Abstract
We consider a simplified linear hybrid system for the problem of the control of noise in a cavity, consisting of two coupled wave equations of dimensions two and one respectively. A dissipative term is assumed to act in the one-dimensional equation. We prove the existence and the uniqueness of solutions. Each trajectory is proved to converge to an equilibrium as t → ∞. On the other hand we show that the convergence rate of the energy is not exponential. The proof of this result uses a perturbation argument allowing to modify the boundary conditions so that separation of variables applies.
Partially supported by Grant 5006/1996 (Romania) and CHRX-CT94-0471 of the European Union.
Supported by grant PB93-1203 of the DGICYT (Spain) and CHRX-CT94-0471 of the European Union.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
G. Avalos, The Exponential Stability of a a Coupled Hyperbolic/Parabolic System Arising in Structural Acoustics, Abstract Appl. Analysis, to appear.
G. Avalos and I. LasiecKA, The Strong Stability of a Semigroup Arising from a Coupled Hyperbolic/Parabolic System, Semigroup Forum, to appear.
H. T. Banks, W. Fang, R. J. Silcox and R. C. Smith, Approximation Methods or Control of Acustic/Structure Models with Piezoceramic Actuators, Journal of Intelligent Material Systems and Structures, 4 (1993), pp. 98–116.
H. T. Banks and R. C. Smith, Well-Posedness of a Model for Structural Acoustic Coupling in a Cavity Enclosed by a Thin Cylindrical Shell, J. Math. Analysis and Applications, 191 (1995), pp. 1–25.
H. T. Banks, R. C. Smith and Y. Wang, Smart Material Structures. Modeling, estimation and control, RAM, John Wiley & Sons, Masson, 1996.
C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), pp. 1024–1065.
T. Cazenave And A. Haraux, Introduction aux problémes d’évolution semi-linéaires, Mathématiques et Applications, 1, Ellipses, Paris, 1990.
P. Grisvard, Elliptic Problems in Non-smooth Domains, Pitman, 1985.
S. Hansen and E. Zuazua, Exact Controllability and Stabilization of a Vibrating String with an Interior Point Mass, SIAM J. Control Optim., 33, 5 (1995), pp. 1357–1391.
L. Hörmander, The Analysis of Linear Partial Differential Operators I, Springer-Verlag, 1990.
L. D. Landau and E. M. Lifshitz, Fluid Mechanics, Pergamon Press, 1987.
J. L. Lions, Contrôlabilité exacte, perturbations et stabilisation de systèmes distribués, Tome 1: Contrôlabilité exacte, Masson, RMA, Paris, 1988.
W. Littman And L. Marcus, Some Recent Results on Control and Stabilization of Flexible Structures, Univ. Minn., Mathematics Report 87–139.
W. Littman and L. Marcus, Exact Boundary Controllability of a Hybrid System of Elasticity, Archive Rat. Mech. Anal., 103, 3 (1988), pp. 193–236.
S. Micu, Análisis de un modelo hábrido bidimensional fluido-estructura, Ph. D. dissertation at Universidad Complutense de Madrid, 1996.
S. Micu and E. Zuazua, Propriétés qualitatives d’un modèle hybride bi-dimensionnel intervenant dans le contrôle du bruit, C. R. Acad. Sci. Paris, 319 (1994), pp. 1263–1268.
S. Micu and E. Zuazua, Asymptotics for the spectrum of a fluid/structure hybrid system arising in the control of noise, preprint.
S. Micu and E. Zuazua, Stabilization and Periodic Solutions of a Hybrid System Arising in the Control of Noise, Proceedings of the IFIP TC7/WG-7.2 International Conference, Laredo, Espana, Lecture Notes in Pure and Applied Mathematics, Vol. 174, Marcel Dekker, New York, 1996, pp. 219–230.
F. W. J. Olver, Asymptotics and Special Functions, Academic Press, 1974.
A. Pazy, Semi-groups of Linear Operators and Applicatios to Partial Differential Equations, Springer-Verlag, 1983.
J. Ralston, Solutions of the wave equation with localized energy, Comm. Pure Appl. Math. 22 (1969), pp. 807–823.
B. Rao. Uniform Stabilization of a Hybrid System of Elasticity, SIAM J. Cont. Optim., 33, 2 (1995), pp. 440–454.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1998 Springer Basel AG
About this paper
Cite this paper
Micu, S., Zuazua, E. (1998). On a Weakly Damped System Arising in the Control of Noise. In: Desch, W., Kappel, F., Kunisch, K. (eds) Control and Estimation of Distributed Parameter Systems. International Series of Numerical Mathematics, vol 126. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8849-3_16
Download citation
DOI: https://doi.org/10.1007/978-3-0348-8849-3_16
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-9800-3
Online ISBN: 978-3-0348-8849-3
eBook Packages: Springer Book Archive