  This paper establishes the equivalence between robust safety and the
existence of a barrier function certificate for differential inclusions. More
precisely, for a robustly-safe differential inclusion, a barrier function is
constructed as the time-to-impact function with respect to a
specifically-constructed reachable set. Using techniques from set-valued and
nonsmooth analysis, we show that such a function, although being possibly
discontinuous, certifies robust safety by verifying a condition involving the
system's solutions. Furthermore, we refine this construction, using smoothing
techniques from the literature of converse Lyapunov theory, to provide a smooth
barrier certificate that certifies robust safety by verifying a condition
involving only the barrier function and the system's dynamics. In comparison
with existing converse robust-safety theorems, our results are more general as
they allow the safety region to be unbounded, the dynamics to be a general
continuous set-valued map, and the solutions to be non-unique.
