Solutions to hybrid inclusions via set and graphical convergence with stability theory applications

Motivated by questions in stability theory for hybrid dynamical systems, we establish some fundamental properties of the set of solutions to such systems. Using the notion of a hybrid time domain and general results on set and graphical convergence, we establish under weak regularity and local boundedness assumptions that the set of solutions is sequentially compact and “upper semicontinuous” with respect to initial conditions and system perturbations. The general facts are then used to establish several results for the behavior of hybrid systems that have asymptotically stable compact sets. These results parallel what is already known for differential inclusions and difference inclusions. For example, the basin of attraction for a compact attractor is (relatively) open, the attractivity is uniform from compact subsets of the basin of attraction, and asymptotic stability is robust with respect to small perturbations.