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：本文提出了在扰动分布未知的情况下，证明动力系统Lyapunov稳定性的方法。我们假设仅有一组有限的扰动样本可用，且实际在线扰动实现可能来自不同于给定样本的分布。我们构建了一个优化问题以搜索平方和（SOS）Lyapunov函数，并引入了Lyapunov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，从而确保Lyapunov函数的搜索仍属于SOS多项式优化问题范畴。对于一般系统，我们提出了神经网络Lyapunov函数搜索的分布鲁棒机会约束形式。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高效率。