In this section, we state and prove conditions for the approximate controllability of semilinear fractional control integro-differential systems. To do this, we first prove the existence of a fixed point of the operator ${\mathrm{\Lambda}}_{\epsilon}$ defined below using Krasnoselskii’s fixed-point theorem. Secondly, in Theorem 11, we show that under the uniform boundedness of *f* and *g* the approximate controllability of fractional systems (1) is implied by the approximate controllability of the corresponding linear system (4).

Let $x(T;{x}_{0},u)$ be the state value of (1) at terminal time *T* corresponding to the control *u* and the initial value ${x}_{0}$. Introduce the set $\mathrm{\Re}(T,{x}_{0})=\{x(T;{x}_{0},u):u\in {L}_{2}([0,T],U)\}$, which is called the reachable set of system (1) at terminal time *T*, its closure in ${X}_{\alpha}$ is denoted by $\overline{\mathrm{\Re}(T,{x}_{0})}$.

**Definition 6** The system (1) is said to be approximately controllable on $[0,T]$ if $\overline{\mathrm{\Re}(T,{x}_{0})}={X}_{\alpha}$, that is, given an arbitrary $\epsilon >0$ it is possible to steer from the point ${x}_{0}$ to within a distance *ε* from all points in the state space ${X}_{\alpha}$ at time *T*.

Consider the following linear fractional differential system:

$\begin{array}{r}{D}_{t}^{\beta}x(t)=Ax(t)+Bu(t),\phantom{\rule{1em}{0ex}}t\in [0,T],\\ x(0)={x}_{0}.\end{array}$

(4)

The approximate controllability for linear fractional system (4) is a natural generalization of approximate controllability of linear first order control system [

9,

10,

12]. It is convenient at this point to introduce the controllability and resolvent operators associated with (4) as

respectively, where ${B}^{\ast}$ denotes the adjoint of *B* and ${P}_{\beta}^{\ast}(t)$ is the adjoint of ${P}_{\beta}(t)$. It is straightforward that the operator ${\mathrm{\Gamma}}_{0}^{T}$ is a linear bounded operator.

**Theorem 7** [10]

*Let* *Z* *be a separable reflexive Banach space and let* ${Z}^{\ast}$ *stands for its dual space*.

*Assume that* $\mathrm{\Gamma}:{Z}^{\ast}\to Z$ *is symmetric*.

*Then the following two conditions are equivalent*:

- 1.
$\mathrm{\Gamma}:{Z}^{\ast}\to Z$ *is positive*, *that is*, $\u3008{z}^{\ast},\mathrm{\Gamma}{z}^{\ast}\u3009>0$ *for all nonzero* ${z}^{\ast}\in {Z}^{\ast}$.

- 2.
*For all* $h\in Z{z}_{\epsilon}(h)=\epsilon {(\epsilon I+\mathrm{\Gamma}J)}^{-1}(h)$ *strongly converges to zero as* $\epsilon \to {0}^{+}$. *Here*, *J* *is the duality mapping of* *Z* *into* ${Z}^{\ast}$.

**Lemma 8** *The linear fractional control system* (4) *is approximately controllable on* $[0,T]$ *if and only if* $\epsilon R(\epsilon ,{\mathrm{\Gamma}}_{0}^{T})\to 0$ *as* $\epsilon \to {0}^{+}$ *in the strong operator topology*.

*Proof* The lemma is a straightforward consequence of Theorem 7. Indeed, the system (4) is approximately controllable on $[0,T]$ if and only if $\u3008{\mathrm{\Gamma}}_{0}^{T}x,x\u3009>0$ for all nonzero $x\in X$, see [7]. By Theorem 7, $\parallel \epsilon {(\epsilon I+{\mathrm{\Gamma}}_{0}^{T})}^{-1}(h)\parallel \to 0$ as $\epsilon \to {0}^{+}$ for all $h\in X$. □

**Remark 9** Notice that positivity of ${\mathrm{\Gamma}}_{0}^{T}$ is equivalent to $\u3008{\mathrm{\Gamma}}_{0}^{T}x,x\u3009=0\u27f9x=0$. In other words, since $\u3008{\mathrm{\Gamma}}_{0}^{T}x,x\u3009={\int}_{0}^{T}{(T-s)}^{\beta -1}{\parallel {B}^{\ast}{P}_{\beta}^{\ast}(T-s)x\parallel}^{2}\phantom{\rule{0.2em}{0ex}}ds$, approximate controllability of the linear system (4) is equivalent to ${B}^{\ast}{P}_{\beta}^{\ast}(T-s)x=0$, $0\le s<T\u27f9x=0$.

