\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small 2003 Colloquium on Differential Equations and Applications, Maracaibo, Venezuela.\newline {\em Electronic Journal of Differential Equations}, Conference 13, 2005, pp. 75--88.\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 2005 Texas State University - San Marcos.} \vspace{9mm}} \setcounter{page}{75} \begin{document} \title[\hfilneg EJDE/Conf/13 \hfil Controllability of a generalized wave equation] {Exact controllability of a non-linear generalized damped wave equation: Application to the Sine-Gordon equation} \author[H. Leiva \hfil EJDE/Conf/13 \hfilneg] {Hugo Leiva} \address{Department of Mathematics, Universidad de los Andes \\ Merida 5101, Venezuela} \email{hleiva@ula.ve} \date{} \thanks{Published May 30, 2005.} \subjclass[2000]{34G10, 35B40} \keywords{Non-linear generalized wave equations; strongly continuous groups; \hfill\break\indent exact controllability; Sine-Gordon equation} \begin{abstract} In this paper, we give a sufficient conditions for the exact controllability of the non-linear generalized damped wave equation $$\ddot{w}+ \eta \dot{w} + \gamma A^{\beta} w = u(t) + f(t,w,u(t)),$$ on a Hilbert space. The distributed control $u \in L^{2}$ and the operator $A$ is positive definite self-adjoint unbounded with compact resolvent. The non-linear term $f$ is a continuous function on $t$ and globally Lipschitz in the other variables. We prove that the linear system and the non-linear system are both exactly controllable; that is to say, the controllability of the linear system is preserved under the non-linear perturbation $f$. As an application of this result one can prove the exact controllability of the Sine-Gordon equation. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \newtheorem{propostition}[theorem]{Proposition} \newtheorem{definition}[theorem]{Definition} \newtheorem{lemma}[theorem]{Lemma} \allowdisplaybreaks \section{Introduction} In this paper, we give sufficient conditions for the exact controllability of the following non-linear generalized damped wave equation on a Hilbert space $X$, $$\label{C1} \ddot{w}+ \eta \dot{w} + \gamma A^{\beta} w = u(t) + f(t,w,u(t)), \quad t \geq 0,$$ where $\gamma >0$, $\eta >0$, $\beta \geq 0$, the distributed control $u$ is in $L^{2}(0,t_1; X)$, and $A : D(A) \subset X \to X$ is a positive definite self-adjoint unbounded linear operator in $X$ with compact resolvent. This implies the following spectral decomposition of the operator $A$: $$Ax = \sum_{n = 1}^{\infty} \lambda_{n} \sum_{k = 1}^{\gamma_n} \langle x, \phi_{n,k}\rangle \phi_{n,k}=\sum_{n = 1}^{\infty} \lambda_{n}E_{n}x, \quad x \in D(A).$$ The non-linear term $f:[0,t_1]\times X \times X \to X$ is a continuous function on $t$ and globally Lipschitz in the other variables. i.e., there exists a constant $l >0$ such that for all $x_1, x_2, u_1, u_2 \in X$ we have $$\label{GL1} \|f(t,x_2, u_2 ) - f(t, x_1, u_1) \| \leq l \left\{ \|x_2 -x_1 \| + \|u_2 -u_1 \| \right\}, \quad t \in [0,t_1].$$ We consider the operator $$\label{A1} \mathcal{A} = \begin{bmatrix} 0 & I_{X} \\ -\gamma A^{\beta} & -\eta I \end{bmatrix}$$ which corresponds to the equation $\ddot{w}+ \eta \dot{w} + \gamma A^{\beta} w = 0$ written as a first order system in the space $D(A^{\beta /2}) \times X$. Then we prove the following statements: \begin{itemize} \item[(I)] $\mathcal{A}$ generates a strongly continuous group $\{ T(t) \}_{t \in \mathbb{R}}$ on $D(A^{\beta /2} ) \times X$ such that $\|T(t) \| \leq M(\eta, \gamma) e^{-{\eta \over 2} t}, \quad t \geq 0$. \item[(II)] The linear system (\ref{CO1}) ($f=0$) is exactly controllable on $[0, t_1]$. \item[(III)] The non-linear system (\ref{C1}) is also exactly controllable on $[0, t_1]$. \end{itemize} Moreover, each of the following statements are equivalent to the exact controllability of the linear system $$\label{CO1} \ddot{w}+ \eta \dot{w} + \gamma A^{\beta} w = u(t) \quad t \geq 0,$$ \begin{itemize} \item[(a)] Each of the following finite dimensional systems is controllable on $[0,t_1]$, $$\label{C141} y'= A_{j}P_{j}y + P_{j}Bu, \quad y \in \mathcal{R}(P_j); \quad j=1,2, \dots, \infty.