\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2006(2006), No. 20, pp. 1--12.\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 2006 Texas State University - San Marcos.} \vspace{9mm}} \begin{document} \title[\hfilneg EJDE-2006/20\hfil Boundary stabilization] {Boundary stabilization of a coupled system of nondissipative Schrodinger equations} \author[H. Ilhem\hfil EJDE-2006/20\hfilneg] {Hamchi Ilhem} % in alphabetical order \address{Hamchi Ilhem \hfil\break Department of Mathematics, University of Batna, 05000, Algeria} \email{hamchi\_ilhem@yahoo.fr} \date{} \thanks{Submitted June 20, 2005. Published February 9, 2006.} \subjclass[2000]{93D15, 35Q40, 42B15} \keywords{Boundary stabilization; Schrodinger equation; multiplier method} \begin{abstract} We use the multiplier method and the approach in \cite{a1} to study the problem of exponential stabilization of a coupled system of two Schrodinger equations. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{lemma}[theorem]{Lemma} \newtheorem{remark}[theorem]{Remark} \section{Introduction} Let $\Omega$ be a bounded domain of $\mathbb{R}^{n}$ ($n\in \mathbb{N}^{\ast}$) with smooth boundary $\Gamma$, and let $(\Gamma _{0},\Gamma_{1})$ be a partition of $\Gamma$. Consider the boundary feedback system: \begin{gather} iy_{t}+\Delta y+F_{1}(y,\nabla y)+P_{1}(z)=0\quad\text{in }Q, \label{e1.1.a1} \\ iz_{t}+\Delta z+F_{2}(z,\nabla z)+P_{2}(y)=0\quad\text{in }Q, \label{e1.1.a2} \\ y =z=0\quad\text{on }\Sigma _{0}, \label{e1.1.b1} \\ \frac{\partial y}{\partial \upsilon }+g(y_{t})=0,\quad \frac{\partial z}{\partial \upsilon }+h(z_{t})=0\quad\text{on } \Sigma _{1}, \label{e1.1.b2}\\ y(0)=y_{0},\quad z(0)=z_{0}\quad\text{in }\Omega . \label{e1.1.c} \end{gather} where $T>0$, $Q=\Omega \times ]0,T[$, $\Sigma _{l}=\Gamma_{l}\times ]0,T[$ $(l=0,1)$, $\upsilon$ is the outward unit normal to $\Gamma$, $\frac{\partial }{\partial \upsilon }$ denotes the normal derivative, $y_{t}$ and $z_{t}$ the time derivative, $\nabla$ and $\Delta$ are the Gradient and the Laplacian in the space variables. Operators $F_{l}$, $P_{l}$ $(l=1,2)$ and the functions $g$ and $h$ are defined in \S 2. Our goal in this paper is to obtain, under a suitable geometrical conditions on $(\Omega ,\Gamma _{0},\Gamma _{1})$, the exponential stability of the system \eqref{e1.1.a1}--\eqref{e1.1.c}: For any initial data $(y_{0},z_{0})$ in some space, the energy $E$ (see \eqref{e2.8}) of the solution tends to zero exponentially as $t\to +\infty$. In the case of one equation and where $F=0$ and $g$ is linear of the form $g(s)=(h.\upsilon )s$, $h$ represents a vector field satisfying some geometrical conditions, a lot of work have been (see for example \cite{m1,m2}). Recently, in \cite{l1} they have obtain a result of stabilization in the space $L^{2}(\Omega )$ of one equation where $F=0$ and linear feedback in the form $iy$. In these situations the system is dissipative (the usual energy is decreasing). Notice, that this property (the decrease of energy) specially $E(T)\leq E(0)$ has a crucial role in studying the stability of the solution. So, the approach in \cite{k1} can not be applied to the non dissipative system. To the best of our best knowledge, nonlinear boundary stabilization for the coupled system of two Schrodinger equations with lower order terms has not been considered in the literature. In this case the energy is not decreasing; we do not have any information about the sign of the derivative of the energy (see \eqref{e4.1}). This requires a careful treatment. Our paper is divided in 3 sections. In \S 2, we give notation and assumptions. In \S 3, we give the result of existence of solution. In \S 4 we give some formulas witch are needed for proving our result. In \S 5 we use the multiplier method to show, that the energy of the solution of system \eqref{e1.1.a1}--\eqref{e1.1.c} satisfies the inequalities: $$\begin{gathered} \int_{0}^{T}E(t)\leq \lambda _{1}(E(T)+E(0)) +\lambda _{2}(E(0)-E(T)),\quad\text{for all }T>0 \\ E'(t)\leq \lambda _{3}E(t),\quad \text{for all }t>0\,, \end{gathered} \label{e1.2}$$ where $\lambda _{1}\leq \lambda _{2}$. $E'$ is the time derivative of the energy. As in \cite{a1}, we have $$E(t)\leq cE(0)e^{-\omega t},\quad\text{for all }t>0. \label{e1.3}$$ \section{Notation and Assumptions} Let $x_{0}$ be a fixed point in of $\mathbb{R}^{n}$. Set $h=h(x)=x-x_{0}$ and $R=\sup \{|h(x)|:x\in \overline{\Omega }\}$. Assume that for some constant $h_{0}>0$, we have $$\Gamma _{0}=\left\{ x\in \Gamma :h\upsilon \leq 0\right\},\quad \Gamma_{1}=\left\{ x\in \Gamma :h\upsilon \geq h_{0}\right\} . \label{e2.1}$$ Assume that $F_{1}$, $F_{2}$ are linear differential operators of order one in the space variables with $L_{\infty }(\overline{Q})$-coefficients and $P_{1},P_{2}$ are linear operators of zero order with $L_{\infty }(\overline{Q})$-coefficients: \begin{gather} F_{1}=F_{1}(y,\nabla y)=v_{1}(x).