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
where the state variable x takes values in a separable reflexive Banach space X; is the Caputo fractional derivative of order ; A is the infinitesimal generator of a semigroup of bounded operators on X; the control function u is given in , U is a Hilbert space; B is a bounded linear operator from U into X, and , are continuous bounded functions and .
Definition 5 The fractional integral of order α with the lower limit 0 for a function f is defined as
provided the right-hand side is pointwise defined on , where γ is the gamma function.
Definition 6 Riemann-Liouville derivative of order α with the lower limit 0 for a function can be written as
Definition 7 The Caputo derivative of order α for a function can be written as
If , then
The Caputo derivative of a constant is equal to zero.
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 .
In order to define the concept of a mild solution for problem (10), we associate problem (10) to the integral equation
and is a probability density function defined on , that is, , and .
Lemma 9 
For any fixed , the operators and are linear compact and bounded operators, i.e., for any , and .
Definition 10 A solution is said to be a mild solution of (10) if for any and the integral equation (11) is satisfied.
Let be the state value of (10) at terminal time b corresponding to the control u and the initial value . Introduce the set , which is called the reachable set of system (10) at terminal time b, its closure in X is denoted by .
Definition 11 System (10) is said to be approximately controllable on J if , that is, given an arbitrary , it is possible to steer from the point to within a distance ϵ from all points in the state space X at time b.
Consider the following linear fractional differential system:
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
where denotes the adjoint of B and is the adjoint of . It is straightforward that the operator is a linear bounded operator. By Theorem 2, linear fractional control system (12) is approximately controllable on if and only if for any , converges strongly to zero as .
Proposition 12 If , , are compact operators and , then the operator
is compact from into .
Proof According to the infinite dimensional version of the Ascoli-Arzela theorem, we need to show that
for arbitrary , the set is relatively compact in ;
for arbitrary , there exists such that
To prove (i), fix and define for and operators from into X
Since , , is a compact operator, the operators are compact. Moreover, we have
One can estimate and as follows:
where we have used the equality
Consequently, in the operator norm so that is compact and (i) follows immediately.
To prove (ii), note that, for and , we have
Applying the Hölder inequality, we obtain
It is clear that as . On the other hand, the compactness of , (and consequently ), implies the continuity of , , in the uniform operator topology. Then, by the Lebesque dominated convergence theorem, as . Thus the proof of (ii), and therefore the proof of the proposition, is complete. □
Theorem 13 Suppose , , is compact and . Then system (10) is approximately controllable on if the corresponding linear system is approximately controllable on .
Proof Let , , and . Define the linear operators Q, L, and the nonlinear operator F by
for , . 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. □