Before proving the main results, let us first introduce our basic assumptions.

(H

_{1})

$f,g:[0,T]\times {X}_{\alpha}\times {X}_{\alpha}\to X$ are continuous and for each

$r\in \mathbb{N}$, there exists a constant

$\gamma \in [0,\beta (1-\alpha )]$ and functions

${\phi}_{r}\in {L}^{1/\gamma}([0,T];{\mathbb{R}}^{+})$,

${\psi}_{r}\in {L}^{\mathrm{\infty}}([0,T];{\mathbb{R}}^{+})$ such that

(H_{2}) $h:C([0,T];{X}_{\alpha})\to {X}_{\alpha}$ is a Lipschitz function with Lipschitz constant ${L}_{h}$.

(H_{
c
}) The linear system (4) is approximately controllable on $[0,T]$.

Using the hypothesis (H

_{
c
}), for an arbitrary function

$x\in C([0,T];{X}_{\alpha})$, we choose the feedback control function as follows:

$\begin{array}{rcl}{u}_{\epsilon}(t,x)& =& {B}^{\ast}{P}_{\beta}^{\ast}(T-t){(\epsilon I+{\mathrm{\Gamma}}_{0}^{T})}^{-1}[{S}_{\beta}(T)({x}_{0}+h(x))\\ -{\int}_{0}^{T}{(T-s)}^{\beta -1}{P}_{\beta}(T-s)[f(s,x(s))+{\int}_{0}^{s}K(s,r)g(r,x(r))\phantom{\rule{0.2em}{0ex}}dr]\phantom{\rule{0.2em}{0ex}}ds].\end{array}$

(5)

Let

${B}_{r}=\{x\in C([0,T];{X}_{\alpha}):{\parallel x\parallel}_{\alpha}\le r\}$, where

*r* is a positive constant. Then

${B}_{r}$ is clearly a bounded closed and convex subset in

$C([0,T];{X}_{\alpha})$. We will show that when using the above control the operator

${\mathrm{\Lambda}}_{\epsilon}:{B}_{k}\to {B}_{k}$ defined by

$({\mathrm{\Lambda}}_{\epsilon}x)(t):=({\mathrm{\Phi}}_{\epsilon}x)(t)+({\mathrm{\Pi}}_{\epsilon}x)(t),\phantom{\rule{1em}{0ex}}t\in [0,T],$

has a fixed point in $C([0,T];{X}_{\alpha})$.

**Theorem 10** *Let the assumptions* (H

_{1})

*and* (H

_{2})

*be satisfied*.

*Then for* ${x}_{0}\in {X}_{\alpha}$,

*the fractional Cauchy problem* (1)

*with* $u={u}_{\epsilon}(t,x)$ *has at least one mild solution provided that* ${L}_{C}+\frac{{C}_{\alpha}{T}^{(1-\alpha )\beta}}{\epsilon (1-\alpha )\beta}\frac{M}{\mathrm{\Gamma}(\beta )}{L}_{B}^{2}{L}_{C}<1,$

(6)

*Proof* It is easy to see that for any $\epsilon >0$ the operator ${\mathrm{\Lambda}}_{\epsilon}$ maps $C([0,T];{X}_{\alpha})$ into itself.

Let

$x\in {B}_{r}$ and

$0\le t\le T$. Using assumption (H

_{1}) yield the following estimations,

From (6) and the assumption (H

_{2}), it follows that for any

$\epsilon >0$ there exists

$r(\epsilon )>0$ such that

Therefore, from (7) and (8), it follows that for any $\epsilon >0$ there exists $r(\epsilon )>0$ such that ${\mathrm{\Phi}}_{\epsilon}y+{\mathrm{\Pi}}_{\epsilon}x\in {B}_{r(\epsilon )}$ for every $x,y\in {B}_{r(\epsilon )}$. Therefore, for any $\epsilon >0$ the fractional Cauchy problem (1) with the control (5) has a mild solution if and only if the operator ${\mathrm{\Phi}}_{\epsilon}+{\mathrm{\Pi}}_{\epsilon}$ has a fixed point in ${B}_{r(\epsilon )}$.

In what follows, we will show that ${\mathrm{\Phi}}_{\epsilon}$ and ${\mathrm{\Pi}}_{\epsilon}$ satisfy the conditions of Krasnoselskii’s fixed-point theorem. From (H_{2}) and (6), we infer that ${\mathrm{\Phi}}_{\epsilon}$ is a contraction. Next, we show that ${\mathrm{\Pi}}_{\epsilon}$ is completely continuous on ${B}_{r(\epsilon )}$.

