
Previous Article
Characterizations of integral inputtostate stability for bilinear systems in infinite dimensions
 MCRF Home
 This Issue

Next Article
Determining the waveguide conductivity in a hyperbolic equation from a single measurement on the lateral boundary
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 
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), 127150. doi: 10.1007/s0002800280830. 
[3] 
F. AlabauBoussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511541 (electronic). doi: 10.1137/S0363012901385368. 
[4] 
F. AlabauBoussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semidiscretized vibrating damped systems, J. Differential Equations, 249 (2010), 11451178. doi: 10.1016/j.jde.2009.12.005. 
[5] 
F. AlabauBoussouira, Z. Wang and L. Yu, A onestep 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), 10241065. 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), 213243. 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), 4356. 
[12] 
V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, volume 36 of ParisChicester, MassonJohn Wiley, 1994. 
[13] 
J. P. LaSalle, Some extensions of Liapunov's second method, IRE Trans., CT7 (1960), 520527. 
[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), 139160. doi: 10.1007/s1140101207548. 
[15] 
T. Li, B. Rao and L. Hu, Exact boundary synchronization for a coupled system of 1D wave equations, ESAIM Control Optim. Calc. Var., 20 (2014), 339361. 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), 630644. doi: 10.1007/s0003300430734. 
[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), 860881. 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), 399414. 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, SpringerVerlag, New York, 1983. doi: 10.1007/9781461255611. 
[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), 847857. 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), 127150. doi: 10.1007/s0002800280830. 
[3] 
F. AlabauBoussouira, Indirect boundary stabilization of weakly coupled hyperbolic systems, SIAM J. Control Optim., 41 (2002), 511541 (electronic). doi: 10.1137/S0363012901385368. 
[4] 
F. AlabauBoussouira, A unified approach via convexity for optimal energy decay rates of finite and infinite dimensional vibrating damped systems with applications to semidiscretized vibrating damped systems, J. Differential Equations, 249 (2010), 11451178. doi: 10.1016/j.jde.2009.12.005. 
[5] 
F. AlabauBoussouira, Z. Wang and L. Yu, A onestep 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), 10241065. 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), 213243. 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), 4356. 
[12] 
V. Komornik, Exact Controllability and Stabilization: The Multiplier Method, volume 36 of ParisChicester, MassonJohn Wiley, 1994. 
[13] 
J. P. LaSalle, Some extensions of Liapunov's second method, IRE Trans., CT7 (1960), 520527. 
[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), 139160. doi: 10.1007/s1140101207548. 
[15] 
T. Li, B. Rao and L. Hu, Exact boundary synchronization for a coupled system of 1D wave equations, ESAIM Control Optim. Calc. Var., 20 (2014), 339361. 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), 630644. doi: 10.1007/s0003300430734. 
[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), 860881. 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), 399414. 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, SpringerVerlag, New York, 1983. doi: 10.1007/9781461255611. 
[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), 847857. doi: 10.2307/1999112. 
