This paper develops methods for proving Lyapunov stability of dynamical systems subject to disturbances with an unknown distribution. We assume only a finite set of disturbance samples is available and that the true online disturbance realization may be drawn from a different distribution than the given samples. We formulate an optimization problem to search for a sum-of-squares (SOS) Lyapunov function and introduce a distributionally robust version of the Lyapunov function derivative constraint. We show that this constraint may be reformulated as several SOS constraints, ensuring that the search for a Lyapunov function remains in the class of SOS polynomial optimization problems. For general systems, we provide a distributionally robust chance-constrained formulation for neural network Lyapunov function search. Simulations demonstrate the validity and efficiency of either formulation on non-linear uncertain dynamical systems.

翻译：本文开发了针对扰动分布未知的动力系统李雅普诺夫稳定性证明方法。我们假设仅能获取有限扰动样本集，且真实在线扰动实现可能服从与给定样本不同的分布。通过构建优化问题来搜索平方和（SOS）型李雅普诺夫函数，并引入李雅普诺夫函数导数约束的分布鲁棒版本。研究表明该约束可重构为多个SOS约束，确保李雅普诺夫函数搜索仍属于SOS多项式优化问题范畴。针对一般系统，我们提出基于神经网络李雅普诺夫函数搜索的分布鲁棒机会约束方法。仿真结果验证了两种方法在非线性不确定动力系统中的有效性与计算效率。