The purpose of this section is to establish sufficient conditions for the approximate controllability of certain classes of abstract fractional integrodifferential equations of the form

$\{\begin{array}{c}{}^{c}D_{t}^{\alpha}x(t)=Ax(t)+Bu(t)+f(t,x(t),{\int}_{0}^{t}g(t,s,x(s))\phantom{\rule{0.2em}{0ex}}ds),\phantom{\rule{1em}{0ex}}t\in [0,b],\hfill \\ x(0)={x}_{0},\hfill \end{array}$

(10)

where the state variable *x* takes values in a separable reflexive Banach space *X*; ${}^{c}D^{\alpha}$ is the Caputo fractional derivative of order $\frac{1}{2}<\alpha <1$; *A* is the infinitesimal generator of a ${C}_{0}$ semigroup $S(t)$ of bounded operators on *X*; the control function *u* is given in ${L}_{2}([0,b],U)$, *U* is a Hilbert space; *B* is a bounded linear operator from *U* into *X*, $\mathrm{\Delta}=\{(t,s):0\le s\le t\le T\}$ and $g:\mathrm{\Delta}\times X\to X$, $f:I\times X\times X\to X$ are continuous bounded functions and ${x}_{0}\in X$.

**Definition 5** The fractional integral of order

*α* with the lower limit 0 for a function

*f* is defined as

${I}^{\alpha}f(t)=\frac{1}{\gamma (\alpha )}{\int}_{0}^{t}\frac{f(s)}{{(t-s)}^{1-\alpha}}\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t>0,\alpha >0,$

provided the right-hand side is pointwise defined on $[0,\mathrm{\infty})$, where *γ* is the gamma function.

**Definition 6** Riemann-Liouville derivative of order

*α* with the lower limit 0 for a function

$f:[0,\mathrm{\infty})\to R$ can be written as

${}^{L}D^{\alpha}f(t)=\frac{1}{\gamma (n-\alpha )}\frac{{d}^{n}}{d{t}^{n}}{\int}_{0}^{t}\frac{f(s)}{{(t-s)}^{\alpha +1}}\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t>0,n-1<\alpha <n.$

**Definition 7** The Caputo derivative of order

*α* for a function

$f:[0,\mathrm{\infty})\to R$ can be written as

${}^{c}D^{\alpha}f(t){=}^{L}{D}^{\alpha}(f(t)-\sum _{k=0}^{n-1}\frac{{t}^{k}}{k!}{f}^{(k)}(0)),\phantom{\rule{1em}{0ex}}t>0,n-1<\alpha <n.$

**Remark 8**
- (1)
If

$f(t)\in {C}^{n}[0,\mathrm{\infty})$, then

${}^{c}D^{\alpha}f(t)=\frac{1}{\gamma (n-\alpha )}{\int}_{0}^{t}\frac{{f}^{(n)}(s)}{{(t-s)}^{\alpha +1-n}}\phantom{\rule{0.2em}{0ex}}ds={I}^{n-\alpha}{f}^{(n)}(t),\phantom{\rule{1em}{0ex}}t>0,n-1<\alpha <n.$

- (2)
The Caputo derivative of a constant is equal to zero.

- (3)
If *f* is an abstract function with values in *X*, then the integrals which appear in the above definitions are taken in Bochner’s sense.

For basic facts about fractional integrals and fractional derivatives, one can refer to [49].

In order to define the concept of a mild solution for problem (10), we associate problem (10) to the integral equation

$\begin{array}{rcl}x(t)& =& {\stackrel{\u02c6}{S}}_{\alpha}(t){x}_{0}+{\int}_{0}^{t}{(t-s)}^{q-1}{S}_{\alpha}(t-s)f(s,x(s),{\int}_{0}^{s}g(s,r,x(r))\phantom{\rule{0.2em}{0ex}}dr)\phantom{\rule{0.2em}{0ex}}ds\\ +{\int}_{0}^{t}{(t-s)}^{q-1}{S}_{\alpha}(t-s)Bu(s)\phantom{\rule{0.2em}{0ex}}ds,\end{array}$

(11)

and ${\eta}_{\alpha}$ is a probability density function defined on $(0,\mathrm{\infty})$, that is, ${\eta}_{\alpha}(\theta )\ge 0$, $\theta \in (0,\mathrm{\infty})$ and ${\int}_{0}^{\mathrm{\infty}}{\eta}_{\alpha}(\theta )\phantom{\rule{0.2em}{0ex}}d\theta =1$.

**Lemma 9** [34]

*For any fixed* $t\ge 0$, *the operators* ${\stackrel{\u02c6}{S}}_{\alpha}(t)$ *and* ${S}_{\alpha}(t)$ *are linear compact and bounded operators*, *i*.*e*., *for any* $x\in X$, $\parallel {\stackrel{\u02c6}{S}}_{\alpha}(t)x\parallel \le M\parallel x\parallel $ *and* $\parallel {S}_{\alpha}(t)x\parallel \le \frac{M}{\mathrm{\Gamma}(\alpha )}\parallel x\parallel $.