\nabla y-q_{1}(x)y, \label{e2.2.a} \\ F_{2}=F_{2}(z,\nabla z)=v_{2}(x).\nabla z-q_{2}(x)z, \label{e2.2.b} \\ P_{1}=P_{1}(z)=\alpha _{1}(x)z, \label{e2.2.c} \\ P_{2}=P_{2}(y)=\alpha _{2}(x)y. \label{e2.2.d} \end{gather} Where in \eqref{e2.2.c}--\eqref{e2.2.d}, $\alpha _{1}$ and $\alpha _{2}$ are a bounded complex valued functions such that \begin{gather} \mathop{\rm Im}(\overline{\alpha _{1}\alpha _{2}})\geq 0\,, \label{e2.3.a} \\ |\alpha _{1}|=|\alpha _{1}(x)|_{L^{\infty }( \Omega )}\leq \frac{\min (1,\frac{n}{2})}{( n+R)C+R} \label{e2.3.b} \end{gather} where $q_{l}$ $(l=1,2)$ are a positives functions satisfying $$\max_{l=1,2}(|q_{l}|)\leq \frac{\min (1,\frac{n}{2})-|\alpha _{1}|(( n+R)C+R)}{2R(C+1)}. \label{e2.4}$$ Where $C$ is the positive constant satisfying \begin{equation*} \int_{\Omega }|y|^{2}\leq C\int_{\Omega }|\nabla y|^{2}. \end{equation*} We assume that for all $l=1,2$ $\ v_{l}=v_{l}(x)$ is a complex $n$-vector field with $|v_{l}|\in L_{\infty }(\overline{Q})$, the Hessian matrix $V_{l}$ of $v_{l}$ satisfies $|V_{l}|\in L_{\infty }(\overline{Q})$ and the following properties for $\mathop{\rm Im}v_{l}$ are achieved: \begin{gather} |\mathop{\rm Im}v_{l}.\mu |\leq \mathop{\rm Im}v_{l}.\upsilon \quad\text{on }\Gamma _{1}, \label{e2.5.a} \\ \mathop{\rm Im}v_{l}.\upsilon = 0\quad\text{on }\Gamma _{0}, \label{e2.5.b} \end{gather} where $\mu$ represent the tangential unit vector on $\Gamma$. Put \begin{equation*} \beta =\max_{l=1,2}\big\{|\alpha _{2}-\overline{\alpha _{1} }|,|v_{l}|,|\nabla (\alpha _{2}-\overline{ \alpha _{1}})|,|V_{l}|,|\mathop{\rm div}(\mathop{\rm Im} v_{l})|\big\}. \end{equation*} Assume that $g$ and $h$ are a complex valued functions such that there exists a constants $g_{\ast },h_{\ast },g^{\ast },h^{\ast }>0$ such that for all $s\in \mathbb{C}$ we have \begin{gather} g_{\ast }|s|\leq |g(s)|\leq g^{\ast }| s|\quad\text{and}\quad g(s)\overline{s}\geq 0, \label{e2.6}\\ h_{\ast }|s|\leq |h(s)|\leq h^{\ast }| s|\quad\text{and}\quad h(s)\overline{s}\geq 0. \label{e2.7} \end{gather} Put \begin{equation*} H_{\Gamma _{0}}^{1}(\Omega )=\big\{ u\in H^{1}(\Omega ):u=0\quad\text{on } \Gamma _{0}\big\} . \end{equation*} We define the energy $E$ of \eqref{e1.1.a1}--\eqref{e1.1.c} by $$E(t)=\frac 12\Big(\int_{\Omega }|\nabla y| ^{2}+\int_{\Omega }|\nabla z|^{2}+\int_{\Omega }q_{1}|y|^{2}+\int_{\Omega }q_{2}|z| ^{2}dx\Big)-\mathop{\rm Re}\int_{\Omega }\overline{\alpha _{1}}y \overline{z}. \label{e2.8}$$ \begin{remark} \label{rmk1} \rm We note that, because \eqref{e2.3.b}, we have \begin{equation*} \zeta _{1}\Big(\int_{\Omega }|\nabla y| ^{2}+\int_{\Omega }|\nabla z|^{2}\Big)\leq \| (y,z)\| _{H_{\Gamma _{0}}^{1}(\Omega )}^{2}\leq \zeta _{2}\Big(\int_{\Omega }|\nabla y| ^{2}+\int_{\Omega }|\nabla z|^{2}\Big), \end{equation*} where \begin{equation*} \left\| (y,z)\right\| _{H_{\Gamma _{0}}^{1}(\Omega )}^{2}=\int_{\Omega }|\nabla y|^{2}+\int_{\Omega }|\nabla z|^{2}+\int_{\Omega }q_{1}|y| ^{2}+\int_{\Omega }q_{2}|z|^{2}dx-2\mathop{\rm Re} \int_{\Omega }\alpha _{1}y\overline{z}. \end{equation*} \end{remark} \section{Existence of solutions} We define an operator \begin{equation*} A(u_{1},u_{2})=(i\Delta u_{1}+iF_{1}(u_{1},\nabla u_{1})+iP_{1}(u_{2}),i\Delta u_{2}+iF_{2}(u_{2},\nabla u_{2})+iP_{2}(u_{1})) \end{equation*} with domain \begin{align*} D(A)=\big\{&(u_{1},u_{2})\in V\times V: \text{for }l=1,2\;\Delta u_{l}\in H_{\Gamma _{0}}^{1}(\Omega ), \\ &[\frac{\partial u_{1}}{\partial \upsilon }+g(i\Delta u_{1}+iF_{1}+iP_{1})]_{\Gamma _{1}}=0, \\ &[\frac{\partial u_{2}}{\partial \upsilon }+h( i\Delta u_{2}+iF_{2}+iP_{2})]_{\Gamma _{1}}=0 \big\} \end{align*} Where $V=H^{2}(\Omega )\cap H_{\Gamma _{0}}^{1}(\Omega )$. Let $U=(u_{1},u_{2})$, we may rewrite system \eqref{e1.1.a1}--\eqref{e1.1.c} as $$\begin{gathered} U_{t}=AU \\ U(0)=U_{0}. \end{gathered} \label{e3.1}$$ then the solvability of \eqref{e1.1.a1}--\eqref{e1.1.c} is equivalent to the one of \eqref{e3.1}. We can prove that, if $g$ and $h$ are globally lipschitz, there exists a positive constant $k$ such that the operator $A-kI$ is maximal dissipative on $(H_{\Gamma _{0}}^{1}(\Omega ))^{2}$. Therefore, we have the following theorem \cite{k1,l2}. \begin{theorem} \label{thm2} 1. For every $(y_{0},z_{0})\in D(A)$ the system \eqref{e1.1.a1}--\eqref{e1.1.c} has a unique solution $(y,z)\in L^{\infty}([0,+\infty );D(A))\cap W^{1,\infty }([0,+\infty ) ;(H_{\Gamma _{0}}^{1}(\Omega ))^{2})$. \noindent 2. For every $(y_{0},z_{0})\in (H_{\Gamma _{0}}^{1}(\Omega ))^{2}$ the system \eqref{e1.1.a1}--\eqref{e1.1.c} has a unique solution $(y,z)\in C([0,+\infty );(H_{\Gamma _{0}}^{1}(\Omega ))^{2})$. \end{theorem} \section{Preliminary results} We begin by establishing a formula concerning the derivative of the energy of the system \eqref{e1.1.a1}--\eqref{e1.1.c} and two estimates concerning $\int_{\Sigma _{1}}(g(y_{t})\overline{y_{t}}+h( z_{t})\overline{z_{t}})$ and $\mathop{\rm Re}\int_{\Omega }( v_{1}.\nabla y\overline{y_{t}}+(v_{2}.\nabla z+(\alpha _{2}- \overline{\alpha _{1}})y)\overline{z_{t}})$. \begin{lemma} \label{lem3} For each solution $y$ of system \eqref{e1.1.a1}--\eqref{e1.1.c} we have $$E'(t)=\mathop{\rm Re}\int_{\Omega }(v_{1}.\nabla y\overline{y_{t} }+(v_{2}.\nabla z+(\alpha _{2}-\overline{\alpha _{1}}) y)\overline{z_{t}})-\int_{\Sigma _{1}}(g( y_{t})\overline{y_{t}}+h(z_{t})\overline{z_{t}}). \label{e4.1}$$ \end{lemma} \begin{proof} We multiply both side \eqref{e1.1.a1} by $\overline{y}_{t}$ and use Green's formula to obtain \begin{equation*} i\int_{\Omega }|y_{t}|^{2}+\int_{\Gamma }\frac{\partial y}{ \partial \upsilon }\overline{y_{t}}-\int_{\Omega }\nabla y.\nabla \overline{ y_{t}}+\int_{\Omega }(F_{1}+P_{1})\overline{y_{t}}=0. \end{equation*} Taking the real part, we obtain \begin{equation*} \mathop{\rm Re}\int_{\Omega }\nabla y.\nabla \overline{y_{t}}+\mathop{\rm Re} \int_{\Omega }q_{1}y\overline{y_{t}}=\mathop{\rm Re}\int_{\Gamma }\frac{\partial y }{\partial \upsilon }\overline{y_{t}}+\mathop{\rm Re}\int_{\Omega }( v_{1}.\nabla y+\alpha _{1}z)\overline{y_{t}}. \end{equation*} But, by \eqref{e1.1.b1}--\eqref{e1.1.b2} and \eqref{e2.6} we find that \begin{equation*} \mathop{\rm Re}\int_{\Gamma }\frac{\partial y}{\partial \upsilon }\overline{y_{t}} =-\mathop{\rm Re}\int_{\Gamma _{1}}g(y_{t})\overline{y_{t}} =-\int_{\Gamma _{1}}g(y_{t})\overline{y_{t}} \end{equation*} and $$\mathop{\rm Re}\int_{\Omega }\nabla y.\nabla \overline{y_{t}}+\mathop{\rm Re} \int_{\Omega }q_{1}y\overline{y_{t}}=-\int_{\Gamma _{1}}g(y_{t}) \overline{y_{t}}+\mathop{\rm Re}\int_{\Omega }(v_{1}.\nabla y+\alpha _{1}z)\overline{y_{t}}. \label{e4.2.a}$$ We obtain similar identity for $z$: $$\mathop{\rm Re}\int_{\Omega }\nabla z.\nabla \overline{z_{t}}+\mathop{\rm Re} \int_{\Omega }q_{2}z\overline{z_{t}}=-\int_{\Gamma _{1}}h(z_{t}) \overline{z_{t}}+\mathop{\rm Re}\int_{\Omega }(v_{2}.\nabla z+\alpha _{2}y)\overline{z_{t}}. \label{e4.2.b}$$ However, \begin{equation*} \mathop{\rm Re}\int_{\Omega }(\nabla y.\nabla \overline{y_{t}}+\nabla z.\nabla \overline{z_{t}}+q_{1}y\overline{y_{t}}+q_{2}z\overline{z_{t}}- \overline{\alpha _{1}}(\overline{z}y_{t}+\overline{z_{t}}y) )=E'(t)\,. \end{equation*} Then by \eqref{e4.2.a}--\eqref{e4.2.b}, we find \eqref{e4.1}. \end{proof} \begin{lemma} \label{lem4} There exists a constant $\overline{C}>0$ such that we have for all $t>0$, \begin{aligned} &\mathop{\rm Re}\int_{\Omega }(v_{1}.\nabla y\overline{y_{t}} +(v_{2}.\nabla z+(\alpha _{2}-\overline{\alpha _{1}})y) \overline{z_{t}}) \\ &\leq 2\beta \max (g^{\ast },h^{\ast })\Big(\int_{\Gamma _{1}}g(y_{t})\overline{y_{t}}+\int_{\Gamma _{1}}h( z_{t})\overline{z_{t}}\Big) \\ &\quad +\frac{\beta \overline{C}}{2}\Big(\int_{\Omega }|\nabla z| ^{2}+\int_{\Omega }|\nabla y|^{2}+\int_{\Omega }q_{1}| y|^{2}+\int_{\Omega }q_{2}|z|^{2}\Big). \end{aligned}\label{e4.3} Moreover, if $$\beta <\frac{1}{4\max (h^{\ast },g^{\ast })} \label{e4.4}$$ then, for all $T>0$, we have \begin{aligned} &\int_{\Sigma _{1}}g(y_{t})\overline{y_{t}}+\int_{\Sigma _{1}}h(z_{t})\overline{z_{t}}\\ & \leq 2(E(0)-E(T)) +\frac{\beta \overline{C}}{2} \Big(\int_{\Omega }|\nabla z| ^{2}+\int_{\Omega }|\nabla y|^{2}+\int_{\Omega }q_{1}| y|^{2}+\int_{\Omega }q_{2}|z|^{2}\Big). \end{aligned}\label{e4.5} \end{lemma} \begin{proof} We start with the proof of \eqref{e4.3}. Using \eqref{e1.1.a1}, we have \begin{equation*} \mathop{\rm Re}\int_{\Omega }v_{1}.\nabla y\overline{y_{t}}=-\mathop{\rm Re}\int_{\Omega }iv_{1}.\nabla y(\Delta \overline{y}+\overline{v_{1}.\nabla y}-q \overline{y}+\overline{\alpha _{1}z})\,. \end{equation*} Then \begin{equation*} \mathop{\rm Re}\int_{\Omega }v_{1}.\nabla y\overline{y_{t}} =\mathop{\rm Im}\int_{\Omega }v_{1}.\nabla y\Delta \overline{y}-\mathop{\rm Im}\int_{\Omega }v_{1}.