[1] 
Florian Monteghetti, Ghislain Haine, Denis Matignon. Asymptotic stability of the multidimensional wave equation coupled with classes of positivereal impedance boundary conditions. Mathematical Control and Related Fields, 2019, 9 (4) : 759791. doi: 10.3934/mcrf.2019049 
[2] 
Yuriy Golovaty, Anna MarciniakCzochra, Mariya Ptashnyk. Stability of nonconstant stationary solutions in a reactiondiffusion equation coupled to the system of ordinary differential equations. Communications on Pure and Applied Analysis, 2012, 11 (1) : 229241. doi: 10.3934/cpaa.2012.11.229 
[3] 
Xiaojie Hou, Wei Feng. Traveling waves and their stability in a coupled reaction diffusion system. Communications on Pure and Applied Analysis, 2011, 10 (1) : 141160. doi: 10.3934/cpaa.2011.10.141 
[4] 
Hichem Kasri, Amar Heminna. Exponential stability of a coupled system with Wentzell conditions. Evolution Equations and Control Theory, 2016, 5 (2) : 235250. doi: 10.3934/eect.2016003 
[5] 
Serge Nicaise. Stability and asymptotic properties of dissipative evolution equations coupled with ordinary differential equations. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021057 
[6] 
Jason Metcalfe, David Spencer. Global existence for a coupled wave system related to the Strauss conjecture. Communications on Pure and Applied Analysis, 2018, 17 (2) : 593604. doi: 10.3934/cpaa.2018032 
[7] 
Tatsien Li, Bopeng Rao, Yimin Wei. Generalized exact boundary synchronization for a coupled system of wave equations. Discrete and Continuous Dynamical Systems, 2014, 34 (7) : 28932905. doi: 10.3934/dcds.2014.34.2893 
[8] 
Bopeng Rao, Zhuangyi Liu. A spectral approach to the indirect boundary control of a system of weakly coupled wave equations. Discrete and Continuous Dynamical Systems, 2009, 23 (1&2) : 399414. doi: 10.3934/dcds.2009.23.399 
[9] 
Takashi Narazaki. Global solutions to the Cauchy problem for the weakly coupled system of damped wave equations. Conference Publications, 2009, 2009 (Special) : 592601. doi: 10.3934/proc.2009.2009.592 
[10] 
José Manuel Palacios. Orbital and asymptotic stability of a train of peakons for the Novikov equation. Discrete and Continuous Dynamical Systems, 2021, 41 (5) : 24752518. doi: 10.3934/dcds.2020372 
[11] 
Dominika Pilarczyk. Asymptotic stability of singular solution to nonlinear heat equation. Discrete and Continuous Dynamical Systems, 2009, 25 (3) : 9911001. doi: 10.3934/dcds.2009.25.991 
[12] 
Alexander Komech, Elena Kopylova, David Stuart. On asymptotic stability of solitons in a nonlinear Schrödinger equation. Communications on Pure and Applied Analysis, 2012, 11 (3) : 10631079. doi: 10.3934/cpaa.2012.11.1063 
[13] 
Denis Matignon, Christophe Prieur. Asymptotic stability of WebsterLokshin equation. Mathematical Control and Related Fields, 2014, 4 (4) : 481500. doi: 10.3934/mcrf.2014.4.481 
[14] 
Long Hu, Tatsien Li, Bopeng Rao. Exact boundary synchronization for a coupled system of 1D wave equations with coupled boundary conditions of dissipative type. Communications on Pure and Applied Analysis, 2014, 13 (2) : 881901. doi: 10.3934/cpaa.2014.13.881 
[15] 
Olha P. Kupenko, Rosanna Manzo. Shape stability of optimal control problems in coefficients for coupled system of Hammerstein type. Discrete and Continuous Dynamical Systems  B, 2015, 20 (9) : 29672992. doi: 10.3934/dcdsb.2015.20.2967 
[16] 
Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control and Related Fields, 2015, 5 (2) : 305320. doi: 10.3934/mcrf.2015.5.305 
[17] 
Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure and Applied Analysis, 2005, 4 (4) : 861869. doi: 10.3934/cpaa.2005.4.861 
[18] 
Sergey Zelik. Asymptotic regularity of solutions of a nonautonomous damped wave equation with a critical growth exponent. Communications on Pure and Applied Analysis, 2004, 3 (4) : 921934. doi: 10.3934/cpaa.2004.3.921 
[19] 
Guanggan Chen, Jian Zhang. Asymptotic behavior for a stochastic wave equation with dynamical boundary conditions. Discrete and Continuous Dynamical Systems  B, 2012, 17 (5) : 14411453. doi: 10.3934/dcdsb.2012.17.1441 
[20] 
Xiaoxiao Zheng, Hui Wu. Orbital stability of periodic traveling wave solutions to the coupled compound KdV and MKdV equations with two components. Mathematical Foundations of Computing, 2020, 3 (1) : 1124. doi: 10.3934/mfc.2020002 
2020 Impact Factor: 1.284
Tools
Metrics
Other articles
by authors
[Back to Top]