Learning Regions of Attraction in Unknown Dynamical Systems via Zubov-Koopman Lifting: Regularities and Convergence

• 成果类型: Article
• 署名作者: Meng, Yiming; Zhou, Ruikun; Liu, Jun
• 署名单位: University of Illinois System; University of Illinois Urbana-Champaign; University of Waterloo
• 刊物名称: IEEE TRANSACTIONS ON AUTOMATIC CONTROL
• ISSN/ISSBN: 0018-9286
• DOI: 10.1109/TAC.2025.3560653
• 发表日期: 2025
• 页码: 6672-6687
• 关键词: Lyapunov methods; estimation; asymptotic stability; trajectory; Stability criteria; vectors; training; Numerical stability; Neural Networks; Eigenvalues and eigenfunctions; Region of attraction (ROA); regularity analysis; unknown nonlinear systems; Viscosity solution; Zubov's equation; Zubov-Koopman operators

摘要

The estimation for the region of attraction of an asymptotically stable equilibrium point is crucial in the analysis of nonlinear systems. There has been a recent surge of interest in estimating the solution to Zubov's equation, whose nontrivial sublevel sets form the exact ROA. In this article, we propose a lifting approach to map observable data into an infinite-dimensional function space, which generates a flow governed by the proposed Zubov-Koopman operators. By learning a Zubov-Koopman operator over a fixed time interval, we can indirectly approximate the solution to Zubov's equation by iteratively applying the learned operator on certain functions. We also demonstrate that a transformation of such an approximator can be readily utilized as a near-maximal Lyapunov function. We approach our goal through a comprehensive investigation of the regularities of Zubov-Koopman operators and their associated quantities. Based on these findings, we present an algorithm for learning Zubov-Koopman operators that exhibit strong convergence to the true operator. We show that this approach reduces the amount of required data and can yield desirable estimation results, as demonstrated through numerical examples.