Ergodicity and stability of hybrid systems with piecewise constant type state-dependent switching

TL;DR

This paper introduces an approximation method for stochastic hybrid systems with state-dependent switching, providing explicit criteria for their ergodicity and stability, and demonstrating the method's effectiveness through examples.

Contribution

It develops a new approximation approach using piecewise constant switching and establishes explicit ergodicity and stability criteria for such systems.

Findings

Convergence rate in Wasserstein distance depends on transition rate matrix differences.

Explicit criteria for ergodicity and stability are derived.

Examples demonstrate the sharpness of the proposed criteria.

Abstract

To deal with stochastic hybrid systems with general state-dependent switching, we propose an approximation method by a sequence of stochastic hybrid systems with piecewise constant type switching. The convergence rate in the Wasserstein distance is estimated in terms of the difference between transition rate matrices. Our method is based on an elaborate construction of coupling processes in terms of Skorokhod's representation theorem for jumping processes. Moreover, we establish explicit criteria on the ergodicity and stability for stochastic hybrid systems with piecewise constant type switching. Some examples are given to illustrate the sharpness of these criteria.