The Solvability and Optimal Controls for Some Fractional Impulsive Equations of Order 1<𝛼<2

Abstract

We study the existence of solutions and optimal controls for some fractional impulsive equations of order . By means of Gronwall’s inequality and Leray-Schauder’s fixed point theorem, the sufficient condition for the existence of solutions and optimal controls is presented. Finally, an example is given to illustrate our main results.

1. Introduction

In this paper, we study some fractional evolution equation with finite impulsive: where is the standard Caputo fractional derivative of order , , , and is a sectorial operator of type defined on a complex Banach space . Let be a given function satisfying some assumptions that will be specified later. The function is continuous and ; , where and denote the right and the left limits of at , and respectively, has the similar meaning for . The control function is given in a suitable admissible control set ; is a linear operator from a separable reflexive Banach space into . The associated cost functions to be minimized over the family of admissible state control pairs are given by

For the last decades, fractional differential equations have been receiving intensive attention because they provide an excellent tool for the description of memory and hereditary properties of various materials and processes, such as physics, mechanics, chemistry, and engineering. For more details on fractional calculus theory, one can see the monographs of Miller and Ross [1], Podlubny [2], and Kilbas et al. [3] and the references therein.

Recently, impulsive differential equations have been proved to be valuable tools in the modelling of many phenomena in various fields of engineering, physics, and economics. The reason for the interest in the study of them is that the impulsive differential systems can be used to model processes which are subjected to abrupt changes and which cannot be described by the classical differential problem. For example, Liu and Li [4] utilized the well-known fixed point theorems to investigate the existence and uniqueness of solutions for the nonlinear impulsive fractional differential equations. Shu and Wang [5] studied the existence of mild solutions for fractional differential equations with nonlocal conditions of order : where is Caputo’s fractional derivative of order and is a sectorial operator of type .

In [6], Dabas and Chauhan researched the existence and uniqueness of mild solution which is expressed by Mittag-Leffler functions for an impulsive neutral fractional integro-differential equation with infinite delay: where denotes the Caputo fractional derivative of order . Bazhlekova [7], Li and Peng [8] were concerned with the controllability of nonlocal fractional differential systems of order in Banach spaces. Wang et al. [9] discussed the new concept of solutions and existence results for impulsive fractional evolution equations.

To the best of our knowledge, the system (1) is still untreated in the literature and it is the motivation for the present work. The rest of this paper is organized as follows. In Section 2, some notations and preparations are given. In Section 3, mainly some results of (1) are obtained. At last, an example is given to demonstrate our results.

2. Preliminaries

In this section, we will give some definitions and preliminaries which will be used in the paper.

Firstly, we will define and . The norm of the space will be defined by ; let denote the Banach space of all -value continuous functions from into , the norm . Let another Banach space , , there exist , , , and ; the norm . , , there exist , , , and , the norm . Obviously and are Banach spaces.

We denote by the Banach space of all Lebesgue measurable functions from to with .

Let us recall some known definitions of fractional calculus; for more details, see [1–3, 10].

Let ; then , and is a suitable function.

Definition 1 (Riemann-Liouville fractional integral and derivative operators). The integral operator is defined on by The derivative operator is defined as , where and

Definition 2. Caputo’s fractional derivative of of order is defined as If , we can write the Caputo derivative of the function via the above Riemann-Liouville fractional derivative as

Definition 3 (see [11]). Let be a sectorial operator of type if there exists , , such that the -resolvent of exists outside the sector:

Theorem 4. According to Lemma 2.6 in [4], one can get that if , then

Proof. If , then If , then where with the help of the substitution , the proof is completed.

Lemma 5 (see [12]). Let satisfy the following inequality: where , and are constants. Then where .

Theorem 6 (Hölder’s inequality). Assume that , and ; if and , then and .

Theorem 7 (Arzela-Ascoli theorem). If a sequence in is bounded and equicontinuous, then it has a uniformly convergent subsequence.

Remark 8. A subset of is compact if and only if it is closed, bounded, and equicontinuous.

Theorem 9 (Leray-Schauder’s fixed point theorem). If is a closed bounded and convex subset of Banach space and is completely continuous, then has a fixed point in .

3. 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 Schauder’s fixed point theorem and someone else.

Firstly, we will make the following assumptions.The function satisfies the following.(i) is measurable for all .(ii)There exists a constant such that , for all .(iii)There exist a real function , and a constant , such that , for and all . satisfies the following.(i) and are continuous and map a bounded set to a bounded set.(ii)There exist constants such that  Specially, if ,  We can make , .Operator and bounded, so there exists , .The multivalued maps (where is a class of nonempty closed and convex subsets of ) are measurable and , where is a bounded set of .

Set the admissible control set: Then, (see Proposition and Lemma of [13]). And it is obvious that   for  all  .

