  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.
