Control Barrier Functions for Safe Stabilization via Zubov’s Theorem

Safety guarantee is a fundamental requirement for numerous dynamical systems. In this work, the safe stabilization problem in complex scenarios is considered. First, we introduce the notion of the local safe set, a convex and compact set disjoint from unsafe regions, to construct a finite sequence of such sets. Next, we leverage Zubov’s theorem to propose the Zubov barrier function (Zubov BF), which provides safety guarantees on local safe sets. Compared with the existing ones, the Zubov BF reduces the conservatism and complexity in safety analysis. The relations between the Zubov BF and existing ones are discussed to show the generality of Zubov BF. The Zubov control barrier function (Zubov CBF) is further extended for control synthesis. Sufficient and necessary conditions are provided to verify the existence of the derived controller. To achieve the safe stabilization objective, we provide a modified quadratic programming (QP) framework with feasibility guarantee and demonstrate the continuity of the derived controller in closed form. Finally, a numerical example from the overtaking problem of two autonomous vehicles and a practical experiment on the safe navigation for a quadrotor are provided to illustrate the derived results.