1. Introduction
There are many papers which discuss stability of systems. It is called a stabilization problem when we impose such conditions on a given unstable system to make it stable. There have been rich literatures on this topic, here we only mention [1-4]. It is talked about suppression of noise in [1,2]. It is showed similar stabilization phenomena in stochastic systems as those in deterministic systems in [3,4]. They all indicate clearly that different structures of environmental noise may have different effects on the deterministic system. On the other hand, there are also many papers related to stabilization of functional systems, such as [5-8]. [5] investigates a stochastic Lotka-Volterra system with infinite delay, whose initial data come from an admissible Banach space C, and show that its unique global positive solution has asymptotic boundedness property by using the exponential martingale inequality. [6] studies existence and uniqueness of the global positive solution of stochastic functional Kolmogorov-type system and its asymptotic bound properties and moment average boundedness in time under the traditionally diagonally dominant condition. [7] studies the same problems as [6] under some other conditions. [8] discusses stabilization of a given unstable nonlinear functional system by introducing two Brownian noise.
Many practical systems may experience abrupt changes in their structure and parameters caused by phenomena such as component failures or repairs, changing subsystem interconnections, and abrupt environmental disturbances. The hybrid systems have been used to desribe such situations. Along the trajectories of the Markovian jump system, the mode switches from one value to another in a random way are governed by a Markov process with discrete state space. [9,10] studied the stability of a jump system. Feng et al. [11] systematically studied stochastic stability properties of jump linear systems and the relationship among various moment and sample path stability properties. Shen and Wang [12] presented new exponential stability results for recurrent neural networks with Markovian switching. Wang et al. [13] dealt with the problem of state estimation for a class of delayed neural networks with Markovian jumping parameters without the traditional monotonicity and smoothness assumptions on the activation function.
Taking both the environmental noise and jump into account, the system under consideration becomes a stochastic differential system with Markovian switching (SDSwMS), which has received a lot of attention (see [14-24]) recently. [17] provided some useful conditions on the exponential stability for general nonlinear SDSwMSs, which was improved by himself in Mao et al. [19]. Yuan and Lygeros [20] investigated almost sure exponential stability for a class of switching diffusion processes. [25] discusses the asymptotic stability and exponential stability of SDSwMSs, whose coefficients are assumed to satisfy the local Lipschitz condition and the polynomial growth condition.
Motivated by [25,26] and some other literatures, we will investigate suppression and stabilization by noise of functional differential system with Markov chains, whose coefficient satisfies the local Lipschitz condition and the one-sided polynomial growth condition. For a given unstable functional system with Markovian switching
(1)
where
![](https://www.scirp.org/html/6-7401907\63e91da3-a90c-41bc-8a24-d23f47f5d06b.jpg)
![](https://www.scirp.org/html/6-7401907\2256f173-d820-4f7c-9d84-90ad209c797e.jpg)
is defined by
, by introducing two independent scalar Brownian noise under some conditions, we get a stochastic functional system which admits a unique global positive solution. Furthermore, choosing appropriate intensity noise, we can get an exponential stable stochastic functional system
(2)
on
, where
is a scalar Brownian motion, and
![](https://www.scirp.org/html/6-7401907\9c6890de-1720-4184-9778-14daa9f853fd.jpg)
![](https://www.scirp.org/html/6-7401907\9840e453-df88-4d5a-832e-b400ee6bcbe1.jpg)
![](https://www.scirp.org/html/6-7401907\51f290c0-ad61-419a-aa35-83abae5e5455.jpg)
In the next section we will give some necessary notations and lemmas. In Section 3, we will give the main results of this paper.
2. Preliminaries
Throughout this paper, unless otherwise specified, let
be the Euclidean norm in
. If
is a vector or matrix, its transpose is denoted by
. If
is a matrixits trace norm is denoted by
. Denote the inner product of
by
or
. Let
be positive integers. Let
denote the maximum of
and
, while
the minimum of
and
. Let
. Denote by
the family of continuous functions from
to Rn with the norm
which forms a Banach space. Let
and
. Let
denote the family of functions
on
which are continuously twice differentiable in
and once in
.
