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 Tr. Mat. Inst. Steklova, 2010, Volume 269, Pages 133–142 (Mi tm2906)

A terminal–boundary value problem that describes the process of damping the vibrations of a rod consisting of two segments with different densities and elasticity coefficients but with identical wave travel times

V. A. Il'inab

a Moscow State University, Moscow, Russia
b Steklov Mathematical Institute, Russian Academy of Sciences, Moscow, Russia

Abstract: In this paper, in terms of a finite-energy weak solution, we study a terminal–boundary value problem describing the complete damping, at a terminal time $T$, of the longitudinal vibrations of a rod consisting of two segments with different densities and elasticity coefficients under the condition that the lengths of the segments are such that the wave travel times along these segments are equal. We find an explicit analytic expression for a solution to this problem and prove its uniqueness. This problem is important for the design of acoustic systems in which one can completely damp an acoustic signal by a terminal time instant by applying boundary controls at the ends of a vibrating rod.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2010, 269, 127–136

Bibliographic databases:

UDC: 517.956.32+517.977

Citation: V. A. Il'in, “A terminal–boundary value problem that describes the process of damping the vibrations of a rod consisting of two segments with different densities and elasticity coefficients but with identical wave travel times”, Function theory and differential equations, Collected papers. Dedicated to Academician Sergei Mikhailovich Nikol'skii on the occasion of his 105th birthday, Tr. Mat. Inst. Steklova, 269, MAIK Nauka/Interperiodica, Moscow, 2010, 133–142; Proc. Steklov Inst. Math., 269 (2010), 127–136

Citation in format AMSBIB
\Bibitem{Ili10} \by V.~A.~Il'in \paper A terminal--boundary value problem that describes the process of damping the vibrations of a~rod consisting of two segments with different densities and elasticity coefficients but with identical wave travel times \inbook Function theory and differential equations \bookinfo Collected papers. Dedicated to Academician Sergei Mikhailovich Nikol'skii on the occasion of his 105th birthday \serial Tr. Mat. Inst. Steklova \yr 2010 \vol 269 \pages 133--142 \publ MAIK Nauka/Interperiodica \publaddr Moscow \mathnet{http://mi.mathnet.ru/tm2906} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2729978} \zmath{https://zbmath.org/?q=an:1202.35322} \elib{http://elibrary.ru/item.asp?id=15109756} \transl \jour Proc. Steklov Inst. Math. \yr 2010 \vol 269 \pages 127--136 \crossref{https://doi.org/10.1134/S0081543810020100} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000281705900010} \elib{http://elibrary.ru/item.asp?id=15325677} \scopus{http://www.scopus.com/record/display.url?origin=inward&eid=2-s2.0-77956628578} 

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. V. A. Il'in, “On the complete damping of oscillations of an inhomogeneous rod by means of a boundary control at one end”, Proc. Steklov Inst. Math. (Suppl.), 276, suppl. 1 (2012), S97–S105
2. Rogozhnikov A.M., “A mixed problem describing oscillations of a rod consisting of several segments with equal wave travel times”, Dokl. Math., 84:3 (2011), 830–832
3. Il'in V.A., “Optimization of the boundary displacement control of vibrations of a rod consisting of two dissimilar parts”, Differ. Equ., 47:7 (2011), 988–996
4. Lomov I.S., “Nonsmooth eigenfunctions in problems of mathematical physics”, Differ. Equ., 47:3 (2011), 355–362
5. Ilin V.A., “Optimizatsiya granichnogo upravleniya kolebaniyami sterzhnya, sostoyaschego iz dvukh raznorodnykh uchastkov”, Dokl. RAN, 440:2 (2011), 159–163
6. Stakun A.A., “Zadacha koshi dlya odnogo klassa uravnenii”, Vestnik Chuvashskogo universiteta, 2011, no. 3, 160–169
7. B. P. Osilenker, “On linear summability methods of fourier series in polynomials orthogonal in a discrete Sobolev space”, Siberian Math. J., 56:2 (2015), 339–351
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