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.

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)

References in corpus (2)

• Smooth Converse Lyapunov-Barrier Theorems for Asymptotic Stability with Safety Constraints and Reach-Avoid-Stay Specifications
• Robust Decidability of Sampled-Data Control of Nonlinear Systems with Temporal Logic Specifications