Lyapunov-Barrier Characterization of Robust Reach-Avoid-Stay Specifications for Hybrid Systems

TL;DR

This paper extends Lyapunov-barrier methods to hybrid systems, providing a necessary and sufficient framework for verifying reach-avoid-stay specifications without state-space discretization, thus improving computational efficiency.

Contribution

It introduces a Lyapunov-barrier characterization for hybrid systems, establishing both sufficiency and necessity under perturbations, advancing formal verification techniques.

Findings

Lyapunov-barrier functions guarantee reach-avoid-stay properties for hybrid systems.

Necessary and sufficient conditions are established for perturbed hybrid systems.

Numerical examples validate the theoretical results.

Abstract

Stability, reachability, and safety are crucial properties of dynamical systems. While verification and control synthesis of reach-avoid-stay objectives can be effectively handled by abstraction-based formal methods, such approaches can be computationally expensive due to the use of state-space discretization. In contrast, Lyapunov methods qualitatively characterize stability and safety properties without any state-space discretization. Recent work on converse Lyapunov-barrier theorems also demonstrates an approximate completeness or verifying reach-avoid-stay specifications of systems modelled by nonlinear differential equations. In this paper, based on the topology of hybrid arcs, we extend the Lyapunov-barrier characterization to more general hybrid systems described by differential and difference inclusions. We show that Lyapunov-barrier functions are not only sufficient to guarantee reach-avoid-stay specifications for well-posed hybrid systems, but also necessary for arbitrarily slightly perturbed systems under mild conditions. Numerical examples are provided to illustrate the main results.