\documentclass[reqno]{amsart}
\usepackage{hyperref}
\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2009(2009), No. 79, pp. 1--6.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2009 Texas State University - San Marcos.}
\vspace{9mm}}
\begin{document}
\title[\hfilneg EJDE-2009/79\hfil Some remarks on controllability]
{Some remarks on controllability of evolution equations in
Banach spaces}
\author[S. K. Ntouyas, D. O'Regan\hfil EJDE-2009/79\hfilneg]
{Sotiris K. Ntouyas, Donal O'Regan} % in alphabetical order
\address{Sotiris K. Ntouyas \newline
Department of Mathematics,
University of Ioannina, 451 10 Ioannina, Greece}
\email{sntouyas@uoi.gr}
\address{Donal O'Regan \newline
Department of Mathematics\\
National University of Ireland, Galway, Ireland}
\email{donal.oregan@nuigalway.ie}
\thanks{Submitted May 1, 2009. Published June 24, 2009.}
\subjclass[2000]{93B05}
\keywords{Semilinear differential equations; controllability;
fixed point theorem}
\begin{abstract}
In almost all papers in the literature, the results on exact
controllability hold only for finite dimensional Banach spaces,
since compactness of the semigroup and the bounded invertibility
of an operator implies finite dimensional.
In this note we show that the existence theory on controllability
in the literature, can trivially be adjusted to include the
infinite dimensional space setting, if we replace the compactness
of operators with the complete continuity of the nonlinearity.
\end{abstract}
\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}{Lemma}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{remark}{Remark}[section]
\section{Introduction \& Preliminaries}
In a long list of papers in the literature on exact controllability of
abstract control systems (e.g. \cite{NO1, NO3, NO4})
the compactness of the linear operators $T(t)$, $t>0$, with other
hypothesis guarantee that the Banach space
$X$ is finite dimensional. This has been pointed out by a number of
authors (see \cite{HO, Tr}). However it is easy to consider
the case when $X$ is infinite dimensional. Simply replace the
compactness of $T(t)$, $t>0$ with the complete continuity of the
nonlinearity. As a result the case when $X$ is infinite dimensional is
trivially extended (the proof
is almost exactly the same as in the literature). For simplicity we will
consider the case in \cite{NO1} (the other papers \cite{NO3}-\cite{NO5}, \cite{NO6}-\cite{NO7}, \cite{NO9}, use exactly the same ideas).
Consider the first order
semilinear controllability problem of the form
\begin{gather}\label{e1-1c}
y'(t)= Ay(t)+f(t,y(t))+(\mathcal{B}u)(t), \quad t\in J:=[0,b], \\
\label{e1-2c}
y(0)=y_{0}.
\end{gather}
Here $J=[0,b]$, $b>0$, $f:J \times X\to X$,
$A: D(A)\subset X\to X$ is the infinitesimal generator of a
$C_0$ semigroup $T(t), t\geq 0$, $y_0\in X$,
and $X$ a real Banach space with norm $|\cdot|$.
Also the control function
$u(\cdot)$ is given in $L^{2}(J,U)$, a Banach space of admissible
control functions with $U$ as a Banach space.
Finally $\mathcal{B}$ is a bounded linear operator from $U$ to $X$.
\begin{definition} \rm
A function $y\in C(J,X)$ is said to be a mild
solution of \eqref{e1-1c}--\eqref{e1-2c} if $y(0)=y_0$ and
$$
y(t)=T(t)y_0+\int_0^tT(t-s)[(\mathcal{B}u)(s)+f(s,y(s))]ds.
$$
\end{definition}
\begin{definition} \rm
The system \eqref{e1-1c}--\eqref{e1-2c} is said to be controllable
on the interval $J$, if for every $y_0, y_{1}\in X$
there exists a control $u\in L^{2}(J,U)$, such that there exists a
mild solution
$y(t)$ of \eqref{e1-1c}--\eqref{e1-2c} satisfying $y(b)=x_{1}$.
\end{definition}
In almost all papers in the literature, including our papers
\cite{NO1}, \cite{NO3}-\cite{NO5}, \cite{NO6}-\cite{NO7}, \cite{NO9},
the study of controllability is based on the compactness of the
operator $T$ and the bounded invertibility on the operator $W$, i.e.:
\begin{itemize}
\item[(HT)] $A$ is the infinitesimal generator of a strongly continuous
semigroup of\\ bounded linear operators
$T(t), t\ge 0$ on $X$, which is compact for $t>0$.
\item[(HW)] $\mathcal{B}$ is a continuous operator from $U$ to $X$
and the linear operator $W:
L^{2}(J,U)\to X$, defined by
$$Wu=\int_{0}^{b} T(b-s)\mathcal{B}u(s)\, ds,$$
has a bounded invertible operator $W^{-1}: X\to L^{2}(J,U)$
such that $\|\mathcal{B}\|\le M_1$ and
$\|W^{-1}\|\le M_2$, for some positive
constants $M_1, M_2$.
