Existence Solution and Controllability of Sobolev Type Delay Nonlinear Fractional Integro-Differential System

Fractional integro-differential equations arise in the mathematical modeling of various physical phenomena like heat conduction in materials with memory, diffusion processes, etc. In this manuscript, we prove the existence of mild solution for Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. We establish the sufficient conditions for the approximate controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. In addition, we prove the exact null controllability of Sobolev type nonlinear impulsive delay integro-differential system with fractional order 1 < q < 2. Finally, an example is given to illustrate the obtained results.


Introduction
Recently, fractional differential equations have been considered greatly by research community in various aspects due to its salient features for real world problems (see [1][2][3][4][5][6][7]). Controllability problems for different kinds of dynamical systems have been studied by several authors (see [8][9][10][11][12][13][14][15]) and references therein.Thus, the dynamical systems must be treated by the weaker concept of controllability, namely approximate controllability (see [16][17][18][19][20][21]). However, up to now, no work has been reported yet regarding the null controllability of Sobolev type nonlinear fractional delay integro-differential system with the impulsive condition of order 1 < q < 2. Motivated by these facts, we study the existence of the mild solution for Sobolev-type impulsive fractional differential equations and we also discuss the sufficient conditions for approximate controllability and null controllability of the same problem.

Preliminaries
Let L(X) be a Banach space of all bounded linear operators from X into X equipped with the norm • L(X) and let C(J, X) be the Banach space of all continuous functions from J to X equipped with the norm • C(J,X) .Moreover, B r (x, X) is the closed ball in X with center at x and radius r > 0. Definition 1. ([22]).The linear space of all functions from (−∞, 0] into the Banach space X with a semi-norm • β is known as phase space β.
(B) For the function x(•) in (A), t → x t is continuous function for t ∈ [0, T].
(C) The space β is complete.
To obtain our results, we assume that the abstract fractional integro-differential problem has an associated q-resolvent operator of bounded linear operator (R q (t)) t≥0 on X.
Here, (I) and (II) together with the closed graph theorem imply the boundedness of the linear operator Aµ −1 : X → X.Furthermore, Aµ −1 generates a strongly continuous semigroup of bonded linear operators in X.

Definition 2. ([23]
).A family of bounded linear operators (R q (t)) t≥0 on X is said to be resolvent operator for Equation (1) if the following conditions are verified: In this paper, we have considered the following conditions.(P1) The operator Aµ −1 : D(Aµ −1 ) ⊂ X → X is a closed linear operator with [D(Aµ −1 )] dense in X.

Existence Solution
In this section, we investigate the existence of mild solution of Sobolev type of fractional integrodifferential equation with finite delay and impulsive conditions in the following form: where the state x takes values in a Banach space X, A, µ and (γ(t)) t∈J are closed linear operators on X, c D q 0 represent the Caputo derivative of order q ∈ (1, 2).The history x t : (−∞, 0] → X given by x t (θ) = x(t + θ) belongs to some abstract phase space β, defined later, F, G and To establish the result, we need the following hypotheses: Hypothesis 1 (H1).Aµ −1 is the infinitesimal generator of a resolvent operator R q (t) in X and there exists constant M > 0 such that

Hypothesis 2 (H2).
The functions F, G : J × β → X and I k : β → X are continuous and there exist positive constants L F , L G and L k such that

Definition 4. ([23]
).We say that x ∈ C(J, X) is a mild solution of the system (3) on J, if it satisfies Theorem 1.If Hypotheses (H1)-(H4) are satisfied, then the system (3) has a unique mild solution on J provided that Proof.Consider the operator ψ on C(J, X) defined as follows: We want to prove that the operator ψ has a fixed point.First, we show that ψ maps B r into itself.For x ∈ B r , Thus, ψ maps B r into itself.Next, we prove that ψ is contraction on B r For, m, n ∈ B r , we obtain Then, ψ is a contraction mapping on B r .Next, we will prove that ψ is completely continuous Let, (x n ) n∈N be a sequence in B r and x ∈ B r such that x n → x, we want to prove that Since the functions γ, F and G are continuous, i.e., then, as n → ∞, the following are satisfied: Therefore, ψ is continuous.Next, we show that (ψx)(t) is equicontinuous on J for any x ∈ B r .Let 0 < t ≤ b and > 0 be sufficiently small; then, It is known that the right-hand side of (5) tends to zero as → 0 .Hence, (ψx)(t) is completely continuous on J.By using a fixed point theorem , ψ has a unique fixed point x(t) on J .Therefore, system (3) has a unique mild solution on J.

