A Converse Robust-Safety Theorem for Differential Inclusions

Abstract: 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.