Suppression and stabilisation of noise

Abstract

In this article, we investigate the stochastic suppression and stabilisation of nonlinear systems. Given an unstable differential equation , 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 . This article shows that appropriate β 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.

Keywords:

• Brownian motion
• suppression
• stabilisation
• the Itô formula

Acknowledgement

The authors would like to thank the financial support from the National Natural Science Foundation of China (Grant No. 50775082).