Approximate Controllability
We will establish a set of sufficient conditions for approximate controllability of impulsive delay fractional differential equation in the following form: where the control function u(•) ∈ L 2 (J, U), the Banach space of admissible control functions with U a Banach space and B is a bounded linear operator from U into X.Definition 5. We say that x ∈ C(J, X) is a mild solution of system (6) if it satisfies In order to study the approximate controllability for system (6), we introduce the following linear fractional differential system: We define the operators associated with (6) as follows: Let x(T, x 0 , u) be the state value of (6) at terminal state T, corresponding to the control u and the initial value x 0 .Denote by R(T, x 0 ) = x(T, x 0 , u) : u ∈ L 2 (J, U) the reachable set of system (6) at terminal time T, its closure in X is denoted by R(T, x 0 ) Definition 6. System (6) is said to be approximately controllable on the interval [0, T] if R(T, x 0 ) = X.Lemma 5. ([24]).The linear system ( 7) is approximate controllable on [0, T] if and only if the operator λR(λ, Γ T 0 ) = λ(λI + Γ T 0 ) −1 → 0 as λ → 0 in the strong operator topology.
To establish the result, we need the following addition.

Hypothesis 5 (H5).
There exist a constant r > 0 such that where Theorem 2. If hypotheses (H1)-(H3) and (H5) are satisfied, then system (6) has a mild solution on J provided that Proof.Consider the operator ψ * on C(J, X) as follows: We want to prove that the operator ψ * has a fixed point.This fixed point is then a mild solution of system (6).We show that ψ * maps B r into itself, x ∈ B r , Thus, ψ * maps B r into itself.Next , for x, y ∈ B r , we obtain Then, ψ * is a contraction mapping and hence there exist a unique fixed point x ∈ B r such that ψ * x(t) = x(t).Hence, any fixed point of ψ * is a mild solution of (6) on J. Theorem 3. Assume that hypotheses (H1)-(H5) hold.Further, if the functions F : J × X × X → X, G : J × X → X are uniformly bounded and the resolvent operator R q and S q compact operators, then system (6) is approximately controllable on J.
Proof.Let x λ (.) be a fixed point of ψ * in B r .Any fixed point of ψ * is a mild solution of (6) on J under the control where and satisfies Thus, the Lebesgue dominated convergence theorem and the compactness of S q (t) yield Then, system (6) is approximate controllable.

Exact Null Controllability
In this section, we investigate the exact null controllability of fractional nonlinear differential equation of the system (6).
To study the exact null controllability of (6), we consider the fractional linear system associated with system (6).Define the operator , where L T 0 u has a bounded inverse operator (L 0 ) −1 with values in L 2 (J, U)/ ker(L T 0 ) and, Definition 7. ( [25].)The linear system ( 14) is said to be exactly null controllable on J if ImL T 0 ⊃ ImN T 0 .
Definition 8. ([26].)System ( 6) is said to be exactly null controllable if there is a u ∈ L 2 (J, U), such that the solution x of the system (6) satisfies x(T) = 0.
In this section, we need the following assumption.
Hypothesis 6 (H6).The linear system ( 14) is exactly null controllable on [0, T].Through this section, set Now, we are able to state and prove our main results.Theorem 4. Assume assumptions (H1)-(H3) and (H6) are satisfied.Then, the system (6) is exactly null controllable provided that: Proof.For an arbitrary x, define the operator ϕ on C(J, X) as follows: (ϕx where It will be shown that the operator ϕ from C(J, X) into itself has a fixed point. Step Step 2. There exist r > 0 such that ϕ sends B r into itself.
Hence, ϕ maps B r into itself.
Step 3. We prove (ϕx)(t) is continuous on J for any x ∈ X.
Step 4. We prove that (ϕx)(t) is a contraction on X.

Conclusions
This paper dealt with a Sobolev type nonlinear fractional delay integro-differential system with an impulsive condition.By using fractional calculus and fixed point theorems with the resolvent operator, we proved the existence of a mild solution for a Sobolev type nonlinear fractional delay integro-differential system with an impulsive condition.In addition, we established the necessary conditions for approximate controllability and null controllability of a Sobolev type nonlinear fractional delay integro-differential system with an impulsive condition.In the end, we provided an example in a fractional integro-partial differential equation to illustrate our results.
P3) There exists a subspace D ⊆ D(A) dense in [D(A)] and a positive constant c 1 such that A(D) ⊆ D(A), Ã1 (λ)(D) ⊆ D(A) and