Abstract
We discuss the approximate controllability of nonlinear fractional integrodifferential system under the assumptions that the corresponding linear system is approximately controllable. Using the fixedpoint technique, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional integrodifferential equations are formulated and proved. The results in this paper are generalization and continuation of the recent results on this issue. An example is provided to show the application of our result.
1 Introduction
Controllability is one of the fundamental concepts in mathematical control theory, which plays an important role in control systems. The controllability of nonlinear systems represented by evolution equations or inclusions in abstract spaces and qualitative theory of fractional differential equations has been extensively studied by several authors. An extensive list of these publications can be found in [144] and the references therein. Recently, the approximate controllability for various kinds of (fractional) differential equations has generated considerable interest. A pioneering work on the approximate controllability of deterministic and stochastic systems has been reported by Bashirov and Mahmudov [5], Dauer and Mahmudov [8] and Mahmudov [10]. Sakthivel et al.[28] studied the approximate controllability of nonlinear deterministic and stochastic evolution systems with unbounded delay in abstract spaces. On the other hand, the fractional differential equation has gained more attention due to its demonstrated applications in numerous seemingly diverse and widespread fields of science and engineering. Yan [45] derived a set of sufficient conditions for the controllability of fractionalorder partial neutral functional integrodifferential inclusions with infinite delay in Banach spaces. Debbouche and Baleanu [1] established the controllability result for a class of fractional evolution nonlocal impulsive quasilinear delay integrodifferential systems in a Banach space using the theory of fractional calculus and fixed point technique. However, there exists only a limited number of papers on the approximate controllability of the fractional nonlinear evolution systems. Sakthivel et al.[28] studied the approximate controllability of deterministic semilinear fractional differential equations in Hilbert spaces. Wang [40] investigated the nonlocal controllability of fractional evolution systems. Surendra Kumar and Sukavanam [33] obtained a new set of sufficient conditions for the approximate controllability of a class of semilinear delay control systems of fractional order using the contraction principle and the Schauder fixedpoint theorem. More recently, Sakthivel et al.[27] derived a new set of sufficient conditions for approximate controllability of fractional stochastic differential equations.
In this paper, we discuss the approximate controllability of nonlinear fractional integrodifferential system under the assumption that the corresponding linear system is approximately controllable. We consider the following fractional integrodifferential control system involving nonlocal conditions,
in
The rest of this paper is organized as follows. In Section 2, we give some preliminary
results on the fractional powers of the generator of an analytic compact semigroup
and introduce the mild solution of system (1). In Section 3, we study the existence
of mild solutions for system (1) under the feedback control
2 Preliminaries
In this section, we introduce some facts about the fractional powers of the generator of a compact analytic semigroup, the Caputo fractional derivative that are used throughout this paper.
We assume that X is a Hilbert space with norm
It follows that each
We denote by
Lemma 1[42]
(i)
(ii)
(iii) For every
(iv)
Let us recall the following known definitions of fractional calculus. For more details, see [43,44].
Definition 2 The fractional integral of order
provided the righthand side is pointwise defined on
Definition 3 The Caputo derivative of order
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 Definitions 2 and 3 are taken in Bochner’s sense.
According to Definitions 2 and 3, it is suitable to rewrite the problem (1) in the equivalent integral equation
provided that the integral in (2) exists. Applying the Laplace transform
to (2) and using the method similar to that used in [38] we get
where
Here,
For
respectively.
The following lemma follows from the results given in [3739].
Lemma 4The operators
(i) For any fixed
(ii) The operators
(iii)
(iv)
(v) For every
(vi) For every
(vii) For all
In this paper, we adopt the following definition of mild solution of equation (1).
Definition 5 A function
is satisfied.
It is clear that
3 Approximate controllability
In this section, we state and prove conditions for the approximate controllability
of semilinear fractional control integrodifferential systems. To do this, we first
prove the existence of a fixed point of the operator
Let
Definition 6 The system (1) is said to be approximately controllable on
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 [9,10,12]. It is convenient at this point to introduce the controllability and resolvent operators associated with (4) as
respectively, where
Theorem 7[10]
LetZbe a separable reflexive Banach space and let
1.
2. For all
Lemma 8The linear fractional control system (4) is approximately controllable on
Proof The lemma is a straightforward consequence of Theorem 7. Indeed, the system (4) is
approximately controllable on
Remark 9 Notice that positivity of
Before proving the main results, let us first introduce our basic assumptions.
(H_{1})
(H_{2})
(H_{c}) The linear system (4) is approximately controllable on
Using the hypothesis (H_{c}), for an arbitrary function
Let
where
has a fixed point in
Theorem 10Let the assumptions (H_{1}) and (H_{2}) be satisfied. Then for
where
Proof It is easy to see that for any
Let
and
From (6) and the assumption (H_{2}), it follows that for any
Therefore, from (7) and (8), it follows that for any
In what follows, we will show that
Step 1: We first prove that
Also, by (H_{1}), we see that
Since
using the Lebesgue dominated convergence theorem that for all
implying that
Step 2.
For the sake of brevity, we write
Let
Therefore, from Lemma 4, we see that for each
approaches to zero as
On the other hand, for
where
Therefore, it follows from (H_{1}) and Lemma 4 that
and
from which it is easy to see that all
and the limit is independent of
Therefore, applying Krasnoselskii’s fixedpoint theorem, we conclude that
Theorem 11Let the assumptions (H_{1}), (H_{2}) and (H_{c}) be satisfied. Moreover, assume the functions
Proof It is clear that all assumptions of Theorem 10 are satisfied with
and satisfies the equality
where
Moreover, by the boundedness of the functions f and g and DunfordPettis theorem, we have that the sequences
It follows that
as
Then from (9), we obtain
as
4 Applications
Example 1 As an application to Theorem 11, we study the following simple example. Consider a control system governed by the fractional partial differential equation of the form
where
Let us take
where
Clearly, the assumption (H_{1}) is satisfied. On the other hand, it can be easily seen that the deterministic linear
system corresponding to (11) is approximately controllable on
The operator
where
Let
1. The functions
2.
3.
Denote by
Therefore,
Define
Then, for each
and
It follows that
Next, we show that the linear system corresponding to (11) is approximately controllable
on
By Remark 9, the linear system corresponding to (11) is approximately controllable
on
Now, we note that the problem (11) can be reformulated as the abstract problem. Thus,
by Theorem 11, the system (11) is approximately controllable on
Competing interests
The authors declare that they have no competing interests.
Authors’ contributions
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
Acknowledgements
Dedicated to Professor Hari M Srivastava.
The authors would like to thank the reviewers for their valuable comments and helpful suggestions that improved the note’s quality.
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