# On the graphical stability of hybrid solutions with non-matching jump times: Extended Paper

## TL;DR

This paper introduces a stability concept for hybrid systems allowing small mismatches in jump times, enabling analysis via Lyapunov methods and avoiding peaking phenomena.

## Contribution

It defines a new graphical stability notion for hybrid solutions with non-matching jump times and links it to existing Lyapunov stability analysis.

## Key findings

- Conditions under which graphical stability is implied by Lyapunov stability
- A distance-like function is used to analyze stability
- Stability analysis can be performed with existing Lyapunov techniques

## Abstract

We investigate stability of a solution of a hybrid system in the sense that the graphs of solutions from nearby initial conditions remain close and tend towards the graph of the given solution. In this manner, a small continuous-time mismatch is allowed between the jump times of neighbouring solutions and the `peaking phenomenon' is avoided. We provide conditions such that this stability notion is implied by stability with respect to a specifically designed distance-like function. Hence, stability of solutions in the graphical sense can be analysed with existing Lyapunov techniques.