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

Although the mean square stabilization of hybrid systems by feedback control based on discretetime observations of state and mode has been studied by several authors since 2013, the corresponding almost sure stabilization problem has received little attention. Recently, Mao was the first to study the almost sure stabilization of a given unstable system ẋ(t) = f(x(t)) by a linear discrete-time 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 consider a much more general problem. That is, we study the almost sure stabilization of a given unstable hybrid system ẋ(t) = f(x(t), r(t)) by nonlinear discrete-time stochastic feedback control u(x([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))dB(t), where B(t) is a multi-dimensional Brownian motion and r(t) is a Markov chain.