Let
be a complete probability space with a filtration
satisfying the usual conditions (i.e. it is increasing and right continuous while
contains all
-null sets). Let
denote the family of
-valued
-measurable random variables
with
. Denote the family of Rn-valued bounded
-measurable random variables by
. If
is an Rn-valued process on
, let
.
If
is a continuous local martingale, denote the quadratic variation of
by
. Let
![](https://www.scirp.org/html/6-7401907\f9395a03-aaf7-49cf-a2a1-140f665a4549.jpg)
be independent scalar Brownian motion defined on the probability space. Let
be a right-continuous Markov chain on the probability space taking values in a definite state space
with the generator
given by
![](https://www.scirp.org/html/6-7401907\07eed1a2-f66b-4cec-8f0f-35114b980498.jpg)
where
is the transition rate from
to
and
if
while
. We assume that the Markov chain
is independent of the Brownian motion
. For any initial value
denote the solution of the corresponding initial value problem by
or simply
on
.
In order to obtain the main results, we need the following assumptions.
(H1) There are some nonnegative constants
such that
(3)
for all
, where
is a probability measure on
and
means some functions satisfying
.
(H2) For every integer
, there is a
such that
(4)
for all
with
.
(H3) There are some nonnegative constants
and probability measure
such that for any
satisfying
,
(5)
(6)
(7)
. (8)
Definition 1: The irreducibility of the Markov chain means that the Markov chain has a unique stationary (probability) distribution
which can be determined by solving the following linear equation
(9)
subject to
(10)
Lemma 1: [27] Let (H2) hold, for any initial value
, system (2) has a unique maximal local strong solution on
, where
is the explosion time.
3. Main Results
Similar to the proof of Theorem 1 in [28], we slightly modify the condition on the coefficient of (1) and obtain the following theorem.
Theorem 1: Let (H1) - (H3) hold, for any initial value
, if
and
then there exists a unique global solution
of system (2) on all
a.s.
Similarly to that in [28], we define the stopping time
(11)
and a C2-function
, for any
.
Using the Itô formula and the Young inequality, for any
, by (H1) and (H3), we get
(12)
where
(13)
These results will be used in the following.
3.1. Boundedness
Theorem 2: Let (H1) - (H3) hold, for any initial value
and
, if
and
, then there exists a constant
such that the global solution
of system (2) has the property that
(14)
where
is dependent on
and independent of the initial value
, that is,
is bounded in moment.
Proof: For any
, applying the Itô formula to
yields
![](https://www.scirp.org/html/6-7401907\c8d537ca-809d-4e48-bc34-b7678b7f4b86.jpg)
By (12) and (13), we have
![](https://www.scirp.org/html/6-7401907\96ac4ace-89c8-48e6-bc6c-7f29e478ca4e.jpg)
By the boundedness of polynomial functions, there exists a constant
such that
which implies
![](https://www.scirp.org/html/6-7401907\66b200ef-2687-453d-9253-a93167ee6f5e.jpg)
Then
![](https://www.scirp.org/html/6-7401907\d08283ed-9e45-4c99-a1d3-06a2c076a8e7.jpg)
That is, the global solution
of system (2) is bounded in
-th moment for any
.
