Approximate controllability of fractional nonlinear neutral stochastic differential inclusion with nonlocal conditions and infinite delay

In this paper we consider a class of fractional nonlinear neutral stochastic evolution inclusions with nonlocal initial conditions in Hilbert space. Using fractional calculus, stochastic analysis theory, operator semigroups and Bohnenblust–Karlin’s fixed point theorem, a new set of sufficient conditions are formulated and proved for the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusions under the assumption that the associated linear part of the system is approximately controllable. An example is provided to illustrate the theory.

Cauchy problems, stochastic differential equations with nonlocal conditions were studied by many authors and some basic results on nonlocal problems have been obtained. Balasubramaniam et al. [3] investigated the approximate controllability of fractional impulsive integro-differential systems with nonlocal conditions in a Hilbert space. Slama and Boudaoui [40] obtained sufficient conditions for the existence of mild solutions for the fractional impulsive stochastic differential equation with nonlocal conditions and infinite delay. For more details see [4,33,39] and the references contained therein.
The controllability is one of the fundamental concepts in linear and nonlinear control theory, and plays a crucial role in both deterministic and stochastic control systems. Moreover, approximate controllable systems are more prevalent and very often approximate controllability is completely adequate in applications (see [37,38]). Approximate controllability for semilinear deterministic and stochastic control systems can be found in Mahmudov [26]. Moreover, there are many researchers discussing the approximate controllability for the stochastic fractional systems; for example, see [4,7,33], and the references therein.
For fractional differential inclusions, Sakthivel et al. [38] formulated and proved a new set of sufficient conditions for the approximate controllability of fractional nonlinear differential inclusions. Yan and Jia [43] investigated the existence of mild solutions for a class of impulsive fractional partial neutral functional integrodifferential inclusions with infinite delay and analytic α-resolvent operators in Banach spaces. Yan and Jia [44] studied the approximate controllability of partial fractional neutral stochastic functional integro-differential inclusions with state-dependent delay under the assumptions that the corresponding linear system is approximately controllable. Guendouzi and Bousmaha [17] investigated the approximate controllability for a class of fractional neutral stochastic functional integro-differential inclusions involving the Caputo derivative in Hilbert spaces. A new set of sufficient conditions are formulated and proved for the approximate controllability of fractional stochastic integro-differential inclusions under the assumption that the associated linear part of system is approximately controllable.
Recently also, Yan and Lu [45] considered the approximate controllability of a class of fractional stochastic neutral integro-differential inclusions with infinite delay in Hilbert spaces. Sakthivel et al. [39] investigated the approximate controllability of fractional stochastic differential inclusions with nonlocal conditions and established the approximate controllability results for the fractional stochastic control system with infinite delay.
However, to the best of our knowledge, so far no work has been reported in the literature about the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusions with nonlocal conditions and infinite delay of the form (2.1). Inspired by the above mentioned works, the aim of this paper is to fill this gap. The purpose of this paper is to show the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusion of the form (2.1) in a Hilbert space under simple and fundamental assumptions on the system operators, in particular that the corresponding linear system is approximate controllable.
The structure of this paper is as follows: In Sect. 2 we briefly present some basic notations and preliminaries. Section 3 is devoted to the existence of solutions for fractional stochastic control system (2.1). In Sect. 4 we establish the approximate controllability of fractional stochastic control system (2.1). An example to illustrate our results is given in Sect. 5. In the last section, concluding remarks are given.

