Controllability of a Stochastic Neutral Functional Differential Equation Driven by a fBm ()
1. 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
(1.1)
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).
2. Preliminaries
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
The mapping
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
(2.1)
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
and
where
is a probability density function defined on
.
Lemma 2.3 Feckan, M. et al. [15] The operators
and
have the following properties:
・ For every
,
and
are linear and bounded, and moreover for every
(2.2)
・
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 [16] )
for
.
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
(2.3)
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
and
.
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
(3.1)
with
.
Consider the corresponding linear fractional deterministic control system to (1.1)
(3.2)
and define the relevant operators
(3.3)
and
(3.4)
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 [17] ):
・
in the strong operator topology, as
.
Lemma 3.1 (Guendouzi and Idrissi [18] ) For any
, there exists
such that
(3.5)
For any
and
, we now define the control function
as follows.
(3.6)
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
. 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
(3.7)
where
denotes the space of all continuous U-valued stochastic process
.
Lemma 3.2 (Li and Liu [20] ) Assume that
, then for all
. Moreover,
(3.8)
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
(3.9)
for
.
We will show that Ф has a fixed point which is a mild solution for system (1.1). For
, define
(3.10)
Then,
. Let
. It is easy to check that
satisfies (1.1) if and only if
and
(3.11)
Denote
and let
be the seminorm in
, defined by
(3.12)
For
we set
. Then,
is a bounded closed convex set in
for each r. According to Lemma 3.2, we get
(3.13)
for
. Define the mapping
by
(3.14)
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
(3.15)
(3.16)
Now, we need to prove the operator
is a contraction map and
is compact.
Step I.
is a contraction map. For
, we have
(3.17)
It follows that
is a contraction map with the assumption
.
Step II. We claim that
is compact. In [14] , 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
(3.18)
Since
is a compact operator, the set
is precompact in U for every
. For each
(3.19)
By using Hölder inequality and the assumption (b1) we have
(3.20)
and
(3.21)
Since
(3.22)
where
(3.23)
it follows that
(3.24)
and
(3.25)
For the last parts
when
, we have
(3.26)
which imply that
(3.27)
and
(3.28)
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
(3.29)
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
(3.30)
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).
4. Conclusion
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 [14] , we show the existence of solution and approximate controllability of (1.1).
Funding
The Project-sponsored by NSFC (No. 11571071).