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Stability, reachability, and safety are crucial properties of dynamicalsystems. While verification and control synthesis of reach-avoid-stayobjectives can be effectively handled by abstraction-based formal methods, suchapproaches can be computationally expensive due to the use of state-spacediscretization. In contrast, Lyapunov methods qualitatively characterizestability and safety properties without any state-space discretization. Recentwork on converse Lyapunov-barrier theorems also demonstrates an approximatecompleteness or verifying reach-avoid-stay specifications of systems modelledby nonlinear differential equations. In this paper, based on the topology ofhybrid arcs, we extend the Lyapunov-barrier characterization to more generalhybrid systems described by differential and difference inclusions. We showthat Lyapunov-barrier functions are not only sufficient to guaranteereach-avoid-stay specifications for well-posed hybrid systems, but alsonecessary for arbitrarily slightly perturbed systems under mild conditions.Numerical examples are provided to illustrate the main results.

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