A Sub-Finsler Problem on the Cartan Group

We study a sub-Finsler geometric problem on the free nilpotent group of rank $2$ and step $3$. Such a group is also called the Cartan group and has a natural structure of Carnot group, which we metrize by considering the $\ell _\infty $ norm on its first layer. We adopt the point of view of time-optimal control theory. We characterize extremal curves via the Pontryagin maximum principle. We describe abnormal and singular arcs and construct the bang–bang flow.