Semicontinuity Properties of Solutions and Reachable Sets of Nominally Well-Posed Hybrid Dynamical Systems

Nominally well-posed hybrid inclusions are a class of hybrid dynamical systems characterized by outer/upper semicontinuous dependence of solutions on initial conditions. In the context of reachable sets of hybrid systems, this property guarantees outer/upper semicontinuous dependence with respect to both initial conditions and time. This article defines a counterpart to the notion of nominal well-posedness, referred to as nominal inner well-posedness, which ensures inner/lower semicontinuous dependence of solutions on initial conditions. Consequently, it is shown that reachable sets of nominally inner well-posed hybrid systems depend inner/lower semicontinuously on initial conditions and time under appropriate assumptions. Sufficient conditions guaranteeing nominal inner well-posedness are provided and demonstrated with an example.