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,xt), we stochastically perturb it into a stochastic functional differential equation dX(t)=f(t,Xt)dt+ΣX(t)dB(t), where Σ is a matrix and B(t) a Brownian motion while Xt={X(t+θ):-τ⩽θ⩽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 τ is sufficiently small, there are many matrices Σ for which the stochastic functional differential equation is almost surely exponentially stable while the corresponding functional differential equation dx(t)/dt=f(t,xt) may be unstable.