\nabla yq_{1} \overline{y}+\mathop{\rm Im}\int_{\Omega }v_{1}. \nabla y\overline{\alpha _{1}z}. \end{equation*} First, we consider the term $\mathop{\rm Im}\int_{\Omega }v_{1}.\nabla y\Delta \overline{y}$. Applying Green's formula, we obtain: \begin{equation*} \mathop{\rm Im}\int_{\Omega }v_{1}.\nabla y\Delta \overline{y}=\mathop{\rm Im} \int_{\Gamma }\frac{\partial \overline{y}}{\partial \upsilon }v_{1}.\nabla y-\mathop{\rm Im}\int_{\Omega }\nabla \overline{y}.\nabla (v_{1}.\nabla y). \end{equation*} Indeed, with $s=v_{1}$, we recall the identity $$\nabla (s.\nabla u)\nabla \overline{u}=S\nabla u.\nabla \overline{u}+\frac{1}{2}s.\nabla (|\nabla u|^{2}), \label{e*}$$ where $S$ denotes the Hessian of $s$. Then $\mathop{\rm Im}\int_{\Omega }v_{1}.\nabla y\Delta \overline{y} =\mathop{\rm Im} \int_{\Gamma _{1}}\frac{\partial \overline{y}} {\partial \upsilon } v_{1}.\nabla y-\mathop{\rm Im} \int_{\Omega }V_{1}\nabla y.\nabla \overline{y} -\frac{1}{2}\int_{\Omega }\mathop{\rm Im}v_{1}\nabla (|\nabla y|)^{2}.$ We invoke the divergence identity: $$\int_{\Omega }s.\nabla \psi =\int_{\Gamma }\psi s.\upsilon -\int_{\Omega }\psi divs, \label{e**}$$ with $\psi =|\nabla y|^{2}$ and $s=v_{1}$. Then \begin{align*} &\mathop{\rm Im}\int_{\Omega }v_{1}.\nabla y\Delta \overline{y} \\ &=\mathop{\rm Im} \int_{\Gamma _{1}} \frac{\partial \overline{y}}{\partial \upsilon } v_{1}.\nabla y-\mathop{\rm Im}\int_{\Omega }V_{1}\nabla y.\nabla \overline{y} -\frac{1}{2}\int_{\Gamma }\mathop{\rm Im}v_{1}.\upsilon |\nabla y| ^{2}+\frac{1}{2}\int_{\Omega }\mathop{\rm Im}(divv_{1})|\nabla y|^{2}. \end{align*} Since, $s=(s.\upsilon ).\upsilon +(s.\mu ).\mu$ for all vector $s$ on $\Gamma$, \begin{equation*} |\nabla y|^{2}=|\frac{\partial y}{\partial \upsilon } |^{2}+|\frac{\partial y}{\partial \mu }|^{2}. \end{equation*} Moreover, \begin{equation*} \frac{\partial y}{\partial \mu }\big|_{\Gamma _{0}}= \nabla y.\mu \big|_{\Gamma _{0}}=0, \quad\mbox{or}\quad |\nabla y|=|\frac{\partial y}{\partial \upsilon }| \quad\text{on }\Gamma _{0,}. \end{equation*} Then $$\mathop{\rm Re}\int_{\Omega }v_{1}.\nabla y\overline{y_{t}}=R_{\Gamma _{0}}+R_{\Gamma _{1}}+R_{\Omega }, \label{e4.6}$$ where \begin{align*} R_{\Gamma _{0}} &= \mathop{\rm Im}\int_{\Gamma _{0}}v_{1}.\upsilon |\nabla y|^{2}-\frac{1}{2}\int_{\Gamma _{0}}(\mathop{\rm Im}v_{1}.\upsilon )|\nabla y|^{2} \\ &= \frac{1}{2}\int_{\Gamma _{0}}(\mathop{\rm Im}v_{1}.\upsilon ) |\nabla y|^{2}\leq 0\quad (\text{by \eqref{e2.5.b}}) \end{align*} and \begin{align*} R_{\Gamma _{1}} &= \int_{\Gamma _{1}}\mathop{\rm Im}v_{1}.\upsilon |\frac{ \partial y}{\partial \upsilon }|^{2}+\mathop{\rm Im}\int_{\Gamma _{1}}v_{1}.\mu \frac{\partial y}{\partial \mu }\frac{\partial \overline{y} }{\partial \upsilon } -\frac{1}{2}\int_{\Gamma _{1}}(\mathop{\rm Im}v_{1}.\upsilon ) |\nabla y|^{2} \\ &= \frac{1}{2}\int_{\Gamma _{1}}\mathop{\rm Im}v_{1}.\upsilon |\frac{ \partial y}{\partial \upsilon }|^{2}+\mathop{\rm Im}\int_{\Gamma _{1}}v_{1}.\mu \frac{\partial y}{\partial \mu }\frac{\partial \overline{y} }{\partial \upsilon } -\frac{1}{2}\int_{\Gamma _{1}}\mathop{\rm Im}v_{1}.\upsilon |\frac{ \partial y}{\partial \mu }|^{2}\,. \end{align*} Then \begin{align*} R_{\Gamma _{1}} &\leq \frac{1}{2}\int_{\Gamma _{1}}\mathop{\rm Im}v_{1}.\upsilon |\frac{\partial y}{\partial \upsilon }|^{2}+\frac{1}{2} \int_{\Gamma _{1}}|\mathop{\rm Im}v_{1}.\mu ||\frac{\partial y }{\partial \upsilon }|^{2} \\ &\quad +\frac{1}{2}\int_{\Gamma _{1}}(|\mathop{\rm Im}v_{1}.\mu |- \mathop{\rm Im}\upsilon _{1}.\upsilon )|\frac{\partial y}{\partial \mu }|^{2}. \end{align*} By \eqref{e2.5.a}, $R_{\Gamma _{1}}\leq \beta \int_{\Gamma _{1}}|\frac{\partial y}{ \partial \upsilon }|^{2}$. Then, by \eqref{e1.1.b2} and \eqref{e2.6}, we obtain \begin{equation*} R_{\Gamma _{1}}\leq \beta g^{\ast }\int_{\Gamma _{1}}g(y_{t}) \overline{y_{t}} \end{equation*} and (see \eqref{e4.6}) \begin{align*} R_{\Omega } &= -\mathop{\rm Im}\int_{\Omega }V_{1}\nabla y.\nabla \overline{y}+ \frac{1}{2}\int_{\Omega }\mathop{\rm Im}(divv_{1})|\nabla y|^{2} \\ &\quad-\mathop{\rm Im}\int_{\Omega }v_{1}.\nabla yq_{1}\overline{y}+\mathop{\rm Im} \int_{\Omega }v_{1}.\nabla y\overline{\alpha _{1}z}. \end{align*} Then \begin{equation*} R_{\Omega }\leq \frac{5}{2}\beta \int_{\Omega }|\nabla y| ^{2}+\beta \frac{|q_{1}|}{2}\int_{\Omega }q_{1}|y| ^{2}+\beta |\alpha _{1}|^{2}\frac{C}{2}\int_{\Omega }| \nabla z|^{2}. \end{equation*} We insert $R_{\Gamma _{l}}$ $(l=0,1)$ and $R_{\Omega }$ in \eqref{e4.6} to obtain \begin{aligned} &\mathop{\rm Re}\int_{\Omega }v_{1}.