Suppression and stabilisation of noise

In this article, we investigate the stochastic suppression and stabilisation of nonlinear systems. Given an unstable differential equation [image omitted], 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 [image omitted]. 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.