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Control of the Wave Equation by Time-Dependent Coefficient

Published online by Cambridge University Press:  15 August 2002

Antonin Chambolle  and
Fadil Santosa
Antonin Chambolle
Affiliation:
CEREMADE, UMR 7534 du CNRS, Université de Paris-Dauphine, 75775 Paris Cedex 16, France; antonin.chambolle@ceremade.dauphine.fr.
Fadil Santosa
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, MN 55455, U.S.A.; santosa@math.umn.edu.
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Abstract

We study an initial boundary-value problem for a wave equation with time-dependent sound speed. In the control problem, we wish to determine a sound-speed function which damps the vibration of the system. We consider the case where the sound speed can take on only two values, and propose a simple control law. We show that if the number of modes in the vibration is finite, and none of the eigenfrequencies are repeated, the proposed control law does lead to energy decay. We illustrate the rich behavior of this problem in numerical examples.

Keywords

Control problemtime dependent wave equationdamping.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 8: A tribute to JL Lions , 2002 , pp. 375 - 392
Copyright
© EDP Sciences, SMAI, 2002

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References

D'Ancona, P. and Spagnolo, S., Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math. 108 (1992) 247-262. CrossRef
Destuynder, P. and Saidi, A., Smart materials and flexible structures. Control Cybernet . 26 (1997) 161-205.
G. Haritos and A. Srinivasan, Smart Structures and Materials. ASME, New York, ASME, AD 24 (1991).
H. Janocha, Adaptronics and Smart Structures. Springer, New York (1999).
Lurié, K., Control in the coefficients of linear hyperbolic equations via spatio-temporal components, in Homogenization. World Science Publishing, River Ridge, NJ, Ser. Adv. Math. Appl. Sci. 50 (1999) 285-315.
Pohozaev, S., On a class of quasilinear hyperbolic equations. Math. USSR Sbornik 25 (1975) 145-158.
J. Restorff, Magnetostrictive materials and devices, in Encyclopedia of Applied Physics, Vol. 9. VCH Publishers (1994).