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

\AtBeginDocument{{\noindent\small
\emph{Electronic Journal of Differential Equations},
Vol. 2013 (2013), No. 275, pp. 1--21.\newline
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
\newline ftp ejde.math.txstate.edu}
\thanks{\copyright 2013 Texas State University - San Marcos.}
\vspace{9mm}}

\begin{document}
\title[\hfilneg EJDE-2013/275\hfil Approximate controllability]
{Approximate controllability of abstract  impulsive fractional
 neutral evolution equations with infinite delay in Banach spaces}

\author[D. N. Chalishajar, K. Malar, K. Karthikeyan \hfil EJDE-2013/275\hfilneg]
{Dimplekumar N. Chalishajar, Kandasamy Malar,\\ Kulandhivel Karthikeyan}  

\address{Dimplekumar N. Chalishajar \newline
Department of Mathematics and Computer Science
Virginia Military Institute (VMI)
417, Mallory Hall, Lexington, VA 24450, USA}
\email{dipu17370@yahoo.com, chalishajardn@vmi.edu}

\address{Kandasamy Malar \newline
Department of Mathematics, Erode Arts and Science College, Erode 638 009,
Tamil Nadu. India}
\email{malarganesaneac@gmail.com}

\address{Kulandhivel Karthikeyan \newline
Department of Mathematics,  K.S.R. College of Technology,  Tiruchengode
637 215,  Tamil Nadu,  India}
\email{karthi\_phd2010@yahoo.co.in}

\thanks{Submitted July 26, 2013. Published December 18, 2013.}
\subjclass[2000]{34A37, 34K30, 47D09, 93C15}
\keywords{Impulsive conditions; approximate controllability;\hfill\break\indent
 neutral evolution equations; phase space}

\begin{abstract}
 In this  article, we study  the approximate controllability of impulsive
 abstract fractional neutral evolution equations in Banach spaces.
 The main results are obtained by using Krasnoselkii's fixed point theorem,
 fractional calculus and methods of controllability theory. An application
 is provided to illustrate the theory. Here we have provided new definition
 of phase space for the impulsive and infinite delay term. Our result is
 new for the approximate controllability with infinite delay in Hilbert space.
\end{abstract}

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



\section{Introduction}

Differential equations of fractional order have proved to be valuable
tools in the modelling of many phenomena in various fields of science
and engineering. Indeed, we can find numerous applications in
viscoelasticity, electrochemistry, control, porous media,
electromagnetic, etc. (see \cite{DNC3,DNC4,DiFr,BaSi,GlNo,Hil,Mai,MeScKiNo,OlPs}).
 Now a days, controllability theory for linear systems has already 
been well established, for finite and infinite dimensional systems (see \cite{rf}). 
Several authors have extended these concepts to infinite dimensional 
systems represented by nonlinear evolution equations in infinite-dimensional 
spaces, (see \cite{ad,X.L,ni1,rs1,rs2,ns,jw}).
On the other hand, approximate controllability problems for fractional 
evolution equations in Hilbert spaces is not yet sufficiently investigated 
and there are only few works on it (see \cite{rs1,rs2,jw}). 
On the other hand, it has been observed
that the existence or the controllability results proved by
different authors are through an axiomatic definition of the phase
space given by Hale and Kato \cite{HK}. However, as remarked by
Hino, Murakami, and Naito \cite{HMN}, it has come to our attention
that these axioms for the phase space are not correct for the
impulsive systems with infinite delay, refer the work in 
\cite{DNC1,DNC2}. Benchohra et al.~\cite{BGN} discussed the
controllability of first and second order neutral functional
differential and integro-differential inclusions in a Banach space
with non-local conditions, without impulse effect. Chang and Li
\cite{CL} obtained the controllability result for functional
integro-differential inclusions on an unbounded domain without
impulse term. Benchohra et al.~\cite{BHNO} studied the existence result
for damped differential inclusion with impulse effect. Hernandez
et al.~\cite{HRH} proved the existence of solutions for impulsive
partial neutral functional differential equations for first and
second order systems with infinite delay. 
In last couple of years, JinRong Wang et al.~\cite{0,2,3,4,5,6,7,8}
 discussed  various development in the theory of boundary value problems, optimal 
feedback control, and new concept of solutions for impulsive fractional 
evolution/differential equations.

Impulsive differential equations have become
important in recent years as mathematical models of phenomena in
both the physical and social sciences. There has been significant
development in impulsive theory especially in the area of impulsive
differential equations with fixed moments; see for instance the
monographs by Benchohra et al.~\cite{BeHeNt1}, Lakshmikantham et al.~\cite{LBS}, 
and Samoilenko and Perestyuk \cite{SaPe}, and the references therein. 
A neutral generalization of impulsive differential equations is abstract 
impulsive differential equations in Banach spaces. For general aspects 
of impulsive differential equations, see  monographs  given in  \cite{yvr1,yvr2}.

The purpose of this paper is to study the approximate controllability of 
impulsive fractional neutral evolution differential system with infinite 
delay using the new definition of the phase space for impulsive term and 
infinite delay term.

In this article, Section 2 provides the definitions and some preliminary 
results to be used in the main theorems stated and proved. 
In Section 3, we focus on existence of the solutions of \eqref{e2.1}.
We study the main results in Section 4. Finally an application is 
given in Section 5 to justify the theory.

\section{Preliminaries}
Recently, in \cite{cc} the existence of solutions for an  impulsive neutral 
functional differential equations of the form
\begin{gather*}
\frac{d}{dt}(u(t)+F(t,u_t))= A(t)u(t)+G(t,u_t) , \quad t\in I,\; t\neq t_i \\
\Delta u(t_i)=I_i(u_{t_i}),\\
u_0=\varphi \in \mathcal{B},
\end{gather*}
was studied  using Leray-Schauder's alternative theorem.
We consider the following impulsive fractional neutral evolution differential 
system with infinite delay:
\begin{equation} \label{e2.1}
\begin{gathered}
 \frac{d^\alpha}{dt^\alpha}[x(t)-h(t,x_t)]= Ax(t)+Bu(t)+f(t,x_t) ,
 \quad t\in[0,T], \; t\neq t_i \\		
 x(t)=\phi(t) \in{{\mathcal{B}}_h},\\
 \Delta x(t_i)=I_i(x_{t_i})
\end{gathered}
\end{equation}
where the state $x$ takes values in a Banach space $X$, the control function
takes values in a Hilbert space $U$. The functions $h, f$ will be specified
in the sequel and $I$  is an interval of the form $[0,T)$;
$0<t_1<t_2<\dots <t_i<\dots <T$ are prefixed numbers.
Let $x_{t}(\cdot)$ denote $x_{t}(\theta)=x(t+\theta)$, $\theta \in (-\infty,0]$.
Assume that $ l:(-\infty,0] \to (-\infty,0)$ is a continuous function
satisfying $l=\int_{-\infty}^0l(t) dt <\infty$.

We present the abstract phase space ${\mathcal{B}}_h$. Assume
that $h:~]-\infty,0]\to ]0,\infty[$ be a continuous
function with $l=\int_{-\infty}^0h(s)ds < +\infty$. Define,
\begin{align*}
{\mathcal{B}}_h&:=\big\{\phi:]-\infty,0]\to X
\text{ such that, for any $r>0$, $\phi(\theta)$ is
bounded and}\\    \\
&\quad \text{ measurable function on $ [-r,0]$ and }
\int_{-\infty}^0h(s)\sup_{s\leq \theta \leq 0}|\phi(\theta)|ds<+\infty\}.
\end{align*}
Here, ${\mathcal{B}}_h$ is endowed with the norm
\[
\|\phi\|_{{\mathcal{B}}_h}=\int_{-\infty}^0h(s)\sup_{s \leq
\theta \leq 0}|\phi(\theta)|ds,\quad  \forall \phi \in {\mathcal{B}}_h.
\]
Then it is easy to show that
$({\mathcal{B}}_h,\|.\|_{{\mathcal{B}}_h})$ is a Banach space.

\begin{lemma} \label{lem2.1}
Suppose $y \in {\mathcal{B}}_h$; then,
for each $t \in J$, $y_{t}\in {\mathcal{B}}_h$. Moreover,
\[
l|y(t)|\leq \|y_{t}\|_{{\mathcal{B}}_h}
\leq l \sup_{s\leq \theta \leq 0}
(|y(s)|+\|y_0\|_{{\mathcal{B}}_h}),
\]
 where $l:=\int_{-\infty}^0h(s)ds < +\infty$.
\end{lemma}

\begin{proof}
 For any $t \in [0,a]$, it is easy to see
that $y_{t}$ is bounded and measurable on $[-a,0]$ for $a>0$, and
\begin{align*}
\|y_{t}\|_{{\mathcal{B}}_h}
&=  \int_{-\infty}^0 h(s)
\sup_{\theta \in [s,0]}|y_{t}(\theta)|ds  \\
&=  \int_{-\infty}^{-t}h(s)\sup_{\theta \in [s,0]}|y(t+\theta)|ds
+ \int_{-t}^0h(s)\sup_{\theta \in [s,0]}|y(t+\theta)|ds\\
&=  \int_{-\infty}^{-t}h(s)\sup_{\theta_{1} \in
[t+s,t]}|y(\theta_{1})|ds
+ \int_{-t}^0h(s)\sup_{\theta_{1} \in [t+s,t]}|y(\theta_{1})|ds\\
&\leq  \int_{-\infty}^{-t}h(s)\Big[\sup_{\theta_{1} \in
[t+s,0]}|y(\theta_{1})| + \sup_{\theta_{1} \in
[0,t]}|y(\theta_{1})|\Big]ds + \int_{-t}^0h(s) \sup_{\theta_{1}
\in [0,t]}|y(\theta_{1})|ds \\
&=  \int_{-\infty}^{-t} h(s) \sup_{\theta_{1} \in
[t+s,0]}|y(\theta_{1})|ds + \int_{-\infty}^0h(s)ds. \sup_{s \in
[0,t]}|y(s)| \\
&\leq  \int_{-\infty}^{-t} h(s)\sup_{\theta_{1}\in
[s,0]}|y(\theta_{1})|ds + l. \sup_{s \in [0,t]}|y(s)|\\
&\leq  \int_{-\infty}^0h(s) \sup_{\theta_{1} \in
[s,0]}|y(\theta_{1})| ds + l \sup_{s \in [0,t]}|y(s)| \\
&=  \int_{-\infty}^0h(s) \sup_{\theta_{1} \in
[s,0]}|y_0(\theta_{1})| ds + l \sup_{s \in [0,t]}|y(s)| \\
&=  l \sup_{s \in [0,t]}|y(s)| + \|y_0\|_{{\mathcal{B}}_h}
\end{align*}
Since $ \phi \in {{\mathcal{B}}_h}$, then $y_{t} \in {{\mathcal{B}}_h}$.
 Moreover,
\[
\|y_{t}\|_{{\mathcal{B}}_h}
=\int_{-\infty}^0h(s) \sup_{\theta \in [s,0]}|y_{t}(\theta)|ds
\geq |y_{t}(\theta)|\int_{-\infty}^0h(s)ds=l |y(t)|
\]
The proof is complete.
\end{proof}

