EXISTENCE AND CONTROLLABILITY FOR IMPULSIVE FRACTIONAL STOCHASTIC EVOLUTION SYSTEMS WITH STATE-DEPENDENT DELAY

This paper is concerned with the impulsive fractional stochastic neutral evolution systems with state-dependent delay and nonlocal condition. First, the existence of solutions of considered evolution systems are obtained by applying the Banach contraction theorem. Then, on the basis of existence of solutions, the controllability concept of the system is investigated. The main aim is to derive some conditions that could be applied to analyze the controllability results for the considered evolution systems involving state-dependent delay. Finally, the efficiency of theoretical analysis is verified by an example.


Introduction
Stochastic differential equations have become more essential when the occurrence of random effects in dynamical systems.Many problems in real time situations are mainly modeled by stochastic equations rather than deterministic.The stochastic analysis technique and methods have attracted a great deal of interest due to their abundant and real application in numerous areas such as applied science and engineering [1,7,30].The controllability concept plays a fundamental role in control theory.The problem of controllability is to prove the presence of a control function, which drives the solution of system from its initial position to final position.This notion leads to some significant conclusions based on the behavior of linear and nonlinear dynamical systems [22,32,33].In [17], sufficient conditions for constrained controllability are formulated and proved.Approximate constrained controllability of mechanical systems have been reported in [19].In [18,20], the constrained controllability of semilinear systems was considered.On the other hand, impulsive differential equation is used to model the dynamical systems of changing processes.These perturbations are often treated in the form of impulses which is negligible in evaluation with the whole process.It should be pointed out that the controllability of impulsive perturbations and stochastic analysis have been treated separately in most existing literature [3,[33][34][35]39].Thus, the problem of controllability analysis for stochastic systems with impulsive effects arises as a research area of primary importance [4,8,9,16,25].
In the past two decades, fractional differential equations (FDE) have received huge consideration because of their potential applications in many areas [27,29].However, these fruitful applications are really dependent on the dynamic behaviors of FDE.Fractional derivatives also appear in the theory of control dynamical systems, when the controlled system and the controller are defined by FDE [31].As is well known, controllability is one of the key properties of FDE, which is an important feature in the design of FDE in dynamical system [24,28,40].On the other hand, it has been realized that the delay effects often occur in several FDE, since the derivative of a function depends on the solution of previous state at any time.Consequently, the controllability analysis for FDE with time delays have been an interesting area of research, where the types of delay can be classified as constant, time-varying, control and infinite [11,14,22,36].However, generally equations with state-dependent delay have less smoothness properties than those representing equations with constant delay.The controllability of FDE involving the Riemann-Liouville and Caputo fractional derivatives with and without state-dependent delay have been paid much attention in [2,26,[37][38][39]41].
In recent years, the controllability analysis problem for FDE with stochastic perturbations and impulsive effects becomes increasingly important, and some researches connected to this analysis have been stated [6,10,13].Recently, Li and Wang [23] examined the relative controllability of fractional systems involving pure delay.In [32], the concept of controllability has been established for fractional differential systems involving state and control delay.The controllability analysis of multi-term time-fractional differential systems with state-dependent delay has been studied in [5].To the best of the authors knowledge, the controllability analysis for impulsive fractional stochastic evolution systems involving nonlocal condition and state-dependent delays where the delay depends only on state has not been well addressed, which still remains interesting and essential.Motivated by the above discussions, in this paper we study the existence and controllability results for impulsive fractional stochastic neutral evolution systems involving state dependent-delay and nonlocal condition.
The outline of this paper is as follows.In Section 2, we recall some preliminary notations and results.Based on stochastic theory and Banach contraction principle, the existence of mild solution of the considered problem is discussed in Section 3. In Section 4, using the Krasnoselskii's fixed point theorem, we also develop the controllability results for the impulsive fractional stochastic evolution systems with nonlocal condition and state-dependent delays.In Section 5, an example is given to illustrate the effectiveness of the derived results.Conclusions are made in Section 6.

