Geodesic
Abnormal Trajectories in the Sub-Riemannian $(2,3,5,8)$ Problem
Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 252-285.

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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. 
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Yu. L. Sachkov; E. F. Sachkova. Abnormal Trajectories in the Sub-Riemannian $(2,3,5,8)$ Problem. Trudy Matematicheskogo Instituta imeni V.A. Steklova, Optimal Control and Dynamical Systems, Tome 321 (2023), pp. 252-285. http://geodesic.mathdoc.fr/item/TM_2023_321_a16/

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