# 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  Preview Text (Mao-etal-IEEE-TOAC-2016-Almost-sure-exponential-stabilization-by-discrete-time-stochastic) Mao_etal_IEEE_TOAC_2016_Almost_sure_exponential_stabilization_by_discrete_time_stochastic.pdf Accepted Author Manuscript Download (251kB)| Preview

## 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.

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• Item type: Article 54009 DateEvent1 June 2016Published16 August 2015Accepted (c) 2015 IEEE. Personal use of this material is permitted. Permission from IEEE must be obtained for all other users, including reprinting/ republishing this material for advertising or promotional purposes, creating new collective works for resale or redistribution to servers or lists, or reuse of any copyrighted components of this work in other works. Brownian motion, stochastic differential delay equations, difference equations , stochastic stabilization, discrete-time feedback control, Probabilities. Mathematical statistics, Control and Systems Engineering, Computer Science Applications, Electrical and Electronic Engineering Science > Mathematics > Probabilities. Mathematical statistics Faculty of Science > Mathematics and Statistics Pure Administrator 19 Aug 2015 09:56 20 Jul 2021 01:34 Journal or Publication https://strathprints.strath.ac.uk/id/eprint/54009