Existence and Controllability Results for Fractional Control Systems in Reflexive Banach Spaces Using Fixed Point Theorem

In this paper, a fixed point theorem of nonexpansive mapping is established to study the existence and sufficient conditions for the controllability of nonlinear fractional control systems in reflexive Banach spaces. The result so obtained have been modified and developed in arbitrary space having Opial’s condition by using fixed point theorem deals with nonexpansive mapping defined on a set has normal structure. An application is provided to show the effectiveness of the obtained result.


Introduction:
Many systems in physics, chemistry, biology, stochastic, and control theory are represented by fractional control systems (FCS). For more details on (FCS) in the control theory one can see, Balachandran and Kokila (1), AL-Jawari and Shaker (2), Li Ding and Nieto (3), Lizzy and Balachandran (4).
One of the important topics in the study (FCS) in the control theory is controllability and it means that it is possible to transfer a (FCS) from an arbitrary initial state to an arbitrary final state by using the admissible controls. Thus controllability plays an important role in the analysis and design of these systems, see references (1)(2)(3)(4)(5)(6).
To study the result of the controllability of (FCS), some techniques of nonlinear functional analysis are used such as, fixed point theorems. Lizzy and Balachandran in (4) studied the controllability of stochastic fractional system in Hilbert spaces (HS) by using the Banach contraction mapping theory. Li Ding and Nieto in (3) discussed the controllability of (FCS) using Schauder's fixed point theory.
Since every (HS) are reflexive Banach space (RBS) and the contraction mapping is nonexpansive mapping, but the converse in general is not true (7) (also see section 2 of this paper), thus the purpose of this paper is to study the controllability of (FCS) in arbitrary (RBS) by using fixed point theorem that deals with nonexpansive mapping. The rest of this article is organized as follows. In section 2, preliminaries are given to study the solutions of (FCS) and then to prove the main result (theorem 4) in section 3. In section 4, an application is presented to illustrate the value of the obtained results.

Preliminaries and (FCS):
In this section, the solution of linear and nonlinear (FCS) is explored and present some definitions with theorems that will be used in the prove the main result of controllability (theorem 4) in section 3.

Definition 1 (8):
Let be a self mapping on a normed space , such that : . Then is contraction if < 1, and is nonexpansive if ≤ 1. It can be shown that, a contraction mapping is nonexpansive and isometry mapping is nonexpansive but not contraction, see (8). Definition 2 (9): Let X be a Banach space such that, if ∀ ∈ and ∀sequence { } converges weakly to ,then lim →∞ ‖ − ‖ > lim →∞ ‖ − ‖, holds ∀ ≠ . Thus the space satisfies Opial's condition. Every finite dimensional Banach space, (Hilbert space) for = 2 and spaces for 1 < < ∞ are satisfies Opial's condition, see (9). Definition 3 (7): A subset of a normed space is called weakly compact if every sequence { } in contains a subsequence which converges weakly in . For example, every nonempty, closed, convex and bounded subset of (RBS) is weakly compact see (8)(9). Definition 4 (8): Let be nonempty, closed, convex and bounded subset of the Banach space . Let a point x ∈ , such that { ‖ − ‖ ∶ ∈ } = , then is called a diametral. Also, has normal structure, if for each nonempty, convex ⊆ with diam > 0, there exist a point ∈ which is not diametral. Example 1 (8): Every compact convex set in a Banach space has normal structure. And every nonempty, closed, convex and bounded subset of uniformly convex Banach space has normal structure. Also Opial's condition implies normal structure, see (10). Theorem 1 (7): Every contraction mapping of a Banach space into itself has a unique fixed point. Theorem 2 (8, 11): Let be nonexpansive from into , where is a nonempty weakly compact convex subset having a normal structure in a Banach space , then has fixed point in . Remark 1: The convexity assumption in theorem 2, is very important, because the nonexpansive map on a non-convex set in Banach space may be has no fixed point. For example: Take = [−2, −1] ∪ [1, 2] ⊂ and : → by ( ) = − , ∈ , then is nonexpansive, but has no fixed points in , (11). Here, the solutions of linear and nonlinear (FCS) are discussed. Suppose that, , > 0, with − 1 < < , − 1 < < and ∈ , [0, ℎ] ⊂ . Let = and = be the and m-dimensional Euclidean spaces. Throughout this paper, the fractional derivative is taken in the Caputo sense and for brevity let us denote the Caputo fractional derivative by , for more details of properties to , see (12). Now, consider the linear control system represented by a (FCS) of the form where 0 < < 1, the state vector ( ) ∈ , the control vector ( ) ∈ and with are matrices of dimensions × , × respectively. The solution of the system Eq.1 can be obtained by using the method of successive approximation, see (12) and given by the following formula where , ( is the Mittag-Leffler function for a square matrix ∈ × , with ,1 ( ) = ( ). The function , ( ) is continuous and it satis- such that the solution of the Eq.1 with (0) = 0 also satisfies (ℎ) = 1 , then saying that the control system Eq.1 is controllable over The control (t) is said to be admissible control, i.e., (t) is transfer the trajectory from 0 to the final state 1 . Thus the controllability of the control system Eq.1 is equivalent to finding ( ) such that Here, suppose that ( ) = { : ∈ , (0) = z 0 , ‖ ( )‖ ≤ , ∀ ∈ } where is a positive constant. Then is a closed, convex and bounded subset of . Since ( ) is closed subset of a (RBS), then ( ) is a (RBS), see (7).
Second, to show that is nonexpansive mapping from ( ) into ( ). Thus for 1 ( ) , 2 ( ) ∈ ( ) and from the definition ( ) ( ) in Eq.6 is gotten that is nonexpansive mapping. Since ( ) is closed, convex, bounded subset of (RBS) and ( ) is weakly compact having normal structure (see example of Definition 3 and example 1), then by theorem 2, there exists a fixed point ∈ ( ) such that ( )( ) = ( ) and hence this fixed point is a solution of Eq.4 on , which satisfied (ℎ) = 1 , therefore the nonlinear control system Eq.4 is controllable on . ∎ Now, from the obtained results in this work, the following important results can deduce. Remark 2 : In this paper, the controllability of the solution ( ) ∈ , ∈ [0, ℎ] of the nonlinear control system Eq.4, where is (RBS) and has Opial's condition (see Definition 2 with example ) have been discussed. Thus, the previous results for control system Eq.4 when it is defined on arbitrary (RBS) having Opial's condition can extend by using the same manner in the proof of theorem 4. Remark 3 : Let be only a Banach space with 0 ≤ < 1 in condition [ 3 ], then by using the same manner of theorem 4, the operator being a contraction mapping. Thus, by theorem 1 a unique fixed point which is a solution to the control system Eq.4 on I = [0, ℎ] is obtained.

Conclusion:
The controllability of (FCS) in arbitrary (RBS) by using fixed point theorem that deals with nonexpansive mapping is examined. For this purpose, then some preliminaries related to the solutions of (FCS) and to prove the main result which guarantees the sufficient condition for the controllability of considered system are given. An application is presented to illustrate the value of the obtained results.