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We examine the controllability problem for a class of neutral fractional integrodifferential equations with impulses and infinite delay. More precisely, a set of sufficient conditions are derived for the exact controllability of nonlinear neutral impulsive fractional functional equation with infinite delay. Further, as a corollary, approximate controllability result is discussed by assuming compactness conditions on solution operator. The results are established by using solution operator, fractional calculations, and fixed point techniques. In particular, the controllability of nonlinear fractional control systems is established under the assumption that the corresponding linear control system is controllable. Finally, an example is given to illustrate the obtained theory.

Control theory is an area of application-oriented mathematics which deals with the analysis and design of control systems. In particular, the concept of controllability plays an important role in various areas of science and engineering. More precisely, the problem of controllability deals with the existence of a control function, which steers the solution of the system from its initial state to a final state, where the initial and final states may vary over the entire space. Control problems for various types of deterministic and stochastic dynamical systems in infinite dimensional systems have been studied in [

On the other hand, the impulsive differential systems can be used to model processes which are subject to abrupt changes, and which cannot be described by the classical differential systems [

Recently, many authors pay their attention to study the controllability of fractional evolution systems [

However, controllability of impulsive fractional integrodifferential equations with infinite delay has not been studied via the theory of solution operator. Motivated by this consideration, in this paper, we investigate the exact controllability of a class of fractional order neutral integrodifferential equations with impulses and infinite delay in the following form:

In this section, we will recall some basic definitions and lemmas which will be used in this paper. Let

Now, we present the abstract space

We assume that the phase space

If

For the function

The space

The Caputo derivative of order

The Mittag-Leffler type function in two arguments is defined by the series expansion

A closed and linear operator

Let

Consider the space

If the functions

Let

The fractional control system (

Assume that the linear fractional differential control system

If the linear fractional system (

In order to define the concept of mild solution for the control problem (

A function

In this section, we formulate and prove a set of sufficient conditions for the exact controllability of impulsive neutral fractional control differential system (

There exists a constant

The function

There exist constants

The linear fractional system (

Assume that the hypotheses (H1)–(H5) are satisfied, then the fractional impulsive system (

For an arbitrary function

Define the function

Let

However, the concept of exact controllability is very limited for many dynamic control systems, and the approximate controllability is more appropriate for these control systems instead of exact controllability. Taking this into account, in this paper, we will also discuss the approximate controllability result of the nonlinear impulsive fractional control system (

The fractional control system (

Assume that the linear fractional control system

Assume that conditions (H1)–(H4) hold and that the family

For each

Let

Now, we present an example to illustrate the abstract results of this paper which do not aim at generality but indicate how our theorem can be applied to concrete problems. Let

Note that the subordination principle of solution operator implies that

Consider the following fractional partial integrodifferential equation with infinite delay and control in the following form:

Let

The work of Yong Ren is supported by the National Natural Science Foundation of China (no. 11371029).