Solutions and Reachable Sets of Hybrid Dynamical Systems: Semicontinuous Dependence on Initial Conditions, Time, and Perturbations

TL;DR

This paper investigates the continuity properties of solutions and reachable sets in hybrid dynamical systems, establishing conditions for semicontinuous dependence on initial conditions, time, and perturbations, with implications for control and simulation.

Contribution

It introduces a novel property related to sequential compactness that ensures lower semicontinuous dependence of reachable sets, expanding understanding of hybrid system behavior.

Findings

Solutions depend upper semicontinuously on initial conditions and perturbations.

Reachable sets depend lower semicontinuously on initial conditions, time, and perturbations.

Conditions for these properties involve geometric, regularity, and tangentiality constraints.

Abstract

The sequential compactness afforded hybrid systems under mild regularity constraints guarantee outer/upper semicontinuous dependence of solutions on initial conditions and perturbations. For reachable sets of hybrid systems, this property leads to upper semicontinuous dependence with respect to initial conditions, time, and perturbations. Motivated by these results, we define a counterpart to sequential compactness and show that it leads to lower semicontinuous dependence of solutions on initial conditions and perturbations. In the sequel, it is shown that under appropriate assumptions, reachable sets of systems possessing this novel property depend lower semicontinuously on initial conditions, time, and perturbations. When those assumptions fail, continuous approximations of reachable sets turn out to be still possible. Necessary and sufficient conditions for the introduced property are given by a combination of geometric constraints, regularity assumptions, and tangentiality conditions. Further applications to simulations and optimal control of hybrid systems are discussed.