Preliminaries
In this section, we introduce some notations and preliminary results, needed to establish our results. Throughout this paper, H, U be two separable Hilbert spaces and L(U, H) be the space of bounded linear operators from U into H. For convenience, we will use the same notation . to denote the norms in H, U and L(U, H), and use ., . to denote the inner product of H and U without any confusion. Let ( , F, {F t } t≥0 , P) be a complete filtered probability space satisfying that usual conditions (i.e., it is increasing and right continuous, while F 0 contains all P-null sets of F), and E(.) denotes the expectation with respect to the measure P. An H-valued random variable is an F-measurable function x(t) : J → H, and the collection of random variables S = {x(t, w) : → H/ t ∈ J } is called a stochastic process. Generally, we just write x(t) instead of x(t, w) and x(t) : J → H in the space of S. Let {e i } ∞ i=1 be a complete orthonormal basis of U. Suppose that W = (W t ) t≥0 is a cylindrical U-valued Wiener process with a finite trace nuclear covariance operator are mutually independent one-dimensional standard Wiener processes. We assume that F t = σ {W (s) : 0 ≤ s ≤ t} is the σ -algebra generated by W and F T = F. Let L(U, H) denote the space of all bounded linear operator from U to H. equipped with the usual operator norm . . For ϕ ∈ L(U, H we define is a Hilbert space with the above norm topology. The collection of all strongly measurable, square integrable, H-valued random variables, denoted by The main aim of the present article is to study the approximate controllability of fractional nonlinear stochastic differential inclusions with nonlocal conditions of the form where D α t is the Caputo fractional derivative of order α, 0 < α < 1, the state variable x(.) takes the value in the separable Hilbert space H; A : D(A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup of a bounded linear operators T (t), t ≥ 0 in the Hilbert space H. The history Let A : D( A) ⊂ H → H is the infinitesimal generator of a strongly continuous semigroup of a bounded linear operators T (t), t ≥ 0 in the Hilbert space H. That is to say, T (t) ≤ M for some constant M ≥ 1 and every t ≥ 0. Without loss of generality, we assume that 0 ∈ ρ(A), the resolvent set of A. Then it is possible to define the fractional power A α for 0 < α ≤ 1, as a closed linear operator on its domain D( A α ) with inverse A −1 .
The nonlocal term g has a better effect on the solution and is more precise for physical measurements than the classical condition x(0) = x 0 alone [38]. For example, g(x) can be written as where c k = 1; 2; . . . ; n) are given constants and 0 < t 1 < · · · < t n ≤ T . Now, we present the abstract space phase B h . Assume that h : is bounded and measurable function on [34,35]. Now we consider the space We endow a seminorm . B b on B b , it is defined by We recall the following lemma Lemma 2.1 [34] Assume that x ∈ B b ; then for t ∈ J, x t ∈ B h . Moreover The following are basic properties of A α : (v) A α is a bounded linear operator for 0 ≤ α ≤ 1 in H.
Let us recall the following known definition. For more details see [19,28,30].
Definition 2.2 [30] The fractional integral of order α with the lower limit 0 for a function f is defined as provided the right-hand side is pointwise defined on [0, ∞), where the is the gamma function.
Definition 2.3 [12] The Caputo derivative of order α for a function f : [0, ∞) → R, which is at least n-times differentiable can be defined as Obviously, the Caputo derivative of a constant is equal to zero. The Laplace transform of the Caputo derivative of order α > 0 is given as If f is an abstract function with values in X , then integrals which appear in the above definitions are taken in Bochner's sense.
We also introduce some basic definitions and results of multivalued maps. For more details on multivalued maps, we refer to [5,14,18].

Definition 2.4 A multivalued map
is bounded in H for any bounded set B of H, that is, If the multivalued map G is completely continuous with nonempty values, then G is u.s.c., if and only if G has a closed graph, i.e., x n → x * , y n → y * ; y n ∈ Gx n imply y In the following, BCC(H) denotes the set of all nonempty bounded, closed and convex subset of H.
for all x 2 ≤ r and for a.e. t ∈ J Lemma 2.9 [20] Let J be a compact real interval, BCC(H) be the set of all nonempty, bounded, closed and convex subset of H and G be a L 2 -Caratheodory multivalued map S G,x = ∅ and let be a linear continuous mapping from L 2 (J, H) to C(J, H). Then, the operator

x is known as the selectors set from G, is given by
We present the definition of mild solution for the system (2.1).

Definition 2.10
is satisfied, where T α (t) and S α (t) are called characteristic solution operators and given by The following lemmas will be used in the proof of our main results.
Lemma 2.12 [46] The operators T α and S α have the following properties: (i) For any fixed t ≥ 0, T α (t) and S α (t) are linear and bounded operators, i.e., for any x ∈ H, , t ≥ 0} are strongly continuous; which means that for x ∈ H and for At the end of this section, we recall the fixed point theorem of Bohnenblust and Karlin's ( [6]) which is used to establish the existence of the mild solution to the system (2.1). [6]) Let D be a nonempty subset of G, which is bounded, closed, and convex. Suppose G :

Existence of solutions for fractional stochastic control system
In this section, we first prove the existence of solutions for fractional control system (2.1) by using Bohnenblust-Karlin's fixed point theorem. Secondly, we show that under certain assumptions, the approximate controllability of the fractional stochastic inclusion (2.1) is implied by the approximate controllability of the associated linear part (3.1). Definition 3.1 Let x T (φ, u) be the state value of (2.1) at the terminal time T corresponding to the control u and the initial value φ. Introduce the set which is called the reachable set of (2.1) at the terminal time b and its closure in H is denoted by R(T, φ). The system (2.1) is said to be approximately controllable on the interval J if R(T, φ) = H; that is, given an arbitrary > 0, it is possible to steer from the point π(0) to within a distance from all points in the state space H at time T .
In order to study the approximate controllability for the fractional control system (2.1), we consider its fractional linear part It is convenient at this point to introduce the controllability and resolvent operators associated with (3.1) as in the strong operator topology. In order to establish the existence result, we need the following hypothesis:  H) is an L 2 -Caratheodory function satisfies the following conditions: for a.e. t ∈ J and the function s The following lemma is required to define the control function.

