Learning Neural Lyapunov Functions to Characterize Stability Regions for Unknown Dynamical Systems

MingXiao, Zhe Wu

Keywords: Lyapunov function, Machine learning, Stability region, Lyapunov neural network, Nonlinear dynamics

TL;DR: This work develops a machine learning method for designing Lyapunov functions and characterizing stability regions in nonlinear dynamical systems with unknown first-principles models.

Abstract: Lyapunov function is often used as a mathematical tool to evaluate the stability of dynamical systems by demonstrating that system trajectories converge to an equilibrium point. This work develops a machine learning method for designing Lyapunov functions and characterizing stability regions in nonlinear dynamical systems with unknown first-principles models. The Lyapunov function is developed as a neural network model with its architecture and loss function designed to ensure that the conditions of a control Lyapunov function are satisfied. The optimal Lyapunov neural network is identified using Bayesian optimization that maximizes the estimated stability region. Theoretical guarantees are provided to ensure that, despite approximation errors, the Lyapunov function and the stability region derived from the data remain valid for the underlying nonlinear system. The proposed method is applied to various nonlinear systems, demonstrating its effectiveness in Lyapunov function design and stability region characterization.

Primary Area: applications to physical sciences (physics, chemistry, biology, etc.)

Submission Number: 12156