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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2023, Volume 321, Pages 194–214
DOI: https://doi.org/10.4213/tm4307 (Mi tm4307) 		 
Asymptotic Control Theory for a Closed String. II

Lev V. Lokutsievskiya, Alexander I. Ovseevichb

a Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
b Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences, pr. Vernadskogo 101-1, Moscow, 119526 Russia
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DOI: https://doi.org/10.4213/tm4307
Abstract: We develop an asymptotic control theory for one of the simplest distributed (infinite-dimensional) oscillating systems, namely, for a closed string under a bounded load applied to a single distinguished point. We give a precise description of the classes of string states that admit complete damping, and find an asymptotically exact value of the required time. By using approximate reachable sets instead of the exact ones, we design a feedback control, which turns out to be asymptotically optimal. The main results are exact algebraic formulas for the asymptotic shape of the reachable sets, for the asymptotically optimal time of motion, and for the asymptotically optimal control thus constructed.
Keywords: maximum principle, reachable sets, linear system, string control.
Funding agency Grant number
Russian Science Foundation 20-11-20169
21-11-00151
The work of L. V. Lokutsievskiy was supported by the Russian Science Foundation under grant no. 20-11-20169, https://rscf.ru/en/project/20-11-20169/, and performed at the Steklov Mathematical Institute of Russian Academy of Sciences. The work of A. I. Ovseevich was supported by the Russian Science Foundation under grant no. 21-11-00151, https://rscf.ru/en/project/21-11-00151/, and performed at the Ishlinsky Institute for Problems in Mechanics of the Russian Academy of Sciences. Sections 1, 3, 5, 7, and 9 were written by L.V.L., and Sections 2, 4, 6, 8, and 10, by A.I.O. All results in this paper are products of the authors' collaborative work.
Received: February 17, 2022
Revised: July 10, 2022
Accepted: November 21, 2022
English version:
Proceedings of the Steklov Institute of Mathematics, 2023, Volume 321, Pages 179–199
DOI: https://doi.org/10.1134/S008154382302013X
Bibliographic databases:
Document Type: Article
UDC: 517.977
Language: Russian
Citation: Lev V. Lokutsievskiy, Alexander I. Ovseevich, “Asymptotic Control Theory for a Closed String. II”, Optimal Control and Dynamical Systems, Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze, Trudy Mat. Inst. Steklova, 321, Steklov Math. Inst., Moscow, 2023, 194–214; Proc. Steklov Inst. Math., 321 (2023), 179–199
Citation in format AMSBIB
\Bibitem{LokOvs23}
\by Lev~V.~Lokutsievskiy, Alexander~I.~Ovseevich
\paper Asymptotic Control Theory for a Closed String. II
\inbook Optimal Control and Dynamical Systems
\bookinfo Collected papers. On the occasion of the 95th birthday of Academician Revaz Valerianovich Gamkrelidze
\serial Trudy Mat. Inst. Steklova
\yr 2023
\vol 321
\pages 194--214
\publ Steklov Math. Inst.
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm4307}
\crossref{https://doi.org/10.4213/tm4307}
\transl
\jour Proc. Steklov Inst. Math.
\yr 2023
\vol 321
\pages 179--199
\crossref{https://doi.org/10.1134/S008154382302013X}
\scopus{https://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-85170830811}
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    		Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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