Lemma 3.3 [25]
For any Now, for any λ > 0 and x T ∈ L 2 (F T , H ), we define the control function Let us now explain and prove the following theorem about the existence of solution for the fractional system (2.1).

Theorem 3.4
Assume that the assumptions (H1)-(H5) hold. Then for each > 0, the system (2.1) has a mild solution on J provided that Proof In order to prove the existence of mild solutions for system (2.1) transform it into a fixed point problem.
For any > 0, we consider the operator where σ ∈ S G,x . For φ ∈ B h , we define φ by It is evident that y satisfies y 0 = 0, t ∈ (−∞, 0] and The set B r is clearly a bounded closed convex set in B 0 b for each r > 0 and For each y ∈ B r . By Lemma 2.1 we have Define the multi-valued map : B r → 2 B r by y the set of z ∈ B r and there exists σ ∈ L 2 (L(U, H)) such that σ ∈ S G,x and Obviously, the operator has a fixed point is equivalent to has one. So, our aim is to show that has a fixed point. For the sake of convenience, we subdivide the proof into several steps.
Step 1: is convex for each y ∈ B r . Let if z 1 , z 2 belong to y, then there exist σ 1 , σ 2 ∈ S G,x such that Let 0 ≤ γ ≤ 1. Then for each t ∈ J , we have Step 2: We show that there exists some r > 0 such that (B r ) ⊆ B r . If it is not true, then there exists λ > 0 such that for every positive number r and t ∈ J , there exists a function y r (.) ∈ B r , but (y r ) / ∈ B r . that is, E ( y r )(t) 2 = { z r 2 B b : z r ∈ y r } ≥ r . For such λ > 0, we can show that for some σ r ∈ S G,x .
Let us estimate each term above I i , i = 1, . . . , 6. By Lemma 2.1 and assumptions (H1)-(H5), we have By a standard calculation involving Lemma 2.12, assumption (H2), Eq. (3.5) and the Holder inequality, we can deduce that . For I 4 , we have where B ≤ M B . By using (H2)-(H5) Holder's inequality, Eq. 3.5 and Lemma 2.12, for some σ r ∈ S G,x , we get u λ (s, y r + φ)) 2 Thus; Together with assumption (H3) and Eq. (3.5), we have A similar argument involves Lemma 2.12, assumption (H4) and Eq. (3.5); we obtain (3.13) Therefore, with these estimates (3.8)-(3.13), (3.7) becomes (3.14) Dividing both sides of (3.14) by r and taking r → ∞, we obtain that which is a contradiction to our assumption. Thus for α > 0, for some positive number r and some Step 3: y is equicontinuous. Let > 0 small, 0 < < t < t + h ≤ T . For each y ∈ B r and z belong to y, there exists σ ∈ S G,x such that for each t ∈ J , we have Applying Lemma 2.12 and the Holder inequality, we obtain Therefore, for sufficiently small, and by the compactness of T (t), S(t), we can verify that the right-hand side of the above inequality tends to zero as h → 0. Thus E z 2 (t + h) − z 1 (t) 2 → 0 as h → 0 for all x ∈ B r . This implies that maps B r into an equicontinuous of functions.
Step 4: Next we show that the set V (t) = {( y)(t) : y ∈ B r } is relatively compact in H. The case t = 0 is trivial. Let t ∈ [0, T ] be fixed and for each ε ∈ (0, t), arbitrary δ > 0 and y ∈ B r , we define the operator ε,δ y the set of z ε,δ ∈ B 0 b such that ∈ (0, t) and for all δ > 0. On the other hand, we have for every y ∈ B r , we have From (3.17) to (3.26), it can be easily seen that J 1 -J 10 tends to zero as ε → 0 and δ → 0. Thus, for Therefore, there are relative compact sets arbitrarily close to the set V (t), t > 0. Hence, the set V (t), t > 0 is also relatively compact in H.

