Approximate controllability of Sobolev type fractional stochastic nonlocal nonlinear differential equations in Hilbert spaces

. We introduce a new notion called fractional stochastic nonlocal condition, and then we study approximate controllability of class of fractional stochastic nonlinear differential equations of Sobolev type in Hilbert spaces. We use Hölder’s inequality, ﬁxed point technique, fractional calculus, stochastic analysis and methods adopted di-rectly from deterministic control problems for the main results. A new set of sufﬁcient conditions is formulated and proved for the fractional stochastic control system to be approximately controllable. An example is given to illustrate the abstract results.


Introduction
We are concerned with the following fractional stochastic nonlocal system of Sobolev type where C D q t and L D 1−q t are the Caputo and Riemann-Liouville fractional derivatives with 0 < q ≤ 1, and t ∈ J = [0, b]. Let X and Y be two Hilbert spaces and let the state x(·) take its values in X. We assume that the operators L and M are defined on domains contained in X and ranges contained in Y, the control function u(·) belongs to the space L 2 Γ (J, U), a Hilbert space of admissible control functions with U as a Hilbert space and B is a bounded linear operator from U into Y. It is also assumed that f : J × X → Y, σ 1 : J × X → L 0 2 and 2 M. Kerboua, A. Debbouche and D. Baleanu σ 2 : J × X → L 0 2 are appropriate functions; x 0 is a Γ 0 measurable X-valued random variable independent of w 1 and w 2 . Here L 0 2 , Γ, Γ 0 , w 1 and w 2 will be specified later. During the past three decades, fractional differential equations and their applications have gained a lot of importance, mainly because this field has become a powerful tool in modeling several complex phenomena in numerous seemingly diverse and widespread fields of science and engineering [2,5,11,16,18,28,29,32]. Recently, there has been a significant development in the existence results for boundary value problems of nonlinear fractional differential equations and inclusions [1,6].
One of the important fundamental concepts in mathematical control theory is controllability, it plays a vital role in both deterministic and stochastic control systems. Since, the controllability notion has extensive industrial and biological applications, in the literature, there are many different notions of controllability, both for linear and nonlinear dynamical systems. Controllability of the deterministic and stochastic dynamical control systems in infinite dimensional spaces is well-developed using different kind of approaches. It should be mentioned that the theory of controllability for nonlinear fractional dynamical systems is still in the initial stage. There are few works in controllability problems for different kind of systems described by fractional differential equations [41,42].
The exact controllability for semilinear fractional order system, when the nonlinear term is independent of the control function, is proved by many authors [3,12,38]. In these papers, the authors have proved the exact controllability by assuming that the controllability operator has an induced inverse on a quotient space. However, if the semigroup associated with the system is compact then the controllability operator is also compact and hence the induced inverse does not exist because the state space is infinite dimensional [46]. Thus, the concept of exact controllability is too strong and has limited applicability and the approximate controllability is a weaker concept than complete controllability and it is completely adequate in applications for these control systems.
In [10,44] the approximate controllability of first order delay control systems has been proved when nonlinear term is a function of both state function and control function by assuming that the corresponding linear system be approximately controllable. To prove the approximate controllability of a first order system, with or without delay, a relation between the reachable set of a semilinear system and that of the corresponding linear system is proved in [4,9,20,21,45]. There are several papers devoted to the approximate controllability for semilinear control systems, when the nonlinear term is independent of control function [25,39,40,43].
Stochastic differential equations have attracted great interest due to its applications in various fields of science and engineering. There are many interesting results on the theory and applications of stochastic differential equations, (see [3,7,8,30,36] and the references therein). To build more realistic models in economics, social sciences, chemistry, finance, physics and other areas, stochastic effects need to be taken into account. Therefore, many real world problems can be modeled by stochastic differential equations. The deterministic models often fluctuate due to noise, so we must move from deterministic control to stochastic control problems.
In the present literature there is only a limited number of papers that deal with the approximate controllability of fractional stochastic systems [27], as well as with the existence and controllability results of fractional evolution equations of Sobolev type [26].
R. Sakthivel et al. [37] studied the approximate controllability of a class of dynamic control systems described by nonlinear fractional stochastic differential equations in Hilbert spaces.
Approximate controllability of Sobolev type FSDE

