Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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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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  • [1] George Avalos, The exponential stability of a coupled hyperbolic/parabolic system arising in structural acoustics, Abstr. Appl. Anal. 1 (1996), no. 2, 203–217. MR 1401615, https://doi.org/10.1155/S1085337596000103
  • [2] 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)
  • [3] H. T. Banks and K. Ito, A unified framework for approximation in inverse problems for distributed parameter systems, Control Theory Adv. Tech. 4 (1988), no. 1, 73–90. MR 941397
  • [4] H. T. Banks, K. Ito, and C. Wang, Exponentially stable approximations of weakly damped wave equations, Estimation and control of distributed parameter systems (Vorau, 1990) Internat. Ser. Numer. Math., vol. 100, Birkhäuser, Basel, 1991, pp. 1–33. MR 1155634, https://doi.org/10.1007/978-3-0348-6418-3_1
  • [5] 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
  • [6] H. T. Banks and C. Wang, Optimal feedback control of infinite-dimensional parabolic evolution systems: approximation techniques, SIAM J. Control Optim. 27 (1989), no. 5, 1182–1219. MR 1009343, https://doi.org/10.1137/0327062
  • [7] Goong Chen, A note on the boundary stabilization of the wave equation, SIAM J. Control Optim. 19 (1981), no. 1, 106–113. MR 603083, https://doi.org/10.1137/0319008
  • [8] J. S. Gibson, The Riccati integral equations for optimal control problems on Hilbert spaces, SIAM J. Control Optim. 17 (1979), no. 4, 537–565. MR 534423, https://doi.org/10.1137/0317039
  • [9] P. Grisvard, Elliptic problems in nonsmooth domains, Monographs and Studies in Mathematics, vol. 24, Pitman (Advanced Publishing Program), Boston, MA, 1985. MR 775683
  • [10] P. Grisvard, Contrôlabilité exacte des solutions de l’équation des ondes en présence de singularités, J. Math. Pures Appl. (9) 68 (1989), no. 2, 215–259 (French, with English summary). MR 1010769
  • [11] S. G. Kreĭn, Linear differential equations in Banach space, American Mathematical Society, Providence, R.I., 1971. Translated from the Russian by J. M. Danskin; Translations of Mathematical Monographs, Vol. 29. MR 0342804
  • [12] John Lagnese, Decay of solutions of wave equations in a bounded region with boundary dissipation, J. Differential Equations 50 (1983), no. 2, 163–182. MR 719445, https://doi.org/10.1016/0022-0396(83)90073-6
  • [13] I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with 𝐿₂(0,∞;𝐿₂(Γ))-feedback control in the Dirichlet boundary conditions, J. Differential Equations 66 (1987), no. 3, 340–390. MR 876804, https://doi.org/10.1016/0022-0396(87)90025-8
  • [14] I. Lasiecka and R. Triggiani, Riccati equations for hyperbolic partial differential equations with 𝐿₂(0,𝑇;𝐿₂(Γ))-Dirichlet boundary terms, SIAM J. Control Optim. 24 (1986), no. 5, 884–925. MR 854062, https://doi.org/10.1137/0324054
  • [15] I. Lasiecka and R. Triggiani, Differential and algebraic Riccati equations with application to boundary/point control problems: continuous theory and approximation theory, Lecture Notes in Control and Information Sciences, vol. 164, Springer-Verlag, Berlin, 1991. MR 1132440
  • [16] P. M. Morse and K. Uno Ingard, Theoretical Acoustics, Princeton University Press, Princeton, New Jersey, 1968
  • [17] A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
  • [18] A. D. Pierce, Acoustics: An Introduction to its Physical Principles and Applications, McGraw-Hill, New York, 1981
  • [19] Laurent Schwartz, Théorie des distributions, Publications de l’Institut de Mathématique de l’Université de Strasbourg, No. IX-X. Nouvelle édition, entiérement corrigée, refondue et augmentée, Hermann, Paris, 1966 (French). MR 0209834
  • [20] Hiroki Tanabe, Equations of evolution, Monographs and Studies in Mathematics, vol. 6, Pitman (Advanced Publishing Program), Boston, Mass.-London, 1979. Translated from the Japanese by N. Mugibayashi and H. Haneda. MR 533824
  • [21] Chunming Wang, Linear quadratic optimal control of a wave equation with boundary damping and pointwise control input, J. Math. Anal. Appl. 192 (1995), no. 2, 562–578. MR 1332228, https://doi.org/10.1006/jmaa.1995.1190
  • [22] J. Wloka, Partial differential equations, Cambridge University Press, Cambridge, 1987. Translated from the German by C. B. Thomas and M. J. Thomas. MR 895589
  • [23] Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913

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DOI: https://doi.org/10.1090/qam/1672195
Article copyright: © Copyright 1999 American Mathematical Society

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