Abstract
This paper considers the stabilization of the transmission problem of wave equations with variable coefficients. By introducing both boundary feedback control and distribute feedback control near the transmission boundary, the author establishes the uniform energy decay rate for the problem.
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W. Liu and G. Williams, The exponential stability of the problem of transmission of the wave equation, Bull. Austral. Math. Soc., 1998, 57(2): 305–327.
W. Liu, Stabilization and controllability for the transmission wave equation, IEEE Transactions on Automation Control, 2001, 46(12): 1900–1907.
S. Chai and K. Liu, Boundary stabilization of the transmission of wave equations with variable coefficients, Chinese Ann. Math. Ser.A, 2005, 26(5): 605–612.
J. Lagnese, Boundary controllability in problems of transmission for a class of second order hyperbolic systems, ESAIM Control Optim. Calc., 1997, 2: 343–357.
M. Aassila, Exact boundary controllability of the plate equation, Differential Integral Equations, 2000, 13(10–12): 1413–1428.
H. P. Oquendo, Nonlinear boundary stabilization for a tranmission problem in elasticity, Nonlinear Analysis, 2003, 52(4): 1331–1354.
P. F. Yao, On the observability inequality for exact controllability of wave equations with variable coefficients, SIAM J. Contr. Optim., 1999, 37(5): 1568–1599.
S. J. Feng and D. X. Feng, Boundary stabilization of wave equations with variable coefficients, Science in China A, 2001, 44(3): 345–350.
S. J. Feng and D. X. Feng, Nonlinear boundary stabilization of wave equations with variable coefficients, Chinese Ann. Math. Ser.B, 2003, 24(2): 239–248.
H. Wu, C. L. Shen, and Y. L. Yu, An Introduction to Riemannian Geometry, Peking University Press, Beijing, 1989.
I. Lasiecka and R. Triggiani, Uniform stabilization of the wave equations with Dirichlet or Neumann feedback control without geometric conditions, Appl. Math. Optim., 1992, 25(2): 189–224.
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This research is supported by the National Natural Science Foundation of China under Grant Nos. 10571161 and 60774014.
This paper was recommended for publication by Editor Dexing FENG.
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Chai, S. Uniform decay rate for the transmission wave equations with variable coefficients. J Syst Sci Complex 24, 253–260 (2011). https://doi.org/10.1007/s11424-011-8009-4
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DOI: https://doi.org/10.1007/s11424-011-8009-4