# Sufficient conditions for forward invariance and contractivity in hybrid inclusions using barrier functions

## TL;DR

This paper establishes sufficient conditions using barrier functions to ensure forward invariance and contractivity in hybrid systems, addressing nonuniqueness and premature solution termination.

## Contribution

It introduces multiple barrier functions and derives the tightest conditions for invariance and contractivity in hybrid inclusions with complex solution behaviors.

## Key findings

- Derived infinitesimal conditions for invariance and contractivity
- Extended barrier function techniques to hybrid systems with nonuniqueness
- Provided illustrative examples demonstrating theoretical results

## Abstract

This paper studies set invariance and contractivity in hybrid systems modeled by hybrid inclusions using barrier functions. After introducing the notion of a multiple barrier functions, we investigate the tightest possible sufficient conditions to guarantee different forward invariance and contractivity notions of a closed set for hybrid systems with nonuniqueness of solutions and solutions terminating prematurely. More precisely, we consider forward (pre-)invariance of sets, which guarantees solutions to stay in a set, and (pre-)contractivity, which further requires solutions that reach the boundary of the set to evolve (continuously or discretely) towards its interior. Our conditions for forward invariance and contractivity involve infinitesimal conditions in terms of multiple barrier functions. Examples illustrate the results. Keywords: Forward invariance, contractivity, barrier functions, hybrid dynamical systems.