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Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2021, Volume 31, Issue 3, Pages 471–486
DOI: https://doi.org/10.35634/vm210309 (Mi vuu782) 		 
This article is cited in 2 scientific papers (total in 2 papers)

MATHEMATICS

On the structure of the singular set of solutions in one class of 3D time-optimal control problems

A. A. Uspenskii, P. D. Lebedev

Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, ul. S. Kovalevskoi, 16, Yekaterinburg, 620219, Russia
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DOI: https://doi.org/10.35634/vm210309
Abstract: A class of time-optimal control problems in terms of speed in three-dimensional space with a spherical velocity vector is considered. A smooth regular curve $\Gamma$ was chosen as the target set. Pseudo-vertices — characteristic points on $\Gamma,$ responsible for the appearance of a singularity in the optimal result function, are selected. The characteristic features of the structure of a singular set belonging to the family of bisectors are revealed. An analytical representation is found for the extreme points of the bisector corresponding to a fixed pseudo-vertex. As an illustration of the effectiveness of the developed methods for solving nonsmooth dynamic problems, an example of the numerical-analytical construction of resolving structures of a control problem in terms of speed is given.
Keywords: time-optimal problem, dispersing surface, bisector, pseudo-vertex, extreme point, curvature, singular set, Frene's trihedron.
Funding agency Grant number
Ministry of Science and Higher Education of the Russian Federation 075-02-2021-1383
The work was performed as part of research conducted in the Ural Mathematical Center with the financial support of the Ministry of Science and Higher Education of the Russian Federation (Agreement number 075-02-2021-1383).
Received: 19.07.2021
Bibliographic databases:
Document Type: Article
UDC: 517.977
MSC: 35A18, 14H20, 14J17
Language: Russian
Citation: A. A. Uspenskii, P. D. Lebedev, “On the structure of the singular set of solutions in one class of 3D time-optimal control problems”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 31:3 (2021), 471–486
Citation in format AMSBIB
\Bibitem{UspLeb21}
\by A.~A.~Uspenskii, P.~D.~Lebedev
\paper On the structure of the singular set of solutions in one class of 3D time-optimal control problems
\jour Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki
\yr 2021
\vol 31
\issue 3
\pages 471--486
\mathnet{http://mi.mathnet.ru/vuu782}
\crossref{https://doi.org/10.35634/vm210309}
\elib{https://elibrary.ru/item.asp?id=46616518}
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