A new model for acoustic-structure interaction and its exponential stability
Authors:
Fariba Fahroo and Chunming Wang
Journal:
Quart. Appl. Math. 57 (1999), 157-179
MSC:
Primary 76Q05; Secondary 74F10, 93C20
DOI:
https://doi.org/10.1090/qam/1672195
MathSciNet review:
MR1672195
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Abstract: A new model for the interaction between the acoustic wave in an enclosed air cavity and the transversal motion of a flexible beam is proposed in this paper. This new boundary condition for the coupled wave and Euler-Bernoulli beam equations introduces sufficient damping of the energy of the system to gain uniform exponential stability. Careful physical justification of the boundary condition is based upon well-established theoretical results in acoustics. The estimate of the energy decay rate is obtained using a multiplier technique.
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H. T. Banks, W. Fang, R. J. Silcox, and R. C. Smith, Approximation methods for control of structural acoustic models with piezoceramic actuators, Journal of Intelligent Material Systems and Structures, Vol. 4, pp. 98–116 (1993)
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H. T. Banks and K. Ito, A unified framework for approximations in inverse problems for distributed parameter systems, Control Theory and Advanced Technology 4, 73–90 (1988)
H. T. Banks, K. Ito, and C. Wang, Exponentially stable approximations of weakly damped wave equations, in Estimation and Control of Distributed Parameter Systems (W. Desch, F. Kappel, and K. Kunisch, Eds.), Birkhäuser, 1991, pp. 1–33
H. T. Banks, S. L. Keeling, R. J. Silcox, and C. Wang, Linear quadratic tracking problem in Hilbert space: Application to optimal active noise suppression, in “Proc. 5th IFAC Sympos. on Control of DPS” (A. El-Jai and M. Amouroux, Eds.), pp. 17–22, Perpignan, France, June, 1989
H. T. Banks and C. Wang, Optimal feedback control of infinite-dimensional parabolic evolution systems: approximation techniques, SIAM J. Control and Optim. 27, 1181–1219 (1989)
G. Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control and Optim. 19, 106–113 (1981)
J. S. Gibson, The Riccati integral equations for optimal control problems on Hilbert space, SIAM J. Control and Optim. 17, 537–565 (1979)
P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, London, 1985
P. Grisvard, Contrǒlabilité exacte des solutions de certains problémes mixtes pour l’équation des ondes dans un polygone et polyèdre, Math. Pures et Appl. 68, 215–259 (1989)
M. G. Krein, Linear Differential Equations in Banach Space, Transl. Math. Monographs, Vol. 29, American Mathematical Society, Providence, RI, 1971
J. E. Lagnese, Decay of solutions of the wave equation in a bounded region with boundary dissipation, J. Diff. Equations 50, 163–182 (1983)
I. Lasiecka and R. Triggiani, Exponential uniform energy decay rates of the wave equation in a bounded region with ${L_2}\left ( 0, \infty ; {L_2}\left ( \Gamma \right ) \right )$-boundary feedback control in the Dirichlet boundary conditions, J. Differential Equations 66, 340–390 (1987)
I. Lasiecka and R. Triggiani, Riccati equations for hyperbolic partial differential equations with ${L_2}\left ( 0, T; {L_2}\left ( \Gamma \right ) \right )$-Dirichlet boundary controls, SIAM J. Control and Optim. 24, pp. 884–926 (1986)
I. Lasiecka and R. Triggiani, Algebraic Riccati equations with applications to boundary/point control problems: Continuous theory and approximation theory, preprint 1990
P. M. Morse and K. Uno Ingard, Theoretical Acoustics, Princeton University Press, Princeton, New Jersey, 1968
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983
A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, McGraw-Hill, New York, 1981
L. Schwartz, Théorie des Distributions, Hermann, Paris, 1966
H. Tanabe, Equations of Evolution, Pitman, New York, 1979
C. Wang, Linear quadratic optimal control of a wave equation with boundary damping and pointwise control input, J. Math. Analysis and Applications 192, 562–578 (1995)
J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987
K. Yosida, Functional Analysis, 6th Edition, Springer-Verlag, Berlin, 1980
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© Copyright 1999
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