Controllability of fractional order integro-differential inclusions with infinite delay

This paper concerns for controllability of fractional order integro-differential inclusions with infinite delay in Banach spaces. A theorem about the existence of mild solutions to the controllability of fractional order integro-differential inclusions is obtained based on Dhage fixed point theorem. An example is given to illustrate the existence result.

On the other hand, the most important qualitative behavior of a dynamical system is controllability.It is well known that the issue of controllability plays an important role in control theory and engineering [7,8,12,15] because they have close connections to pole assignment, structural decomposition, quadratic optimal control and observer design etc.In D q t x(t) ∈ Ax(t) + Bu(t) + t 0 a(t, s)F(s, x s , x(s)) ds, t ∈ J = [0, T], where D q t is the Caputo fractional derivative of order 0 < q < 1, A generates a compact and uniformly bounded linear semigroup S(•) on X, F : J × B × X −→ P (X) is a multivalued map (P (X) is the family of all nonempty subsets of X), a : D → R (D = {(t, s) ∈ [0, T] × [0, T] : t ≥ s}), φ ∈ B where B is called phase space to be defined in Section 2. B is a bounded linear operator from X into X, the control u ∈ L 2 (J; X), the Banach space of admissible controls.For any function x defined on (−∞, T] and any t ∈ J, we denote by x t the element of B defined by Here x t represents the history of the state up to the present time t.
Our results are based on the Dhage fixed point theorem and the semigroup theory.To our knowledge, very few results are available for controllability for fractional integro-differential inclusions.So the present results complement this literature.
The paper is organized as follows.In Section 2 some preliminary results are introduced.The main result is presented in Section 3, and an example illustrating the abstract theory is presented in Section 4.

Preliminaries
Let (X, • ) be a real Banach space.C = C(J, X) be the space of all X-valued continuous functions on J. L(X) be the Banach space of all linear and bounded operators on X. L 1 (J, X) the space of X-valued Bochner integrable functions on J with the norm L ∞ (J, R) is the Banach space of essentially bounded functions, normed by

Denote by
Controllability of fractional order integro-differential inclusions 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 2.1.The multivalued map F : J × B × X −→ P (X) is said to be an Carathéodory if We need some basic definitions and properties of the fractional calculus theory which are used further in this paper.Definition 2.2.Let α > 0 and f : R + → X be in L 1 (R + , X).Then the Riemann-Liouville integral is given by: where Γ(•) is the Euler gamma function.
For more details on the Riemann-Liouville fractional derivative, we refer the reader to [21].

Definition 2.3 ([40]
).The Caputo derivative of order α for a function f : [0, +∞) → R can be written as Obviously, the Caputo derivative of a constant is equal to zero.
In this paper, we will employ an axiomatic definition for the phase space B which is similar to those introduced by Hale and Kato [29].Specifically, B will be a linear space of functions mapping (−∞, 0] into X endowed with a seminorm • B , and satisfies the following axioms: Let S F,x be a set defined by Lemma 2.5 ([33]).Let X be a Banach space.Let F : J × B × X −→ P cp,c (X) be an L 1 -Carathéodory multivalued map and let Ψ be a linear continuous mapping from L 1 (J, X) to C(J, X), then the operator is a closed graph operator in C(J, X) × C(J, X).
Then either (i) the operator inclusion u ∈ Au + Bu has a solution , or Let Ω be a set defined by Controllability of fractional order integro-differential inclusions 5