3
In [24], the authors proved the approximate controllability of Sobolev type nonlocal fractional stochastic dynamic systems in Hilbert spaces. More recent works can be found in [41,42]. A. Debbouche, D. Baleanu and R. P. Agarwal [13] established a class of fractional nonlocal nonlinear integro-differential equations of Sobolev type using new solution operators. M. Fečkan, J. R. Wang and Y. Zhou [19] presented the controllability results corresponding to two admissible control sets for fractional functional evolution equations of Sobolev type in Banach spaces with the help of two new characteristic solution operators and their properties, such as boundedness and compactness. Debbouche and Torres [14,15] introduced both fractional nonlocal condition and nonlocal control condition for establishing approximate controllability of fractional delay differential equations and inclusions.
In this work, we present a new concept in stochastic analysis that we present a nonlocal condition given in stochastic term together with Riemann-Liouville fractional derivative, then we use this tool to establish the approximate controllability of Sobolev type fractional deterministic nonlocal stochastic control systems in Hilbert spaces.
The paper is organized as follows: in Section 2, we present some essential facts in fractional calculus, semigroup theory, stochastic analysis and control theory that will be used to obtain our main results. In Section 3, we state and prove existence and approximate controllability results for Sobolev type fractional stochastic system (1.1)-(1.2). Finally, in Section 4, as an example, a fractional partial dynamical stochastic control differential equation with a fractional stochastic nonlocal condition is considered.

Preliminaries
In this section we give some basic definitions, notations, properties and lemmas, which will be used throughout the work. In particular, we state main properties of fractional calculus [22,31,34], well known facts in semigroup theory [23,33,49] and elementary principles of stochastic analysis [30,35].
where Γ is the gamma function. If a = 0, we can write and as usual, * denotes the convolution of functions. Moreover, lim α→0 g α (t) = δ(t), with δ the delta Dirac function.
(ii) The Caputo derivative of a constant is equal to zero.
(iii) If f is an abstract function with values in X, then the integrals which appear in Definitions 2.1-2.3 are taken in Bochner's sense.
We introduce the following assumptions on the operators L and M.
(H 1 ) L and M are linear operators, and M is closed.
Remark 2.5. From (H 3 ), we deduce that L −1 is bounded operators, for short, we denote by C = L −1 . Note (H 3 ) also implies that L is closed since the fact: L −1 is closed and injective, then its inverse is also closed. It comes from (H 1 )-(H 3 ) and the closed graph theorem, we obtain the boundedness of the linear operator ML −1 : Y → Y. Consequently, ML −1 generates a semigroup {S(t) := e ML −1 t , t ≥ 0}. We suppose that M 0 := sup t≥0 S(t) < ∞. According to previous definitions, it is suitable to rewrite problem (1.1)-(1.2) as the equivalent integral equation   (c) The function x(0) takes the form x 0 + 1 (d) The explicit and implicit integrals given in (2.1) exist (taken in Bochner's sense).
(i) For any fixed t ≥ 0, S(t) and T (t) are bounded linear operators, i.e., for any x ∈ X, x .
ds is the controllability Gramian, here B * denotes the adjoint of B and T * (t) is the adjoint of T (t).
The following lemma is required to define the control function [37].

Lemma 2.11.
For any x b ∈ L 2 (Γ b , X), there exists ϕ ∈ L 2 Γ (Ω; L 2 (0, b; L 0 2 )) such that Approximate controllability of Sobolev type FSDE 7 Now for any α > 0 and x b ∈ L 2 (Γ b , X), we define the control function in the following form Lemma 2.12. There exist positive real constantsM,N such that for all x, y ∈ H 2 , we have Proof. We start to prove (2.5). Let x, y ∈ H 2 , from the Hölder's inequality, Lemma 2.7 and the assumption on the data, we obtain

M. Kerboua, A. Debbouche and D. Baleanu
The proof of the inequality (2.6) can be established in a similar way to that of (2.5).

