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)}{{(ts)}^{1\alpha}}\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t>0,\alpha >0,
provided the righthand side is pointwise defined on [0,\mathrm{\infty}), where γ is the gamma function.
Definition 6 RiemannLiouville 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)}{{(ts)}^{\alpha +1}}\phantom{\rule{0.2em}{0ex}}ds,\phantom{\rule{1em}{0ex}}t>0,n1<\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}^{n1}\frac{{t}^{k}}{k!}{f}^{(k)}(0)),\phantom{\rule{1em}{0ex}}t>0,n1<\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)}{{(ts)}^{\alpha +1n}}\phantom{\rule{0.2em}{0ex}}ds={I}^{n\alpha}{f}^{(n)}(t),\phantom{\rule{1em}{0ex}}t>0,n1<\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}{(ts)}^{q1}{S}_{\alpha}(ts)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}{(ts)}^{q1}{S}_{\alpha}(ts)Bu(s)\phantom{\rule{0.2em}{0ex}}ds,\end{array}
(11)
where
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 firstorder control system. It is convenient at this point to introduce the controllability operator associated with (12) as
{\mathrm{\Gamma}}_{0}^{b}={\int}_{0}^{b}{(bs)}^{2(\alpha 1)}{S}_{\alpha}(bs)B{B}^{\ast}{S}_{\alpha}^{\ast}(bs)\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}{(ts)}^{\alpha 1}{S}_{\alpha}(ts)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 AscoliArzela 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,ts\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 {(ts)}^{\alpha 1}{\eta}_{\alpha}(\theta )S\left({(ts)}^{\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 {(ts)}^{\alpha 1}{\eta}_{\alpha}(\theta )S({(ts)}^{\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 {(ts)}^{\alpha 1}{\eta}_{\alpha}(\theta )S\left({(ts)}^{\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 {(ts)}^{\alpha 1}{\eta}_{\alpha}(\theta )S\left({(ts)}^{\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}{(ts)}^{\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}{(ts)}^{(\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}{(ts)}^{\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}{(ts)}^{(\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. □