Theorem 3: Let (H1) - (H3) hold, if
and![](https://www.scirp.org/html/6-7401907\f9de4f42-b795-4e22-899f-79dbff355f09.jpg)
then for any initial value
and
, the solution
of system (2) has the property that
(15)
where
![](https://www.scirp.org/html/6-7401907\1c687703-07e1-4534-a7c9-5f8321cf0bb8.jpg)
(16)
Proof: By Theorem 1, there a.s. exists a unique global solution
to system (2) on
a.s. Let
, by the Itô formula, we have
![](https://www.scirp.org/html/6-7401907\5a486e99-39d8-449f-bbc3-c32eb54cb09f.jpg)
where
![](https://www.scirp.org/html/6-7401907\5a4d5606-437f-49c2-9e83-b3f3b987c06d.jpg)
By (H1) and (H3), we have
![](https://www.scirp.org/html/6-7401907\1b2035e9-f958-4e15-a3c0-f1d1e9c5f764.jpg)
Let
be the same stopping time as defined in the proof of Theorem 1. By (13) and (14), we have
![](https://www.scirp.org/html/6-7401907\cef09e58-d5cf-4b39-9d78-e0922437310a.jpg)
Then as
, we have
![](https://www.scirp.org/html/6-7401907\8b268d7b-a4cc-4d4b-904d-a0440778e917.jpg)
That is,
(17)
By the ergodic and irreducibility property of the Markov chain, we have
![](https://www.scirp.org/html/6-7401907\8594d6a9-037d-449f-8968-0b4ea5de601e.jpg)
Hence,
![](https://www.scirp.org/html/6-7401907\d3afef38-d090-4f55-8d60-13e37c8df079.jpg)
as required.
3.2. Stabilization of Noise
The following lemma can be obtained by slightly modifying the proof of Mao [2].
Lemma 2: Let (H1) - (H3) hold, for any initial value
with
, the global solution of system (2) has the property that
(18)
where
is the explosion time.
Theorem 4: Let (H1) - (H3) hold, assume that
and
.
If
(19)
where
(20)
then for any initial value
, satisfying
, the global solution of system (2) has the property that
(21)
That is, the solution to system (2) is a.s. exponentially stable.
Proof: By Lemma 2 and Theorem 1,
a.s. Thus, applying the Itô formula to
yields
![](https://www.scirp.org/html/6-7401907\032aa77a-28fe-47f2-9a66-30f74c113aa4.jpg)
where
is an identity matrix and
![](https://www.scirp.org/html/6-7401907\27880232-de1c-4366-ba48-dc803bf53358.jpg)
Clearly
and
are continuous local martingales with the quadratic variation
![](https://www.scirp.org/html/6-7401907\1415133b-c1d5-43bc-b7e2-f0aee842d6fc.jpg)
By (H3),
![](https://www.scirp.org/html/6-7401907\73e10e48-06fd-4140-a2b7-514739ee4ea0.jpg)
Applying the strong law of large number,
![](https://www.scirp.org/html/6-7401907\a87aaf79-2281-4b61-81ec-c20f56e450fa.jpg)
For any
and each integer
, by the exponential martingale inequality,
![](https://www.scirp.org/html/6-7401907\4564f4f9-e765-4ddc-ad90-1cfef9085619.jpg)
Since
, by the Borel-Cantelli lemma, there exists an
with
such that for any
, when
,
![](https://www.scirp.org/html/6-7401907\a81b1db1-7a48-4675-971f-b8ebabcd2b82.jpg)
From (H1) and (H3),
![](https://www.scirp.org/html/6-7401907\2c6a77bf-585b-4639-a015-1a9534213b1e.jpg)
where
![](https://www.scirp.org/html/6-7401907\2b714d2a-28b9-49ba-98fe-14a9fa10c428.jpg)
By the definition of
in (20),
![](https://www.scirp.org/html/6-7401907\f4c7d750-e6c6-455a-84ef-48f84fb4428a.jpg)
Applying the strong law of large number to the Brownian motion,
![](https://www.scirp.org/html/6-7401907\bd846da7-c4ff-47b9-9d41-f78c49a2f683.jpg)
which implies
![](https://www.scirp.org/html/6-7401907\d16e3593-2f50-403a-a9d6-b7f334a44c58.jpg)
4. Conclusion
In this paper, we study a stochastic functional system with Markovian switching. Motivated by [25,26] and other literatures, we introduce two appropriate intensity Brownian noise to perturb the system so as to suppress its potential explosion and stabilize it. Based on [28], we just slightly modify some conditions on its coefficients and add some contents, then we get some new conclusions about boundedness and stabilization of the system.