Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state

Song, Gongfei and Lu, Zhenyu and Zheng, Bo-Chao and Mao, Xuerong (2018) Almost sure stabilization of hybrid systems by feedback control based on discrete-time observations of mode and state. Science in China Series F - Information Sciences. ISSN 1009-2757

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    Abstract

    Although the mean square stabilisation of hybrid systems by feedback controls based on discretetime observations of state and mode has been studied by several authors since 2013 (see, e.g., [17,19,27,31]), the corresponding almost sure stabilisation problem has little been investigated. Recent Mao [18] is the first to study the almost sure stabilisation of a given unstable system x(t) = f(x(t)) by a linear discretetime stochastic feedback control Ax([t/τ]τ)dB(t) (namely the stochastically controlled system has the form dx(t) = f(x(t))dt + Ax([t/τ]τ)dB(t)), where B(t) is a scalar Brownian, τ > 0 and [t/τ] is the integer part of t/τ. In this paper, we will consider a much more general problem. That is, we will to study the almost sure stabilisation of a given unstable hybrid system x(t) = f(x(t), r(t)) by nonlinear discrete-time stochastic feedback control u(x([t/τ]τ), r([t/τ]τ))dB(t) (so the stochastically controlled system is a hybrid stochastic system of the form dx(t) = f(x(t), r(t))dt + u(x([t/τ]τ), r([t/τ]τ))dB(t)), where B(t) is a multi-dimensional Brownian motion and r(t) is a Markov chain.