\nabla y\overline{y_{t}} \\ &\leq \beta g^{\ast}\int_{\Gamma _{1}}g(y_{t})\overline{y_{t}} +\frac{5}{2}\beta\int_{\Omega }|\nabla y|^{2} +\beta \frac{|q_{1}|}{2}\int_{\Omega }q_{1}|y| ^{2}+\beta |\alpha _{1}|^{2}\frac{C}{2}\int_{\Omega }| \nabla z|^{2}. \end{aligned}\label{e4.7.a} Using the same argument and by \eqref{e2.3.a}, we obtain \begin{aligned} &\mathop{\rm Re}\int_{\Omega }(v_{2}.\nabla z+(\alpha _{2}-\overline{ \alpha _{1}})y)\overline{z_{t}} \\ &\leq 2\beta h^{\ast}\int_{\Gamma _{1}}h(z_{t})\overline{z_{t}} +4\beta \int_{\Omega }|\nabla z|^{2} +|q_{2}|\beta \int_{\Omega }q_{2}|z|^{2}\\ &\quad +\frac{\beta }{2}\big((|\alpha _{2}|^{2}+2) C+C'+1\big)\int_{\Omega }|\nabla y|^{2}, \end{aligned} \label{e4.7.b} where we have put $\beta =\max (\beta ,\beta ^{2})$. However, \eqref{e4.7.a}--\eqref{e4.7.b} gives \begin{align*} &\mathop{\rm Re}\int_{\Omega }(v_{1}.\nabla y\overline{y_{t}}+( v_{2}.\nabla z+(\alpha _{2}-\overline{\alpha _{1}})y) \overline{z_{t}})\\ &\leq 2\beta \max (g^{\ast },h^{\ast })(\int_{\Gamma _{1}}g(y_{t})\overline{y_{t}}+\int_{\Gamma _{1}}h( z_{t})\overline{z_{t}})\\ &\quad +\frac{\beta \overline{C}}{2}(\int_{\Omega }|\nabla z| ^{2}+\int_{\Omega }|\nabla y|^{2}+\int_{\Omega }q_{2}| z|^{2}+\int_{\Omega }q_{1}|y|^{2}), \end{align*} where $\overline{C}=\max \big\{|\alpha _{1}|^{2}C+8,( |\alpha _{2}|^{2}+2)C+C'+6,| q_{1}|,2|q_{2}|\big\}$. Using \eqref{e4.1}, we have \begin{align*} &\int_{\Sigma _{1}}g(y_{t})\overline{y_{t}}+\int_{\Sigma _{1}}h(z_{t})\overline{z_{t}} \\ &= E(0)-E(T) +\mathop{\rm Re}\int_{\Omega }(v_{1}.\nabla y\overline{y_{t}}+( v_{2}.\nabla z+(\alpha _{2}-\overline{\alpha _{1}})y) \overline{z_{t}}). \end{align*} By \eqref{e4.3}, we find \begin{align*} &\int_{\Sigma _{1}}g(y_{t})\overline{y_{t}}+\int_{\Sigma _{1}}h(z_{t})\overline{z_{t}} \\ &\leq E(0)-E(T)+2\beta \max (g^{\ast },h^{\ast })\Big(\int_{\Gamma _{1}}g( y_{t})\overline{y_{t}}+\int_{\Gamma _{1}}h(z_{t}) \overline{z_{t}}\Big)\\ &\quad+\frac{\beta \overline{C}}{2}\Big(\int_{\Omega }|\nabla z| ^{2}+\int_{\Omega }|\nabla y|^{2}+\int_{\Omega }q_{2}| z|^{2}+\int_{\Omega }q_{1}|y|^{2}\Big). \end{align*} Finally, by \eqref{e4.4}, we obtain \eqref{e4.5}. \end{proof} \section{Main result} We begin by stating an identity given by the multiplier method. \begin{lemma} \label{lem5} Every solution $y$ of \eqref{e1.1.a1}--\eqref{e1.1.c} satisfies $$\int_{Q}|\nabla y|^{2}+\int_{Q}|\nabla z|^{2}=I_{\Omega }+I_{\Sigma _{0}}+I_{\Sigma _{1}}+I_{Q}, \label{e5.1}$$ where \begin{gather*} I_{\Omega }= -\frac{1}{2}\mathop{\rm Im}\int_{\Omega }(yh.\nabla \overline{y}+zh.\nabla \overline{z})\big|_{0}^{T}, \\ I_{\Sigma _{0}}=\frac{1}{2}\int_{\Sigma _{0}}\Big(|\frac{\partial y }{\partial \upsilon }|^{2}+|\frac{\partial z}{\partial \upsilon }|^{2}\Big)h.\upsilon , \\ \begin{aligned} I_{\Sigma _{1}} &= \frac{1}{2}\mathop{\rm Im}\int_{\Sigma _{1}}h.\upsilon (y \overline{y_{t}}+z\overline{z_{t}})+\frac{n}{2}\mathop{\rm Re} \int_{\Sigma_{1}}\Big(\frac{\partial \overline{y}}{\partial \upsilon }y+\frac{ \partial \overline{z}}{\partial \upsilon }z\Big)\\ &\quad +\frac{1}{2}\int_{\Sigma _{1}}\Big(|\frac{\partial y}{\partial \upsilon }|^{2}+|\frac{\partial z}{\partial \upsilon }| ^{2}\Big)h.\upsilon +\mathop{\rm Re}\int_{\Sigma _{1}}h.\mu \Big(\frac{ \partial \overline{y}}{\partial \mu }\frac{\partial y}{\partial \upsilon }+ \frac{\partial \overline{z}}{\partial \mu }\frac{\partial z}{\partial \upsilon }\Big)\\ &\quad -\frac{1}{2}\int_{\Sigma _{1}}h.\upsilon \Big(|\frac{\partial y}{ \partial \mu }|^{2}+|\frac{\partial z}{\partial \mu }| ^{2}\Big), \end{aligned} \\ \begin{aligned} I_{Q} &= \mathop{\rm Re}\int_{Q}\big[(F_{1}+P_{1})h.\nabla \overline{y}+(F_{2}+P_{2})h.\nabla \overline{z}\big]\\ &\quad +\frac{n}{2}\mathop{\rm Re}\int_{Q}[(F_{1}+P_{1})\overline{y} +(F_{2}+P_{2})\overline{z}]. \end{aligned} \end{gather*} \end{lemma} \begin{proof} Multiply \eqref{e1.1.a1} by $h.\nabla \overline{y}$ and integrate over $Q$ to obtain \begin{aligned} 0 &= \int_{Q}(iy_{t}+\Delta y+(F_{1}+P_{1}))h.\nabla \overline{y}\\ &=i\int_{Q}y_{t}h.\nabla \overline{y} +\int_{Q}\Delta yh.\nabla \overline{y}+\int_{Q}(F_{1}+P_{1}) h.\nabla \overline{y}. \end{aligned} \label{e5.2} Taking the real part, $$0=-\mathop{\rm Im}\int_{Q}y_{t}h.\nabla \overline{y}+\mathop{\rm Re}\int_{Q}\Delta yh.\nabla \overline{y}+\mathop{\rm Re}\int_{Q}(F_{1}+P_{1})h.\nabla \overline{y}. \label{e5.3}$$ Integrating by parts, \begin{equation*} \int_{Q}y_{t}h.\nabla \overline{y}=\int_{\Omega } yh.\nabla \overline{y }\big|_{0}^{T}-\int_{Q}yh.