\documentclass[reqno]{amsart} \usepackage{hyperref} \AtBeginDocument{{\noindent\small {\em Electronic Journal of Differential Equations}, Vol. 2005(2005), No. 75, pp. 1--9.\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}} \begin{document} \title[\hfilneg EJDE-2005/75\hfil Controllability of integrodifferential equations] {Controllability of semilinear integrodifferential equations with nonlocal conditions} \author[R. Atmania, S. Mazouzi\hfil EJDE-2005/75\hfilneg] {Rahima Atmania, Said Mazouzi} % in alphabetical order \address{Rahima Atmania \hfill \break Department of Mathematics, University of Annaba\\ P. O. Box 12, Annaba 23000, Algeria} \address{Said Mazouzi \hfill \break Department of Mathematics, University of Annaba\\ P. O. Box 12, Annaba 23000, Algeria} \email{mazouzi.s@voila.fr} \date{} \thanks{Submitted April 06, 2005. Published July 8, 2005.} \subjclass[2000]{34A10, 35A05} \keywords{Controllability; nonlocal condition; fixed-point theorem; semigroup} \begin{abstract} We establish sufficient conditions for the controllability of some semilinear integrodifferential systems with nonlocal condition in a Banach space. The results are obtained using the Schaefer fixed-point theorem and semigroup theory. \end{abstract} \maketitle \numberwithin{equation}{section} \newtheorem{theorem}{Theorem}[section] \section{Introduction} The first step in the study of the problem of controllability is to determine if an objective can be reached by some suitable control function The problem of controllability happens when a system described by a state $x( t) $ is controlled by a given law such as a differential equation $x'=G( t,x( t) ,u( t) )$. We discuss the possibility of driving a solution of a given system from an initial state to a final state by an adequate choice of the control function $u$. Several authors have studied the problem of controllability of linear semilinear and nonlinear systems of ordinary differential equations in finite or infinite dimensional Banach spaces with bounded operators. For instance, Naito \cite{n1} studied the controllability of semilinear systems, Yamamoto and Park \cite{p1} discussed this problem for a parabolic equation with uniformly bounded nonlinear terms, Chukwu and Lenhart \cite{c1} studied the controllability of nonlinear systems in abstract spaces, Zhou \cite{z1} discussed the approximate controllability for a class of semilinear abstract equations, Naito \cite{n2} established the controllability for nonlinear Volterra integrodifferential systems. Finally, Balachandran and Sakhtivel \cite{b1,b2} studied the controllability of functional semilinear integrodifferential systems in Banach spaces. In this paper, we study the controllability of some semilinear integrodifferential system subject to nonlocal condition in Banach space whose mild solution has been proved by Mazouzi and Tatar \cite{m1} by using Schaefer fixed-point theorem \cite{d1}. \section{Preliminaries} Consider the following functional semilinear integrodifferential system subject to a nonlocal condition: \begin{equation} \label{e1} \begin{gathered} \begin{aligned} x'( t)&=Ax( t) +Bu( t) \\ &\quad +F\Big( t,x(\delta _{1}(t)),\int_0^{t}g\Big( t,s,x( \delta _{2}( s) ) ,\int_0^{s}k( s,\tau ,x(\delta _{3}( \tau ) ) ) d\tau \Big) ds\Big) \end{aligned}\\ x( 0) +h( t_{1},\dots ,t_{p},x( .) ) =x_0, \\ 01$ and $\omega \in \mathbb{R}^{+}$ such that $\| T( t) \| \leq Me^{\omega t}$, $t\geq 0$. \item[(H5)] $h\in C( I,X)$, and there exists a constant $H>0$ such that $\| h( t_{1},\dots t_{p},x) \| \leq H$, for $x\in B_{r}=\{ x\in X:\| x( t) \| \leq r\}$. Moreover, there exists $H_{1}>0$ such that \[ \| h( t_{1},\dots t_{p},x_{1}( .) ) -h( t_{1},\dots t_{p},x_{2}( .) ) \| \leq H_{1} \sup_{t\in I}\| x_{1}( t) -x_{2}( t) \| \] \item[(H6)] \[ \int_0^{b}\widetilde{Q}( t) dt <\int_{a}^{+\infty} \frac{dz}{\varphi ( z) +z^{\alpha }+z }, \] where $\widetilde{Q}( t) =\max \{ \omega ,\omega MM_{1}M_{2},\omega Mp( t) ,\omega Mq( t) ,h( t)\}$ with \[ h( t) =\frac{1}{\alpha }m_{1}( t,t) +\frac{1}{ \alpha }\int_0^{t}\big| m_{2}( t) m_{3}( t,\tau ) +\frac{\partial m_{1}( t,\tau ) }{\partial t}\big| d\tau , \] and $a^{\alpha }=M^{\alpha }( \| x_0\| +H) ^{\alpha }+N$, with \[ N=\Big( \| x_{1}\| +Me^{\omega b}( \| x_0\| +H) +M\int_0^{b}e^{\omega ( b-\tau ) }\| \phi (\tau ,x)\| d\tau \Big) . \] \item[(H7)] The linear operator $W:\mathbb{L}^{2}( I,U)\to X$ defined by \[ Wu=\int_0^{b}T( b-s) Bu( s) ds \] has an invertible operator $W^{-1}$ which takes values in $\mathbb{L}^{2}(I,U) /\ker W$ and there exist positive constants $M_{1}$, $M_{2}>0$ such that $\| B\| \leq M_{1}$ and $\| W^{-1}\| 0$ such that $v( t) \leq c$, for every $t\in I$. Consequently, $\| x( t) \| \leq c$ for every $t\in I$. In what follows we prove that the operator $\Phi $ is completely continuous. If $y( t) \in V:\| y( t) \| \leq r$, for $r>0$, then \begin{align*} &\big\| F\Big( t,y( t) ,\int_0^{t}g\Big( t,\theta ,y( \theta ) ,\int_0^{\theta }k( \theta ,\tau ,y( \tau ) ) d\tau \Big) d\theta \Big) \big\| \\ &\leq p( t) \| y( t) \| ^{\alpha }+\ q( t) \int_0^{t}m_{1}( t,\theta ) \| y( \theta ) \| ^{\alpha -1}\varphi ( \| y( \theta ) \| ) \\ &\quad +\ m_{2}( \theta ) \int_0^{\theta }m_{3}( \theta ,\tau ) \| y( \tau ) \| ^{\alpha -1}\varphi ( \| y( \tau ) \| ) d\tau \,d\theta \\ &\leq p( t) r^{\alpha }+q( t) r^{\alpha -1}\varphi ( r) \int_0^{t}( m_{1}( t,\theta ) +m_{2}( \theta ) \int_0^{\theta }m_{3}( \theta ,\tau ) d\tau )\, d\theta . \end{align*} We denote the last term of the latter inequality by $F_{r}( t) $. It is obvious that for each $r>0$, $F_{r}$ is summable over $I$. Consider a sequence $( x_{n}) _{n\geq 1}\subset V$ converging to $\widehat{x}\in V$, then $( x_{n}) _{n\geq 1}( t) $ and $\widehat{x}( t) $ must be contained in some closed ball $B(0,r) \subset X$, for all $t\in I$. It follows from hypotheses (H1) and (H2) that \[ \lim_{n\to \infty } \phi ( t,x_{n}) =\phi ( t,\widehat{x}) \quad\text{and}\quad \| \phi ( t,x_{n}) -\phi (t,\widehat{x}) \| \leq 2F_{r}( t) . \] We conclude by\ the dominated convergence theorem that \[ \int_0^{b}\| \phi ( s,x_{n}) -\phi ( s, \widehat{x}) \| ds\to 0,\quad \text{when }n\to \infty . \] Define the sequence $\{ u_{n}\}_{n\geq 1}$ as follows \[ u_{n}( t) =W^{-1}\Big( x_{1}-T( b) ( x_0-h( t_{1},t_{2},\dots ,t_{p},x_{n}) ) -\int_0^{b}T( b-s) \phi ( s,x_{n}) ds\Big) ( t). \] Then \begin{align*} &\| Bu_{n}( s) -Bu( s) \|\\ &\leq \| BW^{-1}\| \Big[ \| T( b) ( h( t_{1},t_{2},\dots ,t_{p},x_{n}) -h( t_{1},t_{2},\dots ,t_{p},\widehat{x} ) ) \| \\ &\quad + \| \int_0^{b}T( b-s) ( \phi ( s,x_{n}) -\phi ( s,\widehat{x}) ) ds\| \Big]\\ &\leq MM_{1}M_{2}e^{\omega b} \Big(H_{1}\sup_{t\in I}\|x_{n} -\widehat{x}\| +\int_0^{b}e^{-\omega s}\| \phi ( s,x_{n}) -\phi ( s,\widehat{x}) \| ds\Big) \to 0, \end{align*} as $n\to \infty $. We infer that \begin{align*} \| \Phi x_{n}-\Phi \widehat{x}\| & \leq \sup_{t\in I}\| T( t) ( h( t_{1},t_{2},\dots ,t_{p},x_{n}) -h(t_{1},t_{2},\dots ,t_{p},\widehat{x}) ) \|\\ &\quad +\sup_{t\in I} \| \int_0^{t}T( t-s) [ ( \phi ( s,x_{n}) -\phi ( s,\widehat{x}) ) +( Bu_{n}( s) -Bu( s) ) ] ds\| \\ &\leq MH_{1}e^{\omega t}\sup_{t\in I}\| x_{n}(t) -\widehat{x}( t) \| \\ &\quad + Me^{\omega b}\Big[ \int_0^{b}( \| \phi ( s,x_{n}) -\phi ( s,\widehat{x}) \| +\| Bu_{n}( s) -Bu( s) \| ) ds\Big] \to 0\,, \end{align*} as $n\to \infty $. This shows that $\Phi $ is continuous. For every positive real number $r$ we set $B_{r,V}=\{ x\in V:\| x( t) \| \leq r\} $. To show that $\Phi (B_{r,V}) $ is precompact in $V$ we only have to check the precompactness of $\Phi ( B_{r,V}) ( t) $ in $V$, for each $t\in I$, according to Arzela -Ascoli theorem. Let $t$ be fixed in $] 0,b] $ and $n\in \mathbb{N}^{*}:\frac{1}{n}0$, there exists $n_0\in \mathbb{N}^{*}$ such that for every $n\geq n_0$, and $x\in B_{r,V}$, we have \[ \| ( T_{n}x) ( t) \| \leq \int_{t-\frac{1}{n}}^{t}\| T( t-s) \| ( M_{1}M_{2}\tilde{N}+F_{r}( s) ) ds < \epsilon \,, \] where \[ \tilde{N}=\Big( \| x_{1}\| +Me^{\omega b}( \| x_0\| +H) +M\int_0^{b}e^{\omega ( b-\tau ) }F_{r}( \tau ) d\tau \Big) . \] Next, we define \begin{align*} &( S_{n}( x) ) ( t)\\ & =T( t) ( x_0-h( t_{1},\dots t_{p},x) ) +T( \frac{1}{n}) \int_0^{t-\frac{1}{n}}T( t-s-\frac{1}{n}) ( Bu( s) +\phi ( s,x) ) ds\,. \end{align*} Following the steps of the proof of the main theorem in \cite{m1} we can show that $\Phi ( B_{r,V}) ( t) $ is compact and consequently the operator $\Phi $ is completely continuous. Therefore, $\Phi $ has a fixed point in $V=C(I,X) $ which is the expected mild solution we are seeking and accordingly the system is controllable on $I$. \end{proof} \section{Example} Consider the problem \begin{equation} \label{e6} \begin{gathered} \begin{aligned} z_{t}( t,y) &=z_{yy}( t,y) +u( t,y) + \frac{z^{2}( t,y) \sin ( z( t,y) ) }{ ( 1+t) ( 1+t^{2}) } \\ &\quad +\int_0^{t}\Big[ \frac{z( s,y) }{( 1+t)( 1+t^{2}) ^{2}( 1+s) ^{2}}\\ &\quad +\frac{1}{( 1+t) ( 1+t^{2}) } \int_0^{s}\frac{z( \tau,y) }{( 1+s) (1+\tau ) }\exp z( \tau ,y) d\tau \Big] ds \end{aligned} \\[3pt] z( t,0) = z( t,1) =0, \quad t\in I=[0,1] \\ z( 0,y) -\sum_{i=1}^p t_{i}z( t_{i},y) =z_0( y) , \quad 00$ such that $|h( t_{1},t_{2},\dots ,t_{p},x( .) )|