Coadjoint orbits of three-step free nilpotent Lie groups and time-optimal control problem

We describe coadjoint orbits for three-step free nilpotent Lie groups. It turns out that two-dimensional orbits have the same structure as coadjoint orbits of the Heisenberg group and the Engel group. We consider a time-optimal problem on three-step free nilpotent Lie groups with a set of admissible velocities in the first level of the Lie algebra. The behavior of normal extremal trajectories with initial covectors lying in two-dimensional coadjoint orbits is studied. Under some broad conditions on the set of admissible velocities (in particular, in the sub-Riemannian case) the corresponding extremal controls are periodic, constant, or asymptotically constant.