 The phase space ${\mathcal{B}}_h$ defined above also satisfies the following 
properties:
\begin{itemize}
\item[(B1)] If $x : (-\infty,\sigma+a] \to X$, $a>0$, $\sigma\in \mathbb{R}$ 
such that $x_{\sigma} \in{ {\mathcal{B}}_h}$, and   
$x[\sigma, \sigma+a] \in PC([\sigma, \sigma+a],X)$, then for every 
$t \in[\sigma, \sigma+a)$   the following conditions  hold:
\begin{itemize}
\item[(i)] $x_t$ is in ${\mathcal{B}}_h$;
\item[(ii)] $\|x(t)\|_X \leq H\|x_t\|_{{\mathcal{B}}_h}$;
\item[(iii)] $ \|x_t\|_{{\mathcal{B}}_h} \leq K(t-\sigma) \sup\{\|x(s)\|_X :
 \sigma \leq s\leq t\} + M(t-\sigma)\|x_\sigma\|{{\mathcal{B}}_h}$,
where $H>0$  is a constant; $K,M : [0,\infty) \to[1,\infty)$, 
$K$  is continuous, $M$ is locally bounded and $H,K,M$ are independent of $x$.
\end{itemize}
\item[(B2)] The space ${\mathcal{B}}_h$ is complete.
\end{itemize}

\begin{example} \label{examp2.2}  \rm
 The   $PC_r \times L^2(g,X)$ be  the phase space.
Let $r>0$  and $g : (-\infty,-r] \to R$ be a non- negative, 
locally Lebesgue integrable function. Assume that there is a non-negative 
measurable, locally  bounded function $\eta(\cdot)$ on $(-\infty,0]$  
such that $g(\xi+\theta) \leq \eta(\xi)g(\theta)$  for all 
$\xi\in (-\infty,0]$ and $\theta \in (-\infty,-r] \backslash  N_{\xi}$, where
$  N_{\xi} \subset(-\infty,-r]$ is a set with Lebesgue measure zero.
We denote by $PC_R \times L^2(g,X)$ the set of all functions 
$\varphi : (-\infty,0] \to X$ such that $\varphi|[-r,0] \in PC([-r,0],X)$ 
and $\int_{-\infty}^{-r} g(\theta)\|\varphi(\theta)\|_{X}^{2} d\theta < +\infty$. 
In $PC_r \times L^2(g,X)$, we consider the seminorm defined by
\[	
 \|\varphi\|_{{\mathcal{B}}_h} 
= \sup_{\theta \in [-r,0]} \|\varphi(\theta)\|_X+ 
\Big( \int_{-\infty}^{-r} g(\theta)\|\varphi(\theta)\|_{X}^{2} d\theta
 \Big)^{1/2}.
\]
 From the proceeding conditions, the space $PC_r \times L^2(g,X)$ satisfies 
 (B1) and (B2). Moreover, when $ r=0$, we can take $ H = 1$,
 $K(t) = \big(1+\int_{-t}^0 g(\theta)\big)^\frac {1}{2}$ and
$M(t) = \eta (-t) $ for  $t\geq 0$. 
\end{example}

Let $0<t_1 <t_2< \dots <t_n< T$ be pre-fixed numbers. We introduce the space 
$PC = PC([0,a]; X)$ formed by functions $u : [0,T] \to X$ such that are 
continuous at $ t\ne t_i$, $u(t_i^-) = u(t_i)$ and $  u(t_i^+$) exists, 
for  $ i = 1,\dots n$. In this paper, we assume that $PC$ is endowed 
with the norm $\|u\|_{PC} = \sup_{s\in [0,a]} \|u(s)\|_X$. 
It is clear that $(PC,\|\cdot\|_{PC})$ is a Banach space; see\cite{eh}  
for more details.

In what follows, we put $t_0=0$, $t_{n+1}= T$, and for $u\in PC$, we denote 
by $\bar{u_i}\in C([t_i,t_{i+1}],X)$, $i = 0,1,2,\dots ,n$, the functions
\[
\tilde{u_i}(t) =
	\begin{cases}
  	u(t_i), &\text{for }  t\in (t,t_{i+1}],\\
    u(t_i^+), &\text{for } t = t_i.
\end{cases}
\]
Moreover, for ${\mathcal{B}}_h\subset {PC}$, we employ the notation 
${\mathcal{B}}_{h_i}$, $i = 1,2,\dots ,n$, for the sets 
 $\bar{\mathcal{B}}_{h_i} = \{\tilde{u_i} : u\in {\mathcal{B}}_h\}$.

\begin{lemma}[\cite{BGN}] \label{lem2.3} 
A set $B\subset PC$ is relatively compact in $PC$ if and only if  the set  
$\tilde{\mathcal{B}}_{h_i}$, is relatively  compact in the space 
$ C([t_i,t_{i+1}];X), $ for every $ i = 0,1,\dots ,n$.
\end{lemma}

We introduced symbols which will be useful throughout this article. 
Let $(X, \|\cdot\|)$ be a separable reflexive Banach space and let 
$(X^*,\| \cdot \|_*)$ stands for its dual space with respect to the 
continuous pairing
$\langle\cdot , \cdot\rangle$. We may assume, without loss of generality,
 that $X$ and $X^*$ are smooth and strictly  convex, by virtue of renorming 
theorem (for example, see   \cite{X.L}). In particular, this implies 
that the duality mapping $J$ of $X$ into $X^*$  given by the following relations
$$
\|J(z)\|_*=\|z\|, \quad \langle J(z),z \rangle=\|z\|^2,\quad \text{for all }
 z\in X
$$
 is bijective, homogeneous, demicontinuous, i.e.,  continuous  
from $X$ with a strong topology into $X^*$  with weak topology  
and strictly  monotonic. Moreover,  $J^{-1} : X^* \to X$  is also duality 
mapping.  Note that our results are new even for the approximate controllbility 
of impulsive fractional neutral differential equations with infinite delay 
in Hilbert spaces.

In this article,  we also assume that $-A : D(A) \subset  X \to X$ 
is the infinitesimal generator of a compact analytic semigroup  $S(t), t>0$,
 of uniformly  bounded linear  operator in $X$, that is, there exists $ M>1$  
such that $\|S(t)\|_{L(X)} \leq M$  for all  $t\geq 0$. Without loss of 
generality, let $0 \in \rho(-A)$, where $\rho(-A)$ is the resolvent set of $-A$. 
 Then for any $\beta>0$,  we can define  $A^{-\beta}$ by
$A^{-\beta} :=\frac{1}{\Gamma(\beta)}\int _0^{\infty} t^{\beta-1} S(t) dt$.

It follows that each $A^{-\beta}$  is an injective continuous and homorphism 
of $ X$.  Hence we can define $A^{\beta} := (A^{-\beta})^{-1}$, 
which is a closed bijective linear operator in $ X$. It can be shown that 
each $A^{\beta}$  has dense  domain and that $D(A^{\gamma}) \subset D(A^{\beta})$ 
for $0\leq \beta \leq \gamma$. Moreover 
$A^{\beta+\gamma}x = A^{\beta}A^{\gamma}x = A^{\gamma}A^{\beta}x $ for  every 
$\beta,\gamma \in \mathbb{R}$ and $x\in D(A^{\mu})$ with 
$\mu := \max (\beta,\gamma, \beta + \gamma)$, where $A^0 = I$, $I$ is 
the identity in $X$.

We denote by $X_{\beta}$  the Banach space of $D(A^{\beta})$  equipped 
 with norm $\|x\|_{\beta} := \|A^{\beta}x\|$ for $x\in D(A^{\beta})$,  
which is equivalent  to the graph norm of $A^{\beta}$. 
Then we have $X_{\gamma} \hookrightarrow X_{\beta}$, for 
$0 \leq \beta \leq \gamma$ (with $X_0=X)$, and the embedding is continuous. 
Moreover, $A^{\beta}$  has the following basic properties.

\begin{lemma}[\cite{ap}] \label{lem2.4}   
$A^{\beta}$  has the following properties
\begin{itemize}
\item[(i)] $ S(t) : X\to X_{\beta}$  for each $t>0$ and $\beta \geq 0$.
\item[(ii)] $A^{\beta}S(t)x= S(t)A^{\beta}x$ for each $x\in D(A^{\beta})$ and 
$t\geq0$.
\item[(iii)] for every $t, ~A^{\beta}S(t)$ is bounded in $ X$ and there 
exists $M_{\beta}>0$  such that
$$
\|A^{\beta}S(t)\| \leq M_{\beta}t^{-\beta}.
$$
\item[(iv)] $A^{-\beta}$  is a bounded linear operator for 
$0 \leq\beta \leq 1$ in X, there exists a constant $C_{\beta}$ such that 
$\|A^{-\beta}\| \leq C_{\beta}$  for $ 0 \leq\beta \leq 1$.
\end{itemize}
\end{lemma}

We recall the following known definitions from fractional calculus. 
For more details, see  \cite{DNC3,Sa}.

\begin{definition} \label{def2.5} \rm
The fractional integral of order $\alpha>0$ with the lower limit $0$ for a
 function $f$ is defined as
\begin{align*}	
  I^{\alpha}f(t) = \frac{1}{\Gamma(\alpha)}\int_0^t
 \frac{f(s)}{(t-s)^{1-\alpha}} ds, \quad t>0, \; \alpha>0,
\end{align*}
provided the right-hand side is pointwise defined  on $ [0,\infty)$, 
where $\Gamma$ is the gamma function.
\end{definition}

\begin{definition} \label{def2.6} \rm
The Riemann- Liouville derivative of order $\alpha$ with the lower 
limit 0 for a function $f : [0,\infty) \to R$ is written as
\[
  ^LD^{\alpha}f(t)= \frac{1}{\Gamma(n-\alpha)}\frac{d^n}{dt^n}
\int_0^t \frac{f^{(n)}(s)}{(t-s)^{\alpha+1-n}} ds, \quad t>0, \; n-1< \alpha<n.
\]
\end{definition}

\begin{definition} \label{def2.7} \rm
The Caputo derivative of order $\alpha$ for a function $f : [0,\infty) \to R $ 
is written as
\begin{align*}	
  	&^cD^{\alpha}f(t)=^LD^{\alpha}\Big(f(t)-\sum_{k=0}^{n-1} 
\frac{t^k}{k!}f^{(k)}(0)\Big), \quad t>0, \; n-1< \alpha<n.
\end{align*}
\end{definition}

For $x\in X$, we define two families ${\mathfrak T_{\alpha}(t) : t \geq0}$  
and ${\mathfrak A_{\alpha}(t) :  t \geq0}$ of operators by
\[	
 \mathfrak T_{\alpha}(t) = \int_0^{\infty}\varPsi_{\alpha}
(\theta)S(t^{\alpha}{\theta})d\theta, \quad
 \mathfrak A_{\alpha}(t) =\alpha \int_0^{\infty}\theta \varPsi_{\alpha}
 (\theta)S(t^{\alpha}{\theta})d\theta,
\]
where
\[	
 \varPsi_{\alpha}(\theta) = \frac{1}{\pi\alpha}
\sum_{n=1}^{\infty}{(-1)^{n-1}}\frac{\Gamma(n\alpha+1)}{n!}\sin(n{\pi}{\alpha}),
 \quad \theta \in (0,\infty)
\]
is the function defined on $ (0,\infty)$  which satisfies
\begin{gather*}	
\varPsi_{\alpha}(\theta) \geq 0, \quad   
 \int_0^{\infty}\varPsi_{\alpha}(\theta)d\theta = 1,\\
 \int_0^{\infty}\theta^{\zeta}\varPsi_{\alpha}(\theta)d\theta 
 = \frac{\Gamma(1+\zeta)}{\Gamma(1+\alpha\zeta)},
\quad \zeta \in (-1, \infty).
\end{gather*}
The following lemma follows from the results in  \cite{yz1}.