**Definition 10** A solution $x(\cdot ;{x}_{0},u)\in C([0,b],X)$ is said to be a mild solution of (10) if for any $u\in {L}_{2}([0,b],U)$ and the integral equation (11) is satisfied.

Let ${x}_{b}({x}_{0};u)$ be the state value of (10) at terminal time *b* corresponding to the control *u* and the initial value ${x}_{0}$. Introduce the set $\mathrm{\Re}(b,{x}_{0})=\{{x}_{b}({x}_{0};u)(0):u\in {L}_{2}([0,b],U)\}$, which is called the reachable set of system (10) at terminal time *b*, its closure in *X* is denoted by $\overline{\mathrm{\Re}(b,{x}_{0})}=X$.

**Definition 11** System (10) is said to be approximately controllable on *J* if $\overline{\mathrm{\Re}(b,{x}_{0})}=X$, that is, given an arbitrary $\u03f5>0$, it is possible to steer from the point ${x}_{0}$ to within a distance *ϵ* from all points in the state space *X* at time *b*.

Consider the following linear fractional differential system:

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

(12)

The approximate controllability for linear fractional system (12) is a natural generalization of the approximate controllability of a linear first-order control system. It is convenient at this point to introduce the controllability operator associated with (12) as

${\mathrm{\Gamma}}_{0}^{b}={\int}_{0}^{b}{(b-s)}^{2(\alpha -1)}{S}_{\alpha}(b-s)B{B}^{\ast}{S}_{\alpha}^{\ast}(b-s)\phantom{\rule{0.2em}{0ex}}ds:X\to X,$

where ${B}^{\ast}$ denotes the adjoint of *B* and ${S}_{\alpha}^{\ast}$ is the adjoint of ${S}_{\alpha}$. It is straightforward that the operator ${\mathrm{\Gamma}}_{0}^{b}$ is a linear bounded operator. By Theorem 2, linear fractional control system (12) is approximately controllable on $[0,b]$ if and only if for any $h\in X$, ${z}_{\epsilon}(h)=\epsilon {(\epsilon I+{\mathrm{\Gamma}}_{0}^{b}J)}^{-1}(h)$ converges strongly to zero as $\epsilon \to {0}^{+}$.

**Proposition 12** *If* $S(t)$,

$t>0$,

*are compact operators and* $0<\frac{1}{p}<\alpha \le 1$,

*then the operator*${L}_{\alpha}f(t)={\int}_{0}^{t}{(t-s)}^{\alpha -1}{S}_{\alpha}(t-s)f(s)\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}f\in {L}^{p}([0,b],X),t\in [0,b],$

*is compact from* ${L}^{p}([0,b],X)$ *into* $C([0,b],X)$.

*Proof* According to the infinite dimensional version of the Ascoli-Arzela theorem, we need to show that

- (i)
for arbitrary $t\in [0,b]$, the set $\{{L}_{\alpha}f(t):{\parallel f\parallel}_{{L}^{p}}\le 1\}$ is relatively compact in $C([0,b],X)$;

- (ii)
for arbitrary

$\eta >0$, there exists

$\delta >0$ such that

$\parallel {L}_{\alpha}f(t)-{L}_{\alpha}f(s)\parallel <\eta \phantom{\rule{1em}{0ex}}\text{if}{\parallel f\parallel}_{{L}^{p}}\le 1,|t-s|\le \delta ,t,s\in [0,b].$

To prove (i), fix

$0<t<b$ and define for

$0<\eta <t$ and

$\delta >0$ operators

${L}_{\alpha}^{\eta ,\delta}$ from

${L}^{p}([0,b],X)$ into

*X* $\begin{array}{rcl}\left({L}_{\alpha}^{\eta ,\delta}f\right)(t)& =& \alpha {\int}_{0}^{t-\lambda}{\int}_{\delta}^{\mathrm{\infty}}\theta {(t-s)}^{\alpha -1}{\eta}_{\alpha}(\theta )S\left({(t-s)}^{\alpha}\theta \right)f(s)\phantom{\rule{0.2em}{0ex}}ds\\ =& \alpha S\left({\lambda}^{\alpha}\delta \right){\int}_{0}^{t-\lambda}{\int}_{\delta}^{\mathrm{\infty}}\theta {(t-s)}^{\alpha -1}{\eta}_{\alpha}(\theta )S({(t-s)}^{\alpha}\theta -{\lambda}^{\alpha}\delta )f(s)\phantom{\rule{0.2em}{0ex}}ds,\\ f\in {L}^{p}([0,b],X).\end{array}$

