Robust Set Stability of Logic Dynamical Systems with respect to Uncertain Switching

TL;DR

This paper develops new definitions and characterizations of robust set stability in logic dynamical systems with uncertain switching, establishing conditions for various stability types and clarifying their relationships.

Contribution

It introduces novel stability definitions for LDSs with uncertain switching and provides necessary and sufficient conditions, correcting previous misconceptions about their implications.

Findings

Robust set stability iff destination set contains all loops.

Uniform robust set stability iff all outside states are unreachable.

Asymptotic set stability iff the largest invariant subset is reachable.

Abstract

This paper proposes several definitions of robust stability for logic dynamical systems (LDSs) with uncertain switching, including robust/uniform robust set stability and asymptotical (or infinitely convergent)/finite-time set stability with ratio one. It is proved herein that an LDS is robustly set stable if and only if the destination set contains all loops (i.e., the paths from each state to itself); an LDS is uniformly robustly set stable, or finite-time set stable with ratio one, if and only if all states outside the destination set are unreachable from any self-reachable state; and an LDS is asymptotically set stable with ratio one if and only if the largest robustly invariant subset (LRIS) in the destination set is reachable from any state. In addition, it is proved that uniform robust set stability implies robust set stability, and robust set stability implies asymptotical set stability with ratio one. However, the inverse claims are not generally true. The relations between robust stability and stability under random switching are revealed, that is, the asymptotical/finite-time set stability with ratio one under uncertain switching is equivalent to asymptotical/finite-time set stability of the LDS under random switching. Furthermore, it is proved that, for uniform set stability and asymptotical/finite-time set stability with ratio one, the set stability is equivalent to the stability with respect to the LRIS in the destination set. However, robust set stability does not imply robust stability with respect to the LRIS in the destination set. This finding corrects a result in a previous study.