\documentclass[reqno]{amsart}
\usepackage{hyperref}

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2017 (2017), No. 111, pp. 1--12.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu}
\thanks{\copyright 2017 Texas State University.}
\vspace{7mm}}

\begin{document}
\title[\hfilneg EJDE-2017/111\hfil Approximate controllability]
{Approximate controllability of nonautonomous nonlocal delay
differential equations with deviating arguments}

\author[R. Haloi \hfil EJDE-2017/111\hfilneg]
{Rajib Haloi}

\address{Rajib Haloi \newline
Department of Mathematical  Sciences,
Tezpur University, Sonitpur,
Assam,  Pin 784028, India}
\email{rajib.haloi@gmail.com, Phone+913712-275511, Fax +913712-267006}

\dedicatory{Communicated by Jesus Ildefonso Diaz}

\thanks{Submitted May 9, 2016. Published April 26, 2017.}
\subjclass[2010]{34G20, 34K30, 35K35, 93C25}
\keywords{Approximate controllability; deviating arguments; 
\hfill\break\indent delay equation;  Krasnoselskii's fixed point theorem}

\begin{abstract}
 The objective of this article is to prove  sufficient conditions for the
 approximate controllability for a class of nonautonomous nonlocal finite delay
 differential equations with deviating arguments in a Hilbert space.
 We also establish  sufficient conditions for the existence of mild solutions.
 The results are established using the fixed point theorem of Krasnoselskii and
 theory of semigroup of bounded linear operators.
 We discuss an example for the application of the analytical  results.
\end{abstract}

\maketitle
\numberwithin{equation}{section}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{remark}[theorem]{Remark}
\newtheorem{definition}[theorem]{Definition}
\allowdisplaybreaks

\section{Introduction}

Let $(X,\|\cdot\|)$ be a complex Hilbert space.
We  study the approximate controllability for the system
consisting of nonautonomuos nonlocal finite
delay differential equation with deviating arguments in $X$,
\begin{equation}
\begin{gathered}{} 
[\frac{d}{dt}+A(t)]x(t)
=f(t,x(t),x([h(x(t),t)]))+Bu(t),\quad t \in J=[0,b],  \\
x(t)= \phi(t)+g(x)(t),\quad t\in[-a,0]. \label{01}
\end{gathered}
\end{equation}
Here, we assume that $-A(t)$, for each $t\geq 0$
generates a compact analytic semigroup of bounded linear operators on $X$,
 $u(\cdot)$  is the control function in $L^2(J,U)$
for a Hilbert space $U$,
$B$ is a bounded linear operator on $U$ into $X$. 
The  functions  $f:J\times X \times X\to X $, 
$h: X\times J \to J $ and $g: C([-a,b],X) \to C([-a,0],X)$ satisfy suitable
conditions in their arguments stated in Section 2.


Differential equations
with deviating arguments are the generalization of differential
equations in which the unknown quantity and its derivative appear in
different values of their arguments\cite{SPB14,DPS15,norkin,ejde2,ejde1,KPB14,MB8}.
They arise as models for the important class of physical phenomenon such as 
self-oscillating systems, the theory of automatic control, the
problems of long-term planning in economics,  the systems in
bio-physics, the study of problems related with combustion in rocket
engines, and many other areas of science and technology, and the
number  is increasing.
In some of the  models, the information is transferred from the input 
to the output after finite time. Such systems are called  the system with 
finite delay. The output is connected with the state space. 
Considering the plentiful  applications of  the differential equation with 
deviating arguments, many author have studied differential equation with 
deviating arguments extensively e.g.\
\cite{SPB14,DPS15,norkin,gal,ejde2,deds,ejde1,KPB14,MB8}.


 The existence of a deviation-delay in time is necessary to
avoid the unstable combustion in liquid rocket engines.
 The delay (in time) in automatic regulator
system  cause the appearance of a self-exciting oscillation,
overregulation, and even of the instability of the system. In this
system,  the delay is  needed to react to the input impulse for the
system. Some of the systems in  mathematical modeling of many real world phenomena,
namely in control theory, population dynamics, biology and epidemiology, 
electro-mechanical and medical domains can be modelled by nonlocal differential 
equations with delay. For more details of such systems,
we refer to \cite{SPB14,fu14,KB,KBS16,W}.

 On the other hand, the concept of controllability is of great importance in
mathematical  theory of control of finite or infinite dynamical systems \cite{CZ}.
 For a nice introduction on control theory of linear systems,
 we refer to \cite{BPDM,CZ}. The main objective of the controllability is to show 
the existence of a control function, which steers the solution of the system from 
its initial state to the final state. Exact  controllability enables to steer 
the system to arbitrary final state. However, there are systems
where it possible to steer the system to arbitrary small neighbourhood of the 
final state.
This is known as approximate controllability. As far as the applications are 
concerned, the approximate controllability is more relevant to dynamical systems 
and  the area  got much attentions in  recent years
\cite{SPB14,DPS15,Mah02,fu11,fu14,RKG,AGKR15,KB,KBS16,Mah03,MH08,Mah14,W,zhu}.


 It  is worth mentioning that  the controllability of the  systems with nonlocal 
conditions are  better than classical Cauchy condition\cite{LB98,Ezz-Fu-Hi-07}. 
So, the approximate controllability of nonlocal systems with delay-deviating 
arguments have practical  importance and   studied much  in the  recent
years by many authors in 
\cite{SPB14,Mah02,fu11,fu14,AGKR15,KB,KBS16,Mah03,MH08,Mah14,MA14,W,zhu}.