\nabla \overline{y_{t}}\,. \end{equation*} Using the divergence identity \eqref{e**} with $\mathop{\rm div} h=n$, we obtain \begin{equation*} \int_{Q}y_{t}h.\nabla \overline{y}=\int_{\Omega } yh.\nabla \overline{y }\big|_{0}^{T}-\int_{\Sigma }y\overline{y_{t}}h.\nu +n\int_{Q}y\overline{ y_{t}}+\int_{Q}\overline{y_{t}}h.\nabla y. \end{equation*} Then \begin{equation*} 2i\mathop{\rm Im}\int_{Q}y_{t}h.\nabla \overline{y}=\int_{\Omega } yh.\nabla \overline{y}\big|_{0}^{T}-\int_{\Sigma }y\overline{y_{t}}h.\nu +n\int_{Q}y \overline{y_{t}}, \end{equation*} so $$2\mathop{\rm Im}\int_{Q}y_{t}h.\nabla \overline{y} =\mathop{\rm Im}\int_{\Omega } yh.\nabla \overline{y}\big|_{0}^{T}-\mathop{\rm Im}\int_{\Sigma }y\overline{ y_{t}}h.\nu +n\mathop{\rm Im}\int_{Q}y\overline{y_{t}}. \label{e5.4}$$ However, \begin{align*} \int_{Q}y\overline{y_{t}} &= i\int_{Q}y(-\Delta \overline{y}-\overline{ F_{1}+P_{1}})\\ &= -i\int_{\Sigma }\frac{\partial \overline{y}}{\partial \upsilon } y+i\int_{Q}|\nabla y|^{2}-i\int_{Q}y\overline{F_{1}+P_{1}}. \end{align*} Then \begin{equation*} \mathop{\rm Im}\int_{Q}y\overline{y_{t}}=-\mathop{\rm Re}\int_{\Sigma }\frac{\partial \overline{y}}{\partial \upsilon }y+\int_{Q}|\nabla y|^{2}-\mathop{\rm Re}\int_{Q}y\overline{F_{1}+P_{1}}\,. \end{equation*} We substitute this expression in \eqref{e5.4} and we use \eqref{e1.1.b1} to obtain \begin{aligned} -\mathop{\rm Im}\int_{Q}y_{t}h\nabla \overline{y} &= -\frac{1}{2}\mathop{\rm Im} \int_{\Omega } yh\nabla \overline{y}\big|_{0}^{T}+\frac{1}{2}\mathop{\rm Im}\int_{\Sigma _{1}}y\overline{y_{t}}h.\nu \\ &\quad +\frac{n}{2}\mathop{\rm Re}\int_{\Sigma _{1}} \frac{\partial \overline{y}}{\partial \upsilon }y -\frac{n}{2}\int_{Q}|\nabla y|^{2} +\frac{n}{2}\mathop{\rm Re}\int_{Q}\overline{F_{1}+P_{1}}y. \end{aligned}\label{e5.5} Concerning the term $\int_{Q}\Delta yh\nabla \overline{y}$, if we use Green's Formula, identity \eqref{e*}, we find \begin{align*} \int_{Q}\Delta yh\nabla \overline{y} &= \int_{\Sigma }\frac{\partial y}{ \partial \upsilon }h\nabla \overline{y}-\int_{Q}\nabla y\nabla ( h\nabla \overline{y})\\ &= \int_{\Sigma }\frac{\partial y}{\partial \upsilon }h\nabla \overline{y} -\int_{Q}|\nabla y|^{2}-\frac{1}{2}\int_{Q}h\nabla ( |\nabla y|^{2})\,. \end{align*} We use the divergence identity \eqref{e**} to find $$\mathop{\rm Re}\int_{Q}\Delta yh\nabla \overline{y}=\mathop{\rm Re}\int_{\Sigma }\frac{ \partial y}{\partial \upsilon }h\nabla \overline{y}-\frac{1}{2}\int_{\Sigma }|\nabla y|^{2}h.\upsilon +(\frac{n}{2}-1) \int_{Q}|\nabla y|^{2}. \label{e5.6}$$ However, in $\Gamma$, \begin{gather*} |\nabla y|^{2}=|\frac{\partial y}{\partial \upsilon } |^{2}+|\frac{\partial y}{\partial \mu }|^{2}, \\ \frac{\partial y}{\partial \upsilon }h\nabla \overline{y}=|\frac{ \partial y}{\partial \upsilon }|^{2}h.\upsilon +\frac{\partial y}{ \partial \upsilon }\frac{\partial \overline{y}}{\partial \mu }h.\mu \end{gather*} and in $\Gamma _{0}$, $\frac{\partial y}{\partial \mu }=0$. Then from \eqref{e5.6}, \begin{aligned} \mathop{\rm Re}\int_{Q}\Delta yh\nabla \overline{y} &= \frac{1}{2}\int_{\Sigma _{0}}|\frac{\partial y}{\partial \upsilon }|^{2}h.\upsilon + \frac{1}{2}\int_{\Sigma _{1}}|\frac{\partial y}{\partial \upsilon } |^{2}h.\upsilon \\ &\quad +\mathop{\rm Re}\int_{\Sigma _{1}}\frac{\partial y}{\partial \upsilon }\frac{ \partial \overline{y}}{\partial \mu }h.\mu -\frac{1}{2}\int_{\Sigma _{1}}|\frac{\partial y}{\partial \mu }|^{2}h.\upsilon +(\frac{n}{2}-1)\int_{Q}|\nabla y|^{2}. \end{aligned} \label{e5.7} Finally, we insert \eqref{e5.5} and \eqref{e5.7} in \eqref{e5.3} to obtain \begin{aligned} \int_{Q}|\nabla y|^{2} &= -\frac{1}{2}\mathop{\rm Im}\int_{\Omega } yh\nabla \overline{y}\big|_{0}^{T}+\frac{1}{2}\int_{\Sigma _{0}}|\frac{\partial y}{\partial \upsilon }|^{2}h.\upsilon + \frac{1}{2}\mathop{\rm Im}\int_{\Sigma _{1}}y\overline{y_{t}}h.\nu \\ &\quad +\frac{n}{2}\mathop{\rm Re}\int_{\Sigma _{1}}\frac{\partial \overline{y}}{ \partial \upsilon }y+\frac{1}{2}\int_{\Sigma _{1}}|\frac{\partial y}{ \partial \upsilon }|^{2}h.\upsilon +\mathop{\rm Re}\int_{\Sigma _{1}} \frac{\partial y}{\partial \upsilon }\frac{\partial \overline{y}}{\partial \mu }h.\mu \\ &\quad -\frac{1}{2}\int_{\Sigma _{1}}|\frac{\partial y}{\partial \mu } |^{2}h.\upsilon +\frac{n}{2}\mathop{\rm Re}\int_{Q}\overline{F_{1}+P_{1}}y+ \mathop{\rm Re}\int_{Q}\overline{F_{1}+P_{1}}h.