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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
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.
Received: 18.05.2021
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
Linking options:
https://www.mathnet.ru/eng/rm10019https://doi.org/10.1070/RM10019 https://www.mathnet.ru/eng/rm/v77/i1/p109
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Abstract page: | 567 | Russian version PDF: | 116 | English version PDF: | 73 | Russian version HTML: | 310 | References: | 81 | First page: | 33 |
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