﻿ Stabilization of Functional System with Markovian Switching

Applied Mathematics
Vol.4 No.11A(2013), Article ID:40118,7 pages DOI:10.4236/am.2013.411A1006

Stabilization of Functional System with Markovian Switching

Lizhu Feng, Qiong Cai

School of Mathematics and Computer Science, Jianghan University, Wuhan, China

Email: fengliahu11@163.com

Copyright © 2013 Lizhu Feng, Qiong Cai. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Received October 14, 2013; revised November 14, 2013; accepted November 21, 2013

Keywords: Stochastic Functional System; Brownian Noise; Markovian Switching; Boundedness; Stabilization

ABSTRACT

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.

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

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

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 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

(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 andthen 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

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 (H1) - (H3) hold, if

and

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

(15)

where

(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

where

By (H1) and (H3), 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,

(17)

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

Hence,

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

where is an identity matrix and

Clearly and are continuous local martingales with the quadratic variation

By (H3),

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 (H1) and (H3),

where

By the definition of in (20),

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

which implies

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.

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