\nabla \overline{y}. \end{aligned} \label{e5.8} We obtain a similar identity for $z$. From such inequality and \eqref{e5.8}, we deduce \eqref{e5.1}. \end{proof} Our main result is the following. \begin{theorem} \label{thm6} The solution of \eqref{e1.1.a1}--\eqref{e1.1.c} is exponentially stable. \end{theorem} \begin{proof} From \eqref{e4.1}, \eqref{e2.6}, \eqref{e2.7}, \eqref{e4.3}, \eqref{e4.4} and Remark \ref{rmk1}, we have \begin{equation*} E'(t)\leq \lambda _{3}E(t),\quad\text{for all }t>0. \end{equation*} Now, we prove that there are two constants $\lambda _{1}$ and $\lambda_{2}$, $(\lambda _{1}\leq \lambda _{2})$, such that \begin{equation*} \int_{0}^{T}E(t)\leq \lambda _{1}(E(T)+E(0)) +\lambda _{2}(E(0)-E(T)),\quad\text{for all }T>0. \end{equation*} We bound the terms: $I_{\Omega }$, $I_{\Sigma _{l}}(l=0,1)$ and $I_{Q}$ in \eqref{e5.1}. For all $\varepsilon _{1}>0$, \begin{equation*} I_{\Omega }\leq R\big(\frac{C}{\varepsilon _{1}}+\varepsilon _{1}\big) \frac{1}{\zeta _{1}}(E(T)+E(0)). \end{equation*} By Cauchy-Scwartz, \begin{align*} I_{Q} &\leq \frac{\beta }{2}(2R+\frac{n}{2}(C+1)) \Big(\int_{Q}|\nabla y|^{2}+\int_{Q}| \nabla z|^{2}\Big)\\ &\quad+\frac{\max_{l=1,2}|q_{l}|}{2} R(C+1)\Big(\int_{Q}|\nabla y|^{2}+\int_{Q}|\nabla z|^{2}\Big)\\ &\quad +\frac{n}{2}\mathop{\rm Re}\int_{Q}(\alpha _{1}z\overline{y} +\alpha _{2}y\overline{z}) +\mathop{\rm Re}\int_{Q}(\alpha _{1}zh.\nabla \overline{y }+\alpha _{2}yh.\nabla \overline{z})\\ &\quad -\frac{n}{2}\Big(\int_{Q}q_{1}|y| ^{2}+\int_{Q}q_{2}|z|^{2}\Big). \end{align*} However, \begin{align*} &\frac{n}{2}\mathop{\rm Re}\int_{Q}(\alpha _{1}z\overline{y}+\alpha _{2}y \overline{z})\\ &= n\mathop{\rm Re}\int_{Q}\alpha _{1}z\overline{y}+\frac{n}{2 }\mathop{\rm Re}\int_{Q}(\alpha _{2}-\overline{\alpha _{1}})y \overline{z} \\ &\leq n|\alpha _{1}|\frac{C}{2}\Big(\int_{Q}| \nabla y|^{2}+\int_{Q}|\nabla z|^{2}\Big) +\frac{n}{2}\beta (C+1)\Big(\int_{Q}|\nabla y|^{2}+\int_{Q}|\nabla z|^{2}\Big) \end{align*} and \begin{align*} &\mathop{\rm Re}\int_{Q}(\alpha _{1}zh.\nabla \overline{z}+\alpha _{2}yh.\nabla \overline{y})\\ &= \mathop{\rm Re}\int_{Q}(\alpha _{1}(zh.\nabla \overline{z}+\overline{y}h.\nabla y)+( \alpha _{2}-\overline{\alpha _{1}})yh.\nabla \overline{y})\\ &\leq R|\alpha _{1}|\frac{(C+1)}{2}\Big( \int_{Q}|\nabla y|^{2}+\int_{Q}|\nabla z|^{2}\Big) +\frac{\beta R(C+1)}{2}\int_{Q}|\nabla y| ^{2}\,. \end{align*} Therefore, \begin{align*} I_{Q} &\leq \frac{\beta }{2}\overline{\overline{C}}\Big( \int_{Q}|\nabla y|^{2}+\int_{Q}|\nabla z|^{2}\Big)+ \frac{\max_{l=1,2}(|q_{l}|)}{2}R(C+1)\Big(\int_{Q}|\nabla y| ^{2}+\int_{Q}|\nabla z|^{2}\Big)\\ &\quad +\frac{|\alpha _{1}|}{2}((n+R)C+R) \Big(\int_{Q}|\nabla y|^{2}+\int_{Q}| \nabla z|^{2}\Big) -\frac{n}{2}(\int_{Q}q_{1}|y|^{2}+\int_{Q}q_{2}|z|^{2}), \end{align*} where $\overline{\overline{C}}=2R+(\frac{3n}{2}+R)(C+1)$. By \eqref{e2.4}, we find \begin{align*} I_{Q} &\leq \frac{\beta }{2}\overline{\overline{C}}\Big( \int_{Q}|\nabla y|^{2}+\int_{Q}|\nabla z|^{2}\Big)\\ &\quad +\Big(\frac{\min (1,\frac{n}{2})}{2}-\frac{|\alpha _{1}|}{2}((n+R)C+R)\Big)\Big( \int_{Q}|\nabla y|^{2}+\int_{Q}|\nabla z|^{2}\Big)\\ &\quad +\frac{|\alpha _{1}|}{2}\big((n+R)C+R\big) \Big(\int_{Q}|\nabla y|^{2}+\int_{Q}| \nabla z|^{2}\Big) -\frac{n}{2}\Big(\int_{Q}q_{1}|y| ^{2}+\int_{Q}q_{2}|z|^{2}\Big)\,. \end{align*} So that \begin{align*} I_{Q} &\leq \Big(\frac{\min (1,\frac{n}{2})}{2}+\frac{\beta }{2}\overline{\overline{C}}\Big)\Big(\int_{Q}|\nabla y|^{2}+\int_{Q}|\nabla z|^{2}+\int_{\Omega }q_{1}|y|^{2}+\int_{\Omega }q_{2}|z|^{2}\Big)\\ &\quad -\frac{n}{2}\Big(\int_{Q}q_{1}|y| ^{2}+\int_{Q}q_{2}|z|^{2}\Big). \end{align*} Concerning the term $I_{\Sigma _{l}}$, we have $I_{\Sigma _{0}}\leq 0$. Put \begin{equation*} I_{\Sigma _{1}}=I_{\Sigma _{1}}(y)+I_{\Sigma _{1}}(z), \end{equation*} where \begin{align*} I_{\Sigma _{1}}(y) &= \frac{1}{2}\mathop{\rm Im}\int_{\Sigma _{1}}h.\upsilon y\overline{y_{t}}+\frac{n}{2}\mathop{\rm Re}\int_{\Sigma _{1}} \frac{\partial \overline{y}}{\partial \upsilon }y+\frac{1}{2}\int_{\Sigma _{1}}|\frac{\partial y}{\partial \upsilon }|^{2}h.\upsilon \\ &\quad +\mathop{\rm Re}\int_{\Sigma _{1}}h.\mu \frac{\partial \overline{y}}{\partial \mu }\frac{\partial y}{\partial \upsilon }-\frac{1}{2}\int_{\Sigma _{1}}h.\upsilon |\frac{\partial y}{\partial \mu }|^{2} \end{align*} and \begin{align*} I_{\Sigma _{1}}(z) &= \frac{1}{2}\mathop{\rm Im}\int_{\Sigma _{1}}h.\upsilon z\overline{z_{t}}+\frac{n}{2}\mathop{\rm Re}\int_{\Sigma _{1}} \frac{\partial \overline{z}}{\partial \upsilon }z+\frac{1}{2}\int_{\Sigma _{1}}|\frac{\partial z}{\partial \upsilon }|^{2}h.\upsilon \\ &\quad +\mathop{\rm Re}\int_{\Sigma _{1}}h.\mu \frac{\partial \overline{z}}{\partial \mu }\frac{\partial z}{\partial \upsilon }-\frac{1}{2}\int_{\Sigma _{1}}h.