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