Characterization of Safe Stabilization and Control Lyapunov-Barrier Functions via Zubov Equation Formulation

Design and analysis of stabilizing controllers with safety guarantees for nonlinear systems have received considerable attention in recent years. Control Lyapunov-barrier functions (CLBFs) provide a powerful framework for simultaneously ensuring stability and safety; however, their construction for nonlinear systems remains challenging. To address this issue, we build on recent advances in PDE-based characterizations of control Lyapunov functions and Lyapunov-barrier functions for autonomous systems, and propose a succinct Zubov-HJB PDE formulation for safe stabilization of nonlinear control-affine systems under a common compatibility assumption. We further show that the viscosity solution of this PDE yields a maximal CLBF, enabling (not necessarily continuous) feedback synthesis with stability and safety guarantees. In light of recent advances in neural-network-based methods for solving Zubov-type PDEs, this theoretical framework also provides a natural interface to emerging numerical approaches.