Controllability for Impulsive Fractional Evolution Inclusions with State-Dependent Delay

In this paper, sufficient conditions are provided for the controllability of impulsive fractional evolution inclusions with state-dependent delay in Banach spaces. We used a fixed-point theorem for condensing maps due to Bohnenblust–Karlin and the theory of semigroup for the achievement of the results. An Illustrative example is presented.


Introduction
Differential inclusions of fractional order have attracted great interest due to their applications in characterizing many problems in physics, biology, mechanics and so on; see, for instance [2,3,4,46,47].The theory of impulsive differential equations is a new and important branch of differential equations, which has an extensive physical background, for instance, we refer to [6,12,14,18,28,33,37,41].
One of the basic qualitative behaviors of a dynamical system is controllability, it means that it is possible to steer a dynamical control system from an arbitrary initial state to an arbitrary final state using the set of admissible controls.As a result of its great application, the controllability of such systems all have received more and more attention, we refer the work for more details [7,9,11,13,15,19,31,32,40,44].Yan [45] established the controllability of fractional-order partial neutral functional integrodifferential inclusions with infinite delay.In [36], the authors provided some sufficient conditions ensuring the existence of mild solution of the problem D α t x(t) = Ax(t) + f (t, x ρ(t,xt) , x(t)), t ∈ J k = (t k , t k+1 ], k = 0, 1, . . ., m, The controllability of fractional integro-differential equation of the form D q t x(t) = Ax(t) + Bu(t) + t 0 a(t, s)f (s, x ρ(s,xs) , x(s))ds, t ∈ J = [0, T ], has been considered by Aissani and Benchohra in [8].Motivated by the papers cited above, in this work, we consider the controllability for a class of impulsive fractional inclusions with state-dependent delay described by where D α t k is the Caputo fractional derivative of order 0 < α < 1, A : D(A) ⊂ E → E is the infinitesimal generator of an α-resolvent family (S α (t)) t≥0 , F : J × B × E −→ P(E) is a multivalued map (P(E) is the family of all nonempty subsets of E) and ρ : J × B → (−∞, T ] are appropriated functions, J = [0, T ], T > 0, B is a bounded linear operator from E into E, the control u ∈ L 2 (J; E), the Banach space of admissible controls.Here, 0 = t 0 < t 1 < . . .< t m < t m+1 = T, I k : E → E, k = 1, 2, . . ., m, are given functions, x(t k − h) denote the right and the left limit of x(t) at t = t k , respectively.We denote by x t the element of B defined by Here x t represents the history up to the present time t of the state x(•).We assume that the histories x t belongs to some abstract phase space B, to be specified later, and φ ∈ B.

Preliminaries
In this section, we state some notations, definitions and preliminary facts about fractional calculus and the multivalued analysis.
Let (E, • ) be a Banach space.C = C(J, E) be the Banach space of continuous functions from J into E with the norm By AC(J, E) we denote the space of absolutely continuous function from L(E) be the Banach space of all linear and bounded operators on E. L 1 (J, E) the space of E−valued Bochner integrable functions on J with the norm G is called upper semi-continuous (u.s.c.) on X if for each x 0 ∈ X the set G(x 0 ) is a nonempty, closed subset of X, and if for each open set U of X containing G(x 0 ), there exists an open neighborhood V of x 0 such that G(V ) ⊆ U.
G is said to be completely continuous if G(B) is relatively compact for every B ∈ P b (X).If the multivalued map G is completely continuous with nonempty compact values, then G is u.s.c.if and only if G has a closed graph (i.e.
Definition 2.2.Let α > 0 and f ∈ L 1 (J, E).The Riemann-Liouville integral is defined by For more details on the Riemann-Liouville fractional derivative, we refer the reader to [20].Definition 2.3.[38].The Caputo derivative of order α for a function f ∈ AC n (J, E) is defined by Obviously, the Caputo derivative of a constant is equal to zero.
In order to defined the mild solution of the problems (3) we recall the following definition.Definition 2.4.A closed and linear operator A is said to be sectorial if there are constants ω ∈ R, θ ∈ [ π 2 , π], M > 0, such that the following two conditions are satisfied: 1. (θ,ω) .
Sectorial operators are well studied in the literature.For details see [25].
Definition 2.5.[10].If A is a closed linear operator with domain D(A) defined on a Banach space E and α > 0, then we say that A is the generator of an α-resolvent family if there exists ω ≥ 0 and a strongly continuous function S α : R + →L(E) such that {λ α : Re(λ) > ω} ⊂ ρ(A) and In this case, S α (t) is called the α-resolvent family generated by A.
Definition 2.6.(see Definition 2.1 in [5]).If A is a closed linear operator with domain D(A) defined on a Banach space E and α > 0, then we say that A is the generator of a solution operator if there exist ω ≥ 0 and a strongly continuous function S α : R + →L(E) such that {λ α : Re(λ) > ω} ⊂ ρ(A) and in this case, S α (t) is called the solution operator generated by A.
In this paper, we will employ an axiomatic definition for the phase space B which is similar to those introduced by Hale and Kato [26].Specifically, B will be a linear space of functions mapping (−∞, 0] into E endowed with a seminorm • B , and satisfies the following axioms: where C > 0 is a constant. (A2) There exist a continuous function C 1 (t) > 0 and a locally bounded function for t ∈ [0, T ] and x as in (A1).
(A3) The space B is complete.
Let S F,x be a set defined by E) be an L 1 -Carathéodory multivalued map and let Ψ be a linear continuous mapping from L 1 (J, E) to C(J, E), then the operator The next result is known as the Bohnenblust-Karlin's fixed point theorem.
Lemma 2.9.(Bohnenblust-Karlin [17]).Let X be a Banach space and D ∈ P cl,c (X).Suppose that the operator G : D → P cl,c (D) is upper semicontinuous and the set G(D) is relatively compact in X.Then G has a fixed point in D.

