# Converse Barrier Functions via Lyapunov Functions

arXiv: https://arxiv.org/abs/2007.11086 · HTML: https://mlsanity.com/paper/2007.11086/

## Key facts

- Cited by 2 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 21 Jul 2020.

## Built on by (most cited first, 2 of 2)

- [Neural Control Barrier Functions from Physics Informed Neural Networks](https://mlsanity.com/paper/2504.11045/) (2025)
- [Learning Geometrically-Informed Lyapunov Functions with Deep Diffeomorphic RBF Networks](https://mlsanity.com/paper/2504.02607/) (2025)

## Abstract (arXiv)

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.