Das et al.\ \cite{SPB14}
used the Schauder fixed point theorem in their study
of  approximate controllability for the following system with deviating 
arguments in a Hilbert space $X$, 
\begin{equation}
\begin{gathered}{} 
[\frac{d}{dt}-A]x(t)=f(t,x_t,x([h(x(t),t)]))+Bu(t),\quad t \in J=[0,b],   \\
x(t)= \phi(t),\quad t\in[-a,0].
\end{gathered} \label{013}
\end{equation}
Here, we assume that $-A(t)$, for each $t\geq 0$ generates a strongly 
continuous semigroup of bounded linear operators on $X$, $u(\cdot)$  
is the control function in $L^2(J,U)$ for a Hilbert space $U$,
$B$ is a bounded linear operator on $U$ into $X$.
The  functions $f:J\times X \times X\to X $ and $h: X \times J \to J $ 
satisfy Lipschitz  conditions in their arguments\cite{SPB14}.

Very recently, Kamaljeet  et al.\ \cite{KBS16}  studied the
approximate controllability for the following   integro-differential
equations with nonlocal condition  in a Hilbert space
$X$,
\begin{equation}
\begin{gathered} 
^cD^qx(t)+Ax(t)=f(t,x_t)+\int_0^tk(t-s)h(s,x_s)ds+Bu(t),\\
 t \in J=[0,b],   \\
x_0= \phi+g(x),\quad \text{on } [-a,0], 
\end{gathered} \label{011}
\end{equation}
where $^cD^q$ is the Caputo fractional derivative of order $0<q<1$, $A$
generates an analytic semigroup of bounded linear operators on $X$, $u(\cdot)$ 
 is the control function in $L^2(J,U)$ for a Hilbert space $U$,
$B$ is a bounded linear operator on $U$ into $X$. The approximate controllability 
results are established by the fixed point argument for
the system \eqref{011}  with appropriate functions $f,g,h$ and the  kernel $k$.

However, the approximate controllability for the nonlocal nonautonomous
 systems with deviating arguments have not studied so far.
In this article, we devote our study of  the approximate controllability  for the
nonautonomous systems with deviating arguments for the system \eqref{01} in 
an arbitrary infinite dimensional Hilbert space.
 The results are new and generalize the results
in \cite{SPB14,AGKR15}.

We organize the article as follows. In Section \ref{preliminaries}, 
we  provide preliminaries,
assumptions and Lemmas that will be needed for proving the main
results. We  prove the local existence of a solution in Section
\ref{existence}. The approximate controllability results are established 
in Section \ref{existence}.
Finally, we  provide an example to illustrate the
application of the abstract results.

\section{Preliminaries}\label{preliminaries}

In this section, we  introduce notation, variuos assumptions and
Lemmas for the use of the remaining part of the article. We briefly
outline the facts concerning evolution family of bounded linear operators, 
controllability, control function and mild solutions. We refer the book 
by Bensoussan \cite{BPDM} and Curtain and Zwart \cite{CZ},
Friedman \cite{AF}, Pazy \cite{pazy}, Tanabe \cite{tanabe} and Yosida\cite{ky} 
for more details.

Let $X$ and $U$ be two complex Hilbert spaces. 
Let $T\in [0,\infty)$  and $\{A(t): 0\leq t \leq T \}$ be a family of closed
linear operators on the Hilbert space $X$. Let $\mathcal L (X)$ denote 
the Banach space of all bounded linear operator on $X$. We assume  the following
hypothesis.
\begin{itemize}
\item[(H1)] For each $0\leq t \leq T$, $-A(t)$ is closed linear operators that 
generates the compact analytic semigroup of bounded linear operator
$\mathcal U: \Delta \to \mathcal L (X)$, where 
$(t,s)\in \Delta=\{(t,s)\in J \times J : 0 \leq s \leq t \leq T\}$.
The domain $D(A)$ of $A(t)$ is dense in $X$ and is independent of
$t$.
\end{itemize}

\begin{remark} \rm
The evolution semigroup $\mathcal U(t,s)$ is strongly continuous on the compact 
set $\Delta$, there exists a constant $M>0$
such that \begin{equation}
           \|\mathcal U(t,s)\| \leq M\quad \text{for all } (t,s)\in \Delta.
          \end{equation}
 \end{remark}

\begin{definition} \rm
 An operator $\mathcal U: \Delta \to \mathcal L (X)$ is said to be a compact 
evolution family if the following holds,
 \begin{itemize}
  \item [(a)] $\mathcal U(s,s)=I$ is the identity operator in $X$ for $s \in J$,
  \item[(b)] $\mathcal U(t,r)U(r,s)=U(t,s)$, $0 \leq s \leq r \leq \leq t \leq T$,
  \item[(c)] $\mathcal U$ is strongly continuous on $\Delta$,
  \item[(d)] $\mathcal U(t,s)$ satisfies
  $$
\frac{\partial \mathcal U(t,s)}{\partial t}+A(t)\mathcal U(t,s)=0, \quad
  \frac{\partial \mathcal U(t,s)}{\partial s}-\mathcal U(t,s)A(s)=0,\quad
 (t,s)\in \Delta,
$$
  \item[(e)] $\mathcal U(t,s)$ are completely continuous for $(t,s)\in \Delta$.
\end{itemize}
\end{definition}

 Let $x(b,\phi,u)$ be the state value of the system \eqref{01} at terminal 
time $b$ corresponding to the initial value $\phi$
  and the control function $u$. We define the following  set 
$$
R(b,\phi)=\{x(b,\phi,u): u \in L^2(J,U)\}.
$$
  The set $R(b,\phi)$ is called the reachable set   of the system \eqref{01} 
at time $b$.

\begin{definition}\rm
(1) A controllability map for the system \eqref{01} on $J$ is the bounded linear 
map $\mathcal B^b : L^2(J,U) \to X$ which is defined as
  \begin{equation}
   \mathcal B^b u:= \int_0^b \mathcal U(b,s)Bu(s)ds,\quad\text{for }
 u \in L^2(J,U).\label{02}
  \end{equation}

(2) The system \eqref{01} is exactly controllable on $J$ if $R(b,\phi)=X$, 
that is for all $y_0,y_1 \in X$, there exists $u \in L^2(J,U)$ such that the 
mild solution to the system \eqref{01} satisfies $x(0,\phi, u)=y_0$ and 
$x(b,\phi, u)=y_1$.

