Controllability of a Stochastic Neutral Functional Differential Equation Driven by a fBm

In this paper, we consider a class of Sobolev-type fractional neutral stochastic differential equations driven by fractional Brownian motion with infinite delay in a Hilbert space. When 1 H α > − , by the technique of Sadovskii’s fixed point theorem, stochastic calculus and the methods adopted directly from deterministic control problems, we study the approximate controllability of the stochastic system.


Introduction
As an important part of mathematical control theory, the research on approximate controllability has attracted more and more attention [1] [2] [3].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 [4] [5].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, [6] [7]).In addition, neutral stochastic differential equations with infinite delay have become very useful in the mathematical models of physical and social sciences [8] [9].So, it is necessarily and significatively to study fractional order neutral differential equations of Sobolev-type ( [10] [11] 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 [12] [13] and references therein for the details of the theory of stochastic calculus for fractional Brownian motion.In [14], 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 • c D α 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 ( ) ( ) • 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).

Preliminaries
In this section, we will introduce some definitions, lemmas and notions which will be used in the next section.

Fractional Brownian Motion
Let ( ) ( ) The fBm H β admits the following integral representation: for all 0 t ≥ , where is a standard Brownian motion and the kernel ( ) with a normalizing constant 0 Let  be the completion of the linear space  generated by the indicator with respect to the inner product is an isometry from  to the Gaussian space generated by H β and it can be extended to  , which is called the Wiener integral with respect to H β .e ∈ is a complete orthogonal basis in W, and

Consider the operator
 is a sequence of independent fBms with the same Hurst • Q is a non-negative self-adjoint trace class operator with finite trace where , L W U is the space of all Hilbert-Schmidt operators from .

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 I α of order α for a function ∫ provided the right side is point-wise defined on [ ) 0, ∞ , where ( ) Γ ⋅ is the gamma function, which is defined by ( ) . Moreover, the Caputo derivative c D α of order α for a function 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 , , U V W is three real separable Hilbert spaces with inner products , U ⋅ ⋅ , , V ⋅ ⋅ and , W ⋅ ⋅ , respectively.We first give some conditions about the three operators , , L A B as follows: (A1) A and L are two linear operators on U such that ( ) and A is closed, From the above assumptions (1)-(3) and the closed graph theorem it follows that the linear operator is bounded, and For x U ∈ , we define two families Lemma 2.3 Feckan, M. et al. [15] The operators ( ) have the following properties: ( ) are linear and bounded, and moreover .
( ) are strong continuous and compact.
We now introduce the abstract phase space.For a continuous function ( ] ( ) Cui and Yan [16]) We present the definition of mild solutions of (1.1).
Definition 2.2 An U-valued stochastic process , iii) ( ) ( ) Finally, in order to prove our main statement, we need some conditions as follows.
(B1) Let the function is continuous and there exist some (B2) For the complete orthogonal basis { } n n e ∈ in W, the function [ ] ( ) . In addition, there exist some 0 t and 0 δ > such that ( ) ( ) ( ) ( ) is continuous and satisfies: (a) there exist some constants 0 and , h ξ η ∈B such that the function AG satisfies the Lipschitz condition ( ) ( ) (B4) There is a constant

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 Consider the corresponding linear fractional deterministic control system to ( ) ( Theorem 3.1 (Daher [19]) 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 ( ) 0 for a convex, closed and bounded set N of X, then Ф has a fixed point in X (where ( ) µ ⋅ denotes Kuratowski's measure of noncompactness).
where ( ) is given in Section 2.
Theorem 3.2 Assume the conditions (B1)-(B4) hold, then for each 0 λ > there exists a mild solution of (1.1) on ( ] We will show that Ф has a fixed point which is a mild solution for system (1.1).
Then, ( ) . It is easy to check that ( ) x t satisfies (1.1) if and only if 0 0 y = and . 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 Now, we need to prove the operator 1 Ψ is a contraction map and 2 Ψ is compact.
It follows that 1 Ψ is a contraction map with the assumption Step II.We claim that 2 Ψ is compact.In [14], we have proved that By using Hölder inequality and the assumption (B1) we have ( ) For the last parts 5 6 , J J when 1 H α > − , we have E 0 as 0, 0. y t y t Therefore, there are relatively compact sets arbitrary close to the set Theorem 3.3 Assume that the conditions of Theorem 3.2 and ( 0 ) hold.In addition, the functions f is uniformly bounded on its domain.Then, the fractional control system (1.1) is approximately controllable on [ ] 0,T .

Conclusion
We consider the following Sobolev-type fractional neutral stochastic differential

TΓ
is clear that the operator 0 is a linear bounded operator.The fact that the linear Sobolev-type fractional control system (3.2) is approximately controllable on [ ] 0,T is equivalent to the next hypothesis (see, for example, define the control function u λ as follows.

Lemma 3 . 2 (
Li and Liu [20]) Assume that a x ∈B , then for all

B
for each r.According to Lemma 3.2, we get

1 Ψ
is a contraction map.For , b y y ∈B , we have

2 Ψ maps bounded sets into bounded sets of b B and 2 Ψ
maps bounded sets into equicontinuous sets of b B .It is enough to prove that 2

(
Sadakovskii's fixed point theorem (Theorem 3.1), the operator Ψ has a fixed point which is a solution to the system (1.1).
a phase space h .