Abstract
Uniform stabilization of a structural acoustic model describing an acoustic chamber with flexible curved wall is addressed. The coupled nonlinear system consists of a variable-coefficient wave equation and a shallow shell model which is used to model the flexible curved wall (interface). The coupling between the wave and the shell takes place on the interface. Derivation of stability estimates for the variable-coefficient coupled system with the shallow shell depends on the Riemannian geometry method and the multiplier technique. The uniform energy decay rates of the overall interactive model are achieved by introducing nonlinear boundary feedbacks applied to the wave equation and the shell model.
Similar content being viewed by others
References
Avalos, G., Toundykov, D.: Boundary stabilization of structural acoustic interactions with interface on a Reissner-Mindlin plate. Nonlinear Anal. RWA 12, 2985–3013 (2011)
Becklin, A.R., Rammaha, M.A.: Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms. Evol. Equ. Control Theory 10(4), 797–836 (2021)
Bongarti, M., Lasiecka, I., Rodrigues, J.H.: Boundary stabilization of the linear MGT equation with partially absorbing boundary data and degenerate viscoelasticity. Discret. Contin. Dyn. Syst. Ser. S 15(6), 1355–1376 (2022)
Camurdan, M., Ji, G.: Uniform feedback stabilization via boundary moments in a three dimensional structural acoustic model. 37 IEEE CDC Proc. 3, 2058–2064 (1998)
Cagnol, J., Lasiecka, I., Lebiedzik, C., Zolesio, J.P.: Uniform stability in structural acoustic models with flexible curved walls. J. Differ. Equ. 186, 88–121 (2002)
Chai, S.G., Guo, Y.X., Yao, P.F.: Boundary feedback stabilization of shallow shells. SIAM J. Control Optim. 42(1), 239–259 (2003)
Dalsen, M.G.-V.: On a structural acoustic model with interface a Reissner-Mindlin plate or a Timoshenko beam. J. Math. Anal. Appl. 320, 121–144 (2006)
Dalsen, M.G.-V.: On a structural acoustic model which incorporates shear and thermal effects in the structural component. J. Math. Anal. Appl. 341(2), 1253–1270 (2008)
Deng, L., Zhang, Z.F.: Controllability for transmission wave/plate equations on Riemannian manifolds. Syst. Control Lett. 19, 48–54 (2016)
Gulliver, R., Lasiecka, I., Littman, W., Triggiani, R.: The case for differential geometry in the control of single and coupled PDEs: the structural acoustic chamber. In: Geometric Methods in Inverse Problems and PDE control. IMA Vol. Math. Appl., vol. 137, pp. 73–181, Springer, New York (2004)
Hao, J.H., Du, F.Q.: Decay rate for viscoelastic wave equation of variable coefficients with delay and dynamic boundary conditions. Math. Methods Appl. Sci. 44(1), 284–302 (2021)
Kaltenbacher, B., Kukavica, I., Lasiecka, I., Triggiani, R., Tuffaha, A., Webster, J. T.: Mathematical Theory of Evolutionary Fluid-Flow Structure Interactions. Lecture Notes from Oberwolfach Seminars. Oberwolfach Seminars 48, Birkhäuser/Springer, Cham (2018)
Lasiecka, I., Triggiani, R.: Uniform stabilization of the wave equations with Dirichlet or Neumann feedback control without geometric conditions. Appl. Math. Optim. 25(2), 189–224 (1992)
Lasiecka, I., Tartaru, D.: Uniform boundary stabilization of semilinear wave equation with nonlinear boundary damping. Differ. Integral Equ. 6(3), 507–533 (1993)
Lasiecka, I., Marchard, R.: Riccati equations arising in acoustic structure interaction with curved walls. Dyn. Control 8, 269–292 (1998)
Lasiecka, I.: Mathematical control theory in structural acoustic problems. Math. Models Methods Appl. Sci. 8(7), 1119–1153 (1998)