(3) The system \eqref{01} is approximately controllable on $J$ if 
$\overline{R(b,\phi)}= X$, that is for given $\epsilon >0$,
 and $y_0,y_1 \in X$, there exists a control $u \in L^2(J,U)$ steers from the 
point $x(0,\phi, u )=y_0$  to all points at time $b$ within a distance of 
$\epsilon$ from $y_1$.
 More precisely, 
$$ 
x(0,\phi, u )=y_0,~~\|x(b,\phi,u)-y_1\|< \epsilon.
$$

(4) The controllability Gramian of the system \eqref{01} on $J$ is defined by
$$
\Gamma_0^b:=\mathcal B^b(\mathcal B^b)^*.
$$
\end{definition}

\begin{lemma}[\cite{AGKR15}] \label{lem2.4}
 The following properties hold for the controllability map:
 \begin{itemize}
  \item [(a)] $(\mathcal B^b)^*z(s)=B^*\mathcal U^*(b,s)z$,
 for $s\in [0,T]$, $z\in X$.

  \item [(b)] $\Gamma_0^b=\mathcal B^b (\mathcal B^b)^* \in \mathcal L(X)$ 
has the  representation
  \begin{equation}
   \Gamma_0^bz=\int_0^b \mathcal U(b,s)BB^*\mathcal U^*(b,s)zds,
\quad \text{for } z\in X
  \end{equation}
and $\Gamma^b_0 \geq 0$, where $B^*$ and $\mathcal U^*$ denote the adjoint 
of $B$ and $\mathcal U$ respectively.
 \end{itemize}
\end{lemma}

We consider the following control system in $X$,
\begin{equation}
\begin{gathered}{}
 [\frac{d}{dt}+A(t)]x(t)=Bu(t),\quad  t\in J,   \\
x(0)= \phi(0).
\end{gathered} \label{03}
\end{equation}
We define the resolvent operator associated with \eqref{03} as
$$
R(\epsilon, \Gamma_0^b)=(\epsilon I+\Gamma_0^b)^{-1}, \epsilon >0.
$$ 
We  use the assumption
\begin{itemize}
\item[(H2)] $\epsilon R (\epsilon, \Gamma_0^b) \to 0$ as $\epsilon \to 0^+$ 
in the strong operator topology.
\end{itemize}

\begin{theorem}[\cite{Mah03}]
 Let $H$ be a separable Banach space with dual $H^*$. The following tow statements
are equivalent for a symmetric operator  $P: H^* \to H$:
 \begin{itemize}
  \item [(i)] $P$ is positive,
  \item [(ii)] $x_\epsilon (h)=\epsilon (\epsilon I+PQ)^{-1}(h)\to 0$  as 
$\epsilon \to 0^+$in the strong operator topology,
  where $Q: H \to H^*$ denotes the duality map.
 \end{itemize}
\end{theorem}

\begin{theorem}[\cite{Mah14}]
 System \eqref{03} is approximately controllable on $J$ if and only if 
the condition {\rm (H2)} holds.
\end{theorem}

It follows from (H2) that  system \eqref{03} is approximately controllable on 
$J$ if and only if
$$
\langle v,\Gamma_0^bv\rangle =\int_0^b \|B^*\mathcal U^*(b,s)v\|^2 ds >0,
\quad \forall v(\neq 0)\in X.
$$
We need the following hypotheses for proving the main results.
\begin{itemize}
\item[(H3)] For every $t\in J$; $x,y, x', y' \in X$,
there exist constants $L_f> 0$ and $M_f>0$  the nonlinear map $f:J\times
X \times X\to X$ satisfies
\begin{equation}\label{f1}
\begin{gathered}
\|f(t,x,x')-f(s,y,y')\| \le L_f (\|x-y\|+ \|x'-y'\| ),\\
\|f(0,x(0), x(h(x(0),0)))\| \leq M_f,\quad \forall t, s\in J \\
f(t,\cdot,\cdot) \text{ is continuous.}
\end{gathered}
\end{equation}

\item[(H4)] There exist constants $L_h> 0$  such that  
$h:X\times J \to J$ satisfies the  condition
\begin{equation}\label{h201}
|h(x,t)-h(y,s)| \le L_h (\|x-y\|),\quad h(\cdot ,0)=0
\end{equation}
for all  $x,y\in X$ and for all $t,s  \in J $.

\item [(H5)] The function $g: \mathcal C \to C([-a,0], X)$ satisfies
\begin{equation}
\begin{aligned}
 \|g(x)-g(y)\|_{C([-a,0], X)} \leq L_g (\|x-y\|_{\mathcal C}),\quad
 \forall x,y \in \mathcal C,\\
 \|g(x)\|_{C([-a,0], X)} \leq L_g (1+\|x\|_{\mathcal C}),\quad
 \forall x \in \mathcal C,
\end{aligned}
\end{equation}
where $\mathcal C=C([-a,b], X)$.
\end{itemize}
For $z \in X$ and $\epsilon >0$, we define the control function
$u_{\epsilon}(t,x)$ for the system \eqref{01} by
\begin{equation}
\begin{aligned}
 u_{\epsilon}(t,x)
&=B^*\mathcal U^*(b,s)R(\epsilon,\Gamma_0^b)\Big\{z-U(b,0)[\phi(0)+g(x)(0)]  \\
& \quad -\int_0^b\mathcal U(b,s)f(s,x(s),x([h(x(s),s)]))ds\Big\}.
\end{aligned}
\end{equation}
We also recall the Krasnoselskii's fixed point theorem.

\begin{theorem}
 Let $P$ be a map from a closed bounded convex subset $S$ of $X$ into $S$. 
Suppose that $Px=P_1x+P_2x$ for $x\in S$
and $P_1u+P_2v \in S$ for every pair $u,v \in S$. If $P_1$ is contraction 
and $P_2$ is compact, then the equation $P_1u+P_2u=u$ has a solution in $S$.
\end{theorem}

\section{Existence of Solution}\label{existence} 

In this section, we establish the existence and uniqueness of a local 
solution to the system \eqref{01} corresponding to a given control function 
$u_\epsilon$. The proof of the theorem is based on  the technique of 
\cite{AGKR15,KBS16}.