Step 5:
has a closed graph. Let y n → y * , as n → ∞, z n ∈ y n for each y n ∈ B r . and z n → z * as n → ∞. We shall show that z * ∈ y * . Since z n ∈ y n , then there exists σ n ∈ S G,y n such that We must prove that there exists σ * ∈ S G,y * such that (3.28) Now, for every t ∈ J , since h is continuous, and from the definition of u λ we get By the conditions (H2), we can choose a sufficiently small positive constant > 0, α + < 1, such that On the other hand, by assumption (H 6), the operator λ(λI + T s ) −1 −→ 0 strongly as λ −→ 0 + for all 0 ≤ s ≤ T , and, moreover, λ(λI + T s ) −1 ≤ 1.Thus, the Lebesgue dominated convergence theorem and the compactness of S α (t) yield x T holds, which shows that the system (2.1) is is approximately controllable and the proof is complete.

Remark 4.2
We notice that, in the case of infinite-dimensional systems, we can distinguish two concepts of controllability: Exact and approximate controllability. Exact controllability means that the system can be steered to an arbitrary final state. Approximate controllability enables us to steer the system to an arbitrary small neighborhood of the final state. In the finite-dimensional case, the notions of approximate and exact controllability coincide. Moreover, Approximate controllable systems are more prevalent and very often approximate controllability is completely adequate in applications (see [37,38]). However, the problems of exact controllability are developed in numerous papers. Ren et al. [36] studied the controllability of a class of impulsive neutral stochastic functional differential inclusions with infinite delay in an abstract space. Sufficient conditions for the controllability are derived with the help of the fixed point theorem for discontinuous multivalued operators due to Dhage [15]. Li and Zou [23] obtained sufficient conditions for the controllability of nonlinear neutral stochastic differential inclusions with infinite delay in a Hilbert space with using a fixed-point theorem for condensing maps due to ORegan [29]. Li and Peng [24], Ganesh Priya and Muthukumar [31] studied the controllability of a class of fractional stochastic functional differential systems. Based on these works, the exact controllability of the system 2.1 can be done by relying on a fixed-point theorem for condensing maps due to ORegan [29] and employing the idea and technique as in Theorem 8 in [23].
Let U = H = L 2 ([0, π]) with the norm . . Now, we present a special phase space B h . Let h(t) = e 2t , t < 0, Then l = 0 −∞ h(s)ds = 1 2 . Let Define an infinite-dimensional space U by U = {u/u = ∞ n=2 u n w n with ∞ n=2 U 2 n < ∞}. The norm in U is defined by u 2 U = ∞ n=2 U 2 n . Now, define a continuous linear mapping B from U into H as Bu = 2u 2 w 1 + ∞ n=2 u n w n for u = ∞ n=2 u n w n ∈ U. We define the operator A by Ax = ∂ 2 x ∂ y 2 . with domain It is well known that A generates an analytic semigroup T (t), t ≥ 0 given by e −n 2 t x, e n e n , x ∈ H , and e n (y) = (2/π) 1/2 sin(ny), n = 1, 2, . . . , is the orthogonal set of eigenvectors of A.  (θ)(x))dθ, Then, the system (5.1) can be rewritten as the abstract form of system (2.1). Thus, under the appropriate conditions on the functions h, f , G and g as those in (H1)-(H6), system (5.1) has a mild solution and is approximately controllable on J .

Concluding remarks
In this paper, we have investigated the approximate controllability of class of fractional neutral stochastic evolution inclusion with nonlocal initial conditions in Hilbert space. Based on a fixed-point theorem, sufficient conditions for the existence of solutions and the approximate controllability of fractional nonlinear stochastic differential inclusions have been derived. Impulsive fractional differential equations and inclusions have become important in recent years as mathematical models of many phenomena in both physical and social sciences [38]. For the basic theory of impulsive differential equations and inclusions the reader can refer to [5,21]. Recently, Debbouche and Baleanu [13] established the controllability result for a class of fractional evolution nonlocal impulsive quasilinear delay integro-differential systems in a Banach space by using the theory of fractional calculus and fixed point technique. More recently, Liu and Li [22] established the controllability of impulsive fractional evolution differential inclusions with initial boundary conditions in Banach spaces by applying the fixed point theorem for multivalued maps due to Dhage association with an evolution system. Upon making some appropriate assumption on system functions, by adapting the techniques and ideas established in the paper of [13] and [22] with suitable modifications, one can prove the existence of solutions and the approximate controllability of fractional stochastic differential inclusions with nonlocal condition and impulses of the form: In our future work we will investigate the existence and controllability results of fractional stochastic differential inclusions driven by fractional Brownian motion.