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：本文针对受未知分布扰动影响的动力系统，提出了证明其李雅普诺夫稳定性的方法。我们假设仅可获得一个有限的扰动样本集，且实际在线扰动可能来自与给定样本不同的分布。我们构建了一个优化问题来搜索平方和（SOS）李雅普诺夫函数，并引入了李雅普诺夫函数导数约束的分布鲁棒版本。我们证明了该约束可重新表述为若干SOS约束，从而确保李雅普诺夫函数的搜索保持在SOS多项式优化问题类别内。对于一般系统，我们为神经网络李雅普诺夫函数搜索提供了分布鲁棒机会约束的表述。仿真实验验证了两种表述在非线性不确定动力系统上的有效性和效率。