$$ \item[(b)] $B^{*}P_{j}^{*}e^{A_{j}^{*}t}y =0$, for all $t \in [0,t_1]$, implies $y=0$ \item[(c)] $\mathop{\rm Rank} \begin{bmatrix}P_jB & A_{j}P_jB & A_{j}^{2}P_jB & \cdots & A_{j}^{2\gamma_{j} -1}P_jB \end{bmatrix} = 2\gamma_{j}$ \item[(d)] The operator $W_{j}(t_1): \mathcal{R}(P_j) \to \mathcal{R}(P_j)$ given by $$\label{W111} W_{j}(t_1) = \int_{0}^{t_1} e^{-A_{j}s}BB^{*}e^{-A^{*}_{j}s}ds,$$ is invertible, where $\lambda_j$ are the eigenvalues of $A$, $\{P_j\}$ are the projections on the corresponding eigenspace, $$B = \begin{bmatrix} 0\\ I_X \end{bmatrix}, \quad A_j=\begin{bmatrix} 0 & 1 \\ -\gamma \lambda^{\beta}_j & -\eta \end{bmatrix} P_j\,,\quad j\geq 1.$$ \end{itemize} The operator, $W_j(t_1)$, allows us to compute explicitly the control $u \in L^{2}(0,t_1; X)$ steering an initial state $z_0$ to a final state $z_1$ in time $t_1 >0$ for the linear system (\ref{CO1}). This control is given by the formula $$\label{fu} u(t) = B^{*}T^{*}(-t)\sum_{j=1}^{\infty} W^{-1}_{j}(t_1)P_{j}(T(-t_1)z_1 - z_0).$$ We use this formula to construct a sequence of controls $u_n$ that converges to a control $u$ that steers an initial state $z_0$ to a final state $z_1$ for the non-linear system (\ref{C1}). That is to say, we proved the exact controllability of this system. As an application of this result we can prove the exact controllability of The Sine-Gordon Equation \label{CKM1} \begin{gathered} w_{tt}+ c w_{t} - d w_{xx} + k\sin{w} = p(t,x), \quad 00$,$c >0$,$k> 0$and$p: \mathbb{R} \times [0,1] \to \mathbb{R}$is continuous and bounded function acting as an external force. The existence of an attractor for the Sine-Gordon equation is proved in \cite{Tem} where we can find a study of this equation, and the existence of bounded solutions for this model (\ref{CKM1}) and others similar one has been carried out recently in \cite{L2}, \cite{L3} and \cite{GL}. To our knowledge, the exact controllability of this model under non-linear action of the control has not been studied before. So, in this paper we give a sufficient conditions for the exact controllability of the system (\ref{C1}) that can be applied to the following controlled Sine-Gordon equation \label{SG2} \begin{gathered} w_{tt}+ c w_{t} - d w_{xx} + k\sin{w} = p(t,x)+ u(t,x) + g(t,w,u(t,x)), \quad 00$ such that for all $x_1, x_2, u_1, u_2 \in \mathbb{R}$ we have $$\label{GL11} \|g(t,x_2, u_2 ) - g(t, x_1, u_1) \| \leq m \left\{ \|x_2 -x_1 \| + \|u_2 -u_1 \| \right\}, \quad t \in [0,t_1].$$ This system can be written in the form of system (\ref{C1}) if we choose $X = L^{2}[0,1]$, $A\phi = - \phi_{xx}$, with domain $D(A) = H^2 \cap H^{1}_{0}$ and $f(t,w,u)= -k \sin{w} + p(t, \cdot) + g(t,w,u)$. Moreover, the exact controllability of (\ref{SG2}) does not depend on the bounded function $p(t, \cdot)$. Also, in \cite{LT} the authors study the exact \emph{null} controllability of the second order linear equation $$\label{C1T} \ddot{w}+ \rho A^{r}\dot{w} + A w = u(t), \quad \rho >0, \; \frac12 \leq r \leq 1, \; t \geq 0,$$ where the distributed control $u \in L^{2}(0,t_1; X)$ and $A : D(A) \subset X \to X$ is a positive definite self-adjoint unbounded linear operator in $X$ with compact resolvent. They prove that if ${1 \over 2} \leq r < 1$, then the system (\ref{C1T}) is exactly \emph{null} controllable on $[0,t_1]$. However, if $\alpha =1$, the system (\ref{C1T}) is not exactly \emph{null} controllable. In \cite[Example 3.27]{CP1} it is shown that exact \emph{null} controllability of an infinite dimensional system does not imply exact controllability of the system. \section{Notation and Preliminaries} The fact that $A : D(A) \subset X \to X$ is a positive definite self-adjoint unbounded linear operator in $X$ with compact resolvent implies the following: \begin{itemize} \item[(a)] The spectrum of $A$ consists of only eigenvalues $$0 < \lambda_1 < \lambda_2 < \cdots < \lambda_n \to \infty,$$ Each $\lambda_j$ has finite multiplicity, $\gamma_n$, equal to the dimension of the corresponding eigenspace. \item[(b)] There exists a complete orthonormal set $\{ \phi_{n,k} \}$ of eigenvectors of $A$. \item[(c)] For all $x \in D(A)$ we have $$\label{p3} Ax = \sum_{n = 1}^{\infty} \lambda_{n} \sum_{k = 1}^{\gamma_n} \langle x, \phi_{n,k}\rangle \phi_{n,k} =\sum_{n = 1}^{\infty} \lambda_{n}E_{n}x,$$ where $\langle \cdot, \cdot\rangle$ is the inner product in $X$ and $$\label{p4} E_{n}x = \sum_{k = 1}^{\gamma_n} \langle x, \phi_{n,k}\rangle \phi_{n,k}.$$ So, $\{ E_n \}$ is a family of complete orthogonal projections in $X$ and $x = \sum_{n = 1}^{\infty}E_{n}x$, $x \in X.