On Approximate Controllability of Second Order Fractional Impulsive Stochastic Differential System with Nonlocal, State-dependent Delay and Poisson Jumps

The main scope of this paper is to focus the approximate controllability of second order (q∈(1,2]) fractional impulsive stochastic differential system with nonlocal, state-dependent delay and Poisson umps in Hilbert spaces. The existence of mild solutions is derived by using Schauder fixed point theorem. Sufficient conditions for the approximate controllability are established by under the assumptions that the corresponding linear system is approximately controllable and it is checked by using Lebesgue dominated convergence theorem. The main results are completly based on the results that the existence and approximate controllability of the fractional stochastic system of order 1<q≤2 and are derived by using stochastic analysis theory, fixed point technique, q-order cosine family {Ca(t)}t≥0, new set of novel sufficient conditions and methods adopted directly from deterministic fractional equations for the second order nonlinear impulsive fractional nonlocal stochastic differential systems with state-dependent delay and Poisson jumps in Hildert space H. Finally an example is added to illustrate the main results.


Introduction
The concept of semigroups of bounded linear operators is taken as an important concept to dealing with differential and integro-differential equations in Banach spaces [8,14,15,17,36]. For more points of interest on this concept, we refer to Pazy [36]. On the other hand, in numerous mathematical models of real world or man made phenomena, we are led to dynamical systems which involve some inherent randomness. These systems are called stochastic systems. Stochastic differential equations [34] have attracted much attention and have played an important role in many ways such as option pricing, forecast of the growth of population, etc [16,18,29,34]. In the last few decades, fractional differential systems (we refer to the monographs [28,31,35,41] and references cited therein) have focused considerable importance in electrochemistry, physics, porous media, control theory, engineering etc., [4,5,6,10,11,44] due to the descriptions of memory and hereditary properties of various materials and processes. Notion of controllability is of great importance in mathematical control theory due to a number of important properties of control systems in engineering. Astrom [1] discussed about introduction to stochastic control theory. In the infinite dimensional systems, two basic concepts of controllability are exact and approximate controllability. Exact controllability enables to steer the system to arbitrary final state while approximate controllability is weaker concept of controllability and it is possible to steer the system to an arbitrary small neighborhood of the final state (see, for example, [2,9,12,46,50]). Impulsive effects [47] exist widely in many evolution process because, the impulsive effects may bring an abrupt change at a certain moments of time involving such fields as economics, mechanics, electronics, telecommunications, medicine and biology, etc. Kexue et al. [27] studied controllability of nonlocal fractional differential systems of order ∈ (1,2] in Banach spaces. Delay fractional differential equations are similar to fractional differential equations, but their evolution involves past value of the state variable. Muthukumar and Thiagu [32] proved the existence of solutions and approximate controllability of fractional nonlocal stochastic differential equations of order ∈ (1,2] with infinite delay and Poisson jumps in Hilbert spaces by using fixed point theory and natural assumption that the corresponding linear system is approximately controllable. Hence problem of existence of approximate controllability for nonlinear fractional impulsive stochastic differential equations with nonlocal conditions and infinite delay has been studied by several authors were received significant attention in modern days (see [10,12,33,40,49] and references therein). Rajivganthi et al. [38] studied existence of solutions and approximate controllability of impulsive fractional stochastic differential systems with infinite delay and poisson jumps.
Many authors (see [21,22,23,30,39]) established the existence and approximate controllability of different types of functional differential equations with state-dependent delay. Fractional differential equations with state-dependent delay appear frequently in applications as models of equations and for this reason the study of this type of equations has been receiving great attention in recent years (see [3,7,43,45] and references therein). Many authors (see [13,19,26,48,51] and references therein) studied the existence and approximate controllability as well as stability of different types of fractional stochastic differential equations with state-dependent delay in Hilbert spaces under different suitable aspects. Selvarasu et al. [41] established approximate controllability of impulsive fractional stochastic integro-differential systems with state-dependent delay and poisson Jumps of order 1 < < 2. Moreover approximate controllability results for fractional impulsive stochastic differential system of order ∈ (1,2] with nonlocal, state-dependent delay and Poisson jumps in Hilbert spaces has not yet been derived in the literature. The main purpose of this paper is to obtain the sufficient conditions of approximate controllability results for fractional impulsive stochastic differential system of order Here, the state variable (⋅) takes values in a real separable Hilbert space D with the inner product 〈⋅,⋅〉 and norm ∥⋅∥ H . The fractional derivative , 1 < ≤ 2 is understood in the Caputo sense. Let 0 = 1 < . < ⋯ < / < /3. = + be the given time points. The control function (⋅) is given in Hilbert space with the above norm topology. Let ‡ be the closed subspace of all continuously differentiable process that belong to the space ‡(', ℒ K (Ω; D)) consisting of U -adapted measurable process such that U 1 -adapted processes 4 , = ∈ ℒ K (Ω, ℬ).
