Robustness of stability through necessary and sufficient Lyapunov-like conditions for systems with a continuum of equilibria

The equivalence between robustness to perturbations and the existence of a continuous Lyapunov-like mapping is established in a setting of multivalued discrete-time dynamics for a property sometimes called semistability. This property involves a set consisting of Lyapunov stable equilibria and surrounded by points from which every solution converges to one of these equilibria. As a consequence of the main results, this property turns out to always be robust for continuous nonlinear dynamics and a compact set of equilibria. Preliminary results on reachable sets, limits of solutions, and set-valued Lyapunov mappings are included.