$ \item[(d)] $-A$ generates an analytic semigroup $\{ e^{-At} \}$ given by $$\label{p5} e^{-At}x = \sum_{n = 1}^{\infty} e^{-\lambda_{n}t}E_{n}x.$$ \item[(e)] The fractional powered spaces $X^{r}$ are given by $$X^{r} =D(A^{r}) = \{x \in X : \sum_{n = 1}^{\infty} (\lambda_{n} )^{2 r} \| E_{n}x \|^2 < \infty \}, \quad r \geq 0,$$ with the norm $$\|x \|_{r} = \|A^{r}x \| = \big\{ \sum_{n = 1}^{\infty} \lambda_{n}^{2 r} \| E_{n}x \|^2 \big\}^{1/2}, \quad x \in X^{r},$$ and $$\label{pp5} A^{r}x = \sum_{n = 1}^{\infty} \lambda_{n}^{ r} E_{n}x.$$ \end{itemize} Also, for $r \geq 0$ we define $Z_{r} = X^{r} \times X$, which is a Hilbert Space endow with the norm $$\Big\| \begin{bmatrix}w \\ v \end{bmatrix} \Big\|^{2}_{Z_{r}} = \|w \|^{2}_{r} + \|v \|^2.$$ Using the change of variables $w' =v$, the second order equation (\ref{C1}) can be written as a first order system of ordinary differential equations in the Hilbert space $Z_{\beta /2} = D(A^{\beta /2}) \times X = X^{\beta /2} \times X$ as $$\label{C11} z' = \mathcal{A}z + Bu + F(t,z,u(t)) \quad z \in Z_{\beta /2}, \; t \geq 0,$$ where $$\label{S22} z = \begin{bmatrix} w \\ v \end{bmatrix}, \quad B = \begin{bmatrix} 0\\ I_X \end{bmatrix},\quad \mathcal{A} = \begin{bmatrix} 0 & I_{X} \\ -\gamma A^{\beta} & -\eta I_X \end{bmatrix}.$$ is an unbounded linear operator with domain $D(\mathcal{A})= D(A^{\beta})\times X$ and $$\label{f1} F(t,z, u)= \begin{bmatrix} 0\\ f(t,w,u) \end{bmatrix},$$ is a function $F: [0,t_1] \times Z_{\beta /2} \times X \to Z$. Since $X^{\beta /2}$ is continuously included in $X$ we obtain for all $z_1, z_2 \in Z_{\beta /2}$ and $u_1, u_2 \in X$ that $$\label{f2} \|F(t,z_2, u_2 ) - F(t, z_1, u_1) \|_{Z_{\beta /2}} \leq L \left\{ \|z_2 -z_1 \| + \|u_2 -u_1 \| \right\}, \quad t \in [0,t_1].$$ In this paper, without lose of generality we shall assume the following condition $$\eta^{2} < 4 \gamma \lambda^{\beta}_{1}.$$ \section{The Uncontrolled Linear Equation} In this section we shall study the well-posedness of the abstract linear Cauchy initial-value problem $$\label{Cau1} \begin{gathered} z' = \mathcal{A}z, \quad (t \in \mathbb{R}) \\ z(0)=z_0 \in D(\mathcal{A}), \end{gathered}$$ which is equivalent to prove that the operator $\mathcal{A }$ generates a strongly continuous group. To this end, we shall use the following Lema from \cite{L}. \begin{lemma}\label{L1} Let $Z$ be a separable Hilbert space and $\{A_n \}_{n \geq 1}$, $\{P_n \}_{n \geq 1}$ two families of bounded linear operators in $Z$ with $\{P_n \}_{n \geq 1}$ being a complete family of orthogonal projections such that $$\label{l11} A_n P_n = P_n A_n, \quad n=1,2,3, \dots$$ Define the family of linear operators $$\label{l2} T(t)z = \sum_{n=1}^{\infty} e^{A_n t}P_n z, \quad t \geq 0.$$ Then \begin{itemize} \item[(a)] $T(t)$ is a linear bounded operator if $$\label{l3} \|e^{A_{n}t} \| \leq g(t), \quad n=1,2,3, \dots$$ for some continuous real-valued function $g(t)$. \item[(b)] Under the condition \eqref{l3} $\{T(t) \}_{t \geq 0}$ is a $C_0$-semigroup in the Hilbert space $Z$ whose infinitesimal generator $\mathcal{A}$ is given by $$\label{L4} \mathcal{A}z = \sum_{n=1}^{\infty} A_n P_n z, \quad z \in D(\mathcal{A})$$ with $%begin{equation}\label{L5} D(\mathcal{A})= \{z \in Z : \sum_{n=1}^{\infty} \|A_{n}P_{n}z \|^2 < \infty \}$ % \item[(c)] the spectrum $\sigma(\mathcal{A})$ of $\mathcal{A}$ is given by $$\label{L6} \sigma(\mathcal{A}) = \overline{\bigcup_{n=1}^{\infty} \sigma(\bar{A}_n)},$$ where $\bar{A}_{n} = A_{n}P_n$. \end{itemize} \end{lemma} \begin{theorem}\label{T1} The operator $\mathcal{A}$ given by \eqref{S22}, is the infinitesimal generator of a strongly continuous group $\{ T(t)\} _{t \mathbb{R}}$ given by $$\label{3.4} %\label{damp} T(t)z =\sum_{n=1}^{\infty}e^{A_{n}t}P_nz, \quad z\in Z_{\beta /2}, \; t \geq 0$$ where $\{ P_n\} _{n\geq 0}$ is a complete family of orthogonal projections in the Hilbert space $Z_{\beta /2}$: $P_n=diag[ E_n,E_n]$, $n\geq 1$, % \label{3.5} and $$A_n = B_{n}P_{n}, \quad B_n=\begin{bmatrix} 0 & 1 \\ -\gamma \lambda^{\beta}_n & -\eta \end{bmatrix}, \; n\geq 1. \label{3.6}$$ This group decays exponentially to zero. In fact, we have the estimate $%\label{TE2} \|T(t) \| \leq M(\eta, \gamma)e^{-{\eta \over 2} t}$, $t \geq 0$, % where $${M(\eta, \gamma) \over 2 \sqrt{2}} = \sup_{n \geq 1} \Big\{ 2\Big| { \eta \pm \sqrt{4 \gamma\lambda^{\beta}_{n} - \eta^2 } \over \sqrt{\eta^2 - 4\gamma \lambda^{\beta}_{n} } } \Big|, \Big| (2+\gamma) \sqrt{{\lambda^{\beta }_{n} \over 4\gamma \lambda^{\beta }_{n} - \eta^{2}}} \Big| \Big\}.