Step 1: We first prove that

${\mathrm{\Pi}}_{\epsilon}$ is continuous on

${B}_{r(\epsilon )}$. Let

${\{{x}_{n}\}}_{n=1}^{\mathrm{\infty}}\subset {B}_{r(\epsilon )}$ be a sequence such that

${x}_{n}\to x$ as

$n\to \mathrm{\infty}$ in

$C([0,T];{X}_{\alpha})$. Therefore, it follows from the continuity of

*f*,

*g* and

${u}_{\epsilon}$ that for each

$t\in [0,T]$,

Also, by (H

_{1}), we see that

using the Lebesgue dominated convergence theorem that for all

$t\in [0,T]$, we conclude

${\parallel ({\mathrm{\Pi}}_{\epsilon}{x}_{n})(t)-({\mathrm{\Pi}}_{\epsilon}x)(t)\parallel}_{\alpha}\to 0,\phantom{\rule{1em}{0ex}}\text{as}n\to \mathrm{\infty},$

implying that ${\parallel {\mathrm{\Pi}}_{\epsilon}{x}_{n}-{\mathrm{\Pi}}_{\epsilon}x\parallel}_{\alpha}\to 0$ as $n\to \mathrm{\infty}$. This proves that ${\mathrm{\Pi}}_{\epsilon}$ is continuous on ${B}_{r(\epsilon )}$.

Step 2. ${\mathrm{\Pi}}_{\epsilon}$ is compact on ${B}_{r(\epsilon )}$.

For the sake of brevity, we write

$N(x(s)):=f(s,x(s))+{\int}_{0}^{s}K(s,r)g(r,x(r))\phantom{\rule{0.2em}{0ex}}dr+B{u}_{\epsilon}(s,x).$

Let

$t\in [0,T]$ be fixed and

$\delta ,\eta >0$ be small enough. For

$x\in {B}_{r(\epsilon )}$, we define the map

$\begin{array}{rcl}\left({\mathrm{\Pi}}_{\epsilon}^{\delta \eta}x\right)(t)& =& {\int}_{0}^{\delta}{\int}_{\eta}^{\mathrm{\infty}}\beta r{(t-s)}^{\beta -1}{\mathrm{\Psi}}_{\beta}(r)S\left({(t-s)}^{\beta}r\right)N(x(s))\phantom{\rule{0.2em}{0ex}}dr\phantom{\rule{0.2em}{0ex}}ds\\ =& S\left({\delta}^{\beta}\eta \right){\int}_{0}^{\delta}{\int}_{\eta}^{\mathrm{\infty}}\beta r{(t-s)}^{\beta -1}{\mathrm{\Psi}}_{\beta}(r)S({(t-s)}^{\beta}r-{\delta}^{\beta}\eta )N(x(s))\phantom{\rule{0.2em}{0ex}}dr\phantom{\rule{0.2em}{0ex}}ds.\end{array}$

Therefore, from Lemma 4, we see that for each

$t\in (0,T]$, the set

$\{({\mathrm{\Pi}}_{\epsilon}^{\delta \eta}x)(t):x\in {B}_{r(\epsilon )}\}$ is relatively compact in

${X}_{\alpha}$. Since

approaches to zero as $\eta \to {0}^{+}$, using the total boundedness, we conclude that for each $t\in [0,T]$, the set $\{({\mathrm{\Pi}}_{\epsilon}^{\delta \eta}x)(t):x\in {B}_{r(\epsilon )}\}$ is relatively compact in ${X}_{\alpha}$.

On the other hand, for

$0<{t}_{1}<{t}_{2}\le T$ and

$\delta >0$ small enough, we have

${\parallel ({\mathrm{\Pi}}_{\epsilon}x)({t}_{1})-({\mathrm{\Pi}}_{\epsilon}x)({t}_{2})\parallel}_{\alpha}\le {I}_{1}+{I}_{2}+{I}_{3}+{I}_{4},$

