Distributionally Robust Optimization (DRO) is a robust framework based on optimal transport theory for solving optimization problems under uncertainty. Recent research provides tractable formulations based on the Wasserstein distance between probability distributions. For the 1-Wasserstein distance, strong duality results hold under convexity and continuity assumptions. Those approaches have proven their interest in deriving distributionally robust counterparts for various problems, such as regressions or supervised learning. We investigate an approach for solving optimal control problems under uncertainty by including distributionally robust formulations. We derive a system of ordinary differential equations leading to a two-point boundary value problem, thanks to Pontryagin's Maximum Principle, which determines the optimal control from the initial conditions of the costates of the problem.