Problem Formulation
Consider the following impulsive fractional neutral stochastic evolution systems with state dependent delay and nonlocal condition: Here c D α t is Caputo fractional derivative of order α, 0 < α < 1.Let A(t) and £ 2 are all closed densely defined evolution operator and is defined by is a continuous function.The state variable x takes values in Hilbert space H, ∆x| t=tj = x(t + j ) − x(t − j ), j = 1, 2, 3, . . ., k and I j = PC → H is a appropriate function with t j (0 < t 1 < . . .< t j < t j+1 < T ).
Consider the space, Let (Ω, F, P) be the complete probability space.A H−valued random variable is a F measurable function x(t) : Ω → H and a collection of random variables Z = x(t, ω) : Ω → H| t∈G is known as stochastic process.The one dimensional standard Brownian motion is denoted by β n (t) n≥1 .Consider then ψ is known as Q− Hilbert Schmidt operator.Here L Q (K, H) denote the space of all Q−Hilbert Schmidt operator.

Definition 2.2 ( [42]
). Caputo derivative of order β for a function g ∈ L(G, H) is defined by

Existence of Mild Solution
In this section, we study the existence of mild solutions for the fractional impulsive neutral stochastic evolution system (2.1) − (2.3) with nonlocal condition and statedependent delay.To prove this, first we define the definition of mild solution for the considered system.
satisfies the following integral equations: To prove the result we always assume that ρ : G×S(B) → (−∞, d] is continuous and that φ ∈ S(B).Also, A(•) generates an evolution operator R(t, s).Now, we assume the following hypotheses: (H1) The mapping t → φ t is continuous and well defined, then there exists a continuous and bounded function (H6) The impulsive function Theorem 3.1.Suppose that the assumptions (H1) − (H6) are satisfied and . Then the fractional impulsive neutral stochastic evolution system (2.1)-( 2.3) has a unique mild solution.
Proof.The mild solution of the considered evolution system (2.1) − (2.3) with state dependent delay and nonlocal condition can be written in the form x(t : φ) = (Γx)(t), where where The operator Γ satisfies the Banach contraction theorem and therefore there is only one fixed point.Hence Γ is the unique mild solution of the fractional impulsive neutral stochastic evolution systems (2.1)−(2.3)with state-dependent delay and nonlocal condition.

