Abstract
We find the three-dimensional subspaces of four-dimensional Lie algebras which generate the algebras, as well as abnormal extremals on the connected Lie groups determined by these algebras and endowed with the left-invariant sub-Finsler quasimetrics defined by seminorms on the subspaces. Using the structure constants of Lie algebras and dual seminorms, we establish a criterion for the strict abnormality of the extremals.
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Acknowledgment
The authors thank the referee for reviewing the paper and making suggestions.
Funding
Berestovskii is supported by the Mathematical Center in Akademgorodok under Agreement 075–15–2022–281 on April 5, 2022 with the Ministry of Science and Higher Education of the Russian Federation. Zubareva’s work was carried out in the framework of the State Task to the Sobolev Institute of Mathematics (Project FWNF–2022–0003).
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Translated from Sibirskii Matematicheskii Zhurnal, 2022, Vol. 63, No. 4, pp. 748–767. https://doi.org/10.33048/smzh.2022.63.403
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Berestovskii, V.N., Zubareva, I.A. Abnormal Extremals of Left-Invariant Sub-Finsler Quasimetrics on Four-Dimensional Lie Groups with Three-Dimensional Generating Distributions. Sib Math J 63, 620–636 (2022). https://doi.org/10.1134/S0037446622040036
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DOI: https://doi.org/10.1134/S0037446622040036
Keywords
- extremal
- left-invariant sub-Finsler quasimetric
- Lie algebra
- optimal control
- polar
- Pontryagin maximum principle
- (strictly) abnormal extremal
- time-optimal control problem