  This article presents novel methods for synthesizing distributionally robust
stabilizing neural controllers and certificates for control systems under model
uncertainty. A key challenge in designing controllers with stability guarantees
for uncertain systems is the accurate determination of and adaptation to shifts
in model parametric uncertainty during online deployment. We tackle this with a
novel distributionally robust formulation of the Lyapunov derivative chance
constraint ensuring a monotonic decrease of the Lyapunov certificate. To avoid
the computational complexity involved in dealing with the space of probability
measures, we identify a sufficient condition in the form of deterministic
convex constraints that ensures the Lyapunov derivative constraint is
satisfied. We integrate this condition into a loss function for training a
neural network-based controller and show that, for the resulting closed-loop
system, the global asymptotic stability of its equilibrium can be certified
with high confidence, even with Out-of-Distribution (OoD) model uncertainties.
To demonstrate the efficacy and efficiency of the proposed methodology, we
compare it with an uncertainty-agnostic baseline approach and several
reinforcement learning approaches in two control problems in simulation.
