  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 system, 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 integral techniques from the literature, to provide a
smooth barrier certificate that certifies robust safety by verifying a
condition involving only the barrier function and the system's right-hand side.
In comparison with existing converse robust-safety theorems, our result is more
general as it allows the safety region to be unbounded, the right-hand side to
be a general continuous set-valued map, and the solutions to be non-unique.

Optimization and Control