Abstract
We obtain two versions of ODEs for the control function of normal geodesics for left-invariant sub-Riemannian metrics on Lie groups, involving only the structure of the Lie algebras of these groups. The first version is applicable to all Lie groups, while the second, to all matrix Lie groups; both versions are different invariant forms of the Hamiltonian system of the Pontryagin maximum principle for a left-invariant time-optimal problem on a Lie group. Basing on the first version, we find sufficient conditions for the normality of all geodesics of a given sub-Finslerian metric on a Lie group; in particular, we show that all three-dimensional Lie groups possess this property. The proofs use simple techniques of linear algebra.
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Dedicated with gratitude to the 85th anniversary of Academician Yuriĭ Grigor-evich Reshetnyak.
Original Russian Text Copyright © 2014 Berestovskiĭ V.N.
The author was partially supported by the Russian Foundation for Basic Research (Grant 14-01-00068-a) and a grant of the Government of the Russian Federation for the State Support of Scientific Research (Agreement 14.B25.31.0029).
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Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 55, No. 5, pp. 959–970, September–October, 2014.
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Berestovskĭ, V.N. Universal methods of the search of normal geodesics on Lie groups with left-invariant sub-Riemannian metric. Sib Math J 55, 783–791 (2014). https://doi.org/10.1134/S0037446614050012
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DOI: https://doi.org/10.1134/S0037446614050012