In this work we will employ an axiomatic definition of the phase space ℬ introduced by Hale and Kato [20]. The axioms of the space ℬ are established for ℱ ? -measurable functions from ' 1 into D , where ' 1 : = (−∞, 0 ], endowed with a seminorm ∥⋅∥ H. We will assume that ℬ satisfies the following axioms: Let ℤbe the closed subspace of all continuous process that belong to the space O(' . , ℒ K (Ω; D)) consisting of U •adapted measurable process and U 1 -adapted processes 4 , = ∈ ℒ K (Ω, ℬ) and the restriction : It is easy to verify that ℤ furnished with the norm topology as defined above is a Banach space. Let (D) be the space of all bounded linear operators on D . Let @ be the identity operator on D . If is a linear operator on D, then '(W, ) = (W@ − ) . denotes the resolvent operator of . We can use the notation The foiiowing important defitions 2.1 -2.8 are carried out from Kexue et al. [27]. Consider a class of fractional differential system with infinite delay ( ) = ( ) + ( ), ∈ [0, +], where ∈ (1,2], is the Caputo fractional derivative, is the infinitesimal generator of a strongly continuous - An operator is said to belong to O (©, ! 1 ), if problem (2) has an q-order cosine family O ( ) satisfying the above inequality.
Assume ∈ O (©, ! 1 ) and let O ( ) be the corresponding -order cosine family. Then we have Based on the result found in Kexue et al. [27], the functi on The solution of system (1) for a given control (⋅) ∈ ℒ K (', L) denoted by (⋅; ). In particular, the state of system (1) at = +, (+; ) is called the terminal state with control .. The set ' g (%) = { (+; ); (⋅) ∈ ℒ K (', L)} is called the reachable set of system (1). In what follows, ' g (%) stands for the closure of ' g (%) in the space D. Definition 2. 9 The system (1) is said to be approximately In order to study the approximate controllability (see [12,32,38,42,51]) of the fractional control system (1), we need to introduce the relevant operator where * denotes the adjoint of and y * (y) is the adjoint of y ( ). It is clear that the operator -1 g is a linear bounded operator.
(H0)¯'(¯, -1 g ) → 0 as ¯→ 0 3 in the strong operator topology. Note that the assumption (H0) is equivalent to the fact that the fractional linear control system (2) is approximately controllable on '.
Definition 2.10 It is clear that under these conditions the  In order to prove the theorems, we need the following hypotheses: ( We can prove the next lemma using the phase space axioms. Lemma 2.11 Let : (−∞, +] → D be a function such that ∈ cO(', ℒ K ).  Proof. Let ℬ g 1 = { ∈ cO(', ℒ K ): 1 ∈ ℬ} be the space endowed with uniform convergence topology. On the space ℬ g , consider a set ℬ Ö = { ∈ ℬ g 1 : ∥ ∥ K ≤ "},
Next we shall prove that arbitrary Ä > 0, the operator Υ maps ℬ Ö into a relatively compact subset of ℬ Ö . First, we prove that the set V( ) = {Υ ( ); ∈ ℬ Ö } is relatively compact in Dfor every ∈ '. It is clear that the case = 0 is obvious. Hence V is equicontinuous on '. Finally, we have to show that the map Υ(⋅) is continuous on ℬ Ö . Let ( / ) /∈ℕ be a sequence in ℬ Ö and ∈ ℬ Ö such that / → in cO(', ℒ K ). From the phase space axiom, it is easy to see that ( / ) Z → Z as : → ∞ uniformly for _ ∈ (−∞, +]. Hence Õ (+) → g holds in D and consequently we get the approximate controllability of system (1).
It is clear that the fractional differential system (2) is approximately controllable on 'for = 2. Hence for = 2 with the above choices, the system (3) can be rewritten to the abstract form (1) and all the conditions of Theorem 3.1 are satisfied. Thus there exists mild solutions for the system (3). Moreover all the conditions of Theorem 3.2 are satisfied and hence the fractional stochastic differential equations with Poisson jumps (3) is approximately controllable on '.
Because of the compactness of y ( ) generated by ,the associated linear system (2) is approximately controllable [30]. Hence by Theorem (3.2), the system (3) is approximately controllable.

Conclusion
This paper has investigated the existence of mild solutions and approximate controllability of second order ( ∈ (1,2) ) fractional impulsive stochastic differential system with nonlocal, state-dependent delay and Poisson umps in Hilbert spaces. For the future work, the controllability and stability results could be extended to study the neutral impulsive fractional stochastic differential systems with state-dependent delay and Poisson jumps satisfying the nonlocal condition according to the method in this paper. Hence our main results are the generalization of the recent results on fractional stochastic control systems with state-dependent delay and Poisson jumps.