\documentclass[reqno]{amsart}
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\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 $0<t\le b$. A similar argument will work if $t=0$.
Let $\epsilon>0$. 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<h<\delta.
\end{equation}
For $y\in B_r$ we have
\begin{align*}
Ky(t+h)-Ky(t)
&=\int_{t}^{t+h}T(t+h-s)[f(s,y(s))+(\mathcal{B}u_y)(s)]\, ds\\
&\quad +\int_0^t[T(t+h-s)-T(t-s)][f(s,y(s))+(\mathcal{B}u_y)(s)]\, ds\\
&=\int_{t}^{t+h}T(t+h-s)[f(s,y(s))+(\mathcal{B}u_y)(s)]\, ds\\
&\quad +(T(h)-I)\int_0^tT(t-s) f(s,y(s))\, ds\\
&=\int_{t}^{t+h}T(t+h-s)[f(s,y(s))+(\mathcal{B}u_y)(s)]\, ds \\
&\quad +(T(h)-I)Ky(t),
\end{align*}
so
\begin{align*}
 \|Ky(t+h)-Ky(t)\|
      &\leq M\int_t^{t+h}[\|f(s,y(s))\|+M_1G_0]ds+\|[T(h)-I]Ky(t)\|\\
      &\leq M \int_t^{t+h}[h_{r}(s)+M_1G_0]ds+\|[T(h)-I]Ky(t)\|.
\end{align*}
The equicontinuity follows from \eqref{eq1}.
\end{proof}


\begin{remark} \rm
 We  can apply the same idea to establish existence results for
the initial value problem  \eqref{e1-1c}--\eqref{e1-2c},
when $\mathcal{B}=0$.
As a result we obtain alternative results (i.e. we replace
the compactness of the operators with the complete continuity of the
nonlinearity) to those in \cite{NO2}-\cite{NO3},
\cite{NO6},  \cite{NO8}-\cite{NO10}.
We have:
  \begin{lemma}
 Assume that the conditions {\rm (HT), (Hf1), (Hf2)} hold.
Then the operator
 $$
K_2y(t)=\int_0^tT(t-s)f(s,y(s))ds, \quad t\in [0,b]
$$
 is completely continuous.
  \end{lemma}
  \end{remark}


\section{Second Order Semilinear Differential Equations}

Consider the semilinear second order differential control system
\begin{gather}\label{se1}
y''(t)=Ay(t)+f(t,y(t))+(\mathcal{B}u)(t),\quad t\in J:=[0,b],\\
\label{se2}
y(0)=x_0, \quad y'(0)=\eta
\end{gather}
where $x_0, y_0\in X$, $A$ is the infinitesimal generator of the
strongly continuous cosine family $C(t), t\in {\mathbb R}$, of bounded
linear operators in $X$, and $f, \mathcal{B}$ are as in
problem \eqref{e1-1c}--\eqref{e1-2c}.

Recall  that a family $\{C(t)\mid t\in {\mathbb R}\}$ of operators
in $B(X)$ is a \emph{strongly continuous cosine family} if
\begin{itemize}
\item[(i)] $C(0)=I$,
\item[(ii)] $C(t+s)+C(t-s)=2C(t)C(s)$, for all $s,t\in {\mathbb R}$,
\item[(iii)] the map $t\mapsto C(t)(x)$ is strongly continuous, for
each $x\in X$.
\end{itemize}

The strongly continuous sine family $\{S(t): t\in {\mathbb R}\}$,
associated to the given strongly continuous cosine family
$\{C(t): t\in {\mathbb R}\}$, is defined by
\begin{equation}\label{sin}
S(t)(y)=\int_0^tC(s)(y)\, ds,\quad x\in X,\; t\in {\mathbb R}.
\end{equation}
The infinitesimal generator $A: X\to X$ of a cosine family
$\{C(t):t\in {\mathbb R}\}$
is defined by
$$
A(y)=\frac{d^2}{dt^2}C(t)(y)\Bigr|_{t=0}.
$$

\begin{definition} \rm
A continuous solution of the integral equation
$$
y(t)=C(t)x_0+S(t)\eta+\int_{0}^{t}S(t-s)[(\mathcal{B}u)(s)
+f(s,y(s))]ds,\quad t\in J
$$
 is said to be a mild solution of the problem \eqref{se1}-\eqref{se2}
 on $J$.
\end{definition}

\begin{definition} \rm
The system \eqref{se1}-\eqref{se2} is said to be  controllable on
the interval $J$, if for every  $x_0, \eta, x_{1}\in X$
 there exists a control $u\in L^{2}(J,U)$, such that the mild solution
$y(t)$ of
\eqref{se1}-\eqref{se2}  satisfies
$y(b)=x_{1}$.
\end{definition}

