The structure of abnormal extremals in a sub-Riemannian problem with growth vector

Abstract

A left-invariant sub-Riemannian problem on a free nilpotent Lie group of step 4 with two generators is considered. The structure of abnormal extremals is described. These extremals are shown to define an abnormal foliation of the annihilator of the square of the distribution, which is given by the intersections of this annihilator with the leaves of the symplectic foliation of the Lie coalgebra. The question of abnormal trajectories being strictly/nonstrictly abnormal is investigated, their projection onto a plane of the distribution are described, estimates for the corank are given and examples of nonsmooth trajectories are constructed. Bibliography: 14 titles.