Therefore, it follows from (H

_{1}) and Lemma 4 that

and

$\begin{array}{rcl}{I}_{4}& \le & {C}_{\alpha}{\int}_{0}^{{t}_{1}}|{({t}_{1}-s)}^{\beta -1-\alpha \beta}-{({t}_{2}-s)}^{\beta -1-\alpha \beta}|\\ \times (({\phi}_{r(\epsilon )}(s)+K{\parallel {\psi}_{r(\epsilon )}\parallel}_{{L}^{\mathrm{\infty}}})+\frac{1}{\epsilon}\frac{M}{\mathrm{\Gamma}(\beta )}{L}_{B}^{2}{L}_{u}(r(\epsilon )))\phantom{\rule{0.2em}{0ex}}ds\\ \le & {C}_{\alpha}{\left(\frac{1-\gamma}{(1-\alpha )\beta -\gamma}\right)}^{1-\gamma}{\parallel {\phi}_{r(\epsilon )}\parallel}_{{L}^{1/\gamma}}[{t}_{1}^{(1-\alpha )\beta -\gamma}-{({t}_{2}^{\frac{(1-\alpha )\beta -\gamma}{1-\gamma}}-{({t}_{2}-{t}_{1})}^{\frac{(1-\alpha )\beta -\gamma}{1-\gamma}})}^{1-\gamma}]\\ +{C}_{\alpha}\frac{2K}{(1-\alpha )\beta}{\parallel {\psi}_{r(\epsilon )}\parallel}_{{L}^{\mathrm{\infty}}}[{t}_{1}^{(1-\alpha )\beta}-{t}_{2}^{(1-\alpha )\beta}-{({t}_{2}-{t}_{1})}^{(1-\alpha )\beta}]\\ +{C}_{\alpha}\frac{2{L}_{B}^{2}{L}_{u}(r(\epsilon ))}{(1-\alpha )\beta}\frac{M}{\mathrm{\Gamma}(\beta )}{\parallel {\psi}_{r(\epsilon )}\parallel}_{{L}^{\mathrm{\infty}}}[{t}_{1}^{(1-\alpha )\beta}-{t}_{2}^{(1-\alpha )\beta}-{({t}_{2}-{t}_{1})}^{(1-\alpha )\beta}],\end{array}$

from which it is easy to see that all

${I}_{i}$,

$i=1,2,3,4$, tend to zero independent of

$x\in {B}_{k}$ as

${t}_{2}-{t}_{1}\to 0$ and

$\delta \to 0$. Thus, we can conclude that

${\parallel ({\mathrm{\Pi}}_{\epsilon}x)({t}_{1})-({\mathrm{\Pi}}_{\epsilon}x)({t}_{2})\parallel}_{\alpha}\to 0\phantom{\rule{1em}{0ex}}\text{as}{t}_{2}-{t}_{1}\to 0,$

and the limit is independent of $x\in {B}_{r(\epsilon )}$. The case ${t}_{1}=0$ is trivial. Consequently, the set $\{({\mathrm{\Pi}}_{\epsilon}x)(t):t\in [0,T],x\in {B}_{r(\epsilon )}\}$ is equicontinuous. Now applying the Arzela-Ascoli theorem, it results that ${\mathrm{\Pi}}_{\epsilon}$ is compact on ${B}_{r(\epsilon )}$.

Therefore, applying Krasnoselskii’s fixed-point theorem, we conclude that ${\mathrm{\Lambda}}_{\epsilon}$ has a fixed point, which gives rise to a mild solution of Cauchy problem (1) with control given in (5). This completes the proof. □

**Theorem 11** *Let the assumptions* (H_{1}), (H_{2}) *and* (H_{
c
}) *be satisfied*. *Moreover*, *assume the functions* $f,g:[0,T]\times {X}_{\alpha}\times {X}_{\alpha}\to X$ *and* $h:C([0,T];{X}_{\alpha})\to {X}_{\alpha}$ *are bounded and* $M{L}_{h}<1$. *Then the semilinear fractional system* (3) *is approximately controllable on* $[0,T]$.

*Proof* It is clear that all assumptions of Theorem 10 are satisfied with

${\sigma}_{1}={\sigma}_{2}=0$. Let

${x}_{\epsilon}$ be a fixed point of

${F}_{\epsilon}$ in

${B}_{r}$. Any fixed point of

${F}_{\epsilon}$ is a mild solution of (3) under the control

$\begin{array}{rcl}{u}_{\epsilon}(t,{x}_{\epsilon})& =& {B}^{\ast}{P}_{\beta}^{\ast}(T-t)R(\epsilon ,{\mathrm{\Gamma}}_{0}^{T})(h-{S}_{\beta}(T)({x}_{0}+h({x}_{\epsilon}))\\ -{\int}_{0}^{T}{(T-s)}^{\beta -1}{P}_{\beta}(T-s)[f(s,{x}_{\epsilon}(s))+{\int}_{0}^{s}K(s-\tau )g(\tau ,{x}_{\epsilon}(\tau ))\phantom{\rule{0.2em}{0ex}}d\tau ]\phantom{\rule{0.2em}{0ex}}ds)\end{array}$

and satisfies the equality

${x}_{\epsilon}(T)=h-\epsilon R(\epsilon ,{\mathrm{\Gamma}}_{0}^{T})p({x}_{\epsilon}),$