Assume that
\begin{itemize}
\item[(W)] $\mathcal{B}$ is a   continuous operator from $U$ to $X$
and the linear operator $W: L^{2}(J,U)\to X$, defined by
$$
Wu=\int_{0}^{b} S(b-s)\mathcal{B}u(s)\, ds,
$$
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}

Using hypothesis (W) for an arbitrary function $y(\cdot)$
define the control
$$
u_{y}(t)=W^{-1}\Bigl[y_{1}-f(y)-C(b)y_0-S(b)\eta-\int_{0}^{b}
S(b-s)f(s,y(s))ds\Bigr](t).
$$

Then we must show that when using this control, the operator
$N_1:C(J,X)\to C(J,X)$ defined by:
$$
 N_1y(t) =C(t)y_0+S(t)\eta+\int_{0}^{t}S(t-s)
[(\mathcal{B}u_{y})(s)+f(s,y(s))] ds
$$
has a fixed point.

\begin{remark} \rm
We can take $y_0=0, \eta=0$. See Remark \ref{r1}.
\end{remark}

The following lemma is proved as in Lemma \ref{le1}.

\begin{lemma}\label{le2}
Assume {\rm (Hf1), (Hf2), (W)} hold. Then the operator
$$
Ny(t)=\int_{0}^{t}S(t-s)[(\mathcal{B}u_{y})(s)+f(s,y(s))] ds, \quad
t\in J
$$
is completely continuous.
\end{lemma}

 \begin{remark} \rm
 Similar results to those of Lemmas \ref{le1} and \ref{le2} hold
for differential inclusions.
 \end{remark}

\begin{thebibliography}{99}

\bibitem{NO1} L. Gorniewicz, S. K. Ntouyas and D. O'Regan;
 Controllability of semilinear differential equations and inclusions
via semigroup theory in Banach spaces,
\emph{Rep. Math. Phys.} {\bf 56} (2005), 437--470.

\bibitem{NO2} L. Gorniewicz, S. K. Ntouyas and D. O'Regan;
 Existence results for first and second order semilinear
differential inclusions with nonlocal conditions, \emph{J. Comput.
Analysis Appl.} {\bf 9} (2007),  287-310.

\bibitem{NO3} L. Gorniewicz, S. K. Ntouyas and D. O'Regan;
Existence  and controllability results for first and second order
functional semilinear differential inclusions with nonlocal conditions,
\emph{Numer. Funct.  Anal. Optim.} {\bf 28}  (2007), 53-82.

\bibitem{NO4} L. Gorniewicz, S. K. Ntouyas and D. O'Regan;
Controllability results for first and second order evolution
inclusions with nonlocal conditions,
\emph{Ann. Polon. Math.} {\bf 89} (2006), 65-101.

\bibitem{NO5} L. Gorniewicz,  S. K. Ntouyas and D. O'Regan;
Controllability of  evolution inclusions in Banach spaces with
nonlocal conditions, \emph{Nonlinear Anal. Forum}
{\bf 12} (2007), 103-117.

\bibitem{HO} E. Hernandez and D. O'Regan;
 Controllability of Volterra-Fredholm type systems in
Banach spaces, \emph{J. Franklin Inst.} {\bf 346} (2009), 95-101.

\bibitem{NO6}  S. K. Ntouyas and D. O'Regan;
 Existence and controllability results for
first and second order semilinear neutral functional
 differential inclusions with nonlocal conditions,
\emph{Dissc. Math. Differ. Incl.} {\bf 27} (2007), 213-264.

\bibitem{NO7}  S. K. Ntouyas and D. O'Regan, Controllability for
semilinear neutral functional differential inclusions via analytic
semigroups, \emph{J. Optimization Theory Appl.} {\bf 135}
(2007), 491-513.

\bibitem{NO8}  S. K. Ntouyas and D. O'Regan;
 Existence results  for semilinear neutral
functional differential inclusions via analytic semigroups,
\emph{Acta Appl. Math.} {\bf 98} (2007), 223-253.

\bibitem{NO9} S. K. Ntouyas and D. O'Regan;
 Existence results on semi-infinite intervals for nonlocal
evolution equations and inclusions via semigroup theory, \emph{Numer.
Funct.  Anal. Optim.} {\bf  29} (2008),  419-444.

\bibitem{NO10} S. K. Ntouyas and D. O'Regan;
Existence results for semilinear neutral functional
differential   inclusions with nonlocal conditions,
\emph{Differential equations \& Applications} {\bf 1}
(2009), 41-65.

\bibitem{Tr}  R. Triggiani, A note on the lack of exact controllability
for mild solutions in Banach spaces, \emph{SIAM J. Control Optim.} {\bf 15} (1977) 407-411.

\end{thebibliography}

\end{document}