As an important part of mathematical control theory, the research on approximate controllability has attracted more and more attention    . Approximate controllability means that the system can be steered to a small neighborhood of the final state. In fact, the approximate controllability of systems has been studied by several authors   . During the past three decades, the importance of fractional differential equations and their applications are prominent, especially in modeling several complex phenomena such as anomalous diffusion of particles (see, for examples,   ). In addition, neutral stochastic differential equations with infinite delay have become very useful in the mathematical models of physical and social sciences   . So, it is necessarily and significatively to study fractional order neutral differential equations of Sobolev-type (   and references therein).
On the other hand, the properties of long/short-range dependence are widely used in describing many phenomena in fields like hydrology and geophysics as well as economics and telecommunications. As extension of Brownian motion, fractional Brownian motion (fBm) is a self-similar Gaussian process which has the properties of long/short-range dependence. However, fractional Brownian motion is neither a semimartingale nor a Markov process (except for the case when it is a Brownian motion). For this reason, there are a few publications leaning the systems which are driven by this type of noise. We refer   and references therein for the details of the theory of stochastic calculus for fractional Brownian motion. In  , authors consider the approximate controllability of a class of Sobolev-type fractional stochastic equation driven by fractional Brownian motion in a Hilbert space.
Motivated by these results, in this paper, we study the approximate controllability of the Sobolev-type fractional stochastic differential equations of the form
In the above system, we assume that
・ is the Caputo fractional derivative of order ,
・ A, L are two linear operators on a Hilbert space U,
・ B is a bounded linear operator from the Hilbert space V into Hilbert space U,
・ The time history , ,
・ is a control function on ,
・ is a cylindrical fractional Brownian motion with Hurst index ,
・ The functions G, f and σ are Borel functions with some suitable conditions.
The paper is organized as follows. In Section 2, we represent some preliminaries for stochastic integral of fractional Brownian motion in Hilbert space. In Section 3, we obtain the approximate controllability results of the Sobolev-type fractional neutral stochastic system (1.1).
In this section, we will introduce some definitions, lemmas and notions which will be used in the next section.
2.1. Fractional Brownian Motion
Let be a complete filtered probability space. A fractional Brownian motion (fBm) with Hurst index is a mean zero Gaussian process such that and
for all . When , coincides with the standard Brownian motion, and when it is neither a semi-martingale nor a Markov process. The fBm admits the following integral representation:
for all , where is a standard Brownian motion and the kernel satisfies
with a normalizing constant such that . Throughout this paper we assume that is arbitrary but fixed.
Let be the completion of the linear space generated by the indicator functions with respect to the inner product
is an isometry from to the Gaussian space generated by and it can be extended to , which is called the Wiener integral with respect to . Consider the operator from to defined by
for . Then, the operator is an isometry between and which can be also extended to the Hilbert space .
Lemma 2.1 For every , we have
We now recall that the definition of stochastic integral of fBm in the Hilbert space V. Let be a W-valued -adapted fBm defined on with the representation of the form
where is a complete orthogonal basis in W, and
・ is a sequence of independent fBms with the same Hurst index ,
・ is a bounded sequence of non-negative real numbers such that ,
・ Q is a non-negative self-adjoint trace class operator with finite trace
Let such that
where is the space of all Hilbert-Schmidt operators from to U with norm defined by
Definition 2.1 Let satisfy (2.1). We define the stochastic integral by
Lemma 2.2 Let satisfy (2.1). Then, for any with we have
In addition, is uniformly convergent in , then, we have
2.2. Some Assumptions
In this subsection, we recall that some notions of fractional calculus and give some assumptions for the stochastic system (1.1). Recall that the fractional integral of order a for a function is defined as
provided the right side is point-wise defined on , where is the gamma function, which is defined by . Moreover, the Caputo derivative of order a for a function is defined as
If f is an abstract function with values in U, then the integrals appearing in the above definitions are taken in Bochner’s sense.
To study the stochastic system (1.1), we need some assumptions. Throughout this paper we assume that is three real separable Hilbert spaces with inner products , and , respectively. We first give some conditions about the three operators as follows:
(A1) A and L are two linear operators on U such that , , and A is closed,
(A2) and L is bijective,
(A3) is compact,
(A4) B is a bounded linear operator from V into U.
From the above assumptions (A1)-(A3) and the closed graph theorem it follows that the linear operator is bounded, and generates a semigroup in U. Denote , and .
For , we define two families and of operators by
is a probability density function defined on .
