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

TL;DR

This paper introduces a PDE-based Zubov-HJB formulation for constructing control Lyapunov-barrier functions, facilitating safe stabilization of nonlinear systems with potential for neural network-based numerical solutions.

Contribution

It presents a novel Zubov-HJB PDE framework that yields maximal CLBFs, enabling stability and safety guarantees for nonlinear control-affine systems.

Findings

Viscosity solutions of the PDE provide maximal CLBFs.

The framework supports neural-network-based PDE solving methods.

Enables feedback synthesis with safety and stability guarantees.

Abstract

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.