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 Tr. Mat. Inst. Steklova, 2001, Volume 232, Pages 144–155 (Mi tm209)

Boundary Control of Spherically Symmetric Oscillations of a Three-Dimensional Ball

V. A. Il'in

Abstract: The problem of boundary control of radially symmetric oscillations of a 3-ball that are described by a wave equation whose solutions $u(r, t)$ admit the existence of finite energy at every moment of time is studied. The state of an oscillating ball at every fixed moment of time $t$ is characterized by a pair of functions $\{ u (r, t), u_t (r, t) \}$. A minimal time interval $T$ is determined that is sufficient for changing an arbitrary initial state $\{ u (r, 0), u_t (r, 0) \}$ of the oscillation process to an arbitrary preset state $\{ u (r, T), u_t (r, T) \}$ with the use of a boundary control on the ball surface.

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English version:
Proceedings of the Steklov Institute of Mathematics, 2001, 232, 138–149

Bibliographic databases:
UDC: 517.984.5

Citation: V. A. Il'in, “Boundary Control of Spherically Symmetric Oscillations of a Three-Dimensional Ball”, Function spaces, harmonic analysis, and differential equations, Collected papers. Dedicated to the 95th anniversary of academician Sergei Mikhailovich Nikol'skii, Tr. Mat. Inst. Steklova, 232, Nauka, MAIK «Nauka/Inteperiodika», M., 2001, 144–155; Proc. Steklov Inst. Math., 232 (2001), 138–149

Citation in format AMSBIB
\Bibitem{Ili01} \by V.~A.~Il'in \paper Boundary Control of Spherically Symmetric Oscillations of a~Three-Dimensional Ball \inbook Function spaces, harmonic analysis, and differential equations \bookinfo Collected papers. Dedicated to the 95th anniversary of academician Sergei Mikhailovich Nikol'skii \serial Tr. Mat. Inst. Steklova \yr 2001 \vol 232 \pages 144--155 \publ Nauka, MAIK «Nauka/Inteperiodika» \publaddr M. \mathnet{http://mi.mathnet.ru/tm209} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=1851445} \zmath{https://zbmath.org/?q=an:1032.93028} \transl \jour Proc. Steklov Inst. Math. \yr 2001 \vol 232 \pages 138--149 

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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. Il'in V.A., “A boundary control of spherically symmetric oscillations of a ball under the condition of the existence of finite energy”, Dokl. Math., 68:2 (2003), 211–215
2. Belotserkovskii O.M., Emel'yanov S.V., Korovin S.K., Maslov V.P., Moiseev E.I., Nikol'skii S.M., Osipov Y.S., Sadovnichii V.A., Samarskii A.A., Shemyakina T.K., “Vladimir Aleksandrovich Il'in (A tribute in honor of his seventy-fifth birthday)”, Differ. Equ., 39:5 (2003), 609–613
3. Borovskikh A.V., “Formulas of boundary control of an inhomogeneous string. I”, Differ. Equ., 43:1 (2007), 69–95
4. Sergeev S.A., “On the optimal boundary control of vibrations of a spherical layer”, Differ. Equ., 45:10 (2009), 1514–1525
5. Hasanov K.K., Gasumov T.M., “Minimal Energy Control for the Wave Equation with Non-Classical Boundary Condition”, Applied and Computational Mathematics, 9:1 (2010), 47–56
6. Kurkina A.V., “Problem of Radially Symmetric Vibrations of a Three-Dimensional Ball and Its Optimization”, Dokl. Math., 89:3 (2014), 331–334
7. Kurkina A.V., “Optimization of boundary displacement control of radially symmetric vibrations of a three-dimensional ball”, Differ. Equ., 51:11 (2015), 1449–1460
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