We discuss the approximate controllability of semilinear fractional Sobolev-type differential system under the assumption that the corresponding linear system is approximately controllable. Using Schauder fixed point theorem, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional Sobolev-type differential equations, are formulated and proved. We show that our result has no analogue for the concept of complete controllability. The results of the paper are generalization and continuation of the recent results on this issue.

1. Introduction

Many social, physical, biological, and engineering problems can be described by fractional partial differential equations. In fact, fractional differential equations are considered as an alternative model to nonlinear differential equations. In the last two decades, fractional differential equations (see Samko et al. [1] and the references therein) have attracted many scientists, and notable contributions have been made to both theory and applications of fractional differential equations.

Recently, the existence of mild solutions and stability and (approximate) controllability of (fractional) semilinear evolution system in Banach spaces have been reported by many researchers; see [236]. We refer the reader to El-Borai [3, 4], Balachandran and Park [5], Zhou and Jiao [6, 7] Hernández et al. [8], Wang and Zhou [9], Sakthivel et al. [12, 13], Debbouche and Baleanu [14], Wang et al. [1521], Kumar and Sukavanam [22], Li and Yong [37], Dauer and Mahmudov [28], Mahmudov [27, 29], and the references therein. Complete controllability of evolution systems of Sobolev type in Banach spaces has been studied by Balachandran and Dauer [23], Ahmed [24], and Feckan et al. [2]. However, the approximate controllability of fractional evolution equations of Sobolev type has not been studied.

Motivated by the above-mentioned papers, we study the approximate controllability of a class of fractional evolution equations of Sobolev type: where and are linear operators from a Banach space to . The control function takes values in a Hilbert space and . is a linear bounded operator. The function will be specified in the sequel. The fractional derivative , , is understood in the Caputo sense.

Our aim in this paper is to provide a sufficient condition for the approximate controllability for a class of fractional evolution equations of Sobolev type. It is assumed that is compact, and, consequently, the associated linear control system (35) is not exactly controllable. Therefore, our approximate controllability results have no analogue for the concept of complete controllability. In Section 5, we give an example of the system which is not completely controllable, but approximately controllable.

2. Preliminaries

Throughout this paper, unless otherwise specified, the following notations will be used. Let be a separable reflexive Banach space and let stand for its dual space with respect to the continuous pairing . We may assume, without loss of generality, that and are smooth and strictly convex, by virtue of a renorming theorem (see, e.g., [37, 38]). In particular, this implies that the duality mapping of into given by the relations is bijective, homogeneous, demicontinuous, that is, continuous from with a strong topology, into with weak topology, and strictly monotonic. Moreover, is also duality mapping.

The operators and satisfy the following hypotheses:(S1) and are linear operators, and is closed;(S2) and is bijective;(S3) is compact.

The hypotheses (S1)–(S3) and the closed graph theorem imply the boundedness of the linear operator . Consequently, generates a semigroup in . Assume that .

Let us recall the following known definitions in fractional calculus. For more details, see [1].

Definition 1. The fractional integral of order with the lower limit for a function is defined as provided the right-hand side is pointwise defined on , where is the gamma function.

Definition 2. The Caputo derivative of order for a function can be written as
If is an abstract function with values in , then integrals which appear in the above definitions are taken in Bochner’s sense.
For and , we define two families and of operators by where is the function of Wright type defined on , which satisfies

Lemma 3 (see [2]). The operators and have the following properties. (i)For any fixed , and are linear and bounded operators, and (ii) and are compact.

In this paper, we adopt the following definition of mild solution of (1).

Definition 4. A solution is said to be a mild solution of (1) if for any , the integral equation is satisfied.

Let be the state value of (9) at terminal time corresponding to the control . Introduce the set , which is called the reachable set of system (9) at terminal time ; its closure in is denoted by .

Definition 5. 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 from all points in the state space at time .

To investigate the approximate controllability of system (9), we assume the following conditions.(H4)The function satisfies the following:(a) is continuous for each and for each ,   is strongly measurable;(b)there is a positive integrable function and a continuous nondecreasing function such that for every , we have (H5) The following relationship holds: here .(H6) For every converges to zero as in strong topology, where and is a solution of

3. Existence Theorem

In order to formulate the controllability problem in the form suitable for application of fixed point theorem, it is assumed that the corresponding linear system is approximately controllable. Then it will be shown that system (1) is approximately controllable if for all , there exists a continuous function such that where Having noticed this fact, our goal, in this section, is to find conditions for the solvability of (14). It will be shown that the control in (14) drives system (1) from to provided that system (14) has a solution.

