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arXiv · 2511.09523

Towards Learning and Verifying Maximal Lyapunov-Barrier Functions with a Zubov PDE Formulation

Yiming Meng·Jun Liu
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Abstract

Verifying stability and safety guarantees for nonlinear systems has received considerable attention in recent years. This property serves as a fundamental building block for specifying more complex system behaviors and control objectives. However, estimating the domain of attraction under safety constraints and constructing a Lyapunov-barrier function remain challenging tasks for nonlinear systems. To address this problem, we propose a Zubov PDE formulation with a Dirichlet boundary condition for autonomous nonlinear systems and show that a physics-informed neural network solution, once formally verified, can serve as a Lyapunov-barrier function that jointly certifies stability and safety. This approach extends existing converse Lyapunov-barrier theorems by introducing a PDE-based framework with boundary conditions defined on the safe set, yielding a near-optimal certified under-approximation of the true safe domain of attraction.

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Yiming Meng, Jun Liu. 2025-11-12. Towards Learning and Verifying Maximal Lyapunov-Barrier Functions with a Zubov PDE Formulation. https://arxiv.org/abs/2511.09523

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