\begin{lemma} \label{lem2.8} The operators $\mathfrak T_{\alpha}$ and  
$\mathfrak A_{\alpha}$  have the following properties:
\begin{itemize}
\item[(i)] For any fixed $t\geq 0$, any $x\in X_{\beta}$, the operators 
$\mathfrak T_{\alpha}(t)$ and  $\mathfrak A_{\alpha}(t)$ are linear and bounded, 
i.e. for any $x \in X_{\beta}$,
\begin{align*}	
  	\| \mathfrak T_{\alpha}(t)\| \leq M\|x\|_{\beta}, \quad
   \|\mathfrak A_{\alpha}(t)\| \leq \frac{M}{\Gamma(\alpha)}\|x\|_{\beta};
\end{align*}
\item[(ii)] The operators $S_{\alpha}(t)$ and $\mathfrak A_{\alpha}(t)$ are 
strongly continuous  for all $t\geq0$;

\item[(iii)]  $S_{\alpha}(t)$ and $\mathfrak A_{\alpha}(t)$ are norm-continuous  
in $X$ for $t>0$;

\item[(iv)] $S_{\alpha}(t)$ and $\mathfrak A_{\alpha}(t)$ are compact operators 
in $X$ for $t>0$;

\item[(v)] For any $t>0$, the restriction of $S_{\alpha}(t)$ to $X_{\beta}$  
 and the restriction of  $\mathfrak A_{\alpha}(t)$ to $X_{\beta}$ are 
 norm-continuous;

\item[(vi)] For every $t>0$ the restriction of  $S_{\alpha}(t)$ to $X_{\beta}$ 
  and the restriction of  $\mathfrak A_{\alpha}(t)$ to $X_{\beta}$ are compact  
 operators in  $X_{\beta}$;

\item[(vii)] For all $x\in X$ and  $t\in [0,T]$,
$$
\|A^{\beta}\mathfrak A_{\alpha}(t)x\| \leq C_{\beta}t^{-\alpha\beta}\|x\|, \quad
C_{\beta} := \frac{M_{\beta}\alpha\Gamma(2-\beta)}{\Gamma(1+\alpha(1-\beta))}.
$$
\end{itemize}
\end{lemma}

In the following definition, we introduce the concept of a mild solution for 
 \eqref{e2.1}.

\begin{definition} \label{def2.9} \rm
 A function $x(.^., u) \in PC([0,T],X)$ is said to be mild solution of
 \eqref{e2.1} if for any $u \in L_2([0,T],U)$, and $t\in I$  the integral 
equation
\begin{equation}\label{e2.2}
\begin{aligned}
x(t)&= \mathfrak T_{\alpha}(t) [\phi(0)+g(0,\phi)] - g(t,x_t)
 - \int_0^t(t-s)^{\alpha - 1}A\mathfrak A_{\alpha} (t-s)g(s,x_s) ds  \\
&\quad + \int_0^t(t-s)^{\alpha - 1}\mathfrak A_{\alpha} (t-s) [Bu(s)+ f(s,x_s)]ds
 +\sum_{t_i<t}\mathfrak T_{\alpha} (t-t_i) I_i(x_{t_i}) \\
&\quad + \sum_{t_i<t} [g(t,x_t)_{|t_i^+} - g(t,x_t)_{|t_i^-}],
\end{aligned}
\end{equation}
is satisfied.
\end{definition}

Let $x(T,u)$ be the state value of \eqref{e2.1} at terminal time $T$ 
 corresponding to the control $ u$. Introduce the set 
$\mathfrak R(T) = \{x(T,u) : u\in L_2([0,T],U\}$, which is called the reachable 
set of system \eqref{e2.2} at terminal time $T$, its closure in $X$ is
 denoted by $\overline{\mathfrak R(T)}$.

\begin{definition} \label{def2.10} \rm
System \eqref{e2.1} is said to be approximately controllable on $[0,T]$ 
 if $\overline {\mathfrak R(T)} = X$; that is, given an arbitrary 
$\varepsilon> 0$ it is possible to steer the system  from the initial point  
$x_0$ to within a distance $\varepsilon$  from all points in the space $X$ 
at time $T$. 
\end{definition}

To investigate the approximate controllability  of system \eqref{e2.1}, 
we assume the following conditions:
\begin{itemize}
\item[(H1)] $-A$  is the infinitesimal generator of an analytic semigroup 
of bounded linear operators  $S(t)$  in $X$, $0 \in \rho(-A)$,  $S(t)$ 
is compact for $t>0$, and there exists a positive constant $M$ such that 
$\|S(t)\| \leq M$;

\item[(H2)] The function $g : [0,T] \times {\mathcal{B}}_h \to X$ is 
continuous and there  exists some constant $M_g>0$, $0<\beta<1$, such that
 $g$ is $X_{\beta}$-valued and
\begin{gather*}
\|A^{\beta}g(t,x) - A^{\beta}g(t,y)\| \leq M_{g}\|x-y\|_{{\mathcal{B}}_h}, 
\quad x,y \in {{\mathcal{B}}_h}, \; t\in[0,T],  \\
\|A^{\beta}g(t,x)\|  \leq M_{g} (1+ \|x\|_{{\mathcal{B}}_h}).
\end{gather*}

\item[(H3)] The function $f :  [0,T] \times {\mathcal{B}}_h \to X$ satisfies 
following properties: 
\begin{itemize}
\item[(a)] $f(t_1,.) : {\mathcal{B}}_h \to X$  is continuous for each 
 $t \in [0,T]$ and for each $x\in {\mathcal{B}}_h, f(.,x) : [0,T]\to X$ 
 is strongly measurable;
\item[(b)] There is a positive integrable function
 $ n\in L^{\infty}([0,T],[0,+\infty))$  and a continuous nondecreasing function 
 $\Lambda_f : [0,\infty)\to (0,\infty)$ such that for every 
 $(t,x) \in [0,T] \times{{\mathcal{B}}_h}$, we have
\[
\|f(t,x)\| \leq n(t)\Lambda_f(\|x\|_{{\mathcal{B}}_h)},  \quad 
\lim \inf_{r\to \infty}\frac{{\Lambda_f}(r)}{r} = \sigma_f < \infty.
\]
\end{itemize}

\item[(H4)] The following inequality holds
\begin{align*}
&\Big(1+ \frac{1}{\varepsilon}M_B^{2}M_{\mathfrak A}^{2} 
 \frac{T^{2\alpha-1}}{2\alpha-1}\Big) 
\Big(M_g\|A^{-\beta}\|l+K(\alpha,\beta)M_{g} 
 \frac{T^{\alpha\beta}}{\alpha\beta}l \\
&+\frac{M}{\Gamma(\alpha)}\frac{T^{\alpha}}{\alpha}\sigma_f
 \sup_{s\in J} n(s)\Big) + M \sum_{i=1}^{N} \sigma_{i} <1,
\end{align*}
where  $ M_B := \|B\|$, $ M_{\mathfrak A} :=\|\mathfrak A_{\alpha}\|$,
and $K(\alpha,\beta) = \frac{\alpha M_{1-\beta}\Gamma(1+\beta)}
{\Gamma(1+\alpha\beta)}$;

\item[(H5)] The maps $I_i : {\mathcal{B}}_h\to X$ are completely continuous 
and uniformly bounded, $i\in F = \{1,2,\dots,N\}$.
In what follows, we  denote 
$N_i =\sup\{\|I_i(\phi)\| : \phi \in {\mathcal{B}}_h\}$ and
\[
 \liminf_{r\to \infty}\frac{N_i}{r} = \sigma_i < \infty\,;
\]

\item[(H6)] There are positive constants $L_i$  such that
\[
\|I_i(\psi_1) - I_i(\psi_2)\| \leq L_i\|\psi_1 - \psi_2\|_{{\mathcal{B}}_h}, \quad
\psi_1,\psi_2 \in {\mathcal{B}}_h, \; i\in F;
\]

\item[(H7)] For every $h\in X$, 
$z_{\alpha}(h)= \varepsilon(\varepsilon I + \Gamma_0^{T}J)^{-1}(h)$ 
 converges  to zero as $\varepsilon \to 0^+$ in strong topology, where
\[
 \Gamma_0^{T} := \int_0^{T}(T-s)^{2(\alpha-1)}\mathfrak A_{\alpha}(T-s) 
BB^* \mathfrak A_{\alpha}^*(T-s)ds,
\]
and $z_{\varepsilon}(h)$  is a solution of the  equation 
$\varepsilon z_{\varepsilon} +  \Gamma_0^{T}J(z_{\varepsilon}) =\varepsilon h$.
\end{itemize}

Let $PC_T = \{x : x\in PC((-\infty,T],X), \quad x_0 
= \phi\in {\mathcal{B}}_h\}$.
Let $\|\cdot\|$ be the seminorm 
\[
\|x\|_T = \|x_0\|_{{\mathcal{B}}_h} + \sup_{0\leq s \leq T} \|x(s)\|, \quad
x\in PC_T.
\]

\section{Existence theorem}

To formulate the controllability problem in a form suitable for 
applying a fixed point theorem, it is assumed that the corresponding 
linear system is approximately controllable. Then it will be shown that 
the system \eqref{e2.1} is approximately controllable if for all 
$\varepsilon> 0$ there exists a continuous function $x(\cdot) \in PC([0,T],X)$ 
such that
\begin{equation} \label{e3.1}
\begin{gathered}
 u_{\varepsilon}(t,s)
= (T-t)^{\alpha-1}B^* \mathfrak A_{\alpha}^*(T-t)J((\varepsilon I
+ \Gamma_0^{T}J)^{-1}p(x)),\\
\begin{aligned}
 x(t)&= \mathfrak T_{\alpha}(t) [\phi(0)+g(0,\phi)] - g(t,x_t)
 - \int_0^t(t-s)^{\alpha - 1}A\mathfrak A_{\alpha} (t-s)g(s,x_s) ds \\
 &\quad + \int_0^t(t-s)^{\alpha - 1}\mathfrak A_{\alpha} (t-s)
[Bu_{\varepsilon}(s,x) + f(s,x_s)] ds
+\sum_{t_i<t}\mathfrak T_{\alpha} (t-t_i) I_i(x_{t_i})\\
&\quad + \sum_{t_i<t} [g(t,x_t)_{|t_i^+} - g(t,x_t)_{|t_i^-}], \quad t\in I
\end{aligned}
\end{gathered}
\end{equation}
where
\begin{align*}
p(x) &= h- \mathfrak T_{\alpha}(T) [\phi(0)+g(0,\phi)] + g(T,x_T)
 + \int_0^{T}(T-s)^{\alpha - 1}A\mathfrak A_{\alpha} (T-s)g(s,x_s) ds \\
&\quad - \int_0^{T}(T-s)^{\alpha - 1}\mathfrak A_{\alpha} (T-s)
 + f(s,x_s) ds,\quad h\in X.
\end{align*}
The control in \eqref{e3.1} steers the system \eqref{e2.1} from $\phi(0)$ to
 $h -\varepsilon J\big (\varepsilon I + \Gamma_0^{T}J)^{-1}p(x)\big)$
provided that the system \eqref{e3.1} has a solution.

\begin{theorem} \label{thm3.1}
 Assume that Assumptions {\rm(H1)--(H6)} hold and $\frac{1}{2} < \alpha \leq1$. 
Then there exists a solution to the equation \eqref{e3.1}.
\end{theorem}

The proof of the above theorem follows from Lemmas \ref{lem3.2}--\ref{lem3.7} 
and infinite dimensional analogue of Arzela- Ascoli theorem. 

