Almost Sure Exponential Stabilization by Discrete-Time Stochastic Feedback Control

Given an unstable linear scalar differential equation x·(t)=αx(t) (α>0), we will show that the discrete-time stochastic feedback control σx([t/τ]τ)dB(t) can stabilize it. That is, we will show that the stochastically controlled system dx(t)=αx(t)dt+σx([t/τ]τ)dB(t)is almost surely exponentially stable when σ
²
>2α and τ>0 is sufficiently small, where B(t) is a Brownian motion and [t/τ] is the integer part of t/τ. We will also discuss the nonlinear stabilization problem by a discrete-time stochastic feedback control. The reason why we consider the discrete-time stochastic feedback control is because that the state of the given system is in fact observed only at discrete times, say 0,τ,2τ,⋯, for example, where τ>0 is the duration between two consecutive observations. Accordingly, the stochastic feedback control should be designed based on these discrete-time observations, namely the stochastic feedback control should be of the form σx([t/τ]τ)dB(t). From the point of control cost, it is cheaper if one only needs to observe the state less frequently. It is therefore useful to give a bound on τ from below as larger as better.