Converse Barrier Functions via Lyapunov Functions
Abstract
We prove a robust converse barrier function theorem via the converse Lyapunov theory. While the use of a Lyapunov function as a barrier function is straightforward, the existence of a converse Lyapunov function as a barrier function for a given safety set is not. We establish this link by a robustness argument. We show that the closure of the forward reachable set of a robustly safe set must be robustly asymptotically stable under mild technical assumptions. As a result, all robustly safe dynamical systems must admit a robust barrier function in the form of a Lyapunov function for set stability. We present the results in both continuous-time and discrete-time settings and remark on connections with various barrier function conditions.
Cite
@article{arxiv.2007.11086,
title = {Converse Barrier Functions via Lyapunov Functions},
author = {Jun Liu},
journal= {arXiv preprint arXiv:2007.11086},
year = {2026}
}
Comments
Preprint submitted for publication; v2 corrected the non-strict inequality in condition (3) of Definition 5 with a strict inequality (thanks to a comment by Stefan Ratschan)