$$ \end{theorem} \begin{proof} Computing $\mathcal{A}z$ yields, \begin{align*} \mathcal{A}z & = \begin{bmatrix} 0 & I \\ -\gamma A^{\beta} & -\eta \end{bmatrix} \begin{bmatrix} w \\ v \end{bmatrix}\\ &= \begin{bmatrix} v \\ -\gamma A^{\beta}w -\eta v \end{bmatrix}\\ &= \begin{bmatrix} \sum_{n=1}^{\infty} E_{n}v \2pt] -\gamma \sum_{n=1}^{\infty} \lambda^{\beta} _n E_{n}w -\eta \sum_{n=1}^{\infty} E_{n}v \end{bmatrix}\\ & = \sum_{n=1}^{\infty} \begin{bmatrix} E_{n}v \\ -\gamma \lambda^{\beta}_n E_{n}w -\eta E_{n}v \end{bmatrix}\\ & = \sum_{n=1}^{\infty} \begin{bmatrix} 0 & 1 \\ -\gamma \lambda^{\beta}_{n} & -\eta \end{bmatrix} \begin{bmatrix} E_{n} & 0 \\ 0 & E_{n} \end{bmatrix} \begin{bmatrix} w \\ v \end{bmatrix}\\ & = \sum_{n=1}^{\infty} A_{n}P_{n}z. \end{align*} It is clear that A_{n}P_{n} = P_{n}A_{n}. Now, we need to check condition (\ref{l3}) from Lemma \ref{L1}. To this end, compute the spectrum of the matrix B_{n}. The characteristic equation of B_n is given by  \lambda^{2} + \eta \lambda + \gamma \lambda^{\beta}_{n} = 0,  and the eigenvalues \sigma_{1}(n), \sigma_{2}(n) of the matrix B_n are given by  \sigma_{1}(n)= -c + i l_{n}, \quad \sigma_{2}(n)= -c - i l_{n},  where,  c = {\eta \over 2} \quad \mbox{and} \quad l_{n}= {1 \over 2} \sqrt{4 \gamma \lambda^{\beta}_{n} - \eta^{2} }.  Therefore, \begin{align*} e^{B_{n}t} & = e^{-ct} \big\{ \cos{l_{n}t}I + {1 \over l_n} (B_n + cI ) \big\}\\ & = e^{-ct} \begin{bmatrix} \cos{l_{n}t} + {\eta \over 2l_n}\sin{l_{n}t} & {\sin{l_{n}t} \over l_n} \\ -\gamma S(n) \lambda^{\beta /2}_n \sin{l_{n}t} & \cos{l_{n}t} - {\eta \over 2l_n}\sin{l_{n}t} \end{bmatrix}, \end{align*} From the above formulas, we obtain  e^{B_{n}t} = e^{-ct} \begin{bmatrix} a(n) & {b(n)\over l_n }\\ -\gamma S(n) \lambda^{\beta /2}_{n} c(n) & d(n) \end{bmatrix}  where \begin{gather*} a(n) = \cos{l_{n}t} + {\eta \over 2l_n} \sin{l_{n}t}, \quad b(n) = \sin{l_{n}t}, \\ c(n) = \sin{l_{n}t}, \quad d(n) = \cos{l_{n}t} - {\eta \over 2l_n}\sin{l_{n}t},\quad S(n) = \sqrt{\lambda_{n}^{\beta} \over 4\gamma \lambda_{n}^{\beta} - \eta^2 }. \end{gather*} Now, consider z =(z_1, z_2 )^{T} \in Z_{\beta /2} such that \| z \|_{Z_{\beta /2}} = 1. Then  \|z_1 \|^{2}_{ \beta /2} = \sum_{j=1}^{\infty} \lambda^{\beta}_{j} \|E_{j}z_1 \|^2 \leq 1 \quad \mbox{and} \quad \|z_2 \|^{2}_{X} = \sum_{j=1}^{\infty} \|E_{j}z_2 \|^2 \leq 1.  Therefore, \lambda^{ \beta /2}_{j} \|E_{j}z_1 \| \leq 1, \|E_{j}z_2 \| \leq 1, j=1,2, \dots. and so, \begin{align*} \| e^{ A_{n} t}z\|^{2}_{Z_{\beta /2}} & = e^{-2ct} \Big\|\begin{bmatrix} a(n)E_{n}z_1 + {b(n) \over l_{n}}E_{n}z_2 \\ -\gamma S(n)c(n) \lambda_{n}^{\beta \over 2} E_{n}z_1 + d(n) E_{n}z_2 \end{bmatrix} \Big\|^{2}_{Z_{\beta /2}}\\ & = e^{-2ct}\| a(n)E_{n}z_1 + {b(n) \over l_{n}}E_{n}z_2 \|^{2}_{\beta \over 2} + e^{-2ct}\| \\ &\quad - \gamma S(n)c(n) \lambda_{n}^{\beta \over 2} E_{n}z_1 + d(n) E_{n}z_2 \|^{2}_{X}\\ & = e^{-2ct}\sum_{j=1}^{\infty} \lambda^{\beta}_{j} \|E_{j} \big(a(n)E_{n}z_1 + {b(n) \over l_{n}}E_{n}z_2 \big) \|^2 \\ &\quad + e^{-2ct}\sum_{j=1}^{\infty} \|E_{j} \big(-\gamma S(n)c(n) \lambda_{n}^{\beta \over 2} E_{n}z_1 + d(n) E_{n}z_2 \big) \|^2 \\ & = e^{-2ct}\lambda^{\beta}_{n} \|a(n)E_{n}z_1 + {b(n) \over l_{n}}E_{n}z_2 \|^2 + e^{-2ct}\| \\ & \quad - \gamma S(n)c(n) \lambda_{n}^{\beta \over 2} E_{n}z_1 + d(n) E_{n}z_2\|^2 \\ & \leq e^{-2ct}(|a(n)| + |{{\lambda^{\beta \over 2} \over \lambda^{\alpha}_{n}}} b(n)|)^2 + e^{-2ct}(|\gamma S(n)c(n)| + |d(n)|)^2, \end{align*} where \[ |{\lambda^{{\beta \over 2}}_{n} \over l_{n}} b(n)| = \Big| \sqrt{{\lambda^{\beta }_{n} \over \eta^{2} - 4\gamma \lambda^{\beta}_{n}} } \Big|. If we set, \begin{align*} {M(\eta, \gamma) \over 2 \sqrt{2}} & = \sup_{n \geq 1} \Big\{ 2\Big| { \eta \pm \sqrt{4 \gamma\lambda^{\beta}_{n} - \eta^2 } \over \sqrt{\eta^2 - 4\gamma \lambda^{\beta}_{n} } } \Big|, \Big| (2+\gamma) \sqrt{{\lambda^{\beta }_{n} \over 4\gamma \lambda^{\beta }_{n} - \eta^{2}}} \Big| \Big\}, \end{align*} we have, $$\|e^{ A_{n}t}\| \leq M(\eta, \gamma)e^{- c t}, \quad t \geq 0, \; n=1,2, \dots.