(9)

where

$\begin{array}{rcl}p({x}_{\epsilon})& =& (h-{S}_{\beta}(T)({x}_{0}+h({x}_{\epsilon}))\\ -{\int}_{0}^{T}{(T-s)}^{\beta -1}{P}_{\beta}(T-s)[f(s,{x}_{\epsilon}(s))+{\int}_{0}^{s}K(s-\tau )g(\tau ,{x}_{\epsilon}(\tau ))\phantom{\rule{0.2em}{0ex}}d\tau ]\phantom{\rule{0.2em}{0ex}}ds).\end{array}$

Moreover, by the boundedness of the functions

*f* and

*g* and Dunford-Pettis theorem, we have that the sequences

$\{f(s,{x}_{\epsilon}(s))\}$ and

$\{g(s,{x}_{\epsilon}(s))\}$ are weakly compact in

${L}^{2}([0,T];X)$, so there are subsequences still denoted by

$\{f(s,{x}_{\epsilon}(s))\}$ and

$\{g(s,{x}_{\epsilon}(s))\}$, that weakly converge to, say,

*f* and

*g* in

${L}^{2}([0,T];X)$. On the other hand, there exists

$\tilde{h}\in {X}_{\alpha}$ such that

$h({x}_{\epsilon})$ converges to

$\tilde{h}$ weakly in

${X}_{\alpha}$. Denote

$w=h-{S}_{\beta}({x}_{0}+\tilde{h}))-{\int}_{0}^{T}{(T-s)}^{\beta -1}{P}_{\beta}(T-s)[f(s)+{\int}_{0}^{s}K(s-\tau )g(\tau )\phantom{\rule{0.2em}{0ex}}d\tau ]\phantom{\rule{0.2em}{0ex}}ds.$

It follows that

$\begin{array}{rcl}{\parallel p({x}_{\epsilon})-w\parallel}_{\alpha}& \le & {\parallel {S}_{\beta}(T)h({x}_{\epsilon})-{S}_{\beta}(T)\tilde{h}\parallel}_{\alpha}\\ +{\parallel {\int}_{0}^{T}{(T-s)}^{\beta -1}{P}_{\beta}(T-s)(f(s,{x}_{\epsilon}(s))-f(s))\phantom{\rule{0.2em}{0ex}}ds\parallel}_{\alpha}\\ +{\parallel {\int}_{0}^{T}{(T-s)}^{\beta -1}{P}_{\beta}(T-s){\int}_{0}^{s}K(s-\tau )(g(\tau ,{x}_{\epsilon}(\tau )-g(\tau ))\phantom{\rule{0.2em}{0ex}}d\tau \phantom{\rule{0.2em}{0ex}}ds\parallel}_{\alpha}\to 0\end{array}$

as

$\epsilon \to {0}^{+}$ because of compactness of the operator

$l(\cdot )\to {\int}_{0}^{\cdot}{(\cdot -s)}^{\beta -1}{P}_{\beta}(\cdot -s)l(s)\phantom{\rule{0.2em}{0ex}}ds:{L}_{2}([0,T],X)\to C([0,T],{X}_{\alpha}).$

Then from (9), we obtain

$\begin{array}{rcl}{\parallel {x}_{\epsilon}(T)-h\parallel}_{\alpha}& \le & {\parallel \epsilon R(\epsilon ,{\mathrm{\Gamma}}_{0}^{T})(w)\parallel}_{\alpha}+\parallel \epsilon R(\epsilon ,{\mathrm{\Gamma}}_{0}^{T})\parallel {\parallel p({x}_{\epsilon})-w\parallel}_{\alpha}\\ \le & {\parallel \epsilon R(\epsilon ,{\mathrm{\Gamma}}_{0}^{T})(w)\parallel}_{\alpha}+{\parallel p({x}_{\epsilon})-w\parallel}_{\alpha}\to 0\end{array}$

(10)

as $\epsilon \to {0}^{+}$. This proves the approximate controllability of (1). □