Abstract
In this paper we consider a stochastic nonlinear system under regime switching. Given a system x(t)=f(x(t),r(t),t) in which f satisfies so-called one-side polynomial growth condition. We introduce two Brownian noise feedbacks and stochastically perturb this system into dx(t)=(x(t),r(t),t)dt+ σ (r(t))|x(t)|βx(t)dW1(t)+q(r(t))x(t)dW2(t) . It can be proved that appropriate noise intensity may suppress the potentially explode in a finite time and ensure that this system is almost surely exponentially stable although the corresponding system without Brownian noise perturbation may be unstable system.
Highlights
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Another important fact is that the noise can suppress the explosions in population dynamics [6] which means that this fact guarantees the existence of global solutions
From another point of view, let us take a further step by considering another type of environmental noise, namely, color noise, or say telegraph noise which can be illustrated as a switching between two or more regimes of environment, the regime switching and the environmental noise work together to make the system change
Summary
Appleby and Mao examined the stabilization of noise when f satisfies the one-sided linear growth condition [4,5] Another important fact is that the noise can suppress the explosions (in a finite time) in population dynamics [6] which means that this fact guarantees the existence of global solutions. Suppose f only satisfy the one-side polynomial condition and local Lipschitz condition, author [12] introduce Brownian noise feedback to suppress the potential explosion of the deterministic system (1.1) and stabilize the given system. Little is as yet known about the properties of system satisfying the one-side polynomial growth condition under regime switching, it is the motivation of our present paper to consider the system subjected to both white noise and color telegraph (hybrid system). The section we will show that appropriate may guarantee this system (1.3) or (1.4) exists a unique global solution the corresponding hybrid system (1.2) may explode in a finite time
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