$$ Hence, applying Lemma \ref{L1} we obtain that $\mathcal{A}$ generates a strongly continuous group given by (\ref{3.4}). Next, we will prove this group decays exponentially to zero. In fact, \begin{align*} \| T(t)z \|^{2} & \leq \sum_{n=1}^{\infty} \|e^{A_{n}t}P_nz \|^{2}\\ & \leq \sum_{n=1}^{\infty} \|e^{A_{n}t} \|^{2} \|P_nz \|^{2}\\ & \leq M^{2}(\eta, \gamma) e^{-2 ct} \sum_{n=1}^{\infty} \|P_nz \|^{2}\\ & = M^{2}(\eta, \gamma)e^{-2 ct} \|z \|^{2}. \end{align*} Therefore, $\| T(t) \| \leq M(\eta, \gamma)e^{-ct}$, $t \geq 0$. \end{proof} \section{Exact Controllability of the Linear System} Now, we shall give the definition of controllability in terms of the linear systems $$\label{C12} z' = \mathcal{A}z + Bu \quad z \in Z_{\beta /2}, \; t \geq 0,$$ where $$\label{S222} z = \begin{bmatrix} w \\ v \end{bmatrix}, \quad B = \begin{bmatrix} 0\\ I_X \end{bmatrix}, \quad \mathcal{A} = \begin{bmatrix} 0 & I_{X} \\ -\gamma A^{\beta} & -\eta I_X \end{bmatrix}.$$ For all $z_0 \in Z_{\beta /2}$ equation (\ref{C12}) has a unique mild solution given by $$\label{C1211} z(t) = T(t)z_0 + \int_{0}^{t} T(t-s)Bu(s)ds, \quad 0 \leq t \leq t_1.$$ The following definition of exact controllability can be found in \cite{CP1}. \begin{definition} \label{D:d1} \rm We say that system (\ref{C12}) is exactly controllable on $[0,t_1]$, $t_1>0$, if for all $z_0, z_1 \in Z_{\beta /2}$ there exists a control $u \in L^{2}(0,t_1;X)$ such that the solution $z(t)$ of (\ref{C1211}) corresponding to $u$, satisfies $z(t_1) = z_1$. \end{definition} Consider the bounded linear operator $$\label{c511} G : L^{2}(0,t_1;U) \to Z_{\beta /2}, \quad Gu =\int_{0}^{t_1} T(-s) B(s)u(s) ds.$$ Then the following proposition is a characterization of the exact controllability of system (\ref{C12}). \begin{propostition} \label{pp1} The system (\ref{C12}) is exactly controllable on $[0,t_1]$ if and only if, the operator $G$ is surjective, that is to say $$G L^{2}(0,t_1;X) = \mathop{\rm Range}(G)= Z_{\beta /2}.$$ \end{propostition} Now, consider the family of finite dimensional systems $$\label{C14} y'= A_{j}P_{j}y + P_{j}Bu, \quad y \in \mathcal{R}(P_j); \; j=1,2, \dots, \infty.$$ Then the following proposition can be shown as in \cite[Lemma 1]{LZ}. \begin{propostition}\label{O1} The following statements are equivalent: \begin{itemize} \item[(a)] System \eqref{C14} is controllable on $[0,t_1]$ \item[(b)] $B^{*}P_{j}^{*}e^{A_{j}^{*}t}y =0$, for all $t \in [0,t_1]$, implies $y=0$ \item[(c)] $\mathop{\rm Rank} \begin{bmatrix} P_jB & A_{j}P_jB & A_{j}^{2}P_jB & \cdots &A_{j}^{2\gamma_{j} -1}P_jB \end{bmatrix} = 2\gamma_{j}$ \item[(d)] The operator $W_{j}(t_1): \mathcal{R}(P_j) \to \mathcal{R}(P_j)$ given by $$\label{W11} W_{j}(t_1) = \int_{0}^{t_1} e^{-A_{j}s}BB^{*}e^{-A^{*}_{j}s}ds,$$ is invertible. \end{itemize} \end{propostition} Now, we are ready to formulate the main result on exact controllability of the linear system (\ref{C12}). \begin{theorem}\label{T4} The system \eqref{C12} is exactly controllable on $[0,t_1]$. Moreover, the control $u \in L^{2}(0,t_1; X)$ that steers an initial state $z_0$ to a final state $z_1$ at time $t_1 >0$ is given by the formula $$\label{fuu} u(t) = B^{*}T^{*}(-t)\sum_{j=1}^{\infty} W^{-1}_{j}(t_1)P_{j}(T(-t_1)z_1 - z_0).$$ \end{theorem} \begin{proof} Since $\{T(t) \}_{t \geq 0}$ is a group, the operator $G$ in (\ref{c5}) can be replaced by $$\label{c51} G : L^{2}(0,t_1; X) \to Z_{\beta /2}, \quad Gu =\int_{0}^{t_1} T(-s) B(s)u(s) ds.$$ Then system (\ref{C12}) is exactly controllable on $[0,t_1]$ if and only if, the operator $G$ is surjective, that is to say $$G L^{2}(0,t_1; X) = \mathop{\rm Range}(G)= Z_{\beta /2}.$$ First, we shall prove that each of the following finite dimensional systems is controllable on $[0,t_1]$ $$\label{C1411} y'= A_{j}P_{j}y + P_{j}Bu, \quad y \in \mathcal{R}(P_j); \; j=1,2, \dots, \infty.$$ In fact, we can check the condition for controllability of this systems, $$B^{*}P_{j}^{*}e^{A_{j}^{*}t}y =0, \quad \forall t \in [0,t_1], \quad \Rightarrow y=0.$$ In this case the operators $A_j = B_j P_j$ and $\mathcal{A}$ are given by $$B_j=\begin{bmatrix} 0 & 1 \\ -\gamma \lambda^{\beta}_j & -\eta \end{bmatrix}, \quad \mathcal{A} = \begin{bmatrix} 0 & I_{X} \\ -\gamma A^{\beta} & -\eta I \end{bmatrix},$$ and the eigenvalues $\sigma_{1}(j)$, $\sigma_{2}(j)$ of the matrix $B_j$ are given by $\sigma_{1}(j)= -c + i l_{j}$ and $\sigma_{2}(j)= -c - i l_{j}$, where $$c = {\eta \over 2} \quad \mbox{and} \quad l_{j}= {1 \over 2} \sqrt{4 \gamma \lambda^{\beta}_{j} - \eta^{2} }.