Discontinuous Barrier Functions for Piecewise Continuous Dynamics

We propose a novel characterization of piecewise-defined barrier functions for certifying forward invariant sets of piecewise continuous dynamical systems. Forward invariance is established by checking two conditions: the first condition is a usual barrier-type inequality on the interior of each piece, and the second condition imposes an appropriate interaction of the tangent cone and vector field at the boundary between pieces. We then show that this separation is especially well suited for constructing discontinuous barrier functions that are an appropriate generalization of high-order control barrier functions to the piecewise setting and can be used to construct controllers for forward invariance. In particular, the tangent cone condition at the boundary of pieces does not depend on the particular control strategy and can be checked, e.g., offline, while standard online methods can be used to enforce the barrier-type inequality on the interior of pieces.