Characterizing Safety: Minimal Barrier Functions from Scalar Comparison Systems

Verifying set invariance has classical solutions stemming from the seminal work by Nagumo, and defining sets via a smooth barrier function constraint inequality results in computable flow conditions for guaranteeing set invariance. While a majority of these historic results on set invariance consider flow conditions on the boundary, recent results on control barrier functions extended these conditions to the entire set, although they required regularity conditions on the barrier function. This paper fully characterizes set invariance through minimal barrier functions by directly appealing to a comparison result to define a flow condition over the entire domain of the system. A considerable benefit of this approach is the removal of regularity assumptions of the barrier function. This paper also outlines necessary and sufficient conditions for a valid differential inequality condition, giving the minimum conditions for this type of approach. We also show when minimal barrier functions are necessary and sufficient for set invariance.