Almost sure stability of hybrid stochastic systems under asynchronous Markovian switching

Abstract

This paper presents two novel Lyapunov methodologies for almost sure stability of hybrid stochastic systems under asynchronous Markovian switching. One is established by a concave composite Lyapunov function with exponential martingale inequality. The other is derived by the strong law of large numbers, which can explore the coupling between the drift part and diffusion part of the systems, thus fully capturing the stabilizing effect of the stochastic noise. Both of these stability conditions give a quantitative relationship between the size of the detected delay of switching signal, the stationary distribution and the generator of Markov chain. As applications, easy-to-check stability and stabilization criteria are further provided for one-sided growth nonlinear systems and linear systems. Numerical examples illustrate the proposed theoretical results.