Almost sure exponential stabilization by discrete-time stochastic feedback control

Mao, Xuerong (2016) Almost sure exponential stabilization by discrete-time stochastic feedback control. IEEE Transactions on Automatic Control, 61 (6). pp. 1619-1624. ISSN 0018-9286

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    Abstract

    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 σ2 > 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.