We define the 
$$
C_L(J, X)=\{x \in C(J,X): \|x(t)-x(s)\|\leq L|t-s|\text{ for a constant }
 L >0, t,s \in J\}
$$
 and  the space
$$
C_{L_0}([-a,b],X)=\{x \in C([-a,b],X) : x \in C_L(J,X)\}.
$$

\begin{definition}\rm
  A  function $x \in C_{L_0}([-a,b],X) $ is said to be a mild solution to 
problem \eqref{01} if $x(t)$ satisfies
  \begin{equation}
\begin{gathered}
\begin{aligned}
x(t)&=\mathcal U(t,0)[\phi(0)+g(x)(0)]
 + \int_0^t\mathcal U(t,s)f(s,x(s),x([h(x(s),s)]))ds\\
&\quad + \int_0^t\mathcal U(t,s)Bu(s)ds, \quad t \in J=[0,b],
\end{aligned} \\
 x(t)=\phi(t)+g(x)(t),\quad t\in[-a,0].
\end{gathered} 
\end{equation} 
\end{definition}

\begin{theorem}\label{thm1}
System \eqref{01} has a unique mild solution in $C_L(J,X)$ for each control 
$u_\epsilon \in L^2(J,U)$ if the assumptions
{\rm (H1)--(H5)} hold and
  $$
ML_g+ML_f(2+LL_h)b <1.
$$
\end{theorem}

 \begin{proof} 
We consider the  ball
   $$
B_r=\{ x \in C_{L_0}([-a,b], X): \|x\|_{C_{L_0}([-a,b], X)} \leq r\}.
$$
For each $x \in B_r$, we define the map $F_\epsilon$ by
\[
F_\epsilon x(t)
=\begin{cases}
 \mathcal U(t,0)[\phi(0)+g(x)(0)]+\int_0^t\mathcal U(t,s)f(s,x(s),x([h(x(s),s)]))ds
\\
+\int_0^t\mathcal U(t,s)Bu_\epsilon(s,x)ds,\quad\text{if } t \in J=[0,b],\\[4pt]
\phi(t)+g(x)(t),\quad\text{if } t\in[-a,0]
\end{cases}
\]
For simplicity, we denote
\begin{gather*}
\begin{aligned}
l&= \frac{1}{\epsilon}\|BB^*\mathcal U^*(b,s)\|
\Big\{\|z\|+M[\|\phi(0)\|+L_g(1+r)]  \\
&\quad +[2M(L_f(1+LL_h)r+M_fM] b\Big\},
\end{aligned}\\
K=\frac{1}{\epsilon}\|B\|\sup_{t \in J} \|B^* \mathcal U ^*(b,t)\|.
\end{gather*}
  For $t  \in J$, we have the  estimate
\begin{equation}
\begin{aligned}
 &\|Bu_{\epsilon}(t,x)\| \\
 &\leq \frac{1}{\epsilon}\|BB^*\mathcal U^*(b,s)\|
 \Big\{\|z\|+M[\|\phi(0)\|+\|g(x)(0)]\| \\
 &\quad +M\int_0^b \Big[\|f(s,x(s),x([h(x(s),s)]))-f(s,0,x([h(x(0),0)])))\|\\
&\quad +\|f(s,0,x([h(x(0),0)])))\|\Big]ds\Big\} \\
 &\leq K\Big\{\|z\|+M[\|\phi(0)\|+L_g(1+r)] \\
 & \quad +M\int_0^b [(L_f(\|x(s)-x(0)\|+LL_h\|x(s)-x(0)\|)+M_f]ds\Big\} \\
& \leq K\Big\{\|z\|+M[\|\phi(0)\|+L_g(1+r)] +[2M(L_f(1+LL_h)r+M_fM ]b\Big\} \\
 &=l .
\end{aligned}\label{cnt}
\end{equation}  



Let  $t_1,t_2  \in J$ with $t_1< t_2$ and $ x \in X$.
 Using \cite[Lemmas II.14.1 and  14.4]{AF}, we obtain
\begin{align*}
\|F_\epsilon x(t_1)- F_\epsilon x(t_2)\|
&\leq \|\mathcal U(t_1,0) - \mathcal U(t_2,0)\|(\|\phi(0)\|+\|g(x)(0)\|)\\
&\quad +\Big \|\int_0^{t_1}\mathcal U(t_1,s)f(s,x(s),x([h(x(s),s)]))ds \\
&\quad -\int_0^{t_2}\mathcal U(t_2,s)f(s,x(s),x([h(x(s),s)]))ds  \Big\|\\
&\quad + \big\|\int_0^{t_1}\mathcal U(t_1,s)Bu_\epsilon(s,x)ds
 -\int_0^{t_2}\mathcal U(t_2,s)Bu_\epsilon(s,x)ds \big\|\\
& \leq C_1 (t_2-t_1)+C_2( M_f+l) (1+|\log (t_2-t_1)|)(t_2-t_1),
\end{align*}
 where $C_1=C(\|\phi(0)\|+\|g(x)(0)\|)$, $C_2$ and $C_3$ are positive constants.
Thus $F_\epsilon \in C_L(J,X)$.