Main Result
In this section, we prove our main result.We need the following lemma ( [42]).Lemma 3.1.Consider the Cauchy problem where F is a function satisfying the uniform Hölder condition with exponent β ∈ (0, 1] and A is a sectorial operator, then the Cauchy problem (4) has a unique mild solution which is given by where Br denotes the Bromwich path, S α (t) is called the α-resolvent family and T α (t) is the solution operator, generated by A.
Definition 3.4.The problem ( 3) is said to be controllable on the interval J if for every initial function φ ∈ B and x 1 ∈ E there exists a control u ∈ L 2 (J, E) such that the mild solution x(•) of (3) satisfies We always assume that ρ : J × B → (−∞, T ] is continuous.Additionally, we introduce following hypothesis: and there exists a continuous and bounded function Remark 3.5.The condition (H ϕ ), is frequently verified by continuous and bounded functions.For more details see, e.g., [29].
Remark 3.6.In the rest of this section, C * 1 and C * 2 are the constants where Let us list the following assumptions.
(H3) There exist a function µ ∈ L 1 (J, R + ) and a continuous nondecreasing function ψ : (H4) I k : E → E is continuous, and there exists Ω > 0 such that (H5) The linear operator W : L 2 (J, E) → E defined by has a pseudo inverse operator W −1 , which takes values in L 2 (J, E)/ ker W and there exist two positive constants M 1 and M 2 such that Remark 3.8.The question of the existence of the operator W −1 and of its inverse is discussed in the paper by Quinn and Carmichael (see [39]).
For any z ∈ B 2 , we have Thus (B 2 , • B 2 ) is a Banach space.We define the operator P : B 2 −→ P(B 2 ) by : . .., where v(s) ∈ S F,y ρ(s,ys+zs) +z ρ(s,ys+zs) .It is clear that the operator N has a fixed point if and only if P has a fixed point.So let us prove that P has a fixed point.We shall show that the operators P satisfy all conditions of Lemma 2.9.For better readability, we break the proof into a sequence of steps.Choose and consider the set It is clear that D r is a closed, convex, bounded set in B 2 .
Step 1: P is convex for each z ∈ B 2 .
Step 4: The set (P D r )(t) is relatively compact for each t ∈ J, where Let 0 < t ≤ s ≤ t 1 be fixed and let ε be a real number satisfying 0 < ε < t.For z ∈ D r we define where v ∈ S F,y ρ(s,ys+zs ) +z ρ(s,ys+zs) .Using the compactness of S α (t) for t > 0, we deduce that the set Similarly, for any t ∈ (t i , t i+1 ] with i = 1, . . ., m.Let t i < t ≤ s ≤ t i+1 be fixed and let ε be a real number satisfying 0 < ε < t.For z ∈ D r we define where v ∈ S F,y ρ(s,ys +zs ) +z ρ(s,ys+zs) .Since S α (t) is a compact operator, the set On the other hand, using the continuity of the operator T α (t), it follows that (P D r )(t) is relatively compact in E, for every t ∈ [0, T ].
As a consequence of Step 2 to 4 together with Arzelá-Ascoli theorem we can conclude that P is completely continuous.
Step 5: P has a closed graph.
Since z n → z * and h n → h * , it follows, that for every t ∈ [0, t 1 ], h * (t) = Similarly, for any t ∈ (t i , t i+1 ], i = 1, . . ., m, we have We shall prove that there exists v * ∈ S F,y * ρ(s,y * s+z * s)+z * ρ(s,y * s+z * s) such that, for each t ∈ (t i , t i+1 ], Clearly, we have Consider the linear continuous operatorΥ : In view of Lemma 2.8, we deduce that ΥoS F is a closed graph operator.Also, from the definition of Υ, we have that, for every t Therefore P has a closed graph.Hence by Lemma 2.9, P has a fixed point z on D r , which is the mild solution of the system (3), then problem (3) is controllable on (−∞, T ].This completes the proof of the theorem.

An Example
Consider the impulsive fractional integro-differential inclusion: where ω n (x) = 2 π sin(nx), n ∈ N is the orthogonal set of eigenvectors of A. It is well known that A is the infinitesimal generator of an analytic semigroup {T (t)} t≥0 in E and is given by e −n 2 t (ω, ω n )ω n , ∀ω ∈ E, and every t > 0.
For the phase space, we choose B = C 0 × L 2 (g, X), see Example 2.7 for details.Set  Under the above conditions, we can represent the system (7) in the abstract form (3). Assume that the operator W : L 2 (J, E) → E defined by The following result is a direct consequence of Theorem 3.9.
Proposition 4.1.Let ϕ ∈ B be such that (H ϕ ) holds, and assume that the above conditions are fulfilled, then system (7) is controllable on (−∞, T ].