Abstract
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.
Highlights
In recent years, stochastic systems have been considered by many researchers since many practical systems can be modeled using these kinds of systems
We study the almost sure stabilization 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
Markovian jump systems are a special class of hybrid stochastic systems, which can be found in some engineering systems including power systems, manufacturing systems, ecosystems, and so forth
Summary
Stochastic systems have been considered by many researchers since many practical systems can be modeled using these kinds of systems. Many significant results for stochastic systems have been reported (see [1–13]). Markovian jump systems are a special class of hybrid stochastic systems, which can be found in some engineering systems including power systems, manufacturing systems, ecosystems, and so forth. The literature in this area is huge and lots of papers are open access, we only mention a few [14–18]. Shaikhet [19] provided the sufficient conditions of asymptotic mean square stability for Markovian systems with delay. Mao [20] discussed the problem of exponential stability of general nonlinear Markovian jump systems
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