Controllability Result
In this section, we discuss the result on controllability of fractional impulsive neutral stochastic evolution control systems with state-dependent delay and nonlocal condition.Consider the problem Let U be a separable Hilbert space and the admissible control function u(•) is given in L 2 (G, U ). B is a bounded linear operator from U to H.The remaining functions are defined as same in (2.1) − (2.3).In this section we establish the controllability results for the system (4.1)− (4.3).
Then there exist y ∈ M such that Γy + Θy = y.Now, we impose the hypotheses as follows: (H1) The resolvent operator R(t, s) is compact with has an invertible operator W −1 which taking the values in L 2 (G, U ) \ ker W and there exists a positive constant M 4 such that (H4) The function £ 3 : G × PC → H satisfies the following conditions: (iii) There exists a function Ψ £3 : [0, ∞) → (0, ∞) such that, for every (t, ξ) ∈ G, for each e > 0.
(H6) p is continuous and there exist some positive constant M such that E∥p(x)∥ 2 ≤ M.
(H9) Proof.Using (H2), define the control Now we show that when using this control the operator Φ defined by has a fixed point x(•).This fixed point x(•) is the mild solution of the system (4.1)− (4.3).Clearly, x(d) = (Φx)(d) = x 1 , which means that the control u steers the system from the initial function φ to x 1 in time d, provided we can obtain a fixed point of the operator Φ which implies that the system is controllable.For φ(t) ∈ S(B), we define φ by Define the operator Θ and Γ by Obviously, the operator Φ has a fixed point if and only if the operator Γ + Θ has a fixed point.First, we define for every x ∈ B e = B e (0, S(B)) and t ∈ G.
where L is independent of e. Dividing both sides by e and taking the limit as e → ∞ which contradicts hypothesis (H8), and thus condition (i) in Lemma 4.1 is verified.
Hence for some positive number e, (Γ + Θ)B e ⊂ B e .
Step 2: Γ maps B e into an equicontinuous family.
For y ∈ B e , τ 1 , τ 2 ∈ G and 0 < τ 1 < τ 2 ≤ d.We have By hypotheses (H1) − (H7) and Lemma 2.1, the compactness of R(t, s) for t, s > 0 which implies the continuity in the uniform operator topology.The right-hand side tends to zero as τ 2 − τ 1 → 0. Thus Γ maps B e into an equicontinuous family of functions.
Step 3: Γ maps B e into a precompact set in H.
Let us assume that ϵ be a real number and 0 < t ≤ d be fixed which satisfies 0 < ϵ < t.For y ∈ S(B), we define For every ϵ, Y ϵ (t) = {(Γ ϵ )(t) : y ∈ B e } is relatively compact in H. Since R(t, s) is compact operator.We have R(t, s)£ 3 (s, y ρ(s,ys+ φs) + φρ(s,ys+ φs) )ds Step 4: To prove Γ : S(B) → S(B) is continuous.We prove that Γ is continuous on S(B).Let {y (n) } ∞ 0 ⊆ S(B) with y(n) → y in S(B).Then, there exists a positive number e > 0 such that ∥ y (n) (t) ∥ 2 ≤ e for all n and a.e.t ∈ G, so y (n) ∈ B e and y ∈ B e .a complete state, ii).an instantaneous state.As pointed in [21], the controllability results for these two states are descried as absolute controllability for complete states and relative controllability for instantaneous states.Since the considered system in this paper involves state delays, so the proposed controllability results can be viewed as a relative controllability similar to the results in [21].

Conclusion
In this paper, the existence and controllability results for the fractional impulsive neutral stochastic evolution systems with state-dependent delay and nonlocal condition have been established.Firstly, the existence results of the system is obtained by using the Banach contraction theorem.Further, the Krasnoselskii's fixed point theorem is utilized for the controllability results of the fractional impulsive neutral stochastic evolution control systems with state-dependent delay and nonlocal condition.An example is analyzed to illustrate the importance of the obtained results.Furthermore, the obtained results can be extended to stochastic evolution systems with various delay effects like multiple delay, distributed delay and will be considered in future.

Definition 4 . 1 .
A stochastic process x : (−∞, d] → H is called a mild solution of the problem (4.1)
which proves the operator Γ is continuous.From the above analysis, we can conclude that the operator Γ is completely continuous, and thus satisfies condition (ii) in Lemma 4.1 .£ 1 (t, y ρ(s,ys+ φs) + φρ(s,ys+ φs) ) − £ 1 (t, ȳρ(s,ȳs+ φs) + φρ(s,ȳs+ φs) − £ 4 (s, y ρ(s,ys+ φs) + φρ(s,ys+ φs) )2 ≤3 E Controllability results for nonlinear systems in infinite dimension are commonly proposed with sufficient conditions.The hypotheses (H1)-(H9) used in this paper are sufficient and it is still an open problem to prove that these conditions are necessary for controllability of considered system.It is worth pointing that the dynamical systems containing delays (in state variables or in controls), it is necessary to introduce two types of states i).
.10)By hypotheses (H1), (H2) and (H9), and thus operator Θ is contractive operator.Therefore, all the conditions of Krasnoselskii's fixed point theorem are satisfied and thus operator Γ + Θ has a fixed point in B e .From this it follows that the operator Φ has a fixed point and hence the system (4.1)− (4.3) is controllable on G.This completes the proof.Remark 4.1.