  Differential dynamic programming (DDP) is a popular technique for solving
nonlinear optimal control problems with locally quadratic approximations.
However, existing DDP methods are not designed for stochastic systems with
unknown disturbance distributions. To address this limitation, we propose a
novel DDP method that approximately solves the Wasserstein distributionally
robust control (WDRC) problem, where the true disturbance distribution is
unknown but a disturbance sample dataset is given. Our approach aims to develop
a practical and computationally efficient DDP solution. To achieve this, we use
the Kantrovich duality principle to decompose the value function in a novel way
and derive closed-form expressions of the distributionally robust control and
worst-case distribution policies to be used in each iteration of our DDP
algorithm. This characterization makes our method tractable and scalable
without the need for numerically solving any minimax optimization problems. The
superior out-of-sample performance and scalability of our algorithm are
demonstrated through kinematic car navigation and coupled oscillator problems.