For  $\varepsilon>0$ consider the operator 
$\varPhi_{\varepsilon}: PC_T \to PC_T$ defined by
\[	
(\varPhi_{\varepsilon}x)(t) :=
\begin{cases}
\phi(t), \quad  t\in (-\infty,0];\\
 \mathfrak T_{\alpha}(t) [\phi(0)+g(0,\phi)] - g(t,x_t) 
 - \int_0^t(t-s)^{\alpha - 1}A\mathfrak A_{\alpha} (t-s)g(s,x_s) ds \\
+ \int_0^t(t-s)^{\alpha - 1}\mathfrak A_{\alpha} (t-s) 
 [Bu_{\varepsilon}(s,x) + f(s,x_s)] ds \\
+\sum_{t_i<t}\mathfrak T_{\alpha} (t-t_i) I_i(x_{t_i})\\
+ \sum_{t_i<t} [g(t,x_t)_{|t_i^+} - g(t,x_t)_{|t_i^-}], \quad t\in [0,T].
\end{cases}
\]
where
\begin{align*}	
u_{\varepsilon}(t,s) &= (T-t)^{\alpha-1}B^* 
 \mathfrak A_{\alpha}^*(T-t)J((\varepsilon I + \Gamma_0^{T}J)^{-1}p(x))\\
&:=(T-t)^{\alpha-1} v_{\varepsilon}(t,x).
\end{align*}
It will be shown that for all $\varepsilon > 0$  the operator 
$\varPhi_{\varepsilon}$:  $PC_T \to PC_T$ has a fixed  point.

Suppose that $x(t) =\Tilde{\phi}(t) + z(t)$, $t\in (-\infty,T]$,
where
\[
\tilde{\phi}(t) = 	\begin{cases}
 \phi(t), & \text{for } t\in (-\infty,0],\\
\mathfrak T_{\alpha}(t)\phi(0), &\text{for} t \in [0,T].
\end{cases}
\]
 Set
$PC_T^0 = \{z\in C_T : z_0 = 0\in {{\mathcal{B}}_h}\}$.
For any $z\in PC_T^0$, we have
\[
\|z\|_T = \|z_0\|_{{\mathcal{B}}_h} + \sup_{0\leq s\leq T} \|z(s)\|
 =\sup_{0\leq s\leq T} \|z(s)\|\,.
\]
Thus $(PC_T^0,\|\cdot\|_T)$ is a Banach space. For each positive number $r>0$, 
set
\begin{align*}	
B_r := \{z\in PC_T^0 : \|z\|_T \leq r\}.
\end{align*}
It is clear that $B_r$ is bounded closed convex set in $PC_T^0$. 
For any $z\in B_r$ we have
\begin{align*}	
\| \Tilde{\phi}_t+ z_t\|_{{\mathcal{B}}_h} 
&\leq \|\Tilde{\phi}_t\|_{{\mathcal{B}}_h} + \|z_t\|_{{\mathcal{B}}_h}\\
&\leq l \sup_{0\leq s\leq T}\| \Tilde{\phi}(s)\| 
 +  \|\Tilde{\phi_0} \|_{{\mathcal{B}}_h} 
 +  l \sup_{0\leq s\leq T}\|z(s)\| + \|Z_0\|_{{\mathcal{B}}_h}\\
&\leq l (M\|\phi(0)\| + r) + \|\phi\|_{{\mathcal{B}}_h} := R(r).
\end{align*}
Consider the maps $\prod_{\varepsilon}, \Theta_{\varepsilon},
 \varUpsilon_{\varepsilon} : PC_T^0 \to PC_T^0$ defined by
\begin{gather*}
\big(\prod_{\varepsilon}z\big)(t) :=\begin{cases}
0, &t\in (-\infty,0];\\
 \mathfrak T_{\alpha}(t) - g(0,\phi)] - g(t,\Tilde{\phi}_t+z_t)\\
 - \int_0^t(t-s)^{\alpha - 1}A\mathfrak A_{\alpha} (t-s)g(s,\Tilde{\phi}_s+z_s) 
ds, &t\in [0,T]
\end{cases}
\\
\big(\Theta_{\varepsilon}z\big)(t):=\begin{cases}
0, \quad t\in (-\infty,0];\\
 \int_0^t(t-s)^{\alpha - 1}\mathfrak A_{\alpha} (t-s) 
\big[Bu_{\varepsilon}(s,\Tilde{\phi}+z) + f(s,\Tilde{\phi}_s+z_s)\big] ds,
\;\; t\in [0,T]
\end{cases}
\\
\big(\varUpsilon_{\varepsilon}z\big)(t) :=\begin{cases}
0, &  t\in (-\infty,0];\\
\sum_{0 <t_i<t}\mathfrak T_{\alpha} (t-t_i) I_i(\Tilde{\phi}_{t_i}+z_{t_i}),
& t\in [0,T].
\end{cases}
\end{gather*}
Obviously, the operator $\varphi_{\varepsilon}$ has a fixed point if and 
only if operator $\prod_{\varepsilon}+   \Theta_{\varepsilon}+  
\varUpsilon_{\varepsilon} $   has a fixed point. In order to prove that 
 $\prod_{\varepsilon}+   \Theta_{\varepsilon}+  \varUpsilon_{\varepsilon} $ 
has a fixed point. We will employ the Krasnoselkii fixed point theorem.

\begin{lemma} \label{lem3.2} 
Under Assumptions {(H1)--(H6)}, for any $\varepsilon >0$ there exists a 
positive number $r:=r(\varepsilon)$  such that   
($\prod_{\varepsilon}+   \Theta_{\varepsilon}+ \varUpsilon_{\varepsilon}) (B_r) 
\subset B_r$.
\end{lemma}

\begin{proof}
 Let  $\varepsilon >0$  be fixed. If the statement were not true, then for 
each $r>0$, there exists a function $z_r \in B_r$, but
 ($\prod_{\varepsilon}+   \Theta_{\varepsilon}+  \varUpsilon_{\varepsilon})
 (z_r)\notin B_r$. So for some $t = t(r) \in [0,T]$ one can show that
\begin{equation} \label{e3.2}
\begin{aligned}
r &\leq\big \|(( \prod_{\varepsilon}+   \Theta_{\varepsilon} 
+ \varUpsilon_{\varepsilon})z_r)(t)\big\|\\
&\leq \big\|  \mathfrak T_{\alpha}(t)  g(0,\phi)\big\|
 +\big\| g(t,\Tilde{\phi}_t+z_t)\big\|
 +\| \int_0^t(t-s)^{\alpha - 1}A\mathfrak A_{\alpha}
 (t-s)g(s,\Tilde{\phi}_s+z_s) ds \|\\
&\quad +\| \int_0^t(t-s)^{\alpha - 1}\mathfrak A_{\alpha}
 (t-s)f(s,\Tilde{\phi}_s+z_s)  ds\|\\
&\quad +\| \int_0^t(t-s)^{\alpha - 1}\mathfrak A_{\alpha}
 (t-s) Bu_{\varepsilon}(s,\phi+z)  ds\| \\
&+\|\sum_{0 <t_i<t}\mathfrak T_{\alpha} (t-t_i)
 I_i(\Tilde{\phi}_{t_i}+z_{t_i}) \| \\
& =: I_1+ I_2 + I_3 +I_4 + I_5 +I_6.
\end{aligned}
\end{equation} \label{e3.3}
Let us estimate $I_i$, $i = 1,\dots,6$. By Assumption (H2), we have
\begin{equation}
I_1 \leq M\|A^{-\beta}\|\|A^{\beta} g(0,\phi)\|
\leq MM_g \|A^{-\beta}\|(1+\|\phi\|_{{\mathcal{B}}_h}),
\end{equation}
\begin{equation} \label{e3.4}
\begin{aligned}
I_2 &\leq \|A^{-\beta}\| |\|A^{\beta} g(t,\Tilde{\phi}_t+z_t)\\
& \leq  M_g  \|A^{-\beta}\| \big(1+\|\Tilde{\phi}_t
+z_t\|_{{\mathcal{B}}_h}\big) \\
& \leq  M_g  \|A^{-\beta}\| \big(1+R(r)\big).
\end{aligned}
\end{equation}
Using Lemma \ref{lem2.4} and H\"older inequality, one can deduce that
\begin{equation} \label{e3.5}
\begin{aligned}
I_3 &\leq \big\| \int_0^t(t-s)^{\alpha - 1}A^{1-\beta}\mathfrak A_{\alpha} (t-s)
A^{\beta}g(s,\Tilde{\phi}_s+z_s) ds \big\|\\
& \leq  \frac{M_{1-\beta}\alpha \Gamma(1+\beta)}
{\Gamma(1+\alpha \beta)}\int_0^t(t-s)^{\alpha\beta - 1}
\|A^{\beta}g(s,\Tilde{\phi}_s+z_s) \|\,ds\\
& \leq K(\alpha, \beta) \int_0^t(t-s)^{\alpha\beta - 1}
 M_g\big(1+\|\Tilde{\phi_s}+z_s)\|_{{\mathcal{B}}_h}\big)ds\\
& \leq K(\alpha, \beta)M_g \frac{T^{\alpha \beta}} {\alpha \beta}(1+ R(r)).
\end{aligned}
\end{equation}
Using Assumption (H3), we have
\begin{equation} \label{e3.6}
\begin{aligned}
I_4 &\leq \int_0^t \| (t-s)^{\alpha - 1}\mathfrak A_{\alpha}
(t-s)f(s,\Tilde{\phi}_s+z_s)\|ds \\
&\leq \frac{M}{\Gamma(\alpha)}\int_0^t  (t-s)^{\alpha - 1}
\| f(s,\Tilde{\phi}_s+z_s)\| ds\\
&\leq \frac{M}{\Gamma(\alpha)}\int_0^t (t-s)^{\alpha - 1} n(s)
\Lambda_f\big(\|\Tilde{\phi}_s+z_s)\|\big) ds\\
&\leq \frac{M}{\Gamma(\alpha)} \frac{T^\alpha}{\alpha}\Lambda_f
\big(R(r)\big) \sup_{s\in J} n(s).
\end{aligned}
\end{equation}
Combining the estimates \eqref{e3.2}-\eqref{e3.6} yields
\begin{equation} \label{e3.7}
\begin{aligned}
&I_1+ I_2 + I_3 +I_4 \\
&< MM_g \|A^{-\beta}\|(1+\|\phi\|_{{\mathcal{B}}_h})
 + M_g  \|A^{-\beta}\| \big(1+R(r)\big)\\
&\quad + K(\alpha, \beta)M_g \frac{T^{\alpha \beta}} {\alpha \beta}(1+ R(r))
+ \frac{M}{\Gamma(\alpha)}) \frac{T^\alpha}{\alpha}\Lambda_f \big(R(r)\big)
\sup_{s\in J} n(s) := \Lambda.
\end{aligned}
\end{equation}
On the other hand,
\begin{align*} % 3.8
I_5
&\leq \int_0^t \|(t-s)^{\alpha - 1}\mathfrak A_{\alpha} (t-s)
 Bu_{\varepsilon}(s,\phi+z)\|ds \\
&= \int_0^t \|(t-s)^{\alpha - 1}(T-s)^{\alpha - 1}\mathfrak A_{\alpha} (t-s)
  BB^* \mathfrak A_{\alpha}^*(T-t)\\
&\quad\times J((\varepsilon I
 + \Gamma_0^{T}J)^{-1}p(\phi+z))\|ds\\
&\leq  \int_0^t \|(t-s)^{\alpha - 1}(T-s)^{\alpha - 1}\mathfrak A_{\alpha}
 (t-s)  BB^* \mathfrak A_{\alpha}^*(T-t)\| ds\\
&\quad\times \|J((\varepsilon I + \Gamma_0^{T}J)^{-1}p(\phi+z))\|\\
&\leq M_B^{2}M_{\mathfrak A}^{2} \frac{T^{2\alpha-1}}{2\alpha-1}
  \|J((\varepsilon I + \Gamma_0^{T}J)^{-1}p(\phi+z))\|\\
&\leq M_B^{2}M_{\mathfrak A}^{2} \frac{T^{2\alpha-1}}{2\alpha-1}
 \|(\varepsilon I + \Gamma_0^{T}J)^{-1}p(\phi+z)\|\\
&\leq \frac{1}{\varepsilon} M_B^{2}M_{\mathfrak A}^{2}
 \frac{T^{2\alpha-1}}{2\alpha-1}\|p(\phi+z)\|\\
&\leq \frac{1}{\varepsilon} M_B^{2}M_{\mathfrak A}^{2}
 \frac{T^{2\alpha-1}}{2\alpha-1}\Delta
\end{align*}
and
\[
I_6 \leq \|\sum_{0 <t_i<t}\mathfrak T_{\alpha}
(t-t_i) I_i(\Tilde{\phi}_{t_i}+z_{t_i}) \|\leq M\sum_{i=1}^{N} N_i
\]
Thus
\begin{align*}
r &\leq  \|(( \prod_{\varepsilon}+   \Theta_{\varepsilon}
 + \varUpsilon_{\varepsilon}) (z_r)(t) \| \\
&\leq \Delta +  \frac{1}{\varepsilon} M_B^{2}M_{\mathfrak A}^{2}
 \frac{T^{2\alpha-1}}{2\alpha-1}\Delta+M\sum_{i=1}^{N} N_i \\
& =\Big(1 + \frac{1}{\varepsilon} M_B^{2}M_{\mathfrak A}^{2}
\frac{T^{2\alpha-1}}{2\alpha-1}\Big)\Delta+M\sum_{i=1}^{N} N_i
\end{align*}
Dividing both sides  by $r$ and taking $r \to \infty$, we obtain that
\begin{align*}
&\Big(1+ \frac{1}{\varepsilon} M_B^{2}M_{\mathfrak A}^{2}
\frac{T^{2\alpha-1}}{2\alpha-1}\Big)\Big(M_g \|A^{-\beta}l
+M_g  K(\alpha, \beta)M_g \frac{T^{\alpha \beta}} {\alpha \beta}l\\
&+ \frac{M}{\Gamma(\alpha)} \frac{T^\alpha}{\alpha}\sigma_f \sup_{s\in J}
 n(s)\Big) + M \sum_{i=1}^{N} \sigma_{i} \geq1,
\end{align*}
which is a contradiction to Assumption (H4).
Thus $(\prod_{\varepsilon} + \Theta_{\varepsilon}
+ \varUpsilon _{\varepsilon}) (B_r) \subset B_r$ for some $r>0$.
\end{proof}

