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The exponential stability of the problem of transmission of the wave equation

Published online by Cambridge University Press:  17 April 2009

Weijiu Liu  and
Graham Williams
Weijiu Liu
Affiliation:
School of Mathematics and Applied StatisticsUniversity of WollongongNorthfields AveWollongong NSW 2522Australia e-mail: [email protected]@uow.edu.au
Graham Williams
Affiliation:
School of Mathematics and Applied StatisticsUniversity of WollongongNorthfields AveWollongong NSW 2522Australia e-mail: [email protected]@uow.edu.au
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Abstract

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The problem of exponential stability of the problem of transmission of the wave equation with lower-order terms is considered. Making use of the classical energy method and multiplier technique, we prove that this problem of transmission is exponentially stable.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 57 , Issue 2 , April 1998 , pp. 305 - 327
Copyright
Copyright © Australian Mathematical Society 1998

References

[1]Bardos, C., Lebeau, G. and Rauch, J., ‘Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary’, SIAM J. Control Optim. 30 (1992), 1024–1065.CrossRefGoogle Scholar
[2]Chen, G., ‘Energy decay estimates and exact boundary value controllability for the wave equation in a bounded domain’, J. Math. Pures Appl. 58 (1979), 249–273.Google Scholar
[3]Chen, G., ‘Control and stabilization for the wave equation in a bounded domain’, SIAM J. Control Optim. 17 (1979), 66–81.CrossRefGoogle Scholar
[4]Chen, G., ‘A note on the boundary stabilization of the wave equation’, SIAM J. Control Optim. 19 (1981), 106–113.CrossRefGoogle Scholar
[5]Chen, G., ‘Control and stabilization for the wave equation in a bounded domain, part II’, SIAM J. Control Optim. 19 (1981), 114–122.CrossRefGoogle Scholar
[6]Chen, G., ‘Control and stabilization for the wave equation, part III: Domain with moving boundary’, SIAM J. Control Optim. 19 (1981), 123–138.Google Scholar
[7]Garofalo, N. and Lin, F.H., ‘Unique continuation for elliptic operators: A geometric-variational approach’, Comm. Pure Appl. Math. 40 (1987), 347–366.CrossRefGoogle Scholar
[8]Komornik, V., Exact controllability and stabilization: The multiplier method (John Wiley and Sons, Masson, Paris, 1994).Google Scholar
[9]Lagnese, J., ‘Decay of solutions of wave equations in a bounded region with boundary dissipation’, J. Differential Equations 50 (1983), 163–182.CrossRefGoogle Scholar
[10]Lax, P. and Phillips, R.S., ‘Scattering theory for dissipative hyperbolic systems’, J. Funct. Anal. 14 (1973), 172–235.CrossRefGoogle Scholar
[11]Lax, P. and Phillips, R.S., Scattering theory, (Revised Edition) (Academic Press, Inc., Boston, 1989).Google Scholar
[12]Lions, J.L. and Magenes, E., Non-homogeneous boundary value problems and applications, Vol. I and II (Springer-Verlag, Berlin, Heidelberg, New York, 1972).Google Scholar
[13]Morawetz, C.S., ‘Exponential decay of solutions of the wave equation’, Comm. Pure Appl. Math. 19 (1966), 439–444.CrossRefGoogle Scholar
[14]Morawetz, C.S., ‘Decay for solutions of the exterior problem for the wave equation’, Comm. Pure Appl. Math. 28 (1975), 229–264.CrossRefGoogle Scholar
[15]Nicaise, S., ‘Boundary exact controllability of interface problems with singularities: addition of the coefficients singularities’, SIAM J. Control Optim. 34 (1996), 1512–1532.CrossRefGoogle Scholar
[16]Pazy, A., Semigroup of linear operators and applications to partial differential equations (Springer-Verlag, Berlin, Heidelberg, New York, 1983).CrossRefGoogle Scholar
[17]Russell, D.L., ‘Exact boundary value controlability theorems for wave and heat processes in star-complemented regions’, in Differential games and control theory, (Roxin, , Liu, and Sternberg, , Editors) (Marcel Dekker Inc., New York, 1974), pp. 291–319.Google Scholar
[18]Strauss, W.A., ‘Dispersal of waves vanishing on the boundary of an exterior domain’, Comm. Pure Appl. Math. 28 (1975), 65–278.CrossRefGoogle Scholar