Lemma 2.3 Feckan, M. et al.  The operators and have the following properties:
・ For every , and are linear and bounded, and moreover for every
・ and are strong continuous and compact.
We now introduce the abstract phase space. For a continuous function satisfying
we define a phase space associated with h as follows
Clearly, is a Banach space if is endowed with the norm (see, Cui and Yan  )
We present the definition of mild solutions of (1.1).
Definition 2.2 An U-valued stochastic process is a mild solution of (1.1) if the next conditions hold:
i) is measurable and -adapted, and is -valued,
ii) is continuous on and the function is integrable for each such that satisfies the equation
iii) on such that .
Finally, in order to prove our main statement, we need some conditions as follows.
(B1) Let the function is continuous and there exist some constants , such that for and
for all and .
(B2) For the complete orthogonal basis in W, the function satisfy
and is uniformly convergent in . In addition, there exist some and such that
(B3) Let the function is continuous and satisfies:
(a) there exist some constants , for and such that the function AG satisfies the Lipschitz condition
for all and .
(b) there exist constants , such that
for all , and .
(B4) There is a constant such that , where
3. Main Results
In this section, we will show the approximate controllability of the stochastic system (1.1). We need to establish the existence of the solution for the stochastic control system and to show that the corresponding linear part is approximate controllability.
Definition 3.1 The system (1.1) is called to be approximately controllable on if
Consider the corresponding linear fractional deterministic control system to (1.1)
and define the relevant operators
where and denote the adjoint operators of B and , respectively. It is clear that the operator is a linear bounded operator. The fact that the linear Sobolev-type fractional control system (3.2) is approximately controllable on is equivalent to the next hypothesis (see, for example, Mahmudov and Denker  ):
・ in the strong operator topology, as .
Lemma 3.1 (Guendouzi and Idrissi  ) For any , there exists such that
For any and , we now define the control function as follows.
Theorem 3.1 (Daher  ) Let Ф be a condensing operator on a Banach space X, that is, Ф is continuous and takes bounded sets into bounded sets, and for every bounded set B of X with . If for a convex, closed and bounded set N of X, then Ф has a fixed point in X (where denotes Kuratowski’s measure of noncompactness).
Define the space
and let be a seminorm defined by
where denotes the space of all continuous U-valued stochastic process .
Lemma 3.2 (Li and Liu  ) Assume that , then for all . Moreover,
where is given in Section 2.
Theorem 3.2 Assume the conditions (B1)-(B4) hold, then for each there exists a mild solution of (1.1) on , provided that .
Proof. Define the operator by
We will show that Ф has a fixed point which is a mild solution for system (1.1). For , define
Then, . Let . It is easy to check that satisfies (1.1) if and only if and
Denote and let be the seminorm in , defined by
For we set . Then, is a bounded closed convex set in for each r. According to Lemma 3.2, we get
for . Define the mapping by
for . It is evident that the operator Φ has a fixed point if and only if the operator Ψ has a fixed point. Now, we divide Ψ into , where
Now, we need to prove the operator is a contraction map and is compact.
Step I. is a contraction map. For , we have
It follows that is a contraction map with the assumption .
Step II. We claim that is compact. In  , we have proved that maps bounded sets into bounded sets of and maps bounded sets into equicontinuous sets of . It is enough to prove that maps into a precompact set in . Define an operator on by
Since is a compact operator, the set is precompact in U for every . For each
By using Hölder inequality and the assumption (b1) we have
it follows that
For the last parts when , we have
which imply that
Then, for each ,
Therefore, there are relatively compact sets arbitrary close to the set is precompact in . By Arzela-Ascoli’s theorem, is compact. By Sadakovskii's fixed point theorem (Theorem 3.1), the operator Ψ has a fixed point which is a solution to the system (1.1).
Theorem 3.3 Assume that the conditions of Theorem 3.2 and (H0) hold. In addition, the functions f is uniformly bounded on its domain. Then, the fractional control system (1.1) is approximately controllable on .
Proof. Let be a fixed point of the operator . Using the stochastic Fubini theorem, we can get
It follows from the property of that there exists such that and . Then there is a subsequence denoted by weakly converging to . Thus, from the above equation, we obtain
On the other hand, by assumption (H0) for all , the operator strongly as , and moreover . Thus, by the Lebesgue dominated convergence theorem and the compactness of , we can get as . This gives the approximate controllability of (1.1).
We consider the following Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay:
where is a control function. Inspired by  , we show the existence of solution and approximate controllability of (1.1).
The Project-sponsored by NSFC (No. 11571071).
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