\begin{lemma} \label{lem3.3} 
Let Assumptions {\rm (H1)--(H4)} hold. Then $\Theta_1$ is contractive.
\end{lemma}

\begin{proof} Let $x,y \in B_r$. Then
\begin{align*}
&\|(\prod_{\varepsilon}x)(t) - (\prod_{\varepsilon}y)(t)\|\\
&\leq \|g\big(t,\Tilde{\phi}_t+x_t\big) - g\big(t,\Tilde{\phi}_t+y_t\big)\|\\
& \leq \|  \int_0^t (t-s)^{\alpha - 1} A \mathfrak A_{\alpha} (t-s)
 \Big( g\big(s,\Tilde{\phi}_s+x_s\big) - g\big(s,\Tilde{\phi}_s+y_s\big)\Big) 
 \| ds \\
& \leq \|A^{-\beta}\|M_g\|x_t - y_t\|_{{\mathcal{B}}_h} \\
&\quad + K(\alpha, \beta) 
 \int_0^t (t-s)^{\alpha \beta-1}\|A^{\beta}\Big( g\big(s,\Tilde{\phi}_s+x_s\big)
 - g\big(s,\Tilde{\phi}_s+y_s\big)\Big) \|   ds \\
&\leq \|A^{-\beta}\|M_g\|x_t - y_t\|_{{\mathcal{B}}_h}+ K(\alpha, \beta)
 M_g \int_0^t (t-s)^{\alpha \beta-1}\|x_s - y_s\|_{{\mathcal{B}}_h} ds.
\end{align*}
Hence
\[
\|(\prod_{\varepsilon}x)(t)-(\prod_{\varepsilon}y)(t)\| 
\leq M_gl \Big(\|A^{-\beta}\| + K(\alpha, \beta) 
\frac{T^{\alpha \beta}} {\alpha \beta}\Big) 
 \sup_{0 \leq s \leq T}\|x(s) - y(s)\|,
\]
where we have used the fact that $x_0=y_0=0$. Thus
\[
\sup_{0 \leq t \leq T}\|(\prod_{\varepsilon}x)(t) 
- (\prod_{\varepsilon}y)(t)\| \leq M_gl \Big(\|A^{-\beta}\| 
+ K(\alpha, \beta) \frac{T^{\alpha \beta}} {\alpha \beta}\Big)
 \sup_{0 \leq s \leq T}\|x(s) - y(s)\|,
\]
so $\Theta_1$ is a contraction by Assumption (H4).
\end{proof}

\begin{lemma} \label{lem3.4} 
Let Assumptions {\rm (H1)--(H4)} hold. Then $\theta_{\varepsilon}$ 
maps bounded sets to bounded sets in $B_r$.
\end{lemma}

\begin{proof} 
By a similar argument as Lemma \ref{lem3.2}, we obtain
\[
\|(\Theta_{\varepsilon}z)(t)\|
 < \Big(1+ \frac{1}{\varepsilon} M_B^{2}M_{\mathfrak A}^{2} 
\frac{T^{2\alpha-1}}{2\alpha-1}\Big) \frac{M}{\Gamma(\alpha)}
 \frac{T^\alpha}{\alpha}\Lambda_f(R(r)) \sup_{s\in J} n(s) :=r_1(\varepsilon)
\]
which implies that $(\Theta_{\varepsilon}z) \in B_{{r_1}(\varepsilon)}$.
\end{proof}

\begin{lemma} \label{lem3.5} 
Let Assumptions {\rm (H1)--(H4)} hold. Then the set 
$ \{\Theta_{\varepsilon}z) : z \in B_r\}$  is an equicontinuous family 
of functions on [0,T].
\end{lemma}

\begin{proof} 
Let $0< \eta <t < T$  and $\delta > 0$ such that 
$ \|\mathfrak A_{\alpha}(s_1) - \mathfrak A_{\alpha}(s_2)\| < \eta $  
for every $s_1, s_2 \in [0,T]$  with $|{s_1} -{ s_2}| < \delta$. 
For $z \in B_r, 0< |h| <\delta, t+h \in [0,T]$, we have
\begin{align*}
&\|(\Theta_{\varepsilon})(t+h) - (\Theta_{\varepsilon})(t)\|\\
&\leq \| \int _0^t\Big((t+h-s)^{\alpha-1} - (t-s)^{\alpha-1}\Big) 
 \mathfrak A_{\alpha}(t+h-s) \big[(T-s)^{\alpha-1}Bv_{\varepsilon} 
\big(s,\Tilde{\phi} + z\big)\\
&\quad + f\big(s,\Tilde{\phi_s} + z_s\big)\big]ds\| \\
&\leq \| \int _0^{t+h}(t+h-s)^{\alpha-1} \mathfrak A_{\alpha}(t+h-s) 
\big[(T-s)^{\alpha-1}Bv_{\varepsilon} \big(s,\Tilde{\phi} + z\big)\\
&\quad + f\big(s,\Tilde{\phi_s} + z_s\big)\big]ds\| \\
&\leq \| \int _0^t(t-s)^{\alpha-1}\big(\mathfrak A_{\alpha}(t+h-s) 
 -  \mathfrak A_{\alpha}(t-s)\big)  
\big[(T-s)^{\alpha-1}Bv_{\varepsilon} \big(s,\Tilde{\phi} + z\big)\\
&\quad + f\big(s,\Tilde{\phi_s} + z_s\big)\big]ds\|
\end{align*}
Applying Lemma \ref{lem2.4} and H\"older inequality, we have
\begin{equation} \label{e3.12}
\begin{aligned}
&\|(\Theta_{\varepsilon}z)(t+h) - (\Theta_{\varepsilon}z)(t)\|\\
&\leq  \frac{M}{\Gamma(\alpha)} \Lambda_f (R(r)) 
 \int _0^t\Big((t+h-s)^{\alpha-1} - (t-s)^{\alpha-1}\Big)n(s) ds\\
&+  \frac{M}{\Gamma(\alpha)}\frac{1}{\varepsilon} M_B M_{\mathfrak A} 
 \Delta \int _0^t\Big((t+h-s)^{\alpha-1} 
 - (t-s)^{\alpha-1}\Big)(T-s)^{\alpha-1} ds\\
& +  \frac{M}{\Gamma(\alpha)}\Lambda_f (R(r))
 \int _0^t\Big((t-s)^{\alpha-1}n(s)ds\\
 & +  \frac{M}{\Gamma(\alpha)}\frac{1}{\varepsilon} M_B M_{\mathfrak A} 
 \Delta \int _{t}^{t+h}((t+h-s)^{\alpha-1} (T-s)^{\alpha-1} ds\\
 &+\frac{\eta T^{\alpha}}{\alpha} \Lambda_f (R(r))  
 \int _0^t(t-s)^{\alpha-1}n(s)ds\\
 & +\frac{\eta T^{\alpha}}{\alpha}\frac{1}{\varepsilon} 
 M_B M_{\mathfrak A} \Delta \int _0^t(t-s)^{\alpha-1}(T-s)^{\alpha-1} ds
\end{aligned}
\end{equation}
 Therefore, for $\varepsilon$ sufficiently small, the right- hand side 
of \eqref{e3.12} tends to zero as $h \to0$. On the other hand, the compactness 
of  $\mathfrak A_{\alpha}(t), t>0$, implies the continuity in the uniform 
operator topology. Thus, the set $\{\Theta_{\varepsilon}z : z \in B_r\}$ 
is equicontinuous. 
\end{proof}

\begin{lemma} \label{lem3.6}  
Let Assumptions {\rm (H1)--(H6)} hold. Then the set 
$ \{\varUpsilon_{\varepsilon}z) : z \in B_r\}$  is an equicontinuous 
family of functions on $[0,T]$.
\end{lemma}