Using   estimate \eqref{cnt}, we obtain
\begin{align*}
 &\|F_\epsilon x(t)\|\\
 & \leq  M[\|\phi(0)\|+L_g(1+r)]+ \int_0^t [2M(L_f(1+LL_h)r+M_f ]ds+\int_0^tMl\}ds\\
 & \leq   M[\|\phi(0)\|+L_g(1+r)]+ M[2(L_f(1+LL_h)r+M_f ]b+Mlb\\
 & \leq  r,
\end{align*} provided that
\begin{gather*}
  M[\|\phi(0)\|+L_g(1+r)]+ M[2(L_f(1+LL_h)r+M_f ]b+Mlb \leq r, \quad
  \text{or}\\
M\|\phi \|+ML_g+M\{L_g+2L_f(1+LL_h)b\}r +M(M_f+l)b \leq r,\quad \text{or}\\
M\|\phi \|+ML_g +M(M_f+l)b \leq r [1 -M\{L_g+2L_f(1+LL_h)b\}].
\end{gather*}
This is possible only if $2ML_f(1+LL_h)b \leq M\{L_g+2L_f(1+LL_h)b <1$. 
Thus we choose $b$ such that
$$
b <  \frac{1}{2ML_f(1+LL_h)}.
$$
So, $F_\epsilon $ maps $B_r$ into itself. We decompose  $F_\epsilon$ as
$F_\epsilon= F_{\epsilon,1} +F_{\epsilon,2}$, where
\begin{gather*}
F_{\epsilon,1} x(t)=\begin{cases}
 \mathcal U(t,0)[\phi(0)+g(x)(0)]
 +\int_0^t\mathcal U(t,s)f(s,x(s),x([h(x(s),s)]))ds\\
\quad\text{if }t \in J=[0,b],   \\[4pt]
 \phi(t)+g(x)(t), \quad\text{if }t\in[-a,0],
\end{cases}\\
F_{\epsilon,2} x(t)=\begin{cases}
\int_0^t\mathcal U(t,s)Bu_\epsilon(s)ds, \quad\text{if }t \in J=[0,b],  \\
 0,\quad\text{if } t\in[-a,0].
\end{cases} 
\end{gather*}

We begin  by showing that  $ F_{\epsilon,1}$
is a contraction on $B_r$. For $v_1,v_2 \in B_r$ and $t \in J$, we have
\begin{align*}
&\|F_{\epsilon,1} v_1(t)-F_{\epsilon,1} v_2(t)\| \\
& \leq \|U(t,0)[g(v_1)(0)-g(v_2)(0)]\| \\
& \quad +\int_0^t\mathcal U(t,s)\Big[f(s,v_1(s),v_1([h(v_1(s),s)]))
 -f(s,v_2(s),v_2([h(v_2(s),s)]))\Big]ds\\
& \leq M L_g \|v_1-v_2\|_\mathcal C+ b ML_f(2+LL_h) \|v_1-v_2\|_\mathcal C  \\
& \leq [ML_g+bML_f(2+LL_h)]\|v_1-v_2\|_\mathcal C.
\end{align*}
Also for $t \in [-a,0]$, we have
$$
\|F_{\epsilon,1} v_1(t)-F_{\epsilon,1} v_2(t)\| 
\leq L_g\|v_1-v_2\|_\mathcal C.
$$
Thus we conclude that
$$
\|F_{\epsilon,1} v_1-F_{\epsilon,1} v_2\|_\mathcal C \leq \|v_1-v_2\|_\mathcal C.
$$ 
Hence $F_{\epsilon,1}$ is contraction on $B_r$.
We next show that the map $F_{\epsilon,2}$ is completely continuous.
\smallskip

\noindent\textbf{Step I:} 
Let $\{v_n\}$ be a sequence in $B_r$ such that $v_n \to v \in B_r$
 as $n \to \infty$. It follows from $(H3)-(H5)$ that
 \begin{itemize}
  \item [(a)] $\|Bu_\epsilon (s, v_n)-Bu_\epsilon (s, v)\|\to 0 $ as $n \to \infty$,
  \item[(b)] $\|Bu_\epsilon (s, v_n)-Bu_\epsilon (s, v)\| \leq 2l$.
 \end{itemize}
Using the dominated convergence theorem, we obtain that
\begin{align*}
 \|F_{\epsilon,2} v_n(t) - F_{\epsilon,2} v(t) \|
 &\leq \int_0^t\mathcal \|U(t,s)[Bu_\epsilon(s, v_n)-Bu_\epsilon(s, v)]\|ds\\
 & \leq M \int_0^t \|Bu_\epsilon(s, v_n)-Bu_\epsilon(s, v)\| ds 
  \to 0 \quad \text{as } n \to \infty.
 \end{align*}

\noindent\textbf{Step II:} 
Let $t_1,t_2 \in J$ such that $t_1 < t_2$ and $v\in B_r$. It follows 
from \cite[Lemma,II. 14.1, 14.4]{AF} that
\[
 \|F_{\epsilon,2} v(t_2) - F_{\epsilon,2} v(t_1)\| \leq C_4 (t_2-t_1)^\beta,
\]
 for some constants $0\leq \beta \leq 1$ and $C_4>0$. 
Thus $\{F_{\epsilon,2}(B_r)\}$ is equicontinous on $J$.
\smallskip

