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.

Highlights

  • There are many papers which discuss stability of systems

  • There are 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

  • 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 recently. [17] provided some useful conditions on the exponential stability for general nonlinear SDSwMSs, which was improved by himself in Mao et al [19]

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Summary

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. 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.

Preliminaries
Main Results
Boundedness Theorem 2
Stabilization of Noise
Conclusion
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