\begin{proof}  
For $z \in B_r, 0< |h| <\delta$ and $t+h \in [0,T]$, we have
\begin{align*}
&\|(\varUpsilon_{\varepsilon}z) (t+h) - (\{\varUpsilon_{\varepsilon}z) (t)\|\\
&\leq \| \sum _{0 < t_i < t+h}\mathfrak T_{\alpha} (t+h-t_i)
  I_i(\Tilde{\phi}_{t_i}+z_{t_i}) - \sum_{0<t_i<t}\mathfrak T_{\alpha} 
 (t+h-t_i) I_i(\Tilde{\phi}_{t_i}+z_{t_i})  \| \\
&\quad + \| \sum_{0<t_i<t}\mathfrak T_{\alpha} (t+h-t_i) I_i(\Tilde{\phi}_{t_i}
 +z_{t_i}) -\sum_{0<t_i<t} \mathfrak T_{\alpha} (t-t_i) 
 I_i(\Tilde{\phi}_{t_i}+z_{t_i})  \|\\
& \leq  \sum _{0 < t_i < t+h}\| \mathfrak T_{\alpha} (t+h-t_i) 
 I_i(\Tilde{\phi}_{t_i}+z_{t_i})\|\\
&\quad + \sum _{0 < t_i < t}\| \mathfrak T_{\alpha} (t+h-t_i) 
 I_i(\Tilde{\phi}_{t_i}+z_{t_i}) - \mathfrak T_{\alpha} 
 (t-t_i) I_i(\Tilde{\phi}_{t_i}+z_{t_i}) \|,
\end{align*}
from which follows that $\{\varUpsilon_{\varepsilon}z : z \in B_r\}$ is
 equicontinuous on each interval $[0,T]$ due to the equicontinuous of 
$\mathfrak T(t), t>0 $ and Hypothesis (H5). 
\end{proof}

\begin{lemma} \label{lem3.7} 
Let Assumptions {\rm (H1)-(H6)} hold. Then  
$ (\Theta_{\varepsilon} + \varUpsilon_{\varepsilon})$ maps $B_r$ 
onto a precompact set in $B_r$.
\end{lemma}

\begin{proof} 
Let $ 0<t<T$ be fixed and ${\varepsilon}$ be a real number 
satisfying  $ 0<\lambda<t $.  For $\delta >0$ define an operator 
$(\Theta_{\varepsilon}^{\lambda, \delta}z 
+\varUpsilon_{\varepsilon}^{\lambda, \delta}z)$
on $B_r$ by
\begin{align*}
&\Big(\Theta_{\varepsilon}^{\lambda, \delta}z
 +\varUpsilon_{\varepsilon}^{\lambda, \delta}z\Big)(t)\\
&= \alpha \int_0^{t-\lambda} \int_{\delta}^{\infty} \theta(t-s)^{\alpha -1} 
 \eta _{\alpha}(\theta) S(t-s)^{\alpha} \theta) \big[Bu_{\varepsilon}
 \big(s, \Tilde{\phi} + z\big) \\
&\quad + f\big( s, \Tilde{\phi}_s + z_s\big)
 \big]d\theta ds + \sum_{0<\lambda<t} \mathfrak T_{\alpha} 
 (t-\lambda) I_i(\Tilde{\phi}_{\lambda}+z_{\lambda}) \\
& = \alpha S(\lambda^{\alpha} \delta) \int_0^{t-\lambda} \int_{\delta}^{\infty} 
\theta(t-s)^{\alpha -1} \eta _{\alpha}(\theta) S(t-s)^{\alpha} \theta 
- \lambda^{\alpha}\delta) \big[Bu_{\varepsilon}\big(s, \Tilde{\phi}+z\big) \\
&\quad + f\big( s, \Tilde{\phi}_s + z_s\big)\big]d\theta ds
 + \sum_{0<\lambda<t}\mathfrak T_{\alpha} (t-\lambda) 
 I_i(\Tilde{\phi}_{\lambda}+z_{\lambda})
\end{align*}
Since $S(t)$, $t>0$ is a compact operator, the set 
$\Big\{\Big (\Theta_{\varepsilon}^{\lambda, \delta}z 
+\varUpsilon_{\varepsilon}^{\lambda, \delta}z\Big)(t) : z\in B_r\Big\} $ 
is precompact in $H$ for every   $0<\lambda<t, \delta>0$. Moreover, for each
 $z\in B_r$, we have
\begin{align}
&\|\big(\Theta_{\varepsilon}z +\varUpsilon_{\varepsilon}z\big)(t)
 -\big(\Theta_{\varepsilon}^{\lambda, \delta}z 
 +\varUpsilon_{\varepsilon}^{\lambda, \delta}z\big)(t)\|  \nonumber \\  
& \leq   \alpha E\| \int_0^t \int_0^{\delta} \theta(t-s)^{\alpha -1} 
 \eta _{\alpha}(\theta) S(t-s)^{\alpha} \theta)
 \big[Bu_{\varepsilon}\big(s, \Tilde{\phi} + z\big) + f\big( s, \Tilde{\phi}_s 
 + z_s\big)\big]d\theta ds\| \nonumber \\
&\quad + \alpha E\| \int_{t-\lambda}^t \int_{\delta}^{\infty}
  \theta(t-s)^{\alpha -1} \eta _{\alpha}(\theta) S(t-s)^{\alpha} \theta)
\big[Bu_{\varepsilon}\big(s, \Tilde{\phi} + z\big) \nonumber \\
&\quad + f\big( s, \Tilde{\phi}_s + z_s\big)\big]d\theta ds\| \nonumber\\
&\quad +\| \sum_{0<t_i<t}\mathfrak T_{\alpha} (t-t_i) I_i(\Tilde{\phi}_{t_i}
 +z_{t_i})\|
 + \| \sum_{0<\lambda<t} \mathfrak T_{\alpha} (t-\lambda) 
 I_i(\Tilde{\phi}_\lambda+z_{\lambda})\| \nonumber \\
&\leq \alpha  E\| \int_0^t \int_0^{\delta} \theta(t-s)^{\alpha -1} 
 \eta _{\alpha}(\theta) S(t-s)^{\alpha} \theta)
\big[Bu_{\varepsilon}\big(s, \Tilde{\phi} + z\big) 
 + f\big( s, \Tilde{\phi}_s + z_s\big)\big]d\theta ds\| \nonumber\\
&\quad +\alpha E\| \int_{t-\lambda}^t \int_{\delta}^{\infty} 
 \theta(t-s)^{\alpha -1} \eta _{\alpha}(\theta) S(t-s)^{\alpha} \theta)
 \big[Bu_{\varepsilon}\big(s, \Tilde{\phi} + z\big) \nonumber \\
&\quad + f\big( s, \Tilde{\phi}_s  + z_s\big)\big]d\theta ds\|
+2\sum_{0<\lambda <t_i<t}\| \mathfrak T_{\alpha} (t-t_i) 
 I_i(\Tilde{\phi}{t_i}+z_{t_i})\| \nonumber\\
& =: J_1 + J_2 + J_3 \label{e3.13}
\end{align}
By a similar argument as above, we have
\begin{align*}
J_1 & \leq\alpha M \int_0^t (t-s)^{\alpha-1} 
\Big(\big\|Bu_{\varepsilon}\big(s, \Tilde{\phi} + z\big)\big\| 
+\big\| f\big( s, \Tilde{\phi}_s + z_s\big)\big\|\Big)ds \Big(\int_0^{\delta} 
\theta \eta_{\alpha}(\theta) d\theta\Big)\\
&\leq \alpha M \Big(\frac {1}{\varepsilon} M_{B}M_{\mathfrak A} 
\Delta  \int_0^t (t-s)^{\alpha-1}(T-s)^{\alpha-1} ds 
 +  \Lambda_f(R(r)) \int_0^t (t-s)^{\alpha-1} n(s)ds\Big)  \\
&\quad\times \Big(\int_0^{\delta}\theta \eta_{\alpha}(\theta)d\theta\Big),
\\
 J_2 
&\leq \alpha M \int_{t- \lambda}^t (t-s)^{\alpha-1} 
\Big(\big\|Bu_{\varepsilon}\big(s, \Tilde{\phi} + z\big)\big\| 
 +\big\| f\big( s, \Tilde{\phi}_s + z_s\big)\big\|\Big)ds 
 \Big(\int_{\delta}^{\infty} \theta \eta_{\alpha}(\theta) d\theta\Big)\\
& \leq \frac {\alpha M}{\Gamma(1+ \alpha)} \Big(\frac {1}{\varepsilon} 
 M_{B}M_{\mathfrak A} \Delta  \int_{t-\lambda}^t (t-s)^{\alpha-1}
 (T-s)^{\alpha - 1}  ds \\
&\quad +  \Lambda_f(R(r)) \int_{t-\lambda}^t (t-s)^{\alpha-1}
  n(s)ds\Big),
\end{align*} 
and 
\begin{equation} \label{e3.16}
J_3  \leq 2 M\sum_{i=1}^{N} N_i;
\end{equation}
here we have  used the equality
\[
\int_0^{\infty} \theta^{\beta} \eta_{\alpha}(\theta)  d\theta
= \frac {\Gamma(1+\beta)}{\Gamma(1+\alpha\beta)}.
\]
From \eqref{e3.13}-\eqref{e3.16}, one can see that for each $z\in B_r$,
\[
\|\Big (\Theta_{\varepsilon}z +\varUpsilon_{\varepsilon}z\Big)(t)
- \Big (\Theta_{\varepsilon}^{\lambda, \delta}z
+\varUpsilon_{\varepsilon}^{\lambda, \delta}z\Big)(t)\| \to 0\quad
\text{as }\lambda \to 0^+, \; \delta \to 0^+.
\]
Therefore, there are relatively compact sets arbitrary close to the set\\
 $\{(\Theta_{\varepsilon}z + \varUpsilon_{\varepsilon}z)(t) : z \in B_r\}$;
hence, the set
$\big\{(\Theta_{\varepsilon}z + \varUpsilon_{\varepsilon}z\big)(t) :
 z \in B_r\}$ is also precompact in $B_r$.
\end{proof}

\section{Main Results}

Consider the  linear impulsive fractional differential system
\begin{gather} \label{e4.1}
D_{t}^{\alpha} x(t) = Ax(t)+Bu(t), \quad t\in [0,T], \; t\neq t_i,\\
 x(0) = \phi(0), \label{e4.2} \\
\Delta x(t_i)  = I_{i}(x_{t_i}) \label{e4.3}
\end{gather}
The approximate controllability for linear  impulsive fractional differential 
system \eqref{e4.1}-\eqref{e4.3} is a natural generalization of approximate 
controllability of linear first order control system. It is convenient at 
this point to introduce the controllability operators associated with  
\eqref{e4.1}-\eqref{e4.3} as
\begin{gather*}
L_0^{T} = \int_0^{T} (T-s)^{\alpha-1} \mathfrak A_{\alpha}(T-s) Bu(s) ds 
+ \sum_{t_i<t} \mathfrak T_{\alpha} (T-t_i) I_i(x_{t_i})
\\
\begin{aligned}
L_0^{T} &=L_0^{T}(L_0^{T})^* \\
&= \int_0^{T} (T-s)^{2(\alpha-1)} \mathfrak A_{\alpha}(T-s) BB^*  
\mathfrak A_{\alpha}^{*}(T-s)  ds \\
&\quad + \sum_{t_i<t} \mathfrak T_{\alpha}(T-t_i) 
 \mathfrak T_{\alpha}^{*} (T-t_i) I_i(x_{t_i}),
\end{aligned}
\end{gather*}
respectively, where $B^{*}$ denotes the adjoint of $B$,  
$\mathfrak A_{\alpha}^{*}(t)$ is the adjoint of $\mathfrak A_{\alpha}(t)$   
and  $\mathfrak T_{\alpha}^{*}(t)$ is the adjoint of $\mathfrak T_{\alpha}(t)$.   
It is straightforward that the operator $L_0^{T}$ is  a linear bounded operator 
for $1/2 < \alpha \leq 1$.