Approximate controllability
In this section, we formulate and prove conditions for the existence and approximate controllability results of the fractional stochastic nonlocal dynamic control system of Sobolev type (1.1)-(1.2) using the contraction mapping principle. For any α > 0, define the operator We state and prove the following lemma, which will be used for the main results.
Then for any fixed x ∈ H 2 , from (3.1), we have From Lemma 2.7, we begin with the first term Approximate controllability of Sobolev type FSDE 9 The strong continuity of S(t) implies that the right-hand side of the last inequality tends to zero as t 2 − t 1 → 0. Next, it follows from Hölder's inequality and assumptions on the data that Also, we have Furthermore, we use Lemma 2.7 and previous assumptions, we obtain Hence using the strong continuity of T (t) and Lebesgue's dominated convergence theorem, we conclude that the right-hand side of the above inequalities tends to zero as t 2 − t 1 → 0. Thus, we conclude F α (x)(t) is continuous from the right of [0, b). A similar argument shows that it is also continuous from the left of (0, b]. Proof. We prove the existence of a fixed point of the operator F α by using the contraction mapping principle. First, we show that Using assumptions (i)-(ii), Lemma 2.12, and standard computations yield Thus for each α > 0, the operator F α maps H 2 into itself. Next, we use the Banach fixed point theorem to prove that F α has a unique fixed point in H 2 . We claim that there exists a natural n such that F n α is a contraction on H 2 . Indeed, let x, y ∈ H 2 , we have Hence, we obtain a positive real constant γ(α) such that for all t ∈ J and all x, y ∈ H 2 . For any natural number n, it follows from successive iteration of above inequality (3.5) that, by taking the supremum over J, For any fixed α > 0, for sufficiently large n, γ n (α) n! < 1. It follows from (3.6) that F n α is a contraction mapping, so that the contraction principle ensures that the operator F α has a unique fixed point x α in H 2 , which is a mild solution of (1.1)-(1.2). Proof. Let x α be a fixed point of F α . By using the stochastic Fubini theorem, it can be easily seen that It follows from the assumption on f , σ 1 and σ 2 that there existsD > 0 such that for all s ∈ J. Then there is a subsequence still denoted by { f (s, x α (s)), σ 1 (s, x α (s)), σ 2 (s, x α (s))} which converges weakly to some { f (s), σ 1 (s), σ 2 (s)} in Y × L 0 2 × L 0 2 . From the above equation, we have On the other hand, by assumption (iii), for all 0 ≤ s < b the operator α(αI + Ψ b s ) −1 → 0 strongly as α → 0 + and moreover α(αI + Ψ b s ) −1 ≤ 1. Thus, by the Lebesgue dominated convergence theorem and the compactness of both S(t) and T (t) implies that E x α (b) −x b 2 → 0 as α → 0 + . Hence, we conclude the approximate controllability of (1.1)-(1.2).
In order to illustrate the abstract results of this work, we give the following example.
Let It is easy to see that L −1 is compact, bounded with L −1 ≤ 1 and ML −1 generates the above strongly continuous semigroup S(t) on Y with S(t) ≤ e −t ≤ 1. Therefore, with the above choices, the system (4.1)-(4.3) can be written as an abstract formulation of (1.1)-(1.2) and thus Theorem 3.2 can be applied to guarantee the existence of mild solution of (4.1)-(4.3). Moreover, it can be easily seen that Sobolev type deterministic linear fractional control system corresponding to (4.1)-(4.3) is approximately controllable on J, which means that all conditions of Theorem 3.3 are satisfied. Thus, fractional stochastic nonlinear control system of Sobolev type (4.1)-(4.3) is approximately controllable on J.