Solution Concepts and Existence Results for Hybrid Systems with Continuous-time Inputs

TL;DR

This paper develops solution concepts and existence results for hybrid dynamical systems with continuous-time inputs, extending classical autonomous hybrid system theory to include exogenous inputs and leveraging viability theory for explicit conditions.

Contribution

It formalizes solution notions and provides existence and completeness conditions for hybrid systems with continuous inputs, filling a key gap in the literature.

Findings

Established solution existence and forward completeness conditions.

Provided explicit conditions using tangent cone formulations.

Analyzed variants based on input signal regularity.

Abstract

In many scenarios, it is natural to model a plant's dynamical behavior using a hybrid dynamical system influenced by exogenous continuous-time inputs. While solution concepts and analytical tools for existence and completeness are well established for autonomous hybrid systems, corresponding results for hybrid dynamical systems involving continuous-time inputs are generally lacking. This work aims to address this gap. We first formalize notions of a solution for such systems. We then provide conditions that guarantee the existence and forward completeness of solutions. Moreover, we leverage results and ideas from viability theory to present more explicit conditions in terms of various tangent cone formulations. Variants are provided that depend on the regularity of the exogenous input signals.