K\frac{l^{2}\varepsilon _{1}}{2l}(h(x))^{2}+c_{0}(\frac{% Kl^{2}\varepsilon _{1}(2+\overline{c}^{2})}{2}))(\varphi _{x})^{2}dx \\ & \quad +\int_{0}^{l}(\varepsilon _{1}\frac{EI}{2}c_{xx}+\varepsilon _{1}% \frac{(c(x))^{2}}{2a_{2}})(\varphi )^{2}dx \\ & \quad +\int_{0}^{l}(-\frac{\varepsilon _{2}K}{I_{\rho }}+\frac{\varepsilon _{2}a_{6}}{2I_{\rho }}+\varepsilon _{3}Ka_{7})(\varphi -w_{x})^{2}dx \\ & \quad +(\varepsilon _{2}\frac{K}{2\rho }+\varepsilon _{1}\frac{EI}{2}% h(l))(\varphi _{x}(l))^{2}+(\varepsilon _{2}\frac{K}{2\rho }-\varepsilon _{1}% \frac{EI}{2}h(0))(\varphi _{x}(0))^{2} \\ & \quad +(\varepsilon _{2}\frac{K}{2\rho }+\frac{K\varepsilon _{3}}{2}% k(l))(w_{x}(l))^{2}+(\varepsilon _{2}\frac{K}{2\rho }-\frac{K\varepsilon _{3}% }{2}k(0))(w_{x}(0))^{2} \\ & \quad +\varepsilon _{3}\int_{0}^{l}(-\frac{K}{2}k_{x}+K\frac{a_{5}}{2}+% \frac{K\overline{d}^{2}c_{0}}{2a_{7}}+\frac{Kd^{2}}{2a_{7}})(w_{x})^{2}dx \\ & \quad +\int_{0}^{l}(-\frac{\varepsilon _{3}\rho }{2}k_{x}-\varepsilon _{3}\rho d(x)+(\frac{\varepsilon _{1}a_{3}\alpha }{2}+\frac{\varepsilon _{1}a_{4}\beta }{2})\rho ^{2})(\partial _{t}w)^{2}dx, \end{align*}% where $c_{0}$ is the Poincarr\'{e} constant and $\overline{c}=\max (c(x))$ on $[0,l]$, then we choose $\varepsilon _{i}$ ($i=1,2,3$) sufficiently small and we choose the constant $a_{i}$ ($i=1\dots 7$) such that \begin{gather*} -b(x)+\varepsilon _{1}\frac{a_{1}b^{2}}{2}+\varepsilon _{1}\frac{a_{2}b^{2}}{% 2}+\varepsilon _{2}\frac{I_{\rho }b^{2}}{2a_{6}}<0, \\ -\varepsilon _{1}\frac{I_{\rho }}{2}h_{x}+\varepsilon _{1}I_{\rho }c(x)+% \frac{\varepsilon _{1}\alpha }{2a_{3}}+\frac{\varepsilon _{1}\beta c^{2}}{% 2a_{4}}+\varepsilon _{2}<0, \\ -\varepsilon _{1}\frac{EI}{2}h_{x}(x)-\varepsilon _{1}EIc(x)+\varepsilon _{1}% \frac{(h(x))^{2}}{2a_{1}}+K\frac{\varepsilon _{3}}{2a_{5}}(k(x))^{2}+K\frac{% l^{2}\varepsilon _{1}}{2l}(h(x))^{2}<0, \\ -\frac{\varepsilon _{2}K}{I_{\rho }}+\frac{\varepsilon _{2}a_{6}}{2I_{\rho }}% +\frac{\varepsilon _{3}Ka_{7}}{2}<0, \\ -\frac{\varepsilon _{3}\rho }{2}k_{x}-\varepsilon _{3}\rho d(x)+(\frac{% \varepsilon _{1}a_{3}\alpha }{2}+\frac{\varepsilon _{1}a_{4}\beta }{2})\rho ^{2}<0, \\ \frac{EI}{2}c_{xx}+\frac{(c(x))^{2}}{2a_{2}}<0, \\ -\frac{K}{2}k_{x}+K\frac{a_{5}}{2}+\frac{K\overline{d}^{2}c_{0}}{2a_{7}}+% \frac{Kd^{2}}{2a_{7}}<0, \\ \varepsilon _{2}<\min (-\varepsilon _{1}EI\frac{\rho }{K}h(l),\varepsilon _{1}EI\frac{\rho }{K}h(0),-\rho \varepsilon _{3}k(l),\rho \varepsilon _{3}k(0)) \end{gather*}% Then $\phi (Y(t))$ satisfies (\ref{ahm4}). 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