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.
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 [1-44] 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 this paper, we discuss the approximate controllability of nonlinear fractional integro-differential system under the assumption that the corresponding linear system is approximately controllable. We consider the following fractional integro-differential control system involving nonlocal conditions,
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.
We assume that X is a Hilbert space with norm . Let be the Banach space of continuous functions from into X with the norm , here . In this paper, we also assume that is the infinitesimal generator of a compact analytic semigroup , , of uniformly bounded linear operator in X, that is, there exists such that for all . Without loss of generality, let , where is the resolvent set of A. Then for any , we can define by
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.
and have the following properties.
(i) for each and .
(ii) for each and .
(iii) For every , is bounded inXand there exists such that
(iv) is a bounded linear operator for .
Definition 2 The fractional integral of order with the lower limit 0 for a function f is defined as
provided the right-hand side is pointwise defined on , where Γ is the gamma function.
Definition 3 The Caputo derivative of order with the lower limit 0 for a function f can be written as
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  we get
Here, is a probability density function defined on , that is , and .
For , we define two families and of operators by
Lemma 4The operators and have the following properties.
(i) For any fixed , and any , we have the operators and are linear and bounded operators, i.e. for any ,
(ii) The operators and are strongly continuous for all .
(iii) and are norm continuous inXfor .
(iv) and are compact operators inXfor .
(v) For every , the restriction of to and the restriction of to are norm continuous.
(vi) 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).
Definition 5 A function is said to be a mild solution of (1) if for any the integral equation
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.
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 denotes the adjoint of B and is the adjoint of . It is straightforward that the operator is a linear bounded operator.
LetZbe a separable reflexive Banach space and let stands for its dual space. Assume that is symmetric. Then the following two conditions are equivalent:
1. is positive, that is, for all nonzero .
2. For all strongly converges to zero as . Here, Jis the duality mapping ofZinto .
Lemma 8The 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.
(H1) are continuous and for each , there exists a constant and functions , such that
(H2) is a Lipschitz function with Lipschitz constant .
(Hc) The linear system (4) is approximately controllable on .
Using the hypothesis (Hc), for an arbitrary function , we choose the feedback control function as follows:
Let , where r is a positive constant. Then is clearly a bounded closed and convex subset in . We will show that when using the above control the operator defined by
has a fixed point in .
Theorem 10Let the assumptions (H1) and (H2) be satisfied. Then for , the fractional Cauchy problem (1) with has at least one mild solution provided that
Proof It is easy to see that for any the operator maps into itself.
Let and . Using assumption (H1) yield the following estimations,
From (6) and the assumption (H2), it follows that for any there exists such that
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 .
Step 1: We first prove that is continuous on . Let be a sequence such that as in . Therefore, it follows from the continuity of f, g and that for each ,
Also, by (H1), we see that
using the Lebesgue dominated convergence theorem that for all , we conclude
implying that as . This proves that is continuous on .
Step 2. is compact on .
For the sake of brevity, we write
Let be fixed and be small enough. For , we define the map
Therefore, from Lemma 4, we see that for each , the set is relatively compact in . Since
approaches to zero as , using the total boundedness, we conclude that for each , the set is relatively compact in .
On the other hand, for and small enough, we have
Therefore, it follows from (H1) and Lemma 4 that
from which it is easy to see that all , , tend to zero independent of as and . Thus, we can conclude that
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 11Let the assumptions (H1), (H2) and (Hc) be satisfied. Moreover, assume the functions and are bounded and . Then the semilinear fractional system (3) is approximately controllable on .
Proof It is clear that all assumptions of Theorem 10 are satisfied with . Let be a fixed point of in . Any fixed point of is a mild solution of (3) under the control
and satisfies the equality
Moreover, by the boundedness of the functions f and g and Dunford-Pettis theorem, we have that the sequences and are weakly compact in , so there are subsequences still denoted by and , that weakly converge to, say, f and g in . On the other hand, there exists such that converges to weakly in . Denote
It follows that
as because of compactness of the operator
Then from (9), we obtain
as . This proves the approximate controllability of (1). □
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 and define the operator A by with the domain . Then
where , , . Clearly −A generates a compact analytic semigroup , in X and it is given by
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 .
The operator is given by
where and .
Let , where for . Assume that satisfies the following conditions:
1. The functions , are continuous and uniformly bounded.
3. is continuously differentiable, and
Denote by , the Mittag-Leffler special function defined by
Then, for each we have
It follows that is bounded and Lipschitz continuous. On the other hand, it is not difficult to verify that are continuous.
Next, we show that the linear system corresponding to (11) is approximately controllable on . It is clear that is defined as follows:
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 .
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 , provided that
The authors declare that they have no competing interests.
All authors contributed equally and significantly in writing this paper. All authors read and approved the final manuscript.
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.
