Control of the Wave Equation by Time-Dependent Coefficient

Antonin Chambolle; Fadil Santosa

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 8, page 375-392
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Chambolle, Antonin, and Santosa, Fadil. "Control of the Wave Equation by Time-Dependent Coefficient." ESAIM: Control, Optimisation and Calculus of Variations 8 (2010): 375-392. <http://eudml.org/doc/90653>.

@article{Chambolle2010,
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. },
author = {Chambolle, Antonin, Santosa, Fadil},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Control problem; time dependent wave equation; damping.; damping; energy decay},
language = {eng},
month = {3},
pages = {375-392},
publisher = {EDP Sciences},
title = {Control of the Wave Equation by Time-Dependent Coefficient},
url = {http://eudml.org/doc/90653},
volume = {8},
year = {2010},
}

TY - JOUR
AU - Chambolle, Antonin
AU - Santosa, Fadil
TI - Control of the Wave Equation by Time-Dependent Coefficient
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 8
SP - 375
EP - 392
AB - 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.
LA - eng
KW - Control problem; time dependent wave equation; damping.; damping; energy decay
UR - http://eudml.org/doc/90653
ER -

References

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  1. P. D'Ancona and S. Spagnolo, Global solvability for the degenerate Kirchhoff equation with real analytic data. Invent. Math.108 (1992) 247-262.  Zbl0785.35067
  2. P. Destuynder and A. Saidi, Smart materials and flexible structures. Control Cybernet. 26 (1997) 161-205.  Zbl0884.73043
  3. G. Haritos and A. Srinivasan, Smart Structures and Materials. ASME, New York, ASME, AD24 (1991).  
  4. H. Janocha, Adaptronics and Smart Structures. Springer, New York (1999).  
  5. K. Lurié, 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.  Zbl1035.78021
  6. S. Pohozaev, On a class of quasilinear hyperbolic equations. Math. USSR Sbornik25 (1975) 145-158.  Zbl0328.35060
  7. J. Restorff, Magnetostrictive materials and devices, in Encyclopedia of Applied Physics, Vol. 9. VCH Publishers (1994).  

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