Stabilization of elastic plates with variable coefficients and dynamical boundary control
Authors:
Yuxia Guo, Shugen Chai and Pengfei Yao
Journal:
Quart. Appl. Math. 60 (2002), 383-400
MSC:
Primary 93D15; Secondary 74K20, 74M05, 93C20
DOI:
https://doi.org/10.1090/qam/1900498
MathSciNet review:
MR1900498
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Abstract: The aim of this paper is to investigate the stabilization of a hybrid system composed of a plate equation with variable coefficients and two ordinary differential equations under some suitable feedbacks. A rational energy decay rate is established by the multiplier method and the Riemannian geometry method, and the uniform energy decay rate for a simplified system is obtained.
- Bopeng Rao, Stabilization of elastic plates with dynamical boundary control, SIAM J. Control Optim. 36 (1998), no. 1, 148–163. MR 1616545, DOI https://doi.org/10.1137/S0363012996300975
- Peng-Fei Yao, On the observability inequalities for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim. 37 (1999), no. 5, 1568–1599. MR 1710233, DOI https://doi.org/10.1137/S0363012997331482
- Emmanuel Hebey, Sobolev spaces on Riemannian manifolds, Lecture Notes in Mathematics, vol. 1635, Springer-Verlag, Berlin, 1996. MR 1481970
- Michael E. Taylor, Partial differential equations, Texts in Applied Mathematics, vol. 23, Springer-Verlag, New York, 1996. Basic theory. MR 1395147
H. Wu, C. L. Shen, and Y. L. Yu, An Introduction to Riemannian Geometry (in Chinese), Univ. of Beijing, 1989
- Peng-Fei Yao, On shallow shell equations, Discrete Contin. Dyn. Syst. Ser. S 2 (2009), no. 3, 697–722. MR 2525774, DOI https://doi.org/10.3934/dcdss.2009.2.697
- Peng-Fei Yao, Observability inequalities for the Euler-Bernoulli plate with variable coefficients, Differential geometric methods in the control of partial differential equations (Boulder, CO, 1999) Contemp. Math., vol. 268, Amer. Math. Soc., Providence, RI, 2000, pp. 383–406. MR 1804802, DOI https://doi.org/10.1090/conm/268/04320
- Peng-Fei Yao, Observability inequalities for shallow shells, SIAM J. Control Optim. 38 (2000), no. 6, 1729–1756. MR 1776654, DOI https://doi.org/10.1137/S0363012999338692
- Bopeng Rao, Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback, Nonlinear Anal. 20 (1993), no. 6, 605–626. MR 1214731, DOI https://doi.org/10.1016/0362-546X%2893%2990023-L
- V. Komornik, Exact controllability and stabilization, RAM: Research in Applied Mathematics, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. The multiplier method. MR 1359765
- John E. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1061153
- A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486
B. Rao, Stabilization of elastic plate with dynamical boundary control, SIAM J. Control Optim. 36, 148–163 (1998)
P. F. Yao, On the observability inequality for exact controllability of wave equations with variable coefficients, SIAM J. Control Optim. 37, 1568–1599 (1999)
E. Hebey, Sobolev Spaces on Riemannian Manifolds, Lecture Notes in Mathematics, 1635, Springer-Verlag, Berlin, Heidelberg, 1996
M. E. Taylor, Partial Differential Equations I, Springer-Verlag, New York, 1996
H. Wu, C. L. Shen, and Y. L. Yu, An Introduction to Riemannian Geometry (in Chinese), Univ. of Beijing, 1989
P. F. Yao, On the shallow shell equations, to appear
P. F. Yao, Observability inequalities for the Euler-Bernoulli plate with variable coefficients, Contemporary Mathematics 268, 383–406 (2000)
P. F. Yao, Observability inequalities for shallow shells, SIAM J. Control Optim. 38, 1729–1756 (2000)
B. Rao, Stabilization of Kirchhoff plate equation in star-shaped domain by nonlinear boundary feedback, Nonlinear Anal. 20, 605–626 (1993)
V. Komornik, Exact controllability and stabilization, The multiplier method, Masson, Paris, 1994
J. E. Lagnese, Boundary stabilization of thin plates, SIAM, Philadelphia, PA, 1989
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Springer-Verlag, New York, 1983
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© Copyright 2002
American Mathematical Society