$$ Therefore, $A^{*}_{j} = B^{*}_{j}P_{j}$ with $B^{*}_j=\begin{bmatrix} 0 & -1 \\ \gamma \lambda^{\beta}_j & -\eta \end{bmatrix}$ and \begin{align*} e^{B_{j}t} & = e^{-ct} \big\{ \cos{l_{j}t}I + {1 \over l_j} (B_j + cI ) \big\}\\ & = e^{-ct} \begin{bmatrix} \cos{l_{j}t} + {\eta \over 2l_j}\sin{l_{j}t} & {\sin{l_{j}t} \over l_j} \\ -\gamma S(j) \lambda^{\beta /2}_j \sin{l_{j}t} & \cos{l_{j}t} - {\eta \over 2l_j}\sin{l_{j}t} \end{bmatrix}, \end{align*} \begin{align*} e^{B^{*}_{j}t} & = e^{-ct} \big\{ \cos{l_{j}t}I + {1 \over l_j} (B^{*}_j + cI) \big\}\\ & = e^{-ct} \begin{bmatrix} \cos{l_{j}t} + {\eta \over 2l_j}\sin{l_{j}t}& -{\sin{l_{j}t} \over l_j} \\ \gamma S(j) \lambda^{\beta /2}_j \sin{l_{j}t} & \cos{l_{j}t} - {\eta \over 2l_j}\sin{l_{j}t} \end{bmatrix}, \end{align*} $$B= \begin{bmatrix} 0\\ I_{X} \end{bmatrix}, \quad B^{*} = [0, I_{X}] \quad \mbox{and} \quad BB^{*} =\begin{bmatrix} 0 & 0\\ 0 & I_{X} \end{bmatrix}.$$ Now, let $y=(y_1, y_2)^T$ be in $\mathcal{R}(P_j)$ such that $B^{*}P_{j}^{*}e^{A_{j}^{*}t}y =0$ for all $t \in [0,t_1]$. Then $$e^{-ct} \left[ \gamma S(j) \lambda^{\beta /2}_j \sin{l_{j}t}y_1 + \big(\cos{l_{j}t} - {\eta \over 2l_j}\sin{l_{j}t} \big)y_2\\ \right] = 0, \quad \forall t \in [0,t_1],$$ which implies $y=0$. From Proposition \ref{O1} the operator $W_{j}(t_1): \mathcal{R}(P_j) \to \mathcal{R}(P_j)$ given by $$W_{j}(t_1) = \int_{0}^{t_1} e^{-A_{j}s}BB^{*}e^{-A^{*}_{j}s}ds= P_{j}\int_{0}^{t_1} e^{-B_{j}s}BB^{*}e^{-B^{*}_{j}s}ds P_{j} = P_j \overline{W}_{j}(t_1)P_j$$ is invertible. Since \begin{gather*} \|e^{-A_{j}t} \| \leq M(\eta,\gamma) e^{ct}, \quad \| e^{-A^{*}_{j}t} \| \leq M(\eta,\gamma) e^{ct},\\ \| e^{-A_{j}t}BB^{*}e^{-A^{*}_{j}t} \| \leq M^{2}(\eta,\gamma) \|BB^{*} \| e^{2ct}, \end{gather*} we have $$\|W_{j}(t_1) \| \leq M^{2}(\eta,\gamma)\|BB^{*} \| e^{2ct_1} \leq L(\eta, \gamma), \quad j=1,2, \dots.$$ Now, we shall prove that the family of linear operators, $$W^{-1}_{j}(t_1)= \overline{W}^{-1}_{j}(t_1)P_j :Z_{\beta /2} \to Z_{\beta /2}$$ is bounded and $\|W^{-1}_{j}(t_1) \|$ is uniformly bounded. To this end, we shall compute explicitly the matrix $\overline{W}^{-1}_{j}(t_1)$. From the above formulas we obtain that $$e^{B_{j}t} = e^{-ct} \begin{bmatrix} a(j) & b(j)\\ -a(j) & c(j) \end{bmatrix}, \quad e^{B^{*}_{j}t} = e^{-ct} \begin{bmatrix} a(j) & -b(j) \\ d(j) & c(j) \end{bmatrix},$$ where \begin{gather*} a(j) = \cos{l_{j}t} + {\eta \over 2l_j} \sin{l_{j}t}, \quad b(j) = {\sin{l_{j}t} \over l_j},\\ c(j) = \gamma S(j) \lambda^{\beta /2}_j \sin{l_{j}t}, \quad d(j) = \cos{l_{j}t} - {\eta \over 2l_j}\sin{l_{j}t}, \quad S(j) = \sqrt{\lambda_{j}^{\beta} \over 4\gamma \lambda_{j}^{\beta} - \eta^2 }. \end{gather*} Then $$e^{-B_{j}s}BB^{*}e^{-B^{*}_{j}s} = \begin{bmatrix} b(j)c(j) \lambda^{\beta /2}_{j}I & -b(j)d(j)I\\ -d(j)c(j)\lambda^{\beta /2}_{j}I & d^{2}(j)I \end{bmatrix}.$$ Therefore, $$\overline{W}_{j}(t_1) = \begin{bmatrix} {\gamma S(j) \lambda^{\beta /2}_j \over l_j}k_{11}(j) & {1 \over l_j}k_{12}(j)\\ -\gamma S(j) \lambda^{\beta /2}_j k_{2 1}(j) & k_{22}(j) \end{bmatrix},$$ where \begin{gather*} k_{11}(j) = \int_{0}^{t_1} e^{2cs} \sin^{2}{l_{j}s}ds\\ k_{12}(j) = -\int_{0}^{t_1} e^{2cs} \big[ \sin{l_{j}s} \cos{l_j s} - { \eta \sin^{2}{l_{j}s} \over 2 l_j } \big]ds \\ k_{21}(j) = \int_{0}^{t_1} e^{2cs} \big[ \sin{l_{j}s} \cos{l_j s} - { \eta \sin^{2}{l_{j}s} \over 2 l_j } \big]ds \\ k_{22}(j) = \int_{0}^{t_1} e^{2cs} \big[ \cos{l_j s} - { \eta \sin{l_{j}s} \over 2 l_j } \big]^{2}ds. \end{gather*} The determinant $\Delta(j)$ of the matrix $\overline{W}_{j}(t_1)$ is \begin{align*} \Delta(j) & = {\gamma S(j) \lambda^{\beta /2}_j \over l_j} \left[k_{11}(j)k_{22}(j)-k_{12}(j)k_{21}(j) \right]\\ & = {\gamma S(j) \lambda^{\beta /2}_j \over l_j} \big\{ \big( \int_{0}^{t_1} e^{2cs} \sin^{2}{l_{j}s}ds \big) \big( \int_{0}^{t_1} e^{2cs} \big[ \cos{l_j s} - { \eta \sin{l_{j}s} \over 2 l_j } \big]^{2}ds \big)\\ & \quad- \big( \int_{0}^{t_1} e^{2cs} \big[ \sin{l_{j}s} \cos{l_j s} - { \eta \sin^{2}{l_{j}s} \over 2 l_j } \big]ds \big)^{2} \big\}. \end{align*} Passing to the limit as $j$ approaches $\infty$, we obtain $$\lim_{j \to \infty} \Delta(j) = {(e^{2ct_1} -1)( 1 -2e^{ct_1} + e^{2ct_1}) \over 2^{4}c^3 }.