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September  2016, 6(3): 429-446. doi: 10.3934/mcrf.2016010

Asymptotic stability of wave equations coupled by velocities

1. 

School of Mathematical Sciences, Fudan University, Shanghai Key Laboratory for Contemporary Applied Mathematics, Shanghai 200433, China

2. 

School of Mathematical Sciences and Shanghai Key Laboratory for Contemporary Applied Mathematics, Fudan University, Shanghai 200433

Received  June 2015 Revised  October 2015 Published  August 2016

This paper is devoted to study the asymptotic stability of wave equations with constant coefficients coupled by velocities. By using Riesz basis approach, multiplier method and frequency domain approach respectively, we find the sufficient and necessary condition, that the coefficients satisfy, leading to the exponential stability of the system. In addition, we give the optimal decay rate in one dimensional case.
Citation: Yan Cui, Zhiqiang Wang. Asymptotic stability of wave equations coupled by velocities. Mathematical Control and Related Fields, 2016, 6 (3) : 429-446. doi: 10.3934/mcrf.2016010
References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, volume 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second edition, 2003.

[2]

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0.

[3]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541 (electronic). doi: 10.1137/S0363012901385368.

[4]

F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems, J. Differential Equations, 249 (2010), 1145-1178. doi: 10.1016/j.jde.2009.12.005.

[5]

F. Alabau-Boussouira, Z. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, preprint, arXiv:1503.04126, 2015.

[6]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.

[7]

S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Comm. Partial Differential Equations, 19 (1994), 213-243. doi: 10.1080/03605309408821015.

[8]

I. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18. American Mathematical Society, Providence, R.I., 1969.

[9]

G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, third edition, 1996.

[10]

Z. Hu, Asymptotic Synchronization for a Coupled System of Wave Eqution (in Chinese), Master Thesis, 2014.

[11]

F. Huang, Characterization condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Eq., 1 (1985), 43-56.

[12]

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, volume 36 of Paris-Chicester, Masson-John Wiley, 1994.

[13]

J. P. LaSalle, Some extensions of Liapunov's second method, IRE Trans., CT-7 (1960), 520-527.

[14]

T. Li and B. Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, Chin. Ann. Math. Ser. B, 34 (2013), 139-160. doi: 10.1007/s11401-012-0754-8.

[15]

T. Li, B. Rao and L. Hu, Exact boundary synchronization for a coupled system of 1-D wave equations, ESAIM Control Optim. Calc. Var., 20 (2014), 339-361. doi: 10.1051/cocv/2013066.

[16]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.

[17]

Z. Liu and B. Rao, Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl., 335 (2007), 860-881. doi: 10.1016/j.jmaa.2007.02.021.

[18]

Z. Liu and B. Rao, A spectral approach to the indirect boundary control of a system of weakly coupled wave equations, Discrete Contin. Dyn. Syst., 23 (2009), 399-414. doi: 10.3934/dcds.2009.23.399.

[19]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, volume 398 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 1999.

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, volume 130 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1987.

[22]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.

show all references

References:
[1]

R. A. Adams and J. J. F. Fournier, Sobolev Spaces, volume 140 of Pure and Applied Mathematics (Amsterdam), Elsevier/Academic Press, Amsterdam, second edition, 2003.

[2]

F. Alabau, P. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150. doi: 10.1007/s00028-002-8083-0.

[3]

F. Alabau-Boussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511-541 (electronic). doi: 10.1137/S0363012901385368.

[4]

F. Alabau-Boussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semi-discretized vibrating damped systems, J. Differential Equations, 249 (2010), 1145-1178. doi: 10.1016/j.jde.2009.12.005.

[5]

F. Alabau-Boussouira, Z. Wang and L. Yu, A one-step optimal energy decay formula for indirectly nonlinearly damped hyperbolic systems coupled by velocities, preprint, arXiv:1503.04126, 2015.

[6]

C. Bardos, G. Lebeau and J. Rauch, Sharp sufficient conditions for the observation, control, and stabilization of waves from the boundary, SIAM J. Control Optim., 30 (1992), 1024-1065. doi: 10.1137/0330055.

[7]

S. Cox and E. Zuazua, The rate at which energy decays in a damped string, Comm. Partial Differential Equations, 19 (1994), 213-243. doi: 10.1080/03605309408821015.

[8]

I. Gohberg and M. G. Kreĭn, Introduction to the Theory of Linear Nonselfadjoint Operators, Translated from the Russian by A. Feinstein. Translations of Mathematical Monographs, Vol. 18. American Mathematical Society, Providence, R.I., 1969.

[9]

G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Studies in the Mathematical Sciences. Johns Hopkins University Press, Baltimore, MD, third edition, 1996.

[10]

Z. Hu, Asymptotic Synchronization for a Coupled System of Wave Eqution (in Chinese), Master Thesis, 2014.

[11]

F. Huang, Characterization condition for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Diff. Eq., 1 (1985), 43-56.

[12]

V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, volume 36 of Paris-Chicester, Masson-John Wiley, 1994.

[13]

J. P. LaSalle, Some extensions of Liapunov's second method, IRE Trans., CT-7 (1960), 520-527.

[14]

T. Li and B. Rao, Exact synchronization for a coupled system of wave equations with Dirichlet boundary controls, Chin. Ann. Math. Ser. B, 34 (2013), 139-160. doi: 10.1007/s11401-012-0754-8.

[15]

T. Li, B. Rao and L. Hu, Exact boundary synchronization for a coupled system of 1-D wave equations, ESAIM Control Optim. Calc. Var., 20 (2014), 339-361. doi: 10.1051/cocv/2013066.

[16]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644. doi: 10.1007/s00033-004-3073-4.

[17]

Z. Liu and B. Rao, Frequency domain approach for the polynomial stability of a system of partially damped wave equations, J. Math. Anal. Appl., 335 (2007), 860-881. doi: 10.1016/j.jmaa.2007.02.021.

[18]

Z. Liu and B. Rao, A spectral approach to the indirect boundary control of a system of weakly coupled wave equations, Discrete Contin. Dyn. Syst., 23 (2009), 399-414. doi: 10.3934/dcds.2009.23.399.

[19]

Z. Liu and S. Zheng, Semigroups Associated with Dissipative Systems, volume 398 of Chapman & Hall/CRC Research Notes in Mathematics, Chapman & Hall/CRC, Boca Raton, FL, 1999.

[20]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, volume 44 of Applied Mathematical Sciences, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[21]

J. Pöschel and E. Trubowitz, Inverse Spectral Theory, volume 130 of Pure and Applied Mathematics, Academic Press, Inc., Boston, MA, 1987.

[22]

J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.

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