\noindent \textbf{Step III:} 
We show that $\{ F_{\epsilon,2}v(t): v \in B_r\}$ is relatively compact in $X$. 
For $t \in [-a,0]$,
$$
\{ F_{\epsilon,2}v(t): v \in B_r\}=\{0\}.$$ If $0<\eta <t$, then we have
\begin{align*}
 F^\eta_{\epsilon,2}v(t)
&=\int_0^{t-\eta} \mathcal U(t,s)Bu_\epsilon (s, v)ds\\
 &= \mathcal U(t,t-\eta)\int_0^{t-\eta} \mathcal U(t-\eta,s)Bu_\epsilon (s, v)ds\\
 &=\mathcal U(t,t-\eta)I(t,\eta),
\end{align*}
where $I(t, \eta)=\int_0^{t-\eta} \mathcal U(t-\eta,s)Bu_\epsilon (s, v)ds$. 
We note that $I(t,\eta)$ is bounded on $B_r$.
As $\mathcal U(t,s)$ is compact in $X$, so for each $t \in (0,b]$, the set 
$\{ F^\eta_{\epsilon,2}v(t): v \in B_r\}$
is relatively compact in $X$.
Indeed, we have
\begin{align*}
 \| F_{\epsilon,2}v(t)-F^\eta_{\epsilon,2}v(t)\|
&\leq \int_{t-\eta}^t\|\mathcal U(t,s)Bu_\epsilon (s, v)\| ds\\
&\leq Ml\eta  \to 0 \quad \text{as } \eta \to 0^+.
\end{align*}
Thus the set $\{ F_{\epsilon,2}v(t): v \in B_r\}$ is arbitrarily close to the 
relatively compact  set
$\{ F^\eta_{\epsilon,2}v(t): v \in B_r\}$ for each $t \in J$. 
Hence, for all $t \in[-a,b]$ the set
$\{ F_{\epsilon,2}v(t): v \in B_r\}$ is relatively compact in $X$.

By Ascoli-Arzela theorem, the set $\{ F_{\epsilon,2}v: v \in B_r\}$ is relatively 
compact in $C([-a,b],X)$. Thus the map $F_{\epsilon,2}$ is completely continuous 
from $B_r$ to $B_r$.

Thus the map $F_\epsilon$ has fixed point on $B_r$ by Krasnoselskii's fixed 
 point theorem. Hence for each $\epsilon >0$, the system \eqref{01} has a mild
 solution in $B_r$ corresponding to each control $u_\epsilon (s,x)$.
 \end{proof}

\section{Approximate Controllability}

 We prove the following theorem of approximate controllability for 
the system \eqref{01}.

 \begin{theorem}
  Let the assumptions {\rm (H1)--(H5)} hold. Let the functions 
$f:J\times X \times X\to X$, $h: X
\times J \to J $ and    $g: \mathcal C \to C([-a,0], X)$ be  uniformly bounded. Then the system \eqref{01}
  is approximately controllable on $J$.
 \end{theorem}

\begin{proof}
 From Theorem \ref{thm1}, $F_\epsilon$ has fixed point $x_\epsilon$ 
in $B_r \subset C_{L_0}([-a,b],X)$.
 That is,
 $x_\epsilon$ is a mild solution for the control
\[
  u_\epsilon(t,x_\epsilon)
=B^*\mathcal U^*(b,t)R(\epsilon,\Gamma_0^b)p(x_\epsilon),
\]
where,
 \begin{align*}
  p(x_\epsilon)
&=z-\mathcal U(b,0)[\phi(0)+g(x_\epsilon)(0)]\\
&\quad -  \int_0^b\mathcal U(b,s)f(s,x_\epsilon(s),
 x_\epsilon([h(x_\epsilon(s),s)]))ds.
\end{align*}
Further, we have
 \begin{equation}
\begin{aligned}
 x_\epsilon(b)
&= \mathcal U(b,0)[\phi(0)+g(x)(0)]
 +\int_0^b\mathcal U(b,s)f(s,x_\epsilon(s),x_\epsilon([h(x_\epsilon(s),s)]))ds \\
&\quad +\int_0^b\mathcal U(b,s)Bu_\epsilon(s,x_\epsilon)ds,\quad t \in J=[0,b],  \\
&= z-p(x_\epsilon)+\Gamma_0^b R(\epsilon,\Gamma_0^b)p(x_\epsilon)  \\
&= z-\epsilon R(\epsilon,\Gamma_0^b)p(x_\epsilon).
\end{aligned} \label{re}
\end{equation}
Since $f:J\times X \times X\to X $ and $h: X \times J \to J $ are uniformly bounded, 
it follows that $f(s,x_\epsilon(s),x_\epsilon([h(x_\epsilon(s),s)])))$
is bounded in $L^2(J,X)$. 
Thus there exists a subsequence denoted by 
$f(s,x_\epsilon(s),x_\epsilon([h(x_\epsilon(s),s)])) ) $ that converges to
$f(s)$ say. 
It follows from the compactness of $\mathcal U(b,0)$ and the boundedness of $g$ 
that $\mathcal U (b,0)g(x_\epsilon)(0)$ is relatively compact. So, 
there exists a subsequence denoted by itself and converges to $\widetilde{g}$ say. 
We define
\begin{equation*}
 \alpha= z-\mathcal U (b,0)\phi(0)-\widetilde{g}-\int_0^b\mathcal U(b,s)f(s)ds.
\end{equation*}
By the compactness of  $\mathcal U(t,s)$  and  Arzela-Ascoli theorem, we have
\begin{equation}
\begin{aligned}
&\|p(x_\epsilon)-\alpha\| \\
&\leq M\|g(x_\epsilon)(0)-\widetilde{g}\|
+ M\int_0^b\|f(s,x_\epsilon(s),x_\epsilon([h(x_\epsilon(s),s)]))-f(s)\|ds \\
& \to 0 \quad\text{as } \epsilon \to 0+.
\end{aligned} \label{app}
\end{equation}
Again from \eqref{re}, we have
\begin{align*}
 \|x_\epsilon(b)-z\|
& \leq \|\epsilon(\epsilon,\Gamma_0^b)(\alpha)\|
 + \|\epsilon(\epsilon,\Gamma_0^b)(\alpha)\|\|\alpha-p(x_\epsilon)\|\\
 & \leq  \|\epsilon(\epsilon,\Gamma_0^b)(\alpha)\|+\|p(x_\epsilon)-\alpha\|.
\end{align*}
By assumption (H2) and \eqref{app},
we have
$$
\|x_\epsilon(b)-z\| \to 0 \quad\text{as }\epsilon \to 0^+.
$$
This completes the proof.
\end{proof}

