The Solvability and Optimal Controls for Some Fractional Impulsive Equation

This paper is concerned with the existence and uniqueness of mild solution of some fractional impulsive equations. Firstly, we introduce the fractional calculus, Gronwall inequality, and Leray-Schauder’s fixed point theorem. Secondly with the help of them, the sufficient condition for the existence and uniqueness of solutions is presented. Finally we give an example to illustrate our main results.

The fractional calculus and fractional difference equations have attracted lots of authors during the past years, and they gave some outstanding work [1][2][3][4], because they described many phenomena in engineering, physics, science, and controllability. Delay evolution equation allows someone to think after-effect, so it is a relative important equation. There are some significant development; for example, Wang et al. [5,6] consider the following fractional delay nonlinear integrodiffrential controlled system: and they used laplace transform and probability density functions to prove some sufficient conditions of some fractional nonlinear finite time delay evolution equations. Shu et al. [7] used the solution operator of semigroup to investigate the system given by Benchohra et al. [8] deal with existence of mild solutions of some fractional functional evolution equations with infinite 2 Abstract and Applied Analysis delay. Balachandran et al. [9] concerned the relative controllability of fractional dynamical with delays in control. Specially, Cuevas and Lizama [10] studied some sufficient conditions for the existence and uniqueness of almost automorphic mild solutions to the following semilinear fractional differential equation fractional differential equations: Bazhlekova [11] studied the fractional evolution equations in Banach spaces. Xue and Xiong [12] concerned the existence and uniqueness of mild solutions for abstract differential equations given by Motivated by the abovementioned works, we study (1). The rest of this paper is organized as follows. In Section 2, some notation and preparation are given. In Section 3, some mainly results of (1) are obtained. At last, an example is given to demonstrate our results.

Preliminar
In this section, we will give some definitions and preliminar which will be used in the paper. The norm of the space will be defined by ‖ ⋅ ‖ . Let Let us recall some known definitions; for more details, see [2][3][4].
Definition 2. Caputo fractional derivative of ( ) of order is defined as If = 0, we can write the Caputo derivative of the function ( ) ∈ [0, ∞), : [0, ∞) → via the above Riemann-Liouville fractional derivative as Let us recollect the generalized Gronwall inequality which can be found in [13] and will be used in our main result.
where is the Mittag-Leffler function defined by By Lemma 3 and Remark 4, we can establish a useful nonlinear impulsive Gronwall inequality which will be used in calculating.
Abstract and Applied Analysis 3 Specially, if = 1, We also introduce the following theorem that will be used in our mainly result.

Theorem 8 (Leray-Schauder's fixed point theorem). If is a closed bounded and convex subset of Banach space and :
→ is completely continuous, then the has a fixed point in .

Existence and Uniqueness of Mild Solution
In this section, we will investigate the existence and uniqueness for impulsive fractional differential equations with the help of the Leray-Schauder's fixed point theorem and someone else. Without loss of generality, let ∈ ( , +1 ], 1 ≤ ≤ − 1. Firstly, we will make the following assumptions be satisfied on the data of our problem. (iii) there exists a real function ( ) ∈ 1/ ( , + ), ∈ (0, ), and a constant > 0, such that ‖ ( , )‖ ≤ ( ) + ‖ ‖, for a.e. > 0 and all ∈ .
The proof is completed.
Theorem 12. Assume that the hypotheses (1)-(4) are satisfied, and then the problem (1) has an unique mild solution on provided that Proof. Transform the problem (1) into a fixed point theorem. Consider the operator : ( , ) → ( , ) defined by Clearly, the problem of finding mild solutions of (1) is reduced to find the fixed points of the , the proof base on Theorem 8. Now we prove that the operator satisfies all the conditions of the Theorem 8.
Firstly, choose and consider the bounded set = { ∈ : ‖ ‖ ≤ }. Next, for the sake of convenient, we divide the proof into several steps.
Step 2. We show that is continuous. Let { } be a sequence such that → in ( , ) as → ∞. Then, for each ∈ ( , +1 ], 1 ≤ ≤ −1, we obtain as → , and it is easy to see that − → as → ∞; that is, is continuous.
Step 4. Now we show that is compact.
As a result, by the conclusion of Theorem 8, we obtain that has a fixed point on . So system (1) has a unique mild solution on . The proof is completed.

Optimal Control Results
In the following, we will consider the Lagrange problem (P).
Find a control pair ( 0 , 0 ) ∈ ( , ) × such that where and denotes the mild solution of system (1) corresponding to the control ∈ . For the existence of solution for problem (P), we shall introduce the following assumption. (ii) L( , ⋅, ⋅) is sequentially lower semicontinuous on × for almost all ∈ ; (iii) L( , , ⋅) is convex on for each ∈ and almost all ∈ ; (iv) there exist constants ≥ 0, > 0, is nonnegative, and ∈ 1 ( , ) such that Next, we can give the following result on existence of optimal controls for problem (P). Theorem 12 and (6) hold. Suppose that is a strongly continuous operator. Then Lagrange problem ( ) admits at least one optimal pair; that is, there exists an admissible control pair ( 0 , 0 ) ∈ ( , ) × such that

An Example
Consider the following initial-boundary value problem of fractional impulsive parabolic control system where the domain ( ) is given by Then can be written as where ( ) = √2/ sin ( = 1, 2, . . .) is an orthonormal basis of . It is well known that is the infinitesimal It is easy to see that and then Hence, all the conditions of Theorem 12 are satisfied, and system (58) has a unique optimal solution.