  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.