$$ Therefore, there exist constants $R_1, R_2 > 0$ such that $0 < R_1 < | \Delta(j) | < R_2$, $j=1,2,3, \dots$. Hence, $\overline{W}^{-1}(j) = {1 \over \Delta(j)} \begin{bmatrix} k_{22}(j) & -{1 \over l_j}k_{12}(j)\\ \gamma S(j) \lambda^{\beta /2}_j k_{2 1}(j) & {\gamma S(j) \lambda^{\beta /2}_j \over l_j}k_{11}(j) \end{bmatrix} = \begin{bmatrix} b_{11}(j) & b_{12}(j)\\ b_{2 1}(j)\lambda^{\beta /2}_j & b_{22}(j) \end{bmatrix},$ where $b_{n,m}(j)$ are bounded for $n=1,2$; $m=1,2$; $j=1,2, \dots$. Using the same computation as in Theorem \ref{T1} we can prove the existence of a constant $L_{2}(\eta, \gamma)$ such that $$\| W^{-1}_{j}(t_1) \|_{Z_{\beta /2}} \leq L_{2}(\eta, \gamma), \quad j=1,2, \dots.$$ Now, we define the linear bounded operators $W(t_1): Z_{\beta /2} \to Z_{\beta /2}$, $W^{-1}(t_1): Z_{\beta /2} \to Z_{\beta /2}$, by $$W(t_1)z = \sum_{j=1}^{\infty}W_{j}(t_1)P_{j}z, \quad W^{-1}(t_1)z = \sum_{j=1}^{\infty}W^{-1}_{j} (t_1)P_{j}z.$$ Using these definitions we see that $W(t_1)W^{-1}(t_1)z =z$ and $$W(t_1)z = \int_{0}^{t_1} T(-s)BB^{*}T^{*}(-s)zds.$$ Finally, we show that given $z \in Z_{\beta /2}$ there exists a control $u \in L^{2}(0, t_1; X)$ such that $Gu =z$. In fact, let $u$ be the control $$u(t) = B^{*}T^{*}(-t)W^{-1}(t_1)z, \quad t \in [0,t_1].$$ Then \begin{align*} Gu & = \int_{0}^{t_1} T(-s)Bu(s) ds\\ & = \int_{0}^{t_1} T(-s)BB^{*}T^{*}(-s)W^{-1}(t_1)zds\\ & = \Big( \int_{0}^{t_1} T(-s)BB^{*}T^{*}(-s)ds \Big)W^{-1}(t_1)z\\ & = W(t_1)W^{-1}(t_1)z= z. \end{align*} Then the control steering an initial state $z_0$ to a final state $z_1$ in time $t_1 >0$ is given by \begin{align*} u(t) &= B^{*}T^{*}(-t)W^{-1}(t_1)(T(-t_1)z_1 - z_0) \\ & = B^{*}T^{*}(-t)\sum_{j=1}^{\infty} W^{-1}_{j}(t_1)P_{j}(T(-t_1)z_1 - z_0). \end{align*} \end{proof} \section{Exact Controllability of the Non-Linear System} Now, we give the definition of controllability in terms of the non-linear systems $$\label{C121} \begin{gathered} z' = \mathcal{A}z + Bu + F(t, z, u(t)) \quad z \in Z_{\beta /2}, \; t > 0,\\ z(0) = z_0. \end{gathered}$$ For all $z_0 \in Z_{\beta /2}$, equation (\ref{C121}) has a unique mild solution $$\label{C122} z(t) = T(t)z_0 + \int_{0}^{t} T(t)T(-s)[Bu(s) + F(s, z(s), u(s))] ds.$$ \begin{definition} \label{D:d12} \rm We say that system (\ref{C121}) is exactly controllable on $[0,t_1]$, $t_1>0$, if for all $z_0, z_1 \in Z_{\beta /2}$ there exists a control $u \in L^{2}(0,t_1;X)$ such that the solution $z(t)$ of (\ref{C122}) corresponding to $u$, verifies: $z(t_1) = z_1$. \end{definition} Consider the non-linear operator $G_{F}: L^{2}(0,t_1;U) \to Z_{\beta /2}$, \label{c5} given by $$\label{nf1} G_{F}u =\int_{0}^{t_1} T(-s)B(s)u(s) ds + \int_{0}^{t_1} T(-s)F(s, z(s), u(s)) ds,$$ where $z(t)= z(t;z_0,u)$ is the corresponding solution of (\ref{C122}). Then the following proposition is a characterization of the exact controllability of the non-linear system (\ref{C121}). \begin{propostition} \label{pp11} The system \eqref{C121} is exactly controllable on $[0,t_1]$ if and only if, the operator $G_{F}$ is surjective, that is to say $$G_{F} L^{2}(0,t_1;X) = \mathop{\rm Range}(G_F)= Z_{\beta /2}.$$ \end{propostition} \begin{lemma}\label{lF1} Let $u_1, u_2 \in L^{2}(0,t_1; X)$, $z_0 \in Z_{\beta /2}$ and $z_{1}(t;z_0, u_1)$, $z_{2}(t;z_0, u_2)$ the corresponding solutions of (\ref{C122}). Then $$\label{est1} \|z_{1}(t) - z_{2}(t) \|_{ Z_{\beta /2}} \leq M[\|B \| + L]e^{MLt_1}\sqrt{t_1} \|u_1 -u_2 \|_{ L^{2}(0,t_1; X)},$$ where $0 \leq t \leq t_1$ and $$\label{MM} M = \sup_{0 \leq s \leq t \leq t_1}\{ \|T(t) \| \|T(-s) \| \}.$$ \end{lemma} \begin{proof} Let $z_1, z_2$ be solutions of (\ref{C122}) corresponding to $u_1, u_2$ respectively. Then \begin{align*} \|z_{1}(t) - z_{2}(t) \| & \leq \int_{0}^{t} \|T(t) \| \|T(-s) \| \|B \| \|u_{1}(s) -u_{2}(s) \|\\ & \quad + \int_{0}^{t} \|T(t) \| \|T(-s) \| \|F(s, z_{1}(s), u_{1}(s)) -F(s, z_{2}(s), u_{2}(s))\| ds\\ & \leq M[\|B \| + L] \int_{0}^{t} \|u_{1}(s) -u_{2}(s) \| + ML \int_{0}^{t} \|z_{1}(s) - z_{2}(s) \| ds\\ & \leq M[\|B \| + L] \sqrt{t_1} \|u_{1} -u_{2} \| + ML \int_{0}^{t_1} \|z_{1}(s) - z_{2}(s) \| ds. \end{align*} Using Gronwall's inequality, we obtain $$\|z_{1}(t) - z_{2}(t) \|_{ Z_{\beta /2}} \leq M[\|B \| + L]e^{MLt_1}\sqrt{t_1} \|u_1 -u_2 \|_{ L^{2}(0,t_1; X)},$$ for $0 \leq t \leq t_1$. \end{proof} Now, we are ready to formulate and prove the main Theorem of this