\end{itemize}
We remark (see \cite{HO, Tr}) that the above two hypotheses are valid if $X$
is finite dimensional. In this note we show that the existence theory in
the literature can trivially be adjusted to include the infinite
dimensional space setting if we replace the compactness of the linear
operators with the complete continuity of the nonlinearity.
\section{First Order Abstract Semilinear Differential Equations}
Using hypothesis (HW) for an arbitrary function $y(\cdot)$ define
the control
$$
u_y(t)=W^{-1}\Big[y_1-T(b)y_0-\int_0^bT(b-s)f(s,y(s))\, ds\Big](t).
$$
To prove the controllability of the problem \eqref{e1-1c}--\eqref{e1-2c},
we must show that when using this control, the operator
$K_1:C([0,b], X)\to C([0,b], X)$ defined by
$$
K_1y(t)=T(t)y_0+\int_0^tT(t-s)[f(s,y(s))+(\mathcal{B}u_y)(s)]ds, \quad
t\in [0,b]
$$
has a fixed point.
In the following we use the notation:
\begin{gather*}
M=\sup\{\|T(t)\|: t\in [0,b]\}, \\
B_r=\{y\in C([0,b],X): \|y\|\le r\}.
\end{gather*}
Also it is clear that for $y\in B_r$ we have
$$
\|(\mathcal{B}u_y)(s)\|\le M_1 M_2\Big[|y_1+M|y_0|
+\int_0^bh_{\rho}(s)\, ds\Big]:=G_0.
$$
\begin{remark}\label{r1} \rm
In what follows, without loss of generality, we take $y_0=0$,
since if $y_0\ne 0$ and $y(t)=x(t)+T(t)y_0$ it is
easy to see that $x$ satisfies
\begin{gather*}
x'(t)=\int_0^tT(t-s)f(s,x(s)+T(s)y_0)\, ds+(\mathcal{B}u)(t), \quad t\in J\\
x(0)=0.
\end{gather*}
\end{remark}
We concentrate only in proving in detail the complete continuity
of the operator $K_1y(t)$, since all the other steps in
\cite{NO1} remain unchanged. Thus we have the following result.
\begin{lemma}\label{le1}
Suppose {\rm (HW)} holds. In addition assume the following conditions
are satisfied:
\begin{itemize}
\item[(HT)] $T(\cdot)$ is strongly continuous.
\item[(Hf1)] $f:[0,b]\times X\to X$ is an $L^1$-Car\'atheodory function, i.e.
\begin{itemize}
\item[(i)] $t\mapsto f(t,u)$ is measurable for each $u\in X$;
\item[(ii)] $u\mapsto f(t,u)$ is
continuous on $X$ for almost all $t\in J$;
\item[(iii)] For each $\rho>0$, there exists
$h_{\rho} \in L^{1}(J,{\mathbb R}_{+})$ such that for a.e. $t\in J$
$$
\sup_{|u|\le \rho}\|f(t,u)\|\leq h_{\rho}(t).
$$
\end{itemize}
\item[(Hf2)] $f: [0,b]\times X\to X$ is completely continuous.
\end{itemize}
Then the operator
$$
Ky(t)=\int_0^tT(t-s)[f(s,y(s))+(\mathcal{B}u_y)(s)]ds, \quad t\in [0,b]
$$
is completely continuous.
\end{lemma}
\begin{proof}
(I) The set $\{Ky(t): y\in B_r\}$ is precompact in $X$, for every
$t\in [0,b]$.
It follows from the strong continuity of $T(\cdot)$ and conditions (Hf1),
(Hf2) that the set
$\{T(t-s)f(s,y): t,s\in [0,b], y\in B_r\}$ is relatively compact in $X$.
Moreover, for $y\in B_r$, from the mean value theorem for the Bochner
integral, we obtain
$$
Ky(t)\in t\, \overline{\rm conv}\{T(t-s)f(s,y): s\in [0,t], y\in B_r\},
$$
for all $t\in [0,b]$.
As a result we conclude that the set $\{Ky(t): y\in B_r\}$ is
precompact in $X$, for every $t\in [0,b]$.
(II) The set $\{Ky(t): y\in B_r\}$ is equicontinuous on $[0,b]$.
We just do the case $00$. From (I) $(KB_r)(t)$ is
relatively compact for each $t$ and by the strong continuity of
$(T(t))_{t\ge 0}$ we can choose $0<\delta\le b-t$ with
\begin{equation}\label{eq1}
\|T(h)y-y\|<\epsilon\quad \mbox{for } y\in (KB_r)(t) \mbox{ when }
0