\upsilon (|\frac{\partial y}{\partial \mu }| ^{2}+|\frac{\partial z}{\partial \mu }|^{2})\,. \end{align*} We start with $I_{\Sigma _{1}}(y)$. Using Cauchy-Scwartz and \eqref{e2.1}, for $\varepsilon_{2}>0$, we obtain \begin{align*} I_{\Sigma _{1}}(y) &\leq RC'\varepsilon _{2}\int_{Q}|\nabla y|^{2}+R\frac{1}{\varepsilon _{2}} \int_{\Sigma _{1}}|y_{t}|^{2}+n\varepsilon _{2}C'\int_{Q}|\nabla y|^{2}+n\frac{1}{\varepsilon _{2}} \int_{\Sigma _{1}}|\frac{\partial y}{\partial \upsilon }|^{2} \\ &\quad +\frac{R}{2}\int_{\Sigma _{1}}|\frac{\partial y}{\partial \upsilon } |^{2}+\frac{R}{2}\Big(\frac{1}{\tau }\int_{\Sigma _{1}}| \frac{\partial y}{\partial \upsilon }|^{2}+\tau \int_{\Sigma _{1}}|\frac{\partial y}{\partial \mu }|^{2}\Big) -h_{0}\int_{\Sigma _{1}}|\frac{\partial y}{\partial \mu }|^{2}. \end{align*} Choosing $\tau =\frac{2h_{0}}{R}$ and using \eqref{e1.1.b2}, \eqref{e2.6} we obtain \begin{equation*} I_{\Sigma _{1}}(y)\leq C'\varepsilon _{2}( R+n)\int_{Q}|\nabla y|^{2}+\Big(\frac{R}{g_{\ast }\varepsilon _{2}}+\big(\frac{n}{\varepsilon _{2}}+R+\frac{R^{2}}{4h_{0}} \big)g^{\ast }\Big)\int_{\Sigma _{1}}g(y_{t})\overline{y_{t}}, \end{equation*} similarly for $I_{\Sigma _{1}}(z)$, \begin{equation*} I_{\Sigma _{1}}(z)\leq C'\varepsilon _{2}( R+n)\int_{Q}|\nabla z|^{2}+\Big(\frac{R}{h_{\ast }\varepsilon _{2}}+\big(\frac{n}{\varepsilon _{2}}+R+\frac{R^{2}}{4h_{0}} \big)h^{\ast }\Big)\int_{\Sigma _{1}}h(z_{t})\overline{ z_{t}}. \end{equation*} Therefore, \begin{align*} I_{\Sigma } &\leq C'\varepsilon _{2}(R+n)\Big( \int_{Q}|\nabla y|^{2}+\int_{Q}|\nabla z| ^{2}\Big)\\ &\quad+\Big(\frac{R}{\min (g_{\ast },h_{\ast })\varepsilon _{2}} +\big(\frac{n}{\varepsilon _{2}}+R+\frac{R^{2}}{4h_{0}}\big) \max (g^{\ast },h^{\ast })\Big)\int_{\Sigma _{1}}(g( y_{t})\overline{y_{t}}+h(z_{t})\overline{z_{t}})\,. \end{align*} Using \eqref{e4.5} we find \begin{align*} I_{\Sigma } &\leq \lambda_2 (E(0)-E(T))\\ &\quad +\big(C'\varepsilon _{2}(R+n)+\beta \overline{C} \lambda_2 \big) \Big(\int_{\Omega }|\nabla z| ^{2}+\int_{\Omega }|\nabla y|^{2}+\int_{\Omega }q_{1}| y|^{2}+\int_{\Omega }q_{2}|z|^{2}\Big), \end{align*} where \begin{equation*} \lambda_2 =2\Big(\frac{R}{\min (g_{\ast },h_{\ast }) \varepsilon _{2}}+\big(\frac{n}{\varepsilon _{2}}+R+\frac{R^{2}}{4h_{0}} \big)\max (g^{\ast },h^{\ast })\Big). \end{equation*} We insert this in \eqref{e5.1} to obtain \begin{align*} &\frac{1}{2}\Big(\min (1,\frac{n}{2})-\beta (\overline{ C}+\lambda_2 \overline{\overline{C}})-2C'\varepsilon _{2}(R+n)\Big)\\ &\times \Big(\int_{\Omega }|\nabla z| ^{2}+\int_{\Omega }|\nabla y|^{2}+\int_{\Omega }q_{2}| z|^{2}+\int_{\Omega }q_{1}|y|^{2}\Big)\\ &\leq \lambda_1\big(E(T)+E(0)\big))+\lambda_2 \big(E(0)-E(T)\big), \end{align*} where, we have put \begin{equation*} \lambda_1=\frac{R}{\zeta _{1}}(\frac{C}{\varepsilon _{1}} +\varepsilon _{1}). \end{equation*} We choose \begin{equation*} \beta \leq \frac{\min (1,\frac{n}{2})}{\overline{C}+\lambda_2 \overline{\overline{C}}} \end{equation*} and \begin{equation*} \varepsilon _{2}<\min \Big(\frac{\min (1,\frac{n}{2})-\beta (\overline{C}+\lambda_2 \overline{\overline{C}})}{2C'(R+n)},\sqrt{\frac{2\zeta _{1}^{2}n\max (g^{\ast },h^{\ast })}{RC\min (g_{\ast },h_{\ast })}}\Big). \end{equation*} On the other hand, if we choose $\varepsilon _{1}$ such that \begin{equation*} \frac{\varepsilon _{2}CR}{\zeta _{1}\max (g^{\ast },h^{\ast })} <\varepsilon _{1}<\frac{2\zeta _{1}}{\min (g_{\ast },h_{\ast }) \varepsilon _{2}}, \end{equation*} then $\lambda_1\leq \lambda_2$. \end{proof} \begin{thebibliography}{9} \bibitem{a1} A. Guesmia, \emph{A New Approach of Stabilisation of Nondissipative Distributed Systems}, SIAM J. Control Optimi., 42 (2003), no. 1, 24-52. \bibitem{k1} V. Komornick, \emph{Exact Controlability and Stabilisation}. The multiplier method, Masson Paris (1994). \bibitem{l1} I. Lasieka, R. Triggiani and X. Zhang, \emph{Global Uniqueness, Observability and Stabilisation of Nonconservative Schrodinger Equations Via Pointwise Carleman Estimates}. J. INV. III-Posed Problems 11 (2003), no. 3, 1-96. \bibitem{l2} J. L. Lions, \emph{Quelque Methodes de Resolution des Problemes aux Limites Non Lineaire}. Gauthier Villars 1969. \bibitem{m1} E. Machtyngier and E. Zuazua, \emph{Stabilisation of Schrodinger Equation}, Portugaliae Mathematica 51 (1994), no. 2, 243-256. \bibitem{m2} E. Machtyngier, \emph{Exact Controllability for Schrodinger equation}. SIAM J. Cont. and Optim. 32 (1994), 24-34. \end{thebibliography} \end{document}