  In this paper, we study the relationship between systems controlled via
Control Barrier Function (CBF) approaches and a class of discontinuous
dynamical systems, called Projected Dynamical Systems (PDSs). In particular,
under appropriate assumptions, we show that the vector field of CBF-controlled
systems is a Krasovskii-like perturbation of the set-valued map of a
differential inclusion, that abstracts PDSs. This result provides a novel
perspective to analyze and design CBF-based controllers. Specifically, we show
how, in certain cases, it can be employed for designing CBF-based controllers
that, while imposing safety, preserve asymptotic stability and do not introduce
undesired equilibria or limit cycles. Finally, we briefly discuss about how it
enables continuous implementations of certain projection-based controllers,
that are gaining increasing popularity.
