Volume 29, 2023
|Number of page(s)
|31 July 2023
New second-order optimality conditions in sub-riemannian geometry
1 Faculty of Mathematics, Informatics and Mechanics University of Warsaw, Warsaw, Poland
2 Department of Mathematics University of Fribourg, Fribourg, Switzerland
* Corresponding author: firstname.lastname@example.org
Accepted: 15 June 2023
We study the geometry of the second-order expansion of the extended end-point map for the sub-Riemannian geodesic problem. Translating the geometric reality into equations we derive new second-order necessary optimality conditions in sub-Riemannian Geometry. In particular, we find an ODE for velocity of an abnormal sub-Riemannian geodesics. It allows to divide abnormal minimizers into two classes, which we propose to call 2-normal and 2-abnormal extremals. In the 2-normal case the above ODE completely determines the velocity of a curve, while in the 2-abnormal case the velocity is undetermined at some, or at all points. With some enhancement of the presented results it should be possible to prove the regularity of all 2-normal extremals (the 2-abnormal case seems to require study of higher-order conditions) thus making a step towards solving the problem of smoothness of sub-Riemannian abnormal geodesics. As a by-product we present a new derivation of Goh conditions. We also prove that the assumptions weaker than these used in [Boarotto, Monti, Palmurella, 2020] to derive third-order Goh conditions, imply piece-wise-C2 regularity of an abnormal extremal.
Marty me i Zuzi
Mathematics Subject Classification: 53C17 / 49K15
Key words: Sub-Riemannian geometry / geodesic / abnormal minimizer / optimality condtitions / 2-jet / Goh conditions
© The authors. Published by EDP Sciences, SMAI 2023
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