Continuity and Boundedness of Minimum-Norm CBF-Safe Controllers

The existence of a control barrier function (CBF) for a control-affine system provides a powerful design tool to ensure safety. Any controller that satisfies the CBF condition and ensures that the trajectories of the closed-loop system are well defined makes the zero superlevel set forward invariant. Such a controller is referred to as safe. This article studies the regularity properties of the minimum-norm safe controller as a stepping stone toward the design of general continuous safe feedback controllers. We characterize the set of points where the minimum-norm safe controller is discontinuous and show that it depends solely on the safe set and not on the particular CBF that describes it. Our analysis of the controller behavior as we approach a point of discontinuity allows us to identify sufficient conditions determining whether the controller grows unbounded or remains bounded. Examples illustrate our results, providing insight into the conditions that lead to (un)bounded discontinuous minimum-norm controllers.