Stabilization of Functional System with Markovian Switching

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.


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][2][3][4]. It talkes about suppression of noise in [1,2]. It shows 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][6][7][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][15][16][17][18][19][20][21][22][23][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 ex-ponential 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 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 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.

Preliminaries
Throughout this paper, unless otherwise specified, let  be the Euclidean norm in . If n R A is a vector or matrix, its transpose is denoted by T A . If A is a matrix, its trace norm is denoted by be positive integers. Let denote the maximum of and b , while the minimum of and . Let . Denote by satisfying the usual conditions (i.e. it is increasing and right continuous while con- If M is a continuous local martingale, denote the quadratic variation of M by , be independent scalar Brownian motion defined on the probability space. Let  , 0 r t t  be a right-continuous Markov chain on the probability space taking values in a definite state space Similarly to that in [28], we define the stopping time Lemma 1: [27] Let (H 2 ) hold, for any initial value

Main Results
Using the Itô formula and the Young inequality, for any   0,1 p  , by (H 1 ) and (H 3 ), we get Similar to the proof of Theorem 1 in [28], we slightly These results will be used in the following.

Boundedness
where p M is dependent on and independent of the p initial value  , that is,   x t is bounded in moment.
Proof: For any 0   , applying the Itô formula to (12) and (13), we have By the boundedness of polynomial functions, there exists a constant M  such that  be the same stopping time as defined in the proof of Theorem 1. By (13) and (14)

Stabilization of Noise
The following lemma can be obtained by slightly modifying the proof of Mao [2].
where e  is the explosion time.