Converse Barrier Functions via Lyapunov Functions

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.