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Trudy Matematicheskogo Instituta imeni V.A. Steklova, 2019, Volume 304, Pages 32–48
DOI: https://doi.org/10.4213/tm3960 (Mi tm3960) 		 
This article is cited in 2 scientific papers (total in 2 papers)

Spectrum of the Second Variation

A. A. Agrachevabc

a Scuola Internazionale Superiore di Studi Avanzati (SISSA), via Bonomea 265, 34136 Trieste, Italy
b Steklov Mathematical Institute of Russian Academy of Sciences, ul. Gubkina 8, Moscow, 119991 Russia
c Program Systems Institute of Russian Academy of Sciences, Pereslavl-Zalessky, Yaroslavl Region, 152020 Russia
Full-text PDF (264 kB) Citations (2)
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DOI: https://doi.org/10.4213/tm3960
Abstract: Second variation of a smooth optimal control problem at a regular extremal is a symmetric Fredholm operator. We study the asymptotics of the spectrum of this operator and give an explicit expression for its determinant in terms of solutions of the Jacobi equation. In the case of the least action principle for the harmonic oscillator, we obtain the classical Euler identity ∏n=1∞(1−x2/(πn)2)=sin⁡x/x. The general case may serve as a rich source of new nice identities.
Funding agency Grant number
Russian Science Foundation 17-11-01387
This work is supported by the Russian Science Foundation under grant 17-11-01387.
Received: September 6, 2018
Revised: October 9, 2018
Accepted: December 19, 2018
English version:
Proceedings of the Steklov Institute of Mathematics, 2019, Volume 304, Pages 26–41
DOI: https://doi.org/10.1134/S0081543819010036
Bibliographic databases:
Document Type: Article
UDC: 517.984.5+517.977
Language: Russian
Citation: A. A. Agrachev, “Spectrum of the Second Variation”, Optimal control and differential equations, Collected papers. On the occasion of the 110th anniversary of the birth of Academician Lev Semenovich Pontryagin, Trudy Mat. Inst. Steklova, 304, Steklov Math. Inst. RAS, Moscow, 2019, 32–48; Proc. Steklov Inst. Math., 304 (2019), 26–41
Citation in format AMSBIB
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\by A.~A.~Agrachev
\paper Spectrum of the Second Variation
\inbook Optimal control and differential equations
\bookinfo Collected papers. On the occasion of the 110th anniversary of the birth of Academician Lev Semenovich Pontryagin
\serial Trudy Mat. Inst. Steklova
\yr 2019
\vol 304
\pages 32--48
\publ Steklov Math. Inst. RAS
\publaddr Moscow
\mathnet{http://mi.mathnet.ru/tm3960}
\crossref{https://doi.org/10.4213/tm3960}
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\jour Proc. Steklov Inst. Math.
\yr 2019
\vol 304
\pages 26--41
\crossref{https://doi.org/10.1134/S0081543819010036}
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    • This publication is cited in the following 2 articles:
    1. S. Baranzini, “Functional determinants for the second variation”, J. Fixed Point Theory Appl., 26:2 (2024)  crossref  mathscinet
    2. S. Baranzini, “Operators arising as second variation of optimal control problems and their spectral asymptotics”, J. Dyn. Control Syst., 29:3 (2023), 659  crossref  mathscinet
    Citing articles in Google Scholar: Russian citations, English citations
    Related articles in Google Scholar: Russian articles, English articles
    		Труды Математического института имени В. А. Стеклова Proceedings of the Steklov Institute of Mathematics
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