Lasiecka, I., Lebiedzik, C.: Uniform stability in structural acoustic systems with thermal effiects and nonlinear boundary damping. Control Cyber. (special invited volume on “Control of PDE’s”) 28, 557-581 (1999)
Lasiecka, I.: Boundary stabilization of a 3-dimensional structural acoustic model. J. Math. Pure Appl. 78, 203–232 (1999)
Lasiecka, I., Lebiedzik, C.: Decay rates of interactive hyperbolic-parabolic PDE models with thermal effects on the interface. Appl. Math. Optim. 142, 127–167 (2000)
Lasiecka, I., Lebiedzik, C.: Asymptotic behaviour of nonlinear structural acoustic interactions with thermal effects on the interface. Nonlinear Anal. TMA 49, 703–735 (2002)
Lasiecka, I., Triggiani, R.: Uniform stabilization of a shallow shell model with nonlinear boundary feedbacks. J. Math. Anal. Appl. 269, 642–688 (2002)
Lasiecka, I., Rodrigues, J.H.: Weak and strong semigroups in structural acoustic Kirchhoff-Boussinesq interactions with boundary feedback. J. Differ. Equ. 298, 387–429 (2021)
Li, J., Chai, S.G.: Energy decay for a nonlinear wave equation of variable coefficients with acoustic boundary conditions and a time-varying delay in the boundary feedback. Nonlinear Anal. TMA 112, 105–117 (2015)
Li, J., Chai, S.G.: Existence and energy decay rates of solutions to the variable-coefficient Euler-Bernoulli plate with a delay inlocalized nonlinear internal feedback. J. Math. Anal. Appl. 443, 981–1006 (2016)
Li, J., Chai, S.G.: Stabilization of the variable-coefficient structural acoustic model with curved middle surface and delay effects in the structural component. J. Math. Anal. Appl. 454, 510–532 (2017)
Li, S., Yao, P.F.: Stabilization of the Euler-Bernoulli plate with variable coefficients by nonlinear internal feedback. Automatica 50, 2225–2233 (2014)
Liu, Y.X., Yao, P.F.: Energy decay rate of the wave equations on Riemannian manifolds with critical potential. Appl. Math. Optim. 78, 61–101 (2018)
Morse, P.M., Ingard, K.U.: Theoretical Acoustics. McGraw-Hill, New York (1968)
Wu, H., Shen, C.L., Yu, Y.L.: An Introduction to Riemannian Geometry. Peking University Press, Beijing (1989)
Yao, P.F.: On the observatility inequality for exact controllability of wave equations with variable coefficients. SIAM J. Control Optim. 37(5), 1568–1599 (1999)
Yao, P.F.: Observability inequalities for shallow shells. SIAM J. Control Optim. 38(6), 1729–1756 (2000)
Yao, P.F.: Modeling and Control in Vibrational and Structural Dynamics: A Differential Geometric Approach. CRC Press, Boca Raton (2011)
Yang, F.Y., Yao, P.F., Chen, G.: Boundary controllability of structural acoustic systems with variable coefficients and curved walls. Math. Control Signals Syst. 30, 5 (2018)
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Competing Interests
The authors have not disclosed any competing interests.
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
This work is supported by National Nature Science Foundation of China for the Youth (11801339), National Natural Science Foundation of China (62273217,12131008), Nature Science Foundation of Shanxi Province (201901D111042, 201901D211162), Shanxi Scholarship Council of China (2020-006).
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Li, J., Chai, S. Uniform Decay Rates for a Variable-Coefficient Structural Acoustic Model with Curved Interface on a Shallow Shell. Appl Math Optim 87, 56 (2023). https://doi.org/10.1007/s00245-023-09968-2
Accepted:
Published:
DOI: https://doi.org/10.1007/s00245-023-09968-2
Keywords
- Structural acoustic model
- Shallow shell model
- Nonlinear boundary feedbacks
- Uniform decay rates
- The Riemannian geometry method