Stochastic stabilisation of functional differential equations

In this paper we investigate the problem of stochastic stabilisation for a general nonlinear functional differential equation. Given an unstable functional differential equation dx(t)/dt = f (t, x(1)), we stochastically perturb it into a stochastic functional differential equation dX(t)=f (t, X-t) dt+Sigma X(t) dB(t), where Sigma is a matrix and B(t) a Brownian motion while X-t = {X(t+0) : -tau <= 0 <= 0}. Under the condition that f satisfies the local Lipschitz condition and obeys the one-side linear bound, we show that if the time lag tau is sufficiently small, there are many matrices Sigma for which the stochastic functional differential equation is almost surely exponentially stable while the corresponding functional differential equation dx(t)/dt = f(t, x(t)) may be unstable. (c) 2005 Elsevier B.V. All rights reserved.