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 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 be the state value of (1) at terminal time T corresponding to the control u and the initial value . Introduce the set , which is called the reachable set of system (1) at terminal time T, its closure in is denoted by .
Definition 6 The system (1) is said to be approximately controllable on 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 at time T.
Consider the following linear fractional differential system:
The approximate controllability for linear fractional system (4) is a natural generalization of approximate controllability of linear first order control system [9
]. It is convenient at this point to introduce the controllability and resolvent operators associated with (4) as
respectively, where denotes the adjoint of B and is the adjoint of . It is straightforward that the operator is a linear bounded operator.
Theorem 7 
Let Z be a separable reflexive Banach space and let stands for its dual space
. Assume that is symmetric
. Then the following two conditions are equivalent
Lemma 8 The linear fractional control system (4) is approximately controllable on if and only if as in the strong operator topology.
Proof The lemma is a straightforward consequence of Theorem 7. Indeed, the system (4) is approximately controllable on if and only if for all nonzero , see . By Theorem 7, as for all . □
Remark 9 Notice that positivity of is equivalent to . In other words, since , approximate controllability of the linear system (4) is equivalent to , .
Before proving the main results, let us first introduce our basic assumptions.
are continuous and for each
, there exists a constant
(H2) is a Lipschitz function with Lipschitz constant .
) The linear system (4) is approximately controllable on .
Using the hypothesis (H
), for an arbitrary function
, we choose the feedback control function as follows:
, where r
is a positive constant. Then
is clearly a bounded closed and convex subset in
. We will show that when using the above control the operator
has a fixed point in .
Theorem 10 Let the assumptions
) be satisfied
. Then for
, the fractional Cauchy problem
(1) with has at least one mild solution provided that
Proof It is easy to see that for any the operator maps into itself.
. Using assumption (H1
) yield the following estimations,
From (6) and the assumption (H2
), it follows that for any
Therefore, from (7) and (8), it follows that for any there exists such that for every . Therefore, for any the fractional Cauchy problem (1) with the control (5) has a mild solution if and only if the operator has a fixed point in .
In what follows, we will show that and satisfy the conditions of Krasnoselskii’s fixed-point theorem. From (H2) and (6), we infer that is a contraction. Next, we show that is completely continuous on .
Step 1: We first prove that
is continuous on
be a sequence such that
. Therefore, it follows from the continuity of f
that for each
Also, by (H1
), we see that
using the Lebesgue dominated convergence theorem that for all
, we conclude
implying that as . This proves that is continuous on .
Step 2. is compact on .
For the sake of brevity, we write
be fixed and
be small enough. For
, we define the map
Therefore, from Lemma 4, we see that for each
, the set
is relatively compact in
approaches to zero as , using the total boundedness, we conclude that for each , the set is relatively compact in .
On the other hand, for
small enough, we have
Therefore, it follows from (H1
) and Lemma 4 that
from which it is easy to see that all
, tend to zero independent of
. Thus, we can conclude that
and the limit is independent of . The case is trivial. Consequently, the set is equicontinuous. Now applying the Arzela-Ascoli theorem, it results that is compact on .
Therefore, applying Krasnoselskii’s fixed-point theorem, we conclude that 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 (H1), (H2) and (H
) be satisfied. Moreover, assume the functions and are bounded and . Then the semilinear fractional system (3) is approximately controllable on .
It is clear that all assumptions of Theorem 10 are satisfied with
be a fixed point of
. Any fixed point of
is a mild solution of (3) under the control
and satisfies the equality
Moreover, by the boundedness of the functions f
and Dunford-Pettis theorem, we have that the sequences
are weakly compact in
, so there are subsequences still denoted by
, that weakly converge to, say, f
. On the other hand, there exists
It follows that
because of compactness of the operator
Then from (9), we obtain
as . This proves the approximate controllability of (1). □