发表机构
University of Waterloo（滑铁卢大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究将向量场映射到Lyapunov函数的解算子，证明其良定义性、唯一性和连续性，并采用傅里叶神经算子进行数据驱动逼近，实验表明单个算子可准确逼近参数化系统族的Lyapunov函数。
AI 中文摘要
为非线性动力系统构造Lyapunov函数是稳定性分析中的一个核心问题，但至今仍具挑战性。Lyapunov函数通常被表征为一阶偏微分方程（PDE）的解，但这些解通常是为单个系统获得的，限制了它们在不同系统中的复用。在本文中，我们研究Lyapunov解算子，该算子将向量场映射到由基于耗散的Lyapunov PDE定义的相应Lyapunov函数。我们证明，在吸引域的紧子集上且满足指数稳定性假设时，该算子是良定义的、唯一的，并且关于向量场和耗散函数的扰动是连续的。这些结果为在非线性系统族上一致逼近Lyapunov函数提供了理论基础。基于这些理论基础，我们采用傅里叶神经算子（FNOs）作为Lyapunov解算子的数据驱动逼近。数值实验表明，单个训练好的算子可以在参数化的动力学族上准确逼近数值Lyapunov函数。这展示了神经算子在逼近Lyapunov函数方面的潜力。
英文摘要
Constructing Lyapunov functions for nonlinear dynamical systems is a central problem in stability analysis, yet remains challenging. Lyapunov functions are commonly characterized as solutions to first-order partial differential equations (PDEs), but these solutions are typically obtained for single systems, limiting their reuse across systems. In this paper, we study the Lyapunov solution operator that maps a vector field to the corresponding Lyapunov function defined by a dissipation-based Lyapunov PDE. We establish that, on compact subsets of the domain of attraction and under exponential stability assumptions, this operator is well-defined, unique, and continuous with respect to perturbations of both the vector field and the dissipation function. These results provide a theoretical foundation for approximating Lyapunov functions uniformly over families of nonlinear systems. Building on these theoretical foundations, we employ Fourier Neural Operators (FNOs) as a data-driven approximation of the Lyapunov solution operator. Numerical experiments demonstrate that a single trained operator can accurately approximate the numerical Lyapunov functions across parameterized families of dynamics. This illustrates the potential of neural operators for approximating Lyapunov functions.
Journal refIEEE Conference on Decision and Control (CDC), 2026
