Approximate Controllability of Fractional Integrodifferential Evolution Equations

This paper addresses the issue of approximate controllability for a class of control systemwhich is represented bynonlinear fractional integrodifferential equations with nonlocal conditions. By using semigroup theory, p-mean continuity and fractional calculations, a set of sufficient conditions, are formulated and proved for the nonlinear fractional control systems. More precisely, the results are established under the assumption that the corresponding linear system is approximately controllable and functions satisfy nonLipschitz conditions. The results generalize and improve some known results.


Introduction
In recent years, use theory of fractional calculus and fractional differential equations has gained importance and popularity due to its applications in various fields of science and engineering.Various physical phenomena in science and engineering can be successfully modeled by using fractional calculus theory.Due to its tremendous scope and applications, several papers have been devoted to study the existence of mild solutions of fractional differential equations (see [1][2][3][4] and references therein).On the other hand, controllability is an important property of a control system which plays an important role in the analysis and design of control systems [5][6][7][8].Most literatures in this direction so far have been concerned with controllability of nonlinear differential equations in infinite-dimensional spaces without fractional derivatives (see [9] and references therein).Using generalized open mapping theorem, a set of sufficient conditions for constrained local relative controllability near the origin are formulated and proved for the semilinear systems with delayed controls in [10,11].
Recently, only few papers deal with the controllability of fractional dynamical systems [12][13][14].Klamka [15,16] derived a set of sufficient conditions for the local controllability of finite-dimensional fractional discrete-time semilinear systems.However, the problem of controllability for fractional systems has not been fully investigated, and there is still room open for further research in this area [17].Moreover, the approximate controllable systems are more prevalent, and very often, approximate controllability is completely adequate in applications (see [18][19][20][21] and references therein).Therefore, it is important, in fact necessary, to study the weaker concept of controllability, namely, approximate controllability for nonlinear fractional integrodifferential systems.Motivated by this fact, in this paper, we consider the approximate controllability of the fractional nonlinear integrodifferential evolution equations with nonlocal initial condition in the following form: where the state (⋅) takes the values in a Hilbert space ,    denotes Caputo derivative, − : () →  is the infinitesimal generator of an analytic semigroup {(),  ≥ 0} on ; the control function (⋅) is given in  2 (, );  is a

Journal of Applied Mathematics
Hilbert space;  is a bounded linear operator from  into ; the operator  is defined by ()() = ∫  0 ℎ(, , ()); the nonlinear term  :  ×   ×   → (or   ) is a given function, where here  = [0, ]; and   = (  ) (0 <  < 1) is a Hilbert space with the norm ‖‖  = ‖  ‖ for  ∈   .The functions , ℎ, and  will be specified later.In fact, our results in this paper are motivated by the recent work of [20], and the fractional integrodifferential equations are studied in [4].The main objective of this paper is to derive conditions for the approximate controllability of (1) with non-Lipschitz coefficients, and the associated linear system is approximately controllable.
Let us recall the following known results.The fractional integral of order  with the lower limit 0 for a function  is defined as provided the right-hand side is pointwise defined on [0, ∞), where Γ(⋅) is the gamma function.
Riemann-Liouville derivative of order  with lower limit zero for a function  : [0, ∞) →  can be written as The Caputo derivative of order  for a function  : [0, ∞) →  can be written as Remark 1 (see [4]).(i) If () ∈   [0, ∞), then (ii) The Caputo derivative of a constant is equal to zero.(iii) If  is an abstract function with values in , then integrals which appear in the above results are taken in Bochner's sense.
For additional details concerning the fractional derivative, we refer the reader to [3].
is called the reachable set of system (1) at terminal time , and its closure in  is denoted by R(,  0 ); let   ( 0 ; ) be the state value of (1) at terminal time  corresponding to the control  and the initial value  0 ∈ .
Consider the following linear fractional differential control system The approximate controllability for the system ( 8) is a natural generalization of approximate controllability for the linear first order control system (see [18]).It is convenient at this point to introduce the controllability operator associated with the linear system where  * denotes the adjoint of  and S * () is the adjoint of S().It is straightforward that the operator Γ  0 is a linear bounded operator.Let (, Γ  0 ) = ( + Γ  0 ) −1 for  > 0.
Lemma 4. The linear fractional control system (8) is approximately controllable on  if and only if (, Γ  0 ) → 0 as  → 0 + in the strong operator topology.

Main Result
In this section, we present our main result on approximate controllability of control system (1).We prove that under certain conditions, approximate controllability of the linear system (8) implies the approximate controllability of nonlinear fractional system (1).In order to establish the result, we require the following assumptions.( 4 ) The linear fractional control system ( 8) is approximately controllable.
Proof.The main aim in this section is to find conditions for solvability of systems ( 16) and (17) for  > 0. In the Banach space (,   ), consider a set where  is the positive constant.Now, it will be shown that, using Schauder's fixed point theorem, for all  > 0, the operator   : (,   ) → (,   ) has a fixed point.First, we prove that for an arbitrary  > 0 and there is a positive constant  0 =  0 () such that   :   0 →   0 .
Note.The considered system (1) is of the more general form, and in particular, if functions  and ℎ have various physical meanings, it is important to note that (1) has a great diversity.The result in this paper assumes that the linear system has a compact semigroup and consequently is not completely controllable.Moreover, the functions with Lipschitz condition are considerably strong when one discusses various applications in the real-world problems.Such an assumption is removed from this paper.