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.

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. [

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 [

Motivated by the above-mentioned papers, we study the approximate controllability of a class of fractional evolution equations of Sobolev type:

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

Throughout this paper, unless otherwise specified, the following notations will be used. Let

The operators

The hypotheses (S1)–(S3) and the closed graph theorem imply the boundedness of the linear operator

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

The fractional integral of order

The Caputo derivative of order

If

For

The operators

For any fixed

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

A solution

Let

System (

To investigate the approximate controllability of system (

The function

there is a positive integrable function

The following relationship holds:

For every

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 (

Assume that assumptions (S1)–(S3), (H4), (H5) hold and

The proof of Theorem

For all

It will be shown that for all

Under assumptions (S1)–(S3), (H4), (H5), for any

Let

Let assumptions (S1)–(S3), (H4), (H5) hold. Then the set

Let

Let assumptions (S1)–(S3), (H4), (H5) hold. Then

Let

Consider the following linear fractional differential system:

The following three conditions are equivalent.

For all

For all

It is known that Theorem

Let

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

Let

Then system (

Let

Theorem

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 [

Let

Define

Next, we suppose

(H6)

Define