Stability Analysis for Stochastic Hybrid Inclusions

TL;DR

This paper provides a comprehensive stability analysis framework for stochastic hybrid inclusions (SHIs), addressing non-uniqueness of solutions, various stability conditions, and robustness, serving as a foundation for future research in control and filtering.

Contribution

It introduces new stability conditions, converse theorems, and robustness analysis for SHIs, advancing the theoretical understanding of stochastic hybrid systems.

Findings

Established Lyapunov-based stability criteria for SHIs

Proved converse theorems with robustness guarantees

Analyzed uniformity and causality in stability properties

Abstract

Stochastic hybrid inclusions (SHIs) address situations with the stochastic continuous evolution in a stochastic differential inclusions and random jumps in the difference inclusions due to the forced (the state reaching a boundary in the state space) and/or spontaneous (the state vector may occur spontaneously) transitions. An obvious characteristic of SHIs is the non-uniqueness of random solutions, which can be ensured by the mild regularity conditions, as well as nominal robustness. Basic sufficient conditions for stability/recurrence in probability are usually expressed based on different types of Lyapunov functions, including Lagrange/Lyapunov/Lyapunov-Forster functions respectively for Lagrange/Lyapunov/asymptotical stability in probability and Foster/Lagrange-Forster functions for recurrence, (weaker) relaxed Lyapunov-based sufficient conditions including Matrosov-Foster functions and the stochastic invariance principle, as well as Lyapunov-based necessary and sufficient conditions for asymptotical stability in probability or recurrence (i.e.,converse theorems), etc. The converse theorems involving smooth Lyapunov functions are guaranteed by the sequential compactness and thus robustness. In addition, the uniformity property and causality are analyzed for the stabilities in probability. Hence, serving as a partial roadmap for the theoretical development of SHIs, also serving as inspiration, we anticipate that many of the open questions, including the prediction problem, the filtering problem and the control problem, will be resolved based on the techniques of SHIs.