Complete Controllability for Fractional Evolution Equations

and Applied Analysis 3 A operator A is said to belong to Cα(X,M, ω) or Cα(M, ω), if system (8) has solution operator {S α (t)} t≥0 satisfying |S α (t)| ≤ Me , t ≥ 0. Denote Cα(ω) = ⋃{Cα(M, ω),M ≥ 1} and Cα = ⋃{Cα(ω), ω ≥ 0}. Definition 9 (see [7]). A solution operator {S α (t)} t≥0 of system (8) is called analytic, if {S α (t)} t≥0 admits an analytic extension to a sectorial ∑ θ0 = {λ ∈ C \ {0} : | arg λ| < θ 0 } for some θ 0 ∈ (0, π/2]. An analytic solution operator is said to be of analyticity type (θ 0 , ω 0 ), if, for each θ < θ 0 and ω > ω 0 , there is anM = M(θ, ω) such that |S α (t)| ≤ e , t ∈ ∑ θ = {t ∈ C \ {0} : | arg t| < θ}. Denote A α (θ 0 , ω 0 ) = {A ∈ C α : A generates analytic solution S α (t) of type (θ 0 , ω 0 )} . (12) Lemma 10 (see [7]). Let α ∈ (0, 2); a linear closed densely defined operatorA belongs toA(θ 0 , ω 0 ), if λα ∈ ρ(A), for each λ ∈ Σ θ0+π/2 and, for any ω > ω 0 , θ < θ 0 , there is a constant C = C(θ, ω) such that 󵄩󵄩󵄩󵄩 λ α−1 R (λ α , A) 󵄩󵄩󵄩󵄩 ≤ C |λ − ω| for λ ∈ Σ θ+π/2 (ω) . (13) Next, we consider the definition of the mild solution of system (1). According to Definitions 1 and 2, it is suitable to rewrite the nonlocal Cauchy problem (1) in the equivalent integral equation x (t) = x 0 − g (x) + 1 Γ (α)


Introduction
Fractional differential equations have recently been proved to be valuable tools in the modeling of many phenomena in various fields of science and engineering.It draws a great application in nonlinear oscillations of earthquakes and many physical phenomena such as seepage flow in porous media and in fluid dynamic traffic model.There has been a significant development in fractional differential equations in recent years, see the monographs of Kilbas et al. [1], Miller and Ross [2], Podlubny [3], Lakshmikantham et al. [4], and the papers [5][6][7][8][9][10][11][12][13][14] and the references therein.Some recent papers investigated the problem of the existence of a mild solution for abstract differential equation with fractional derivative [15][16][17][18][19][20][21][22][23].However, the results in [15,16,18,19] are incorrect since the considered variation of constant formulas is not appropriate [17].Zhou and Jiao [22,23] introduced two characteristic solution operators and gave a suitable concept on a mild solution by applying Laplace transform and probability density functions.But the condition that the analytic semigroup {()} ≥0 was uniformly bounded was too strong.Shu et al. [20] researched the existence of mild solutions for impulsive fractional partial differential equation.But, Fe c kan et al. [24] had pointed out that the definition of solution of impulsive fractional differential equation was not correct.By using Laplace transform, Shu and Wang [25] gave a definition of mild solution for fractional differential equation with order 1 <  < 2 and investigated its existence.Agarwal et al. [26] studied the existence and dimension of the set for mild solutions of semilinear fractional differential equations inclusions.
In 1960, Kalman first introduced the concept of controllability which leads to some very important results regarding the behavior of linear and nonlinear dynamical systems.There are various works of complete controllability of systems represented by differential equations, integrodifferential equations, differential inclusions, neutral functional differential equations, and impulsive differential inclusions in Banach spaces (see [8,[27][28][29] and the references therein).Recently, more and more researchers also pay attention to study the controllability of fractional order evolution systems (see [21,30,31] and the references and therein).Unfortunately, the concept of mild solutions used in [30,31] was not suitable for fractional evolution systems at all and the corresponding definition of mild solutions is only a simple extension of the mild solutions of integer order systems.Wang and Zhou [21] investigated the complete controllability of fractional evolution systems with two characteristic solution operators introduced by them.
The nonlocal condition can be applied in physics with better effect than the classical initial condition (0) =  0 .Nonlocal condition was initiated by Byszewski [32] when he proved the existence and uniqueness of mild and classical solutions of nonlocal Cauchy problems.As remarked by Byszewski and Lakshmikantham [33], the nonlocal condition can be more useful than the standard initial condition to describe some physical phenomena.
where     is the Caputo fractional derivative of order  ∈ (0, 1], the state (⋅) takes value in a Banach space , the control function (⋅) is given in  2 (, ), with  as a Banach space,  is a bounded linear operator from  into ,  is a sectorial operator on ,  :  ×  →  and  :  →  are given functions satisfying some assumptions, and  0 ∈ .
The rest of this paper is organized as follows.In Section 2, some notations and preparations are given.A suitable concept on a mild solution for our problem is introduced.In Section 3, the complete controllability results are obtained by using fixed point theorems.Some conclusions are given in Section 4.

Preliminaries
In this section, we will firstly introduce fractional integral and derivative, some notations about sectorial operators, solution operators, and analytic solution operators and then give the definition of a mild solution of system (1).
Throughout this paper, R, C denote the sets of real and complex numbers, respectively, and R + = [0, ∞).By (, ), we denote the space of all continuous functions from  to .L() is the space of all bounded linear operators from  to . () denotes domain of , while () means resolvent set of  and (, ) = ( − ) −1 stands for the resolvent operator of .
Definition 1 (see [3]).The fractional integral of order  with the lower limit  for a function  : [, ∞) → R is defined as provided that the right side is point-wise defined on [, ∞), where Γ(⋅) is the gamma function.
(c)   () is a solution of the following integral equation: for all  ∈ () and  ≥ 0.
According to Definitions 1 and 2, it is suitable to rewrite the nonlocal Cauchy problem (1) in the equivalent integral equation provided that the integral in (14) exists.
The following Lemma 11 is discussed in [20]; for the sake of completeness, we outline its proof here.Lemma 11.If (14) holds and  is a sectorial operator, then we have where and  is a suitable path such that   ∉ Σ  (), for  ∈ .
Proof.By applying the Laplace transform to ( 14), we have Since (   − ) −1 exists, that is,   ∈ (), from the above equation, we obtain Therefore, by the Laplace inverse transform, we have choose the integration path  as follows: such that  is oriented counterclockwise, where  ∈ (0,  0 ),  >  0 , and  > 0.
Remark 18.It is easy to verify that a classical solution of system ( 1) is a mild solution of the same system.
In this paper, we assume the following.

Complete Controllability Results
Theorem 20.Suppose that (H 1 ), (H 2 ), and (H 4 ) are satisfied; then system (1) is completely controllable on , provided that  ∈ A  ( 0 ,  0 ) and Proof.Using hypothesis (H 4 ) for an arbitrary function  ∈ (, ), we defined the control function   () by We show that using this control, the operator  on (, ) by has a fixed point , which is a mild solution of system (1).
It is obvious that ()() =  1 , which means that   steers the mild  from  0 to  1 in finite time .This implies that system (1) is completely controllable on .Next, we will prove that  has a fixed point on (, ).