Extremal trajectories in a nilpotent sub-Riemannian problem on the Engel group (Q2880043)

The authors provide a comprehensive study of the geodesic problem in the sub-Riemannian Engel group, a stratified nilpotent Lie group of step three with two dimensional horizontal space. The problem is of interest for several distinct reasons. For instance, the setting is the simplest situation wherein nontrivial abnormal extremal trajectories exist. Moreover, the Engel group is the simplest nilpotent Lie group with nonsubanalytic sub-Riemannian spheres. The Engel group serves as a nilpotent approximation to the physical system describing planar motions of a wheeled conveyance with a single trailer.NEWLINENEWLINEThe problem is addressed from the framework of geometric control theory. The authors use Pontryagin's maximal principle to formulate a Hamiltonian system for the costate variables. Next, they present a parameterization of the extremal trajectories via a suitable exponential map. They study the symmetries of this exponential map. Finally, by considering the fixed points of these symmetries, they give an upper estimate on the cut time of extremal trajectories.