^{1}

^{*}

^{1}

There are many papers related to stability, some on suppression or on stabilization are one type of them. Functional differential systems are common and important in practice. They are special situations of neutral differential systems and generalization of ordinary differential systems. We discussed conditions on suppression on functional system with Markovian switching in our previous work: “Suppression of Functional System with Markovian Switching”. Based on it, by slightly modifying and adding some conditions, we get this paper. In this paper, we will study a functional system whose coefficient satisfies the local Lipschitz condition and the one-sided polynomial growth condition under Markovian switching. By introducing two appropriate intensity Brownian noise, we find the potential explosion system stabilized.

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]. [

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. [

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. [

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 R^{n} 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 R^{n}-valued bounded -measurable random variables by. If is an R^{n}-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.

(H_{1}) There are some nonnegative constants such that

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

(H_{2}) For every integer, there is a such that

for all with.

(H_{3}) 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: [_{2}) 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 [

Theorem 1: Let (H_{1}) - (H_{3}) hold, for any initial value, if andthen there exists a unique global solution of system (2) on all a.s.

Similarly to that in [

and a C^{2}-function, for any.

Using the Itô formula and the Young inequality, for any, by (H_{1}) and (H_{3}), we get

where

These results will be used in the following.

Theorem 2: Let (H_{1}) - (H_{3}) 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

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

By (12) and (13), we have

By the boundedness of polynomial functions, there exists a constant such that which implies

Then

That is, the global solution of system (2) is bounded in -th moment for any.

Theorem 3: Let (H_{1}) - (H_{3}) hold, if

and

then for any initial value and, the solution of system (2) has the property that

where

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

where

By (H_{1}) and (H_{3}), we have

Let be the same stopping time as defined in the proof of Theorem 1. By (13) and (14), we have

Then as, we have

That is,

By the ergodic and irreducibility property of the Markov chain, we have

Hence,

as required.

The following lemma can be obtained by slightly modifying the proof of Mao [

Lemma 2: Let (H_{1}) - (H_{3}) hold, for any initial value with, the global solution of system (2) has the property that

where is the explosion time.

Theorem 4: Let (H_{1}) - (H_{3}) hold, assume that

and.

If

where

then for any initial value, satisfying, the global solution of system (2) has the property that

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

where is an identity matrix and

Clearly and are continuous local martingales with the quadratic variation

By (H_{3}),

Applying the strong law of large number,

For any and each integer, by the exponential martingale inequality,

Since, by the Borel-Cantelli lemma, there exists an with such that for any, when,

From (H_{1}) and (H_{3}),

where

By the definition of in (20),

Applying the strong law of large number to the Brownian motion,

which implies

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 [