\begin{theorem}[\cite{ni1}] \label{thm4.1}  
The following three conditions are equivalent:
\begin{itemize}
\item[(i)] $\Gamma_0^{T}$ is positive, that is,
 $\langle z^{*},\Gamma_0^{T} z^* \rangle>0 $ for all nonzero $z^* \in X^*$;

\item[(ii)] For all $h \in X, J(z_{\varepsilon}(h))$ coverges to the zero as 
$\varepsilon \to 0^+$ in the weak topology, where
 $z_{\varepsilon}(h) = \varepsilon(\varepsilon I 
+  \Gamma_0^{T}J)^{-1}(h)$ is a solution of the equation 
$\varepsilon z_{\varepsilon} +  \Gamma_0^{T}J(z_{\varepsilon}(h) ) = \alpha h$;

\item[(iii)] For all $h\in X,  z_{\varepsilon}(h)
 = \varepsilon(\varepsilon I +  \Gamma_0^{T}J)^{-1}(h)$  
converges  to the zero as $\varepsilon \to 0^+$ in the strong topology.
\end{itemize}
\end{theorem}

\begin{remark} \label{rmk4.2} \rm
 It is known that Theorem \ref{thm4.1}(i) holds if and only if
 $\overline{\operatorname{Im} L_0^{T}} = X$. In other words, 
Theorem  \ref{thm4.1}(i) holds if and only if the corresponding linear system 
is approximately controllable on $[0,T]$.
\end{remark}

\begin{theorem}[\cite{ni1}] \label{thm4.3}  
Let $p: X\to X$ be a nonlinear operator, Assume $z_{\varepsilon}$ 
is a solution of the following equation 
$\varepsilon z_{\varepsilon} +  \Gamma_0^{T}J(z_{\varepsilon}(h) ) 
= \alpha p(z_{\varepsilon})$
and $\| p(z_{\varepsilon}) -p\| \to 0$ as $\varepsilon \to 0^{+}, p\in X$. 
Then there exists a subsequence of the sequence $\{z_{\varepsilon}\}$ 
strongly converging to  zero as $ \varepsilon \to 0^{+}$.
\end{theorem}

We are now in a position to state and prove our main  result.

\begin{theorem} \label{thm4.4}
 Let $1/2 <\alpha \leq1$. Suppose that Assumptions {\rm (H1)--(H7)} are satisfied.
Also assume that
\begin{itemize}
\item[(H8)] $g: [0,T] \times X \to X$ and $A^{\beta} g(T,\cdot)$ 
is continuous from the weak topology of $X$;
\item[(H9)]  There  exists $N\in L^{\infty}([0,T],[0,+\infty))$ such that
$$
\sup_{x\in B_{t}} \|f(t,x)\| +\sup _{y \in X} \|A^{\beta} g(t,y)\| 
\leq N(t), \quad \text{for  a.e. } t\in [0,T].
$$
\end{itemize}
Then  system \eqref{e2.1} is approximately controllable on $[0,T]$.
\end{theorem}

\begin{proof} 
Let $x^{\varepsilon}$ be a fixed point of $\Phi_{\varepsilon}$ 
in $B_{r(\varepsilon)}$. Then  $x^{\varepsilon}$ is a mild solution 
of \eqref{e2.1} $[0,T]$ under the control given by
\begin{gather*}
u_{\varepsilon} (t,x^{\varepsilon}) 
= (T-t)^{\alpha -1} B^{*} S^{*} (T-t) J 
\big(((\varepsilon I +  \Gamma_0^{T}J)^{-1} p(x^{\varepsilon})\big)\\
\begin{aligned}
 p(x^{\varepsilon}) 
&= h -  \mathfrak T_{\alpha}(T) [\phi(0)+g(0,\phi)] + g(T,x^{\varepsilon}(T))\\
&\quad + \int_0^{T}(T-s)^{\alpha - 1}A\mathfrak A_{\alpha}
   (T-s)g(s,x_{s}^{\varepsilon}) ds\\
&\quad - \int_0^{T}(T-s)^{\alpha - 1}\mathfrak A_{\alpha} (T-s) 
 + f(s,x_{s}^{\varepsilon}) ds
\end{aligned}
\end{gather*}
and satisfies the  equality
\begin{align*}
&x^{\varepsilon}(T) \\
&=  \mathfrak T_{\alpha}(T) [\phi(0)+g(0,\phi)] - g(T,x^{\varepsilon}(T))\\
&\quad -\int_0^{T}(T-s)^{\alpha - 1}A\mathfrak A_{\alpha} (T-s)
 g(s,x_{s}^{\varepsilon}) ds \\
&\quad + \int_0^{T}(T-s)^{\alpha - 1}\mathfrak A_{\alpha} (T-s)[Bu_{\varepsilon}(s,x) 
 + f(s,x_{s}^{\varepsilon})] ds\\
&\quad +\sum_{t_i <T}\mathfrak T_{\alpha} (T-t_i) I_i(x_{t_i}^{\varepsilon})
  + \sum_{t_i<T} [g(T,x^{\varepsilon}(T))_{|t_i^+}
 -g(T,x^{\varepsilon}(T))_{|t_i^-}]\\
&=\mathfrak T_{\alpha}(T) [\phi(0)+g(0,\phi)] - g(T,x^{\varepsilon}(T))
 -\int_0^{T}(T-s)^{\alpha - 1}A\mathfrak A_{\alpha} (T-s)h(s,x_{s}^{\varepsilon}) 
 ds \\
&\quad +(-\varepsilon I+\varepsilon I +\Gamma_0^{T}J)
 \big((\varepsilon I +\Gamma_0^{T}J)^{-1} p(x^{\varepsilon})\big) \\
&\quad  +  \int_0^{T}(T-s)^{\alpha - 1}\mathfrak A_{\alpha} (T-s)
 + f(s,x_{s}^{\varepsilon}) ds\\
&=h-\varepsilon  \big((\varepsilon I +\Gamma_0^{T}J)^{-1} p(x^{\varepsilon})
 +\sum_{t_i <T}\mathfrak T_{\alpha} (T-t_i) I_i(x_{t_i}^{\varepsilon}) \\
&\quad + \sum_{t_i<T} [g(T,x^{\varepsilon}(T))_{|t_i^+}-g(T,x^{\varepsilon}
 (T))_{|t_i^-}]
\end{align*}
In other words, $z_{\varepsilon} = h- x^{\varepsilon}(T)$  
is a solution of the equation
$\varepsilon \big(\varepsilon I +\Gamma_0^{T}J(z_{\varepsilon}) 
= \varepsilon p(x^{\varepsilon})$.
Now it follows that
\begin{gather}
\varepsilon\langle J(z_{\varepsilon}), z_{\varepsilon}\rangle 
 +\langle J(z_{\varepsilon}),\Gamma_0^{T}J(z_{\varepsilon})\rangle 
= \varepsilon\langle J(z_{\varepsilon}),p(x^{\varepsilon})\rangle,
\nonumber \\
\varepsilon \|z_\varepsilon\|^2 + \langle J(z_{\varepsilon}),
 \Gamma_0^{T}J(  z_{\varepsilon})\rangle 
=  \varepsilon\langle J(z_{\varepsilon}),p(x^{\varepsilon})\rangle,
\nonumber\\
\varepsilon \|z_\varepsilon\|^2 
\leq \varepsilon \langle J(z_{\varepsilon}),p(x^{\varepsilon})\rangle 
 \leq \varepsilon \|z_\varepsilon\| \|p(x^{\varepsilon})\|, \label{e4.4}
\\
\|z_\varepsilon\|=  \|J(z_{\varepsilon})\| \leq \|p(x^{\varepsilon})\|. \label{e4.5}
\end{gather}
On the other hand, by (H9),
\begin{align*} % 4.6
\|p(x^{\varepsilon})\| 
&\leq \|h\| + M \|\phi (0)\| + N(T)
 + \frac {M_{1- \beta} \alpha \Gamma (1+\beta)}{\Gamma(1+\alpha \beta)}
  \int_0^{T}(T-s)^{\alpha \beta -1} N(s) ds \\
&\quad + \frac{M}{\Gamma(\alpha)} \int_0^{T} (T-s)^{\alpha-1} N(s) ds.
\end{align*}
From \eqref{e4.4}-\eqref{e4.5}, it follows that  
$x^{\varepsilon}(T) \to \tilde{x}$ converges weakly as $\varepsilon \to 0^{+}$ 
and by the Assumption (H8),
 $A^{\beta} g(T,x^{\varepsilon}(T)) \to A^{\beta}g (T, \tilde {x})$ 
converges strongly as  $\varepsilon \to 0^{+}$. Moreover, Assumption
 (H9) implies that
\[
\int_0^{T} \|f(s,x_{s}^{\varepsilon})\| ^2 ds 
+ \int_0^{T} \|A^{\beta} g(s,x_{s}^{\varepsilon})\|^2 ds 
\leq \int_0^{T} N(s) ds.
\]
Consequently, the sequence 
$\{f(.,x^{\varepsilon}), A^{\beta} g(.,x^{\varepsilon})$\} is bonded. 
Then there is a subsequence denoted by 
$\{f(.,x^{\varepsilon}, A^{\beta} g(.,x^{\varepsilon})\}$ weakly convergent to, 
say, $(f(.),g(.))$ in $L_{2}([0,T],X)$. Then
\begin{align*}
&\|p(x^{\varepsilon}) - p\|\\
&=\| g(T,x^{\varepsilon}(T))- g(T,\tilde{x})\|
 +\| \int_0^{T}(T-s)^{\alpha - 1}A^{1-\beta}\mathfrak A_{\alpha} (T-s)
 [A^{\beta} g(s,x_{s}^{\varepsilon}) - g(s)] ds\|\\
&\quad + \|  \int_0^{T}(T-s)^{\alpha - 1}\mathfrak A_{\alpha} 
 (T-s)[f(s,x_{s}^{\varepsilon}) - f(s)] ds\|\\
&\leq \|A^{-\beta} (A^{\beta} g (T,x^{\varepsilon}(T))-A^{\beta} 
 g (T,\tilde{x}) \| \\
&\quad + \sup_{0 \leq t \leq T} \| 
 \int_0^t(t-s)^{\alpha - 1}A^{1-\beta}\mathfrak A_{\alpha} (T-s)[A^{\beta} 
 g(s,x_{s}^{\varepsilon}) - g(s)] ds\|\\
&\quad + \sup_{0 \leq t \leq T} \|  \int_0^t (T-s)^{\alpha - 1}
 \mathfrak A_{\alpha} (t-s)[f(s,x_{s}^{\varepsilon}) - f(s)] ds\|  \to 0,
\end{align*}
where
\begin{align*}
p(x) &= h- \mathfrak T_{\alpha}(T) [\phi(0)+g(0,\phi(0))] + g(T,\Tilde{x})\\
 &\quad + \int_0^{T}(T-s)^{\alpha - 1}A^{1-\beta}\mathfrak A_{\alpha} (T-s)g(s) ds
- \int_0^{T}(T-s)^{\alpha - 1}\mathfrak A_{\alpha}(T-s)f(s) ds
\end{align*}
as $\varepsilon \to 0^{+}$ because of compactness of an operator   
$$
f(.) \to \int_0^{.} (.-s)^{\alpha -1} \mathfrak A_{\alpha}(.-s)f(s) ds 
:\quad  L_2([0,T],X) \to PC([0,T],X).
$$ 
Then by Theorem \ref{thm4.3},
$\|(x^{\varepsilon}(T) - h\| = \|z_{\varepsilon}\| \to 0$ as  
$\varepsilon \to 0^{+}$.
 This gives the approximate controllability. 
\end{proof}

\begin{remark} \label{rmk4.5} \rm
Theorem \ref{thm4.4} assumes that the operator $A$ generates a compact semigroup and, 
consequently, the associated linear control system \eqref{e4.1}-\eqref{e4.3}  
is not exactly controllable. Therefore Theorem \ref{thm4.4}  has no analogue for 
the concept of exact controllability.
\end{remark}

