Towards Learning and Verifying Maximal Neural Lyapunov Functions
For control theorists and engineers, this work offers a computational method to obtain Lyapunov functions with near-maximal sub-level sets, but it is incremental as it builds on existing PINN and SMT techniques.
This paper presents a physics-informed neural network approach to learn a nearly maximal Lyapunov function for nonlinear systems, using Zubov's equation to train the function on the domain of attraction. The method provides theoretical guarantees and demonstrates effectiveness through numerical examples.
The search for Lyapunov functions is a crucial task in the analysis of nonlinear systems. In this paper, we present a physics-informed neural network (PINN) approach to learning a Lyapunov function that is nearly maximal for a given stable set. A Lyapunov function is considered nearly maximal if its sub-level sets can be made arbitrarily close to the boundary of the domain of attraction. We use Zubov's equation to train a maximal Lyapunov function defined on the domain of attraction. Additionally, we propose conditions that can be readily verified by satisfiability modulo theories (SMT) solvers for both local and global stability. We provide theoretical guarantees on the existence of maximal Lyapunov functions and demonstrate the effectiveness of our computational approach through numerical examples.