Since

$S(t)$,

$t>0$, is a compact operator, the operators

${L}_{\alpha}^{\eta ,\delta}$ are compact. Moreover, we have

$\begin{array}{rcl}\parallel ({L}_{\alpha}f)(t)-\left({L}_{\alpha}^{\eta ,\delta}f\right)(t)\parallel & \le & \alpha \parallel {\int}_{0}^{t}{\int}_{0}^{\delta}\theta {(t-s)}^{\alpha -1}{\eta}_{\alpha}(\theta )S\left({(t-s)}^{\alpha}\theta \right)f(s)\phantom{\rule{0.2em}{0ex}}d\theta \phantom{\rule{0.2em}{0ex}}ds\parallel \\ +\alpha \parallel {\int}_{t-\lambda}^{t}{\int}_{\delta}^{\mathrm{\infty}}\theta {(t-s)}^{\alpha -1}{\eta}_{\alpha}(\theta )S\left({(t-s)}^{\alpha}\theta \right)f(s)\phantom{\rule{0.2em}{0ex}}d\theta \phantom{\rule{0.2em}{0ex}}ds\parallel \\ =:& {J}_{1}+{J}_{2}.\end{array}$

One can estimate

${J}_{1}$ and

${J}_{2}$ as follows:

$\begin{array}{rl}{J}_{1}& \le \alpha M{\int}_{0}^{t}{(t-s)}^{\alpha -1}\parallel f(s)\parallel \phantom{\rule{0.2em}{0ex}}ds({\int}_{0}^{\delta}\theta {\eta}_{\alpha}(\theta )\phantom{\rule{0.2em}{0ex}}d\theta )\\ \le \alpha M{\left({\int}_{0}^{t}{(t-s)}^{(\alpha -1)q}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1/q}{\parallel f\parallel}_{{L}^{p}}({\int}_{0}^{\delta}\theta {\eta}_{\alpha}(\theta )\phantom{\rule{0.2em}{0ex}}d\theta ),\end{array}$

and

$\begin{array}{rl}{J}_{2}& \le \alpha M{\int}_{t-\lambda}^{t}{(t-s)}^{\alpha -1}\parallel f(s)\parallel \phantom{\rule{0.2em}{0ex}}ds({\int}_{\delta}^{\mathrm{\infty}}\theta {\eta}_{\alpha}(\theta )\phantom{\rule{0.2em}{0ex}}d\theta )\\ \le \frac{\alpha M}{\gamma (1+\alpha )}{\left({\int}_{0}^{t}{(t-s)}^{(\alpha -1)q}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1/q}{\left({\int}_{0}^{t}{\parallel f(s)\parallel}^{p}\phantom{\rule{0.2em}{0ex}}ds\right)}^{1/p}\\ =\frac{\alpha M}{\gamma (1+\alpha )}{\left(\frac{{\eta}^{(\alpha -1)q+1}}{(\alpha -1)q+1}\right)}^{1/q}{\parallel f\parallel}_{{L}^{p}},\end{array}$

where we have used the equality

${\int}_{0}^{\mathrm{\infty}}{\theta}^{\beta}{\eta}_{\alpha}(\theta )\phantom{\rule{0.2em}{0ex}}d\theta =\frac{\gamma (1+\beta )}{\gamma (1+\alpha \beta )}.$

Consequently, ${L}_{\alpha}^{\eta ,\delta}\to {L}_{\alpha}$ in the operator norm so that ${L}_{\alpha}$ is compact and (i) follows immediately.

To prove (ii), note that, for

$0\le t\le t+h\le b$ and

${\parallel f\parallel}_{{L}^{p}}\le 1$, we have

Applying the Hölder inequality, we obtain

It is clear that ${I}_{1},{I}_{2}\to 0$ as $h\to 0$. On the other hand, the compactness of $S(t)$, $t>0$ (and consequently ${S}_{\alpha}(t)$), implies the continuity of ${S}_{\alpha}(t)$, $t>0$, in the uniform operator topology. Then, by the Lebesque dominated convergence theorem, ${I}_{3}\to 0$ as $h\to 0$. Thus the proof of (ii), and therefore the proof of the proposition, is complete. □

**Theorem 13** *Suppose* $S(t)$, $t>0$, *is compact and* $\frac{1}{2}<\alpha \le 1$. *Then system* (10) *is approximately controllable on* $[0,b]$ *if the corresponding linear system is approximately controllable on* $[0,b]$.

*Proof* Let

$Y={L}_{2}([0,b],X)$,

$V={L}_{2}([0,b],U)$, and

${y}_{0}={S}_{\alpha}(\cdot ){x}_{0}\in Y$. Define the linear operators

*Q*,

*L*,

${L}_{1}$ and the nonlinear operator

*F* by

for $y\in Y$, $v\in V$. It is easy to see that by Proposition 12 all the conditions of Theorem 4 are satisfied and (10) is approximately controllable. This completes the proof. □