\section{Applications} 


Let $X = L_2[0,\pi]$ and $Az = z''$, with domain 
$D(A) = \{z\in X | z, dz/d\xi  $ are absolutely continuous, 
$d^2{z}/d \xi^{2} \in X$ and $z(0) = z(\pi) = 0\}$,
where $A$ is the infinitesimal generator of a strongly continuous 
semigroup $S(t), t>0$, on $X$ which is analytic compact and self-adjoint, 
the eigenvalues are $-n^2$, $n \in\mathbb{N}$, with corresponding 
normalized eigenvectors $e_n (\xi) := (2/{ \pi})^{1/2}\sin(n \xi)$ and
\[
S(t) e_n =  e^{-n^{2}t} e_n,\quad  n=1,2,\dots .
\]
Moreover the  following statements hold: 
\begin{itemize}
\item[(a)] $\{e_n : n \in\mathbb{N}\}$ is an orthonormal basis of $X$;
\item[(b)] If $z \in D(A)$ then 
$ A(z) = - \sum_{n=1}^{\infty} n^2 \langle{ z,e_n} \rangle{ e_n}$;
\item[(c)] For $z\in \mathbb{H}$,  
$(-A) ^{-1/2} z = \sum_{n=1}^{\infty} \frac {1}{n} \langle{z,e_n} \rangle {e_n}$;
\item[(d)] The operator $(-A)^{1/2}$ is given as 
$(-A)^{1/2} z = \sum_{n=1}^{\infty} n\langle{ z,e_n }\rangle e_n$ 
on the space 
$D\Big[(-A)^{1/2}\Big] = \{ z \in X : \sum_{n=1}^{\infty}
 n\langle{ z,e_n }\rangle e_n \in X\}$
\end{itemize}
Consider the neutral system
\begin{gather} 
\frac {\partial^{\alpha}}{\partial t^{\alpha}}\Big[ x(t, \xi) 
+ \int_0^{\pi} b(\theta, \xi) x(t, \theta) d \theta \Big]
= \frac {\partial^{2}}{\partial {\xi}^{2}}x(t, \xi) + p(t,x(t, \xi)) 
+ Bu(t, \xi), \label{e5.1}\\
x(t,0) = x(t, \pi) = 0, \quad t \geq0,   \label{e5.2} \\
 x(t,\xi) = \phi(\xi), \quad 0 \leq \xi \leq \pi,   \label{e5.3} \\
\Delta x(t_i,\cdot) = x(t_i^{+},\cdot) -  x(\cdot,t_i^{-})
= \int _0^{\pi} (\xi, x(t_i,s)) ds,  \label{e5.4}
\end{gather}
where $p : [0,T] \times R \to R$ is continuous functions and 
$ (t_i)_{i \in\mathbb{N}}$  is a strictly increasing sequence 
of positive real numbers. $B$ is a linear continuous mapping from 
\[
 U = \Big\{u= \sum_{n=2}^{\infty} u_n {e_n} | \|u\|_{U}^{2}
= \sum_{n=2}^{\infty} u_n^2 < \infty\Big\}
\]
to $X$ as follows
\[
Bu = 2u_{2} + \sum_{n=2}^{\infty} u_n {e_n}.
\]
To write problem \eqref{e5.1}-\eqref{e5.4} in the abstract form, we assume 
the following:
\begin{itemize}
\item[(A1)] The function $b$ is measurable and
\[
\int_0^{\pi}\int_0^{\pi} b^2(\theta, \xi) d\theta d\xi <\infty.
\]

\item[(A2)] The function $\frac{\partial}{\partial \xi} b(\theta, \xi)$ 
is measurable, $b(\theta,0) = b(\theta, \pi) = 0$, and let
\[
L_1 = \Big[\int_0^{\pi}\int_0^{\pi} \Big(\frac{\partial}{\partial \xi} 
b(\theta, \xi)\Big)^{2} d\theta d\xi \Big]^{1/2}.
\]
\item[(A3)] The functions $p_i : [0,\pi] \times \mathbb{R} \to \mathbb{R}$, 
$i \in {\mathbb{N}}$, are continuous and there are positive constants 
$L_i$ such that
\[
|p_i(\xi,s) - p_i(\xi, \bar{s})| \leq L_i|s - \bar{s}|, \quad 
\xi \in [0,\pi], \; s,\bar{s} \in {\mathbb{R}}.
\]
\end{itemize}
We now define the functions $g,f : [0,T] \times X \to X $, 
$ I_i : {X} \to {X}$ by
\begin{gather*}
g(x)(\xi) = \int_0^{\pi} \int_0^{\pi}  b(\theta, \xi) x(\theta) d\theta,
\quad \xi \in[0,\pi],\\
f(t, x)(\xi) = p(t, x(\xi)), \quad t\geq 0, \; \xi \in[0,\pi],\\
 I_i(\phi) \xi = \int_0^{\pi} p_i (\xi), \phi(0,s))ds,\quad i\in {\mathbb{N}},
\quad \xi \in[0,\pi]
\end{gather*}
From (A1), it  is clear that $g$ is bounded linear operator on $X$. 
Furthermore, $g(x) \in D\big[A^{1/2}\big]$, and $\|A^{1/2}\| \leq L_1$. 
In fact from the definition of $g$ and (A2) it follows that
\begin{align*}
\langle g(x),e_n \rangle 
&= \int_0^{\pi}
\Big[ \int_0^{\pi} b(\theta, \xi) x(\theta)d\theta\Big]e_n(\xi)d\xi\\
& = \frac{1}{n} \big(\frac{2}{\pi}\big)^{1/2} \Big{\langle}  
\int_0^{\pi}\frac{\partial}{\partial \xi} b(\theta, \xi)x(\theta) d\theta, 
\cos(n \xi) \Big{\rangle}\\
&= \frac{1}{n} \big(\frac{2}{\pi}\big)^{1/2}\langle g_1(x), \cos(n \xi)\rangle,
\end{align*}
where $g_1(x) =  \int_0^{\pi} b(\theta, \xi) x(\theta)d\theta$. 
From (A2) we know that $g_1 : X \to X $ is a bounded linear operator
 with $\|g_1\| \leq L_1$. Hence $\|A^{1/2} g(x)\| = \|g_1(x)\|$,
 which implies the assertion. Moreover, assume that $f$ and $g$ 
satisfy conditions of Theorem \ref{thm4.3}. Thus the problem \eqref{e5.1}-\eqref{e5.2}
 can be written in the abstract form
\begin{gather*}
\frac{d^ {\alpha}}{dt^{\alpha}}\big(x(t) + g(t,x(t)\big) 
= Ax(t) + f(t,x(t)) + Bu(t),\\ 
x(0) = x_0,\quad t \in [0,T],\\
 \Delta x(t_i) = x(t_i^{+} -  x(t_i^{-}).
\end{gather*}
Now consider the associated linear system
\begin{gather} \label{e5.5}
\frac{d^ {\alpha}}{dt^{\alpha}}x(t) =  Ax(t) + Bu(t),\\
  x(0) = x_0, \quad t \in [0,T], \label{e5.6}\\
 \Delta x(t_i) = x(t_i^{+} -  x(t_i^{-}). \label{e5.7}
\end{gather}
So that it is approximately controllable on $[0,T]$ for
 $1/2< \alpha <1$. It is easy to  see that if
 $z = \sum_{n=1}^{\infty} \langle z,e_n \rangle e_n$ then 
$$
B^* v = (2v_1 + v_2) e_2 + \sum_{n=3}^{\infty}{v_n}{ e_n},
$$ \begin{align*}
&B^* \mathfrak A_{\alpha}^{*} (T-s)z \\
&= B^{*} \alpha \int_0^{\infty} \theta \varPsi_{\alpha} (\theta)S^{*} 
 ((T-s)^{\alpha} \theta)z d\theta\\
&= \alpha  \int_0^{\infty} \theta \varPsi_{\alpha} (\theta)
\Big(\Big(2 \langle{z,e_1}\rangle e^{{-(T-s)}^{{\alpha}\theta}} {e_1}
 + \langle {z,e_2}\rangle e^{{-4(T-s)}^{{\alpha}\theta}} \\
&\quad + \sum _{n=3}^{\infty} e^{-n^{2}{(T-s)}^{{\alpha}{\theta}}}  
 \langle{z,e_n}\rangle {e_n} \Big) d\theta \\
&= \Big(2 \langle{z,e_1}\rangle \alpha \int_0^{\infty}\theta \varPsi_{\alpha}
  (\theta)e^{{-(T-s)}^{{\alpha}\theta}} d\theta+\langle{z,e_2}\rangle 
 \alpha \int_0^{\infty}\theta \varPsi_{\alpha} (\theta)e^{{-4(T-s)}^{{\alpha}
 \theta}} d\theta\Big) e_2\\
&\quad +  \alpha \sum_{n=3}^{\infty} \int_0^{\infty}\theta \varPsi_{\alpha} 
(\theta) e^{-n^{2}{(T-s)}^{{\alpha}{\theta}}}d\theta  \langle{z,e_n}\rangle {e_n},
\end{align*}
\begin{align*}
&\big\|(T-s)^{\alpha-1} B^{*} \mathfrak A_{\alpha}^{*} (T-s)z\big\|^2 \\
&= (T-s)^{2(\alpha -1)}\Big(2 \alpha  \int_0^{\infty} \theta 
 \varPsi_{\alpha} (\theta) e^{{-(T-s)}^{{\alpha}\theta}}d\theta 
  \langle{z,e_1}\rangle\\
&\quad + \alpha  \int_0^{\infty} \theta \varPsi_{\alpha} (\theta) 
 e^{{-4(T-s)}^{{\alpha}\theta}}d\theta  \langle {z,e_2}\rangle\Big)^2\\
&+  (T-s)^{2(\alpha -1)}  \sum_{n=3}^{\infty} \Big(\alpha  \sum_{n=3}^{\infty} 
 \int_0^{\infty}\theta \varPsi_{\alpha} (\theta) 
 e^{-n^{2}{(T-s)}^{{\alpha}{\theta}}} d\theta\Big)^2 \langle {z,e_2}\rangle^2 = 0.
\end{align*}
It follows that $ \langle {z,e_1}\rangle =  \langle {z,e_2}\rangle = \dots 
 =  \langle {z,e_n}\rangle =0$, consequently $z=0$, which means that 
system \eqref{e5.5}-\eqref{e5.7} is approximaely controllable on $[0,T]$. 
Therefore, from Theorem \ref{thm4.4}, system \eqref{e5.1}-\eqref{e5.4} is approximately 
controllable on $[0,T]$.

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\end{document}