An Excursion-Theoretic Approach to Stability of Discrete-Time Stochastic Hybrid Systems

TL;DR

This paper develops an excursion-theoretic framework to analyze the stability of discrete-time stochastic hybrid systems with state-dependent dynamics, providing new conditions for Lyapunov function boundedness and stability in complex, non-time-homogeneous systems.

Contribution

It introduces an excursion-theoretic approach to stability analysis, offering novel conditions for Lyapunov functions and stability in non-homogeneous stochastic hybrid systems.

Findings

Established $L_1$-boundedness conditions for Lyapunov functions.

Provided criteria for almost sure and $L_1$ stability of randomly switched systems.

Applicable to systems with complex, non-time-homogeneous dynamics.

Abstract

We address stability of a class of Markovian discrete-time stochastic hybrid systems. This class of systems is characterized by the state-space of the system being partitioned into a safe or target set and its exterior, and the dynamics of the system being different in each domain. We give conditions for $L_1$-boundedness of Lyapunov functions based on certain negative drift conditions outside the target set, together with some more minor assumptions. We then apply our results to a wide class of randomly switched systems (or iterated function systems), for which we give conditions for global asymptotic stability almost surely and in $L_1$. The systems need not be time-homogeneous, and our results apply to certain systems for which functional-analytic or martingale-based estimates are difficult or impossible to get.