Agarwal, RP, Benchohra, M, Slimani, BA: Existence results for differential equations with fractional order and impulses. Mem. Differ. Equ. Math. Phys.. 44, 1–21 (2008). Publisher Full Text
Abada, N, Benchohra, M, Hammouche, H: Existence and controllability results for nondensely defined impulsive semilinear functional differential inclusions. J. Differ. Equ.. 246, 3834–3863 (2009). Publisher Full Text
Bashirov, AE, Mahmudov, NI: On concepts of controllability for deterministic and stochastic systems. SIAM J. Control Optim.. 37, 1808–1821 (1999). Publisher Full Text
Dauer, JP, Mahmudov, NI: Approximate controllability of semilinear functional equations in Hilbert spaces. J. Math. Anal. Appl.. 273, 310–327 (2002). Publisher Full Text
Mahmudov, NI, Denker, A: On controllability of linear stochastic systems. Int. J. Control. 73, 144–151 (2000). Publisher Full Text
Mahmudov, NI: Approximate controllability of semilinear deterministic and stochastic evolution equations in abstract spaces. SIAM J. Control Optim.. 42, 1604–1622 (2003). Publisher Full Text
Mahmudov, NI: Approximate controllability of evolution systems with nonlocal conditions. Nonlinear Anal.. 68, 536–546 (2008). Publisher Full Text
Mahmudov, NI: Controllability of linear stochastic systems in Hilbert spaces. J. Math. Anal. Appl.. 259, 64–82 (2001). Publisher Full Text
Benchohra, M, Ouahab, A: Controllability results for functional semilinear differential inclusions in Frechet spaces. Nonlinear Anal. TMA. 61, 405–423 (2005). Publisher Full Text
Byszewski, L: Theorems about the existence and uniqueness of solutions of a semilinear evolution nonlocal Cauchy problem. J. Math. Anal. Appl.. 162, 494–505 (1991). Publisher Full Text
Górniewicz, L, Ntouyas, SK, O’Regan, D: Controllability of semilinear differential equations and inclusions via semigroup theory in Banach spaces. Rep. Math. Phys.. 56, 437–470 (2005). Publisher Full Text
Tai, Z, Wang, X: Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces. Appl. Math. Lett.. 22, 1760–1765 (2009). Publisher Full Text
Fu, X: Controllability of non-densely defined functional differential systems in abstract space. Appl. Math. Lett.. 19, 369–377 (2006). Publisher Full Text
Fu, X, Mei, K: Approximate controllability of semilinear partial functional differential systems. J. Dyn. Control Syst.. 15, 425–443 (2009). Publisher Full Text
Klamka, J: Constrained approximate controllability. IEEE Trans. Autom. Control. 45, 1745–1749 (2000). Publisher Full Text
Klamka, J: Constrained controllability of semilinear systems with delays. Nonlinear Dyn.. 56, 169–177 (2009). Publisher Full Text
Liu, H, Chang, J: Existence for a class of partial differential equations with nonlocal conditions. Nonlinear Anal.. 70, 3076–3083 (2009). Publisher Full Text
N’Guérékata, GM: A Cauchy problem for some fractional abstract differential equation with nonlocal conditions. Nonlinear Anal. TMA. 70(5), 1873–1876 (2009). Publisher Full Text
Ren, Y, Hu, L, Sakthivel, R: Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay. J. Comput. Appl. Math.. 235, 2603–2614 (2011). Publisher Full Text
Sakthivel, R, Ren, Y, Mahmudov, NI: Approximate controllability of second-order stochastic differential equations with impulsive effects. Mod. Phys. Lett. B. 24, 1559–1572 (2010). Publisher Full Text
Balachandran, K, Sakthivel, R: Controllability of functional semilinear integrodifferential systems in Banach spaces. J. Math. Anal. Appl.. 255, 447–457 (2001). Publisher Full Text
Sakthivel, R, Ren, Y: Complete controllability of stochastic evolution equations with jumps. Rep. Math. Phys.. 68, 163–174 (2011). Publisher Full Text
Sukavanam, N, Kumar, S: Approximate controllability of fractional order semilinear delay systems. J. Optim. Theory Appl.. 151, 373–384 (2011). Publisher Full Text
Nieto, JJ: Basic theory for nonresonance impulsive periodic problems of first order. J. Math. Anal. Appl.. 205, 423–433 (1997). Publisher Full Text
Zhou, Y, Jiao, F: Nonlocal Cauchy problem for fractional evolution equations. Nonlinear Anal., Real World Appl.. 11, 4465–4475 (2010). Publisher Full Text
Wang, J, Zhou, Y: A class of fractional evolution equations and optimal controls. Nonlinear Anal., Real World Appl.. 12, 262–272 (2011). Publisher Full Text
Wang, J, Fan, Z, Zhou, Y: Nonlocal controllability of semilinear dynamic systems with fractional derivative in Banach spaces. J. Optim. Theory Appl.. 154(1), 292–302 (2012). Publisher Full Text
Yan, Z: Approximate controllability of partial neutral functional differential systems of fractional order with state-dependent delay. Int. J. Control. 85, 1051–1062 (2012). Publisher Full Text