According to Definitions 1–2 and Theorem 4, without loss of generality, let and , by comparson with the fractional differential equations given in [4, 5, 8, 9, 12]; we will define the concept of mild solution for problem (1) as follows.

Definition 10. A function is said to be a solution (mild solution) of the problem (1) such that where with being a suitable path such that for . For more details, one can see [5].

Lemma 11 (see [5]). For any fixed , , and are compacted and bounded operators; that is, for any ,

Theorem 12. If the assumptions , and are satisfied and Lemma 5 and (1) is mildly solvable on [0,b], then there exists a constant such that ,  for all  .

Proof. If (1) can be solvable on [0,b], we may suppose is the mild solution of it, so must satisfy (19).
From Theorem 6, we also get that Let ; then so it follows from Lemma 5 that where the proof is completed.

Theorem 13. Assume that the hypotheses , and are satisfied Theorem 12; then the problem (1) has a unique mild solution on provided that

Proof. Transform problem (1) into a fixed point theorem. Consider the operator defined by Clearly, the problem of finding mild solutions of (1) is reduced to finding the fixed points of the . The proof is based on Theorem 9. Now we prove that the operators satisfy all the conditions of Theorem 9.
Firstly, choose and consider the bounded set . Next, we divide the proof into four steps.
Step  1. We prove that : Hence, we can make . So is a contraction mapping.
Step  2. We show that is continuous.
Let be a sequence such that in as . Then for each , we obtain As , it is easy to see that
Step  3. is equicontinuous on .
Let ; then for each , we obtain Let By Lemma 11, we have By assumption , we obtain Combining the estimations for , and , let ; we know that , which implies that is equicontinuous.
Step  4. Now we show that is compact.
Let be fixed; we show that the set is relatively compact in . From Step  1 and (24), we know that Then the set is uniformly bounded. From Step  3 and Arzela -Ascoli theorem, we know that the set is relatively compact in .
As a result, by the conclusion of Theorem 9, we obtain that has a fixed point on ; therefore system (1) has a unique mild solution on . The proof is completed.

4. Optimal Control Results

In the following, we will consider the Lagrange problem (P).

Find a control pair such that where and denotes the mild solution of system (1) corresponding to the control .

For the existence of solution for problem (P), we will introduce the following assumption.The function satisfies the following.(i)The function is Borel measurable;(ii) is sequentially lower semicontinuous on for almost all .(iii) is convex on for each and almost all .(iv)There exist constants and is nonnegative, and such that

Next, we can give the following result on existence of optimal controls for problem (P).

Theorem 14. Let the assumptions of Theorem 13 and hold. Suppose that is a strongly continuous operator. Then Lagrange problem (P) admits at least one optimal pair; that is, there exists an admissible control pair such that

Proof. If , there is nothing to prove.
Without loss of generality, we assume that . Using , we have . By definition of infimum, there exists a minimizing sequence feasible pair is a mild solution of system (1) corresponding to , such that as . Since is a bounded subset of the separable reflexive Banach space ; there exists a subsequence, relabeled as , and such that
Since is closed and convex, due to Mazur lemma, . Let denote the sequence of solutions of the system (1) corresponding to ; is the mild solution of the system (1) corresponding to and satisfy the following integral equation, respectively:
It follows the boundedness of , and Theorem 12; one can check that there exists a positive number such that .
For , we obtain By (ii), we have Using Lemma 11 and by (ii), one can obtain
Similarly, one has
Since is strongly continuous, we have which implies Thus and by virtue of singular version Gronwall inequality (see Remark 3.2, in [12]), we obtain This yields that
Note that implies that all of the assumptions of Balder (see Theorem 2.1, in [11]) are satisfied. Hence, from Balder’s theorem, we can conclude that is sequentially lower semicontinuous in the strong topology of . Since , is weakly lower semicontinuous on , and since, by (iv), attains its infimum at ; that is, The proof is completed.

5. An Example

We can consider the following initial-boundary value problem of fractional impulsive parabolic system: Take and the operator is defined by where the domain is given by Then can be written as where is an orthonormal basis of . It is well known that is the infinitesimal generator of a compact semigroup in given by From Theorems 3.3 and 3.4 of [5], we can easily get , Denote ; then it is easy to see that

Moreover,

Hence, all the conditions of Theorem 13 are satisfied, system (53) has a unique mild solution.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors thank the referees for their careful reading of the paper and insightful comments that will improve the quality of the paper as well. We would also like to acknowledge the valuable comments and suggestions from the editors that will vastly contribute to improve the presentation of the paper. The project is supported by NNSF of China Grant nos. 11271087 and 61263006 and Guangxi Scientific Experimental (China-ASEAN Research) Centre no. 20120116.