section. \begin{theorem} If in addition of condition \eqref{f2}, $$\label{EST} \|B \|ML \|W^{-1}(t_1) \| K(t_1) t_{1} < 1,$$ where $K(t_1) = M[\|B \| + L]e^{MLt_1}t_1 + 1$, then the non-linear system \eqref{C121} is exactly controllable on $[0,t_1]$. \end{theorem} \begin{proof} Given the initial state $z_0$ and the final state $z_1$, and $u_1 \in L^{2}(0,t_1; X)$, there exists $u_2 \in L^{2}(0,t_1; X)$ such that $$0=z_1 - \int_{0}^{t_1} T(-s)F(s, z_{1}(s), u_{1}(s)) ds - \int_{0}^{t_1} T(-s)Bu{_2}(s)ds,$$ where $z_{1}(t)= z(t;z_0,u_1)$ is the corresponding solution of (\ref{C122}). Moreover, $u_2$ can be chosen as $$u_{2}(t) = B^{*}T^{*}(-t)W^{-1}(t_1) \Big(z_1 - \int_{0}^{t_1} T(-s)F(s, z_{1}(s), u_{1}(s)) ds \Big).$$ For such $u_2$ there exists $u_3 \in L^{2}(0,t_1; X)$ such that $$0=z_1 - \int_{0}^{t_1} T(-s)F(s, z_{2}(s), u_{2}(s)) ds - \int_{0}^{t_1} T(-s)Bu{_3}(s)ds,$$ where $z_{2}(t)= z(t;z_0,u_2)$ is the corresponding solution of (\ref{C122}), and $u_3$ can be taken as follows: $$u_{3}(t) = B^{*}T^{*}(-t)W^{-1}(t_1) \Big(z_1 - \int_{0}^{t_1} T(-s)F(s, z_{2}(s), u_{2}(s)) ds \Big).$$ Following this process we obtain two sequences $$\{u_n \} \subset L^{2}(0,t_1; X), \quad \{z_n \} \subset L^{2}(0,t_1;Z_{\beta /2}), \quad (z_{n}(t)= z(t;z_0,u_n) ) \; n=1,2, \dots,$$ such that \begin{gather}\label{Se1} u_{n+1}(t) = B^{*}T^{*}(-t)W^{-1}(t_1) \Big(z_1 - \int_{0}^{t_1} T(-s)F(s, z_{n}(s), u_{n}(s)) ds \Big)\\ 0 = z_1 - \int_{0}^{t_1} T(-s)F(s, z_{n}(s), u_{n}(s)) ds - \int_{0}^{t_1} T(-s)Bu_{n+1}(s)ds.\label{Se2} \end{gather} Now, we shall prove that $\{z_n \}$ is a Cauchy sequence in $L^{2}(0,t_1;Z_{\beta /2})$. In fact, from formula (\ref{Se1}) we obtain that \begin{align*} & u_{n+1}(t) -u_{n}(t) \\ &= B^{*}T^{*}(-t)W^{-1}(t_1) \Big(\int_{0}^{t_1} T(-s)( F(s, z_{n-1}(s), u_{n-1}(s))- F(s, z_{n}(s), u_{n}(s)) )ds \Big). \end{align*} Hence, using lemma \ref{lF1} we obtain \begin{align*} & \|u_{n+1}(t) -u_{n}(t) \| \\ & \leq \|B \| ML\|W^{-1}(t_1) \| \int_{0}^{t_1} \left( \|z_{n}(s) - z_{n-1}(s) \| + \|u_{n}(s) -u_{n-1}(s) \| \right)ds \\ & \leq \|B \| ML\|W^{-1}(t_1) \| \int_{0}^{t_1} M[\|B \| + L]e^{MLt_1} \sqrt{t_1} \|u_{n}(s) -u_{n-1}(s) \| ds\\ & \quad+ \|B \| ML\|W^{-1}(t_1) \int_{0}^{t_1} \|u_{n}(s) -u_{n-1}(s) \| ds. \end{align*} Using H\'oder's inequality, we obtain $$\label{Hol1} \|u_{n+1}-u_{n} \|_{L^{2}(0,t_1;X)} \leq \|B \|ML \|W^{-1}(t_1) \| K(t_1) t_{1} \|u_{n+1}-u_{n} \|_{L^{2}(0,t_1;X)}.$$ Since $\|B \|ML \|W^{-1}(t_1) \| K(t_1) t_{1} <1$, it follows that $\{u_n \}$ is a Cauchy sequence in $L^{2}(0,t_1;X)$. Therefore, there exists $u \in L^{2}(0,t_1;X)$ such that $\lim_{n \to \infty}u_{n} = u$ in $L^{2}(0,t_1;X)$. Let $z(t) = z(t;z_0, u)$ the corresponding solution of (\ref{C122}). Then we shall prove that $$\lim_{n \to \infty} \int_{0}^{t_1} T(-s)F(s, z_{n}(s), u_{n}(s)) ds = \int_{0}^{t_1} T(-s)F(s, z(s), u(s)) ds.$$ In fact, using lemma \ref{lF1} we obtain that \begin{align*} & \Big\| \int_{0}^{t_1} T(-s)[F(s, z_{n}(s), u_{n}(s)) - F(s, z(s), u(s))] ds \Big\| \\ & \leq \int_{0}^{t_1} ML[\|z_{n}(s) -z(s) \| +\|u_{n}(s) - u(s) \|] ds\\ & \leq \int_{0}^{t_1} ML[M[\|B \| + L]e^{MLt_1}\sqrt{t_1} \|u_n -u \|_{ L^{2}(0,t_1; X)} + \|u_{n}(s) - u(s) \|] ds\\ & \leq MLK(t_1)\sqrt{t_1} \|u_n -u \|_{ L^{2}(0,t_1; X)}. \end{align*} From here we obtain the result. Finally, passing to the limit in (\ref{Se2}) as $n$ approaches $\infty$, we obtain $$0 = z_1 - \int_{0}^{t_1} T(-s)F(s, z(s), u(s)) ds - \int_{0}^{t_1} T(-s)Bu(s)ds.$$ i.e., $G_{F}u = z_1$. \end{proof} \begin{thebibliography}{00} \bibitem {CT} S. Chen, R. Triggiani; Proof of Extensions of two Conjectures on Structural Damping for Elastic Systems'' Pacific Journal of Mathematics Vol. 136, N0. 1, 1989 \bibitem {CP1} R. F. Curtain , A. J. Pritchard; Infinite Dimensional Linear Systems'', Lecture Notes in Control and Information Sciences, Vol. 8. Springer Verlag, Berlin (1978). \bibitem {GL} L. Garcia, H. L; Center Manifold and Exponentially Bounded Solutions of a Forced Newtonian System with Dissipation''E. Journal Differential Equations. conf. 05, 2000, pp. 69-77. \bibitem{LT} I. Lasiecka and R. Triggiani; Exact Null Controllability of Structurally Damped and Thermoelastic Models''. Atti della Accademia dei Lincei, Serie IX, vol IX, pp 43-69, 1998. \bibitem {L2} H. 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