Theorem 6. Assume that assumptions (S1)–(S3), (H4), (H5) hold and . Then there exists a solution to (14).

Proof. The proof of Theorem 6 follows from Lemmas 79, infinite dimensional analogue of Arzela-Ascoli theorem, and the Schauder fixed point theorem.

For all , consider the operator defined as follows: where

It will be shown that for all , the operator has a fixed point. To prove this we will employ the Schauder fixed point theorem.

Lemma 7. Under assumptions (S1)–(S3), (H4), (H5), for any there exists a positive number such that .

Proof. Let be fixed. If it is not true, then for each , there exists a function , but . So for some , one can show that Let us estimate . By the assumption (H4), we have Combining the estimates (19)–(21) yields On the other hand, Thus, Dividing both sides by and taking , we obtain that which is a contradiction by assumption (H5). Thus, for some .

Lemma 8. Let assumptions (S1)–(S3), (H4), (H5) hold. Then the set is an equicontinuous family of functions on .

Proof. Let and such that for every with . For , , we have Applying Lemma 3 and the Holder inequality, we obtain Therefore, for sufficiently small, the right-hand side of (28) tends to zero as . On the other hand, the compactness of , implies the continuity in the uniform operator topology. Thus, the set is equicontinuous.

Lemma 9. Let assumptions (S1)–(S3), (H4), (H5) hold. Then maps onto a precompact set in .

Proof. Let be fixed and let be a real number satisfying . For , define an operator on by Since is a compact operator, the set is precompact in for every , . Moreover, for each , we have A similar argument, as before, is as follows: where we have used the equality From (30)–(32), one can see that for each , Therefore, there are relatively compact sets arbitrary close to the set ; hence, the set is also precompact in .

4. Main Results

Consider the following linear fractional differential system: It is convenient at this point to introduce the controllability and resolvent operators associated with (35) as respectively, where denotes the adjoint of and is the adjoint of . It is straightforward that the operator is a linear bounded operator for .

Theorem 10 (see [27]). The following three conditions are equivalent.(i) is positive; that is, for all nonzero .(ii)For all converges to zero as in the weak topology, where is a solution of (13). (iii)For all converges to zero as in the strong topology.

Remark 11. It is known that Theorem 10 () holds if and only if . In other words, Theorem 10 () holds if and only if the corresponding linear system is approximately controllable on . Consequently, assumption (H6) is equivalent to the approximate controllability of the linear system (35).

Theorem 12 (see [27]). Let   be a nonlinear operator. Assume that is a solution of the following equation: Then there exists a subsequence of the sequence strongly converging to zero as .

We are now in a position to state and prove the main result of the paper.

Theorem 13. Let . Suppose that conditions (S1)–(S3), (H4)–(H5) are satisfied. Besides, assume additionally that there exists such that
Then system (1) is approximately controllable on .

Proof. Let be a fixed point of in . Then is a mild solution of (1) on under the control and satisfies the following equality: In other words, is a solution of By our assumption, Consequently, the sequence is bounded. Then there is a subsequence still denoted by and weakly converges to, say, in . Then where as because of the compactness of an operator   →    →  . Then by Theorem 12 for any , as . This gives the approximate controllability. The theorem is proved.

Remark 14. Theorem 13 assumes that the operator is compact and, consequently, the associated linear control system (35) is not exactly controllable. Therefore, Theorem 13 has no analogue for the concept of exact controllability.

Remark 15. In order to describe various real-world problems in physical and engineering sciences subject to abrupt changes at certain instants during the evolution process, fractional impulsive differential equations have been used for the system model. Our result can be extended to study the complete and approximate controllability of nonlinear fractional impulsive differential equations of Sobolev type; see [35, 36].

5. Applications

Example 16. Let . Consider the following fractional partial differential equation with control:
Define by and by , where each domain, , is given by and can be written as follows: respectively, where , , is the orthonormal set of eigenvalues of . Moreover, for any , we have It is clear that is compact. The linear system corresponding to (47) is completely controllable if and only if there exists such that for all . Assume Then and no such exists which satisfies (51), and hence the linear system corresponding to (47) is never completely controllable. We show that the associated linear system is approximately controllable on . We need to show that . Indeed,
Next, we suppose
(H6) . For each , is measurable and for each , is continuous. Moreover, , for a.e. .
Define by . Now, system (47) can be written in the abstract form (1). Clearly, all the assumptions in Theorem 13 are satisfied if (H6) holds. Then system (47) is approximately controllable on .