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Russian Mathematical Surveys, 2022, Volume 77, Issue 1, Pages 99–163
DOI: https://doi.org/10.1070/RM10019
(Mi rm10019)
 

This article is cited in 17 scientific papers (total in 17 papers)

Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions

Yu. L. Sachkov

Ailamazyan Program Systems Institute of Russian Academy of Sciences
References:
Abstract: Left-invariant optimal control problems on Lie groups are an important class of problems with a large symmetry group. They are theoretically interesting because they can often be investigated in full and general laws can be studied by using these model problems. In particular, problems on nilpotent Lie groups provide a fundamental nilpotent approximation to general problems. Also, left-invariant problems often arise in applications such as classical and quantum mechanics, geometry, robotics, visual perception models, and image processing.
The aim of this paper is to present a survey of the main concepts, methods, and results pertaining to left-invariant optimal control problems on Lie groups that can be integrated by elementary functions. The focus is on describing extremal trajectories and their optimality, the cut time and cut locus, and optimal synthesis. Questions concerning the classification of left-invariant sub-Riemannian problems on Lie groups of dimension three and four are also addressed.
Bibliography: 91 titles.
Keywords: optimal control, geometric control theory, left-invariant problems, sub-Riemannian geometry, Lie groups, optimal synthesis.
Funding agency Grant number
Russian Foundation for Basic Research 20-11-50114
This research was carried out with the support of the Russian Foundation for Basic Research in the framework of research project no. 20-11-50114.
Received: 18.05.2021
Russian version:
Uspekhi Matematicheskikh Nauk, 2022, Volume 77, Issue 1(463), Pages 109–176
DOI: https://doi.org/10.4213/rm10019
Bibliographic databases:
Document Type: Article
UDC: 517.977
MSC: Primary 53C17; Secondary 22E25, 49K15
Language: English
Original paper language: Russian
Citation: Yu. L. Sachkov, “Left-invariant optimal control problems on Lie groups: classification and problems integrable by elementary functions”, Uspekhi Mat. Nauk, 77:1(463) (2022), 109–176; Russian Math. Surveys, 77:1 (2022), 99–163
Citation in format AMSBIB
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  • https://doi.org/10.1070/RM10019
  • https://www.mathnet.ru/eng/rm/v77/i1/p109
  • This publication is cited in the following 17 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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    Успехи математических наук Russian Mathematical Surveys
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    References:81
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