Approximate controllability of fractional integro-differential equations involving nonlocal initial conditions
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We discuss the approximate controllability of nonlinear fractional integro-differential system under the assumptions that the corresponding linear system is approximately controllable. Using the fixed-point technique, fractional calculus and methods of controllability theory, a new set of sufficient conditions for approximate controllability of fractional integro-differential 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.
KeywordsFractional Calculus Mild Solution Fractional Differential Equation Approximate Controllability Caputo Fractional Derivative
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 [1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44] 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 , Dauer and Mahmudov  and Mahmudov . Sakthivel et al.  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  derived a set of sufficient conditions for the controllability of fractional-order partial neutral functional integro-differential inclusions with infinite delay in Banach spaces. Debbouche and Baleanu  established the controllability result for a class of fractional evolution nonlocal impulsive quasi-linear delay integro-differential 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.  studied the approximate controllability of deterministic semilinear fractional differential equations in Hilbert spaces. Wang  investigated the nonlocal controllability of fractional evolution systems. Surendra Kumar and Sukavanam  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 fixed-point theorem. More recently, Sakthivel et al.  derived a new set of sufficient conditions for approximate controllability of fractional stochastic differential equations.
in , where , , stands for the Caputo fractional derivative of order β, and , , , are given functions to be specified later. Here, is the infinitesimal generator of a compact analytic semigroup of bounded linear operators , , on a real Hilbert space X. B is a linear bounded operator from a real Hilbert space U to X.
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 defined in (5). We show that the control system (1) is approximately controllable on provided that the corresponding linear system is approximately controllable. Finally, an example is given to demonstrate the applicability of our result.
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.
It follows that each is an injective continuous endomorphism of X. Hence we can define , which is a closed bijective linear operator in X. It can be shown that each has dense domain and that for . Moreover, for every and with , where , I is the identity in X. (For proofs of these facts, we refer to the literature [15, 20, 22].)
We denote by the Hilbert space of equipped with norm for , which is equivalent to the graph norm of . Then we have , for (with ) and the embedding is continuous. Moreover, has the following basic properties.
Lemma 1 
for each and .
for each and .
- (iii)For every , is bounded in X and there exists such that
is a bounded linear operator for .
provided the right-hand side is pointwise defined on , where Γ is the gamma function.
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.
Here, is a probability density function defined on , that is , and .
- (i)For any fixed , and any , we have the operators and are linear and bounded operators, i.e. for any ,
The operators and are strongly continuous for all .
and are norm continuous in X for .
and are compact operators in X for .
For every , the restriction of to and the restriction of to are norm continuous.
For every , the restriction of to and the restriction of to are compact operators in .
- (vii)For all and ,
In this paper, we adopt the following definition of mild solution of equation (1).
It is clear that is bounded if . In what follows, we assume that .
3 Approximate controllability
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.
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 
is positive, that is, for all nonzero .
For all strongly converges to zero as . Here, J is the duality mapping of Z into .
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.
(H2) is a Lipschitz function with Lipschitz constant .
(H c ) The linear system (4) is approximately controllable on .
has a fixed point in .
Proof It is easy to see that for any the operator maps into itself.
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 .
implying that as . This proves that is continuous on .
Step 2. is compact on .
approaches to zero as , using the total boundedness, we conclude that for each , the set is relatively compact in .
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 c ) be satisfied. Moreover, assume the functions and are bounded and . Then the semilinear fractional system (3) is approximately controllable on .
as . This proves the approximate controllability of (1). □
where , , .
Clearly, the assumption (H1) is satisfied. On the other hand, it can be easily seen that the deterministic linear system corresponding to (11) is approximately controllable on ; see .
where and .
The functions , are continuous and uniformly bounded.
- 3.is continuously differentiable, and
It follows that is bounded and Lipschitz continuous. On the other hand, it is not difficult to verify that are continuous.
By Remark 9, the linear system corresponding to (11) is approximately controllable on if and only if , implies that . This follows from the representation of .
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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