Characterizations and Stability Criteria for the 2-Mode Markovian Switched Nonlinear Systems

This technical note studies characterizations and stability criteria for the 2-mode Markovian switched nonlinear systems. A Markovian switched nonlinear system is a switched system where its switching signal is induced by a continuous-time Markov chain. Two random processes-telegraph process and counting process-are defined on the stochastic switching signal. The evolution of the joint probability density functions of the two random processes conditional on the initial values of the switching signal is then formulated in terms of differential equations. Based on the analysis on the joint probability density functions, characterizations of the Markovian switching signal in terms of the expectations of the telegraph process and counting process are obtained. Moreover, the criteria for global uniformly asymptotic stability in expectation are proposed in this work, which are supported by numerical simulations.