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

where

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

on, where is a scalar Brownian motion, and

In the next section we will give some necessary notations and lemmas. In Section 3, we will give the main results of this paper.

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 normwhich 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

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

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

for all, where is a probability measure on and means some functions satisfying.

(H₂) For every integer, there is a such that

for all with.

(H₃) There are some nonnegative constants and probability measure such that for any satisfying,

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

subject to

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.

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 andthen there exists a unique global solution of system (2) on all a.s.

Similarly to that in [28], we define the stopping time

and a C²-function, for any.

Using the Itô formula and the Young inequality, for any, by (H₁) and (H₃), we get

where

These results will be used in the following.

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.