Main results
In this section, we state and prove the controllability results for the system (1.1).Now we define the mild solution for our problem.
Definition 3.4.The problem (1.1) is said to be controllable on the interval J if for every initial function φ ∈ B and x 1 ∈ X there exists a control u ∈ L 2 (J, X) such that the mild solution x(•) of (1.1) satisfies x(T) = x 1 .
(H2) There exists a function µ ∈ L 1 (J, R + ) and a continuous nondecreasing function ψ : R + → (0, +∞) such that with where and (H3) There exists a function k ∈ L 1 (J, R + ) such that (H5) The linear operator W : L 2 (J, X) → X defined by has an inverse operator W −1 , which takes values in L 2 (J, X)/ ker W and there exist two positive constants M 1 and M 2 such that Proof.We transform the problem (1.1) into a fixed-point problem.Consider the multivalued operator N : Using hypothesis (H5) for an arbitrary function x(•) define the control Controllability of fractional order integro-differential inclusions 7 Obviously, fixed points of the operator N are mild solutions of the problem (1.1).For φ ∈ B, we will define the function y(•) : (−∞, T] −→ X by Then y 0 = φ.For each function z ∈ C(J, X) with z(0) = 0, we denote by z the function defined by If x(•) verifies (3.1), we can decompose it as x(t) = y(t) + z(t), for t ∈ J, which implies x t = y t + z t , for every t ∈ J and the function z(t) satisfies For any z ∈ Z 0 , we have Thus (Z 0 , • Z 0 ) is a Banach space.We define the operator P : Obviously the operator N having a fixed point is equivalent to P having one, so it turns to prove that P has a fixed point.Let r > 0 and consider the set We need the following lemma.
Then for any z ∈ B r we have Proof.Using (2.2), (3.4), (3.6) and (3.7), we obtain The lemma is proved.Now, we define the following multivalued operators P 1 , P 2 : Z 0 −→ P (Z 0 ) as It is clear that P = P 1 + P 2 .The problem of finding solutions of (1.1) is reduced to finding solutions of the operator inclusion z ∈ P 1 (z) + P 2 (z).We shall show that the operators P 1 and P 2 satisfy all conditions of the Theorem 2.8.The proof will be given in several steps.
) is measurable (see Proposition 2.6), there exists a function v * (t), which is a measurable selection for V. So, v * (t) ∈ F(t, y t + z * t , y(t) + z * (t)), and using (A2), for each t ∈ J, we obtain Let us define for each t ∈ J h * (t) = t 0 R(t − s)Bu y+z * (s)ds.
Then we have By an analogous relation, obtained by interchanging the roles of z and z * , it follows that By (3.5), the mapping P 1 is a contraction.
Step 2: P 2 has compact, convex values, and it is completely continuous.This will be given in several claims.Claim 1: P 2 is convex for each z ∈ Z 0 .Indeed, if h 1 and h 2 belong to P 2 , then there exist v 1 , v 2 ∈ S F,y+z such that, for t ∈ J, we have Since S F,y+z is convex (because F has convex values), we have Claim 2: P 2 maps bounded sets into bounded sets in Z 0 .
Indeed, it is enough to show that for any r > 0, there exists a positive constant such that for each z ∈ B r = {z ∈ Z 0 : z Z 0 ≤ r}, we have P 2 (z) Z 0 ≤ .Then for each h ∈ P 2 (z), there exists v ∈ S F,y+z such that Using (H2) and Lemma 3.6 we have for each t ∈ J, Hence P 2 (B r ) is bounded.Claim 3: P 2 maps bounded sets into equicontinuous sets of Z 0 .Let h ∈ P 2 (z) for z ∈ Z 0 and let τ 1 , τ 2 ∈ [0, T], with τ 1 < τ 2 , we have where For I 1 , using (3.2) and (H2), we have Clearly, the first term on the right-hand side of the above inequality tends to zero as τ 2 → τ 1 .
From the continuity of S(t) in the uniform operator topology for t > 0, the second term on the right-hand side of the above inequality tends to zero as τ 2 → τ 1 .
In view of (3.2), we have As τ 2 → τ 1 , I 2 tends to zero.So P 2 (B r ) is equicontinuous.Claim 4: (P 2 B r )(t) is relatively compact for each t ∈ J, where Let 0 < t ≤ T be fixed and let ε be a real number satisfying 0 < ε < t.For arbitrary δ > 0, we define where v ∈ S F,y+z .Since S(t) is a compact operator, the set Therefore, (P We have to prove that there exists v * ∈ S F,y * +z * such that Consider the linear and continuous operator Υ : L 1 (J, X) −→ C(J, X) defined by Let z ∈ E be any element, then there exists v ∈ S F,y+z such that Thus, by (3.8), (H2) and Lemma 3.6, for each t ∈ J we have By the previous inequality, we have Let us take the right-hand side of the above inequality as v(t).Then we have Using the nondecreasing character of ψ we get Integrating from 0 to t we get By a change of variable we get Using the condition (3.3), this implies that for each t ∈ J, we have Thus, for every t ∈ J, there exists a constant Λ such that v(t) ≤ Λ and hence m(t) ≤ Λ.Since z Z 0 ≤ m(t), we have z Z 0 ≤ Λ.This shows that the set E is bounded.As a consequence of Theorem 2.8 we deduce that P 1 + P 2 has a fixed point z defined on the interval (−∞, T] which is the solution of problem (1.1).This completes the proof.

An example
Consider the following integro-differential equation with fractional derivative of the form where 0 < q < 1, µ : It is well known that A is the infinitesimal generator of an analytic semigroup (S(t)) t≥0 on X [43].Furthermore, A has a discrete spectrum with eigenvalues of the form −n 2 , n ∈ N, and the corresponding normalized eigenfunctions are given by u n (x) = 2 π sin(nx).
In addition, {u n : n ∈ N} is an orthogonal basis for X, S(t)u = ∞ ∑ n=1 e −n 2 t (u, u n )u n , for all u ∈ X and every t ≥ 0.
From these expressions it follows that (S(t)) t≥0 is uniformly bounded compact semigroup.
For the phase space, we choose B = B γ defined by With the above choices, we see that the system (4.1) is the abstract formulation of (1.1).Assume that the operator W : L 2 ([0, 1], X) → X defined by has a bounded invertible operator W −1 in L 2 ([0, 1], X)/ ker W. Thus all the conditions of Theorem 3.5 are satisfied.Hence, system (4.1) is controllable on (−∞, T].
is continuous on J and x 0 ∈ B, then x t ∈ B and x t is continuous

Proposition 2.6 (
[13, Proposition III.4]).If Γ 1 and Γ 2 are compact valued measurable multifunctions, then the multifunction t →Γ 1 (t) ∩ Γ 2 (t) is measurable.If (Γ n ) is a sequence of compact valued measurable multifunctions, then t → ∩Γ n (t) is measurable, and if ∪Γ n (t) is compact, then t → ∪Γ n (t) is measurable.Definition 2.7.A multivalued operator N : X → P cl (X) is called a) γ-Lipschitz if and only if there exists γ > 0 such that 2 B r )(t) is relatively compact.As a consequence of Claim 2 to 4 together with the Arzelà-Ascoli theorem we can conclude that P 2 is completely continuous.Claim 5: P 2 has a closed graph.Let z n → z * , h n ∈ P 2 (z n ), and h n → h * .We shall show that h * ∈ P 2 (z * ).h n ∈ P 2 (z n ) means that there exists v n ∈ S F,y n +z n such that s) dτ ds.From Lemma 2.5 it follows that Υ • S F is a closed graph operator and from the definition of Υ one has h n (t) ∈ Υ(S F,y n +z n ).z n → z * and h n → h * , there is a v * ∈ S F,y * +z * such that