\section{Application}\label{application}

   Let $X=L^2([0,\pi]\times [0,b];\mathbb R)$. We  consider the following system
 with deviating arguments in $X$,
\begin{equation}
\begin{gathered}
\begin{aligned}
&\frac{\partial w(x,t)}{\partial t}
 +[\kappa(x,t)+\frac{\partial ^2}{\partial x^2} ]w(x,t)\\
&=Bu(x,t)+f(x,t,w(x,t),w(x,h(w(x,t),t))),\quad b>t>0,\;x\in [0,\pi],
\end{aligned}\\
w(0,t)=0=w(\pi,t),~ 0\leq t\leq b,\\
w(x,\tau)= \psi(x,\tau)+\int_0^bH(s,\tau) \cos(w(s,x))ds,
\quad x\in [0,\pi],\; \tau \in[-a,0],
\end{gathered} \label{ex2011}
\end{equation}
where 
\[
f(x,t,w(x,t),w(x,h(w(x,t),t)))=
\int _0^\pi \beta(y,x)w(y,\chi(t)|w(y,t)|) dy
\]
for all $(x,t)\in [0,\pi]\times[0,b]$,
$\chi:\mathbb{R_{+}}\to \mathbb{R_{+}}$ is locally H\"{o}lder continuous in 
$t$ with $\chi(0)=0$ and 
$\beta\in C^{1}({[0,\pi]}\times {[0,\pi]};\mathbb{R})$, 
$H(s,\tau)$ is $C^1([0,b]\times [-a,0],\mathbb R)$,
$\kappa(x,t)$ are $C^1([0,\pi] \times [0,b], \mathbb R)$.

We write $w(t)(x)=w(x,\tau)$
\[
f(t,w(t),w(h(w(t),t)))(x)=f(x,t,w(x,t),w(x,h(w(x,t),t))),
\]
$\psi(t)(x)=\psi(x,t)$.
With this notation, system \eqref{ex2011} can be put in the form of \eqref{01}.

We define
$$
A(t)v(x)=[\kappa(x,t)+\frac{\partial ^2}{\partial x^2}] v(x,t),
$$ 
where $\frac{\partial ^2}{\partial x^2}$ is
the distributional derivative of $u$. Then
$D(A(t))=H^{2}(0,\pi)\cap H^1_0(0,\pi)$. It is known that
 that $-A(t)$ generates a compact analytic evolution
semigroup of bounded operators $ \mathcal U (t,s)$ on $L^2[0,\pi]$ \cite{AF} 
and is given by
$$
\mathcal U(t,s)v= T(t-s)e^{\int_s^t \kappa(\tau) d\tau}v, \quad v \in D(A(t)).
$$
Here 
\[
T(t)v(\tau)= \sum_{n=1}^{\infty}e^{-n^2\pi^2 t}\langle v,e_n\rangle_{L^2}e_n(\tau)
\]
 with $e_n(\tau)=\sqrt{2}\sin n \tau$, $n=1,2,3,\dots$,
 and $\|T(t)\|\leq e^{-\pi^2 t}, t\ge 0$.
We can show that assumptions (H3) and (H4) are satisfied for  the functions 
$f$ and $h$ respectively. We also note that  $g$ satisfies 
assumption (H5).

We define an infinite dimensional control space 
$$
U=\{u:u=\sum_0^\infty u_n e_n(x), \sum_0^\infty |u_n|^2 <\infty\},
$$
endowed with the norm $\|u\|=(\sum_0^\infty |u_n|^2)^{1/2}$.
 We define $B: U \to X$ by
$$
Bu=3u_2e_1(x)+\sum_{n=2}^\infty u_n e_n(x).
$$ 
Then $B$ is a  bounded linear map  and the adjoint is 
$$
B^*v=(3v_2+2v_2)e_2(x)+\sum_{n=3}^\infty u_n e_n(x).
$$ 
If we assume that 
$B^*U*(t,s)v=0$,
then $v=0$. Thus  system
\eqref{ex2011} is approximately controllable on $[0,b]$.


 \subsection*{Acknowledgements}
 The  author would like to thank Dr. Kamaljeet and Prof. Bahuguna for the 
encouragement and fruitful discussions. The author also thanks
 to Mr. Duranta Chutia for correction of the typos that improved the manuscript.

\begin{thebibliography}{00}

\bibitem{BPDM} A.  Bensoussan, G. Da Prato, M. C.  Delfour, S. K. Mitter;
\emph{Representation and Control of Infinite Dimensional Systems,}
vol. 2. Birkhäuser, Berlin 1993.

\bibitem{LB98} L. Byszewski;
Existence of solutions of semilinear functional differential evolution 
nonlocal problem, \emph{Nonlinear Anal.}
\textbf{34} (1998), 65--72.

\bibitem{chan-Liu-09} J. C. Chang,  H. Liu;
 Existence of solutions for a class of neutral partial differential
equations with nonlocal conditions in the $\alpha$--norm,
\emph{Nonlinear Anal.}, \textbf{71}, no. 9 (2009),  3759--3768.

\bibitem{CZ} R. F. Curtain, H. J. Zwart;
\emph{An introduction to infinite-dimensional linear systems theory,}
Texts in Applied Mathematics, \textbf{21}. Springer-Verlag, New York, 1995.


\bibitem{SPB14} S.  Das, D. N. Pandey,  N. Sukavanam;
 Approximate controllability of a functional  differential equation with
deviated argument,  \emph{Nonlinear Dyn. Syst. Theory},
 \textbf{14}(3) (2014), 266--278.

\bibitem{DPS15} S.  Das, D. N. Pandey,  N. Sukavanam;
 Approximate controllability of an impulsive neutral fractional stochastic
 differential equation with deviated argument and infinite delay.
\emph{Nonlinear Stud.}, \textbf{22}, no. 1 (2015), 115--130.

