Abstract:
Abnormal trajectories are of particular interest for sub-Riemannian geometry, because the most complicated singularities of the sub-Riemannian metric are located just near such trajectories. Important open questions in sub-Riemannian geometry are to establish whether the abnormal length minimizers are smooth and to describe the set filled with abnormal trajectories starting from a fixed point. For example, the Sard conjecture in sub-Riemannian geometry states that this set has measure zero. In this paper, we consider this and other related properties of such a set for the left-invariant sub-Riemannian problem with growth vector $(2,3,5,8)$. We also study the global and local optimality of abnormal trajectories and obtain their explicit parametrization.
Keywords:sub-Riemannian geometry, abnormal trajectories, abnormal set, local and global optimality.
Citation:
Yu. L. Sachkov, E. F. Sachkova, “Abnormal Trajectories in the Sub-Riemannian $(2,3,5,8)$ Problem”, Optimal Control and Dynamical Systems, Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze, Trudy Mat. Inst. Steklova, 321, Steklov Math. Inst., Moscow, 2023, 252–285; Proc. Steklov Inst. Math., 321 (2023), 236–268