Suppression and stabilisation of noise

Abstract

In this article, we investigate the stochastic suppression and stabilisation of nonlinear systems. Given an unstable differential equation _ xðtÞ ¼ f ðxðtÞ, tÞ, in which f satisfies the one-sided polynomial growth condition, we introduce two Brownian noise feedbacks and therefore stochastically perturb this system into the nonlinear stochastic differential equation dxðtÞ ¼ f ðxðtÞ, tÞdt þ qxðtÞdw 1 ðtÞ þ jxðtÞj xðtÞdw 2 ðtÞ. This article shows that appropriate may guarantee that this stochastic system exists as a unique global solution although the corresponding deterministic may explode in a finite time. Then sufficiently large q may ensure that this stochastic system is almost surely exponentially stable.

Analysis of Noise under Regime Switching

“… . In [12], it shows that the trajectory of (4.8) will not tend to 0 although it has global solution.…”

“…Fuke Wu [12] have devoted contributions to improve this work which extend the role of Brownian noise for suppression and stabiliza-tion to cover the wider systems than [6,7]. The authors [12] introduce the following so-called one-side polynomial growth condition they give the further assumption for f: Assumption A. There are some nonnegative numbers , , [12] introduce Brownian noise feedback to suppress the potential explosion of the deterministic system (1.1) and stabilize the given system.…”

“…Fuke Wu [12] have devoted contributions to improve this work which extend the role of Brownian noise for suppression and stabiliza-tion to cover the wider systems than [6,7]. The authors [12] introduce the following so-called one-side polynomial growth condition they give the further assumption for f: Assumption A.…”

“…However, little is as yet known about the properties of system satisfying the one-side polynomial growth condition under regime switching, it is therefore the motivation of our present paper to consider the system subjected to both white noise and color telegraph (hybrid system). More precisely, in this paper we will develop the theory presented in [12] to cope with systems where they are subjected to both white noise and colored noise. Given an unstable hybrid system described by an ordinary differential equation with Markovian switching…”

“…The authors [12] introduce the following so-called one-side polynomial growth condition they give the further assumption for f: Assumption A. There are some nonnegative numbers , , [12] introduce Brownian noise feedback to suppress the potential explosion of the deterministic system (1.1) and stabilize the given system. However, little is as yet known about the properties of system satisfying the one-side polynomial growth condition under regime switching, it is therefore the motivation of our present paper to consider the system subjected to both white noise and color telegraph (hybrid system).…”