\bibitem{Mah02}  J. P. Dauer, N. I. Mahmudov;
  Approximate controllability of semilinear functional equations in Hilbert spaces.
\emph{J. Math. Anal. Appl.}, \textbf{273}, no. 2 (2002), 310--327.


\bibitem{norkin}L. E. El'sgol'ts,  S. B. Norkin;
\emph{Introduction to the theory of differential equations with deviating
arguments}, Academic Press, 1973.

\bibitem{Ezz-Fu-Hi-07} K. Ezzinbi, X. Fu, K.  Hilal;
 Existence and regularity in the $\alpha$ norm for some neutral partial differential
equations with nonlocal conditions, \emph{Nonlinear Anal.},
\textbf{67} (2006),  1613--1622.

\bibitem{AF} A. Friedman;
\emph{Partial Differential Equations}, Holt, Rinehart and Winston, 
Inc., New York-Montreal, Que.-London, 1969.

\bibitem{fu11} X. Fu;
Approximate controllability for neutral impulsive differential inclusions with 
nonlocal conditions,
\emph{J. Dyn. Control Syst.}, \textbf{17}, no. 3 (2011), 359--386.

 \bibitem{fu14} X.  Fu;
 Approximate controllability of semi-linear neutral integro-differential systems
 with finite delay.  \emph{Appl. Math. Comput.}, \textbf{242} (2014), 202--215.

\bibitem{gal}C. G. Gal;
 Nonlinear abstract differential equations with deviated argument,  
\emph{J. Math. Anal. Appl.}  \textbf{333} (2007),  no. 2, 971--983.

\bibitem{RKG} R. K. George;
Approximate controllability of nonautonomous semilinear systems.
\emph{Nonlinear Anal.}, \textbf{24} (1995), no. 9, 1377--1393.

\bibitem{AGKR15} A.  Grudzka, K.  Rykaczewski;
On approximate controllability of functional
impulsive evolution inclusions in a Hilbert space,
\emph{J. Optim. Theory Appl.} \textbf{166} (2015), no. 2, 414--439.

\bibitem{ejde2}  R. Haloi;
 Solutions to quasi-linear differential equations with iterated deviating 
arguments, \emph{Electron. J. Differential Equations}, Vol. 2014 No. 249 (2014), 
 1-13.

\bibitem{deds} R. Haloi, D. N. Pandey, D. Bahuguna;
Existence and Uniqueness of a Solution for a Non-Autonomous Semilinear 
Integro-Differential Equation with Deviated Argument,
\emph{ Differ. Equ. Dyn. Syst.},\textbf{20 (1)}, 2012, 1--16.


\bibitem{ejde1}  R. Haloi, D. N. Pandey,  D. Bahuguna;
Existence and uniqueness of solutions for quasi-linear differential equations 
with deviating arguments,  \emph{Electron. J. Differential Equations},
Vol. 2012, No. 13 (2012), No. 13, pp. 1--10.

\bibitem{KB} Kamaljeet, D. Bahuguna;
Approximate Controllability of Nonlocal Neutral Fractional Integro-Differential 
Equations with Finite Delay, \emph{ J. Dyn. Control Syst.},
 DOI 10.1007/s10883-015-9297--0.

\bibitem{KBS16} Kamaljeet, D. Bahuguna, R. K. Sukla;
 Approximate controllability of finite delay fractional
functional integro-differential equations with nonlocal condition, 
\emph{Differ. Equ. Dyn. Syst.}, DOI 10.1007/s12591-016-0284-z

\bibitem{KPB14} P. Kumar,  D.  N.Pandey,  D.  Bahuguna;
 Approximations of solutions to a fractional differential equation with a 
deviating argument, \emph{Differ. Equ. Dyn. Syst.}, \textbf{22} no. 4, (2014),  
333--352.

\bibitem{Mah03} N. I.  Mahmudov;
Approximate controllability of semilinear deterministic and stochastic 
evolution equations in abstract spaces,
\emph{SIAM J. Control Optim.}, \textbf{42}, no. 5 (2003), 1604--1622.

\bibitem{MH08} N. I.  Mahmudov;
Approximate controllability of evolution systems with nonlocal conditions,
\emph{Nonlinear Anal.}, \textbf{68}, no. 3 (2008), 536--546.

\bibitem{Mah14}  N. I. Mahmudov, S. Zorlu;
 On the approximate controllability of fractional evolution equations with
compact analytic semigroup,
\emph{ J. Comput. Appl. Math.}, \textbf{259}, part A (2014),  194--204.

\bibitem{MB8}  Muslim, M.; Bahuguna;
 D. Existence of solutions to neutral differential equations with deviated argument.
\emph{Electron. J. Qual. Theory Differ. Equ.} 2008, No. 27 (20208), 12 pp.


 \bibitem{MA14} Muslim, Ravi P. Agarwal; 
Exact Controllability of an Integro-differential Equation with Deviated Arguments,
 \emph{Funct. Differ. Equ.}, \textbf{ 21}, no.1-2 (2014),  31-45.

\bibitem{pazy}A.  Pazy;
 \emph{Semigroups of Linear Operators and Applications to Partial Differential 
Equations}, Applied Mathematical Sciences 44,  Springer-Verlag, New York, 1983.

\bibitem{tanabe} H.  Tanabe;
\emph{Equations of Evolution}, Pittman, London 1979.

\bibitem{W} L. Wang;
 Approximate boundary controllability for semilinear delay differential equations.
\emph{J. Appl. Math.}, 2011, Art. ID 587890, 10 pp.


\bibitem{ky} K. Yosida;
{\em Functional Analysis}, Sixth edition, Classics in Mathematics, 
Springer-Verlag, Heidelberg, 1995.

\bibitem{zhu} X. H. Zhou;
 Approximate controllability for a class of semilinear abstract equations.
\emph{SIAM Journal on Control and Optimization} \textbf{21} (1983), 551--565.

\end{thebibliography}




\end{document}