Lyapunov-barrier characterization of robust reach-avoid-stay specifications for hybrid systems

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 for verifying reach-avoid-stay specifications of systems modeled 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 Lyapunovbarrier 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.(c) 2023 Elsevier Ltd. All rights reserved.