\documentclass[twoside]{article} \usepackage{amssymb, amsmath} \pagestyle{myheadings} \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 $$\partial _{tt}u(t)+\mathcal{A}u(t)=h, \label{1}$$ 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 $$E(w,z)(t)=\frac{1}{2}\left( \langle\left(\begin{array}{c} w \\ z \end{array} \right) ,S\left( \begin{array}{c} w \\ z \end{array} \right) \rangle_{H\times G} +\| \partial _tw\| _{H}^2+\| \partial _tz\| _{G}^2\right) . \label{2}$$ 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 $$S=\left( \begin{array}{cc} A & B \\ B^{\ast } & C \end{array} \right) \label{3}$$ The energy $E(w,z)(t)$ is conserved for the second order system $$\left( \begin{array}{c} \partial _{tt}w \\ \partial _{tt}z \end{array} \right) +S\left( \begin{array}{c} w \\ z \end{array} \right) =0. \label{sd}$$ 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 $G$ $$\left( \begin{array}{c} \partial _{tt}w \\ \partial _{tt}z \end{array} \right) +S\left( \begin{array}{c} w \\ z \end{array} \right) +\left( \begin{array}{c} 0 \\ D\partial _tz \end{array} \right) =0. \label{depl}$$ 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 $(u,v)\in D(A)\times D(C)$ \label{e6} \left| (u,Bv)\right| 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 $$\frac{d}{dt}\chi _{\varepsilon }(\left( \begin{array}{c} U(t) \\ \partial _tU(t) \end{array} \right) )\leq -c_{\varepsilon }.\chi _{\varepsilon }(\left( \begin{array}{c} U(t) \\ \partial _tU(t) \end{array} \right) )\,. \label{(11)}$$ 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 $$\begin{gathered} \partial _{tt}u+Au+a.A^{\alpha }v=0 \\ \partial _{tt}v+A^{\beta }v+a.A^{\alpha }u+A^{\gamma }\partial _tv=0\,, \end{gathered} \label{st}$$ 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 $$\begin{gathered} \partial _{tt}u+Au-a.A^{\alpha -\frac{1}{2}}\partial _tv=0 \\ \partial _{tt}v+A^{\beta }(I-a^2A^{2\alpha -\beta -1})v+a.A^{\alpha -\frac{ 1}{2}}\partial _tu+A^{\gamma }\partial _tv=0\,. \end{gathered} \label{st1}$$ 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 $$\begin{gathered} \partial _{tt}u+Au-a.A^{\alpha -\frac{1}{2}}\partial _tv=0 \\ \partial _{tt}v+A^{\beta }v+a.A^{\alpha -\frac{1}{2}}\partial _tu+A^{\gamma }\partial _tv=0\,. \end{gathered} \label{st2}$$ \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 $$\begin{gathered} \rho \partial _{tt}u=K\partial _{xx}u-K\partial _{x}v \quad\mbox{in } ]0,l[\times \mathbb{R}^{+} \\ I_{\rho }\partial _{tt}v=EI\partial _{xx}v+K(\partial _{x}u-v)-b(x)\partial _tv\quad \mbox{in } ]0,l[\times \mathbb{R}^{+} \\ u(0,t)=u(l,t)=v(0,t)=v(l,t)=0\,. \end{gathered} \label{a1}$$ This system is coupled with the initial conditions $$\begin{gathered} u(x;0) =u_{0}(x) \quad \partial _tu(x;0)=u_1(x) \\ v(x;0) =v_{0}(x) \quad \partial _tv(x;0)=v_1(x) \end{gathered} \label{a2}$$ 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 $$\begin{gathered} \left(\begin{array}{c} \partial _{tt}\varphi (t) \\ \partial _{tt}\phi (t) \end{array} \right) =\left(\begin{array}{cc} -A & -B \\ -B^{\ast } & -C \end{array} \right) \left(\begin{array}{c} \varphi (t) \\ \phi (t) \end{array} \right) \\ \left(\begin{array}{c} \varphi (0) \\ \phi (0) \end{array} \right) =\left(\begin{array}{c} \varphi _{0} \\ \phi _{0} \end{array} \right) ,\quad \left(\begin{array}{c} \partial _t\varphi (0) \\ \partial _t\phi (0) \end{array} \right) =\left(\begin{array}{c} \varphi _1 \\ \phi _1 \end{array} \right) \end{gathered}$$ 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 $$\label{e24} \begin{gathered} \partial _{tt}u=-Au+A^{-1/2}B\partial _tv \\ \partial _{tt}v==-B^{\ast }A^{-1/2}\partial _tu-(C-B^{\ast }A^{-1}B)v-D\partial _tv \\ (u(0),v(0)) = (u_{0},v_{0}) \quad (\partial _tu(0),\partial _tv(0)) = (u_1,v_1) \end{gathered}$$ 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\$. 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