Barrier Functions for Robust Safety in Differential Inclusions, Part III: Inner and Outer Perturbations

This paper introduces a new robust-safety notion for differential inclusions. That is, we say that the system is strongly robustly safe if it remains safe in the presence of a continuous and positive perturbation, named robustness margin, added to both the argument and the image of its right-hand side. While in existing literature, including the preceding Parts I [1] and II [2], the perturbation term is added only to the image of the right-hand side, the notion proposed in this paper is shown to be relatively stronger, especially when the right-hand side is a general set-valued map. Furthermore, we show that some of the existing sufficient conditions for robust safety, in term of barrier functions, are strong enough to guarantee strong robust safety, provided that mild assumptions on the right-hand side hold. Following that, we establish the equivalence between strong robust safety and the existence of a smooth barrier certificate.