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Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback
1. | Université de Valenciennes et du Hainaut Cambrésis, LAMAV and FR CNRS 2956, Le Mont Houy, Institut des Sciences et Techniques de Valenciennes, 59313 Valenciennes Cedex 9 |
2. | Dipartimento di Matematica Pura e Applicata, Università di L'Aquila, Via Vetoio, Loc. Coppito, 67010 L'Aquila |
References:
[1] |
K. Ammari and S. Gerbi, Interior feedback stabilization of wave equations with dynamic boundary delay, arXiv:1405.6865. |
[2] |
K. Ammari, S. Nicaise and C. Pignotti, Stability of abstract-wave equation with delay and a Kelvin-Voigt damping, Asymptot. Anal., 95 (2015), 21-38.
doi: 10.3233/ASY-151317. |
[3] |
G. Chen, Control and stabilization for the wave equation in a bounded domain I, SIAM J. Control Optim., 17 (1979), 66-81.
doi: 10.1137/0317007. |
[4] |
G. Chen, Control and stabilization for the wave equation in a bounded domain II, SIAM J. Control Optim., 19 (1981), 114-122.
doi: 10.1137/0319009. |
[5] |
R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.
doi: 10.1137/0326040. |
[6] |
R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.
doi: 10.1137/0324007. |
[7] |
V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5 Springer, Berlin, 1986.
doi: 10.1007/978-3-642-61623-5. |
[8] |
F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. |
[9] |
V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208.
doi: 10.1137/0329011. |
[10] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, 36, Masson, Paris, 1994. |
[11] |
V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54. |
[12] |
J. Lagnese, Decay of solutions of wave equation in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182.
doi: 10.1016/0022-0396(83)90073-6. |
[13] |
J. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control and Optim., 26 (1988), 1250-1256.
doi: 10.1137/0326068. |
[14] |
I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L_2(0,T;L_2(\Sigma))$-feedback control in the Dirichlet boundary conditions, J. Differential Equations, 66 (1987), 340-390.
doi: 10.1016/0022-0396(87)90025-8. |
[15] |
J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] Masson, Paris, 1988. |
[16] |
K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping, Z. angew. Math. Phys., 57 (2006), 419-432.
doi: 10.1007/s00033-005-0029-2. |
[17] |
Ö. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equations, IEEE Trans. Automat. Control., 40 (1995), 1626-1630.
doi: 10.1109/9.412634. |
[18] |
S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
[19] |
S. Nicaise and C. Pignotti, Exponential stability of second-order evolution equations with structural damping and dynamic boundary delay feedback, IMA J. Math. Control Inform., 28 (2011), 417-446.
doi: 10.1093/imamci/dnr012. |
[20] |
J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
[21] |
G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: Control Optim. Calc. Var., 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
[22] |
E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.
doi: 10.1080/03605309908820684. |
show all references
References:
[1] |
K. Ammari and S. Gerbi, Interior feedback stabilization of wave equations with dynamic boundary delay, arXiv:1405.6865. |
[2] |
K. Ammari, S. Nicaise and C. Pignotti, Stability of abstract-wave equation with delay and a Kelvin-Voigt damping, Asymptot. Anal., 95 (2015), 21-38.
doi: 10.3233/ASY-151317. |
[3] |
G. Chen, Control and stabilization for the wave equation in a bounded domain I, SIAM J. Control Optim., 17 (1979), 66-81.
doi: 10.1137/0317007. |
[4] |
G. Chen, Control and stabilization for the wave equation in a bounded domain II, SIAM J. Control Optim., 19 (1981), 114-122.
doi: 10.1137/0319009. |
[5] |
R. Datko, Not all feedback stabilized hyperbolic systems are robust with respect to small time delays in their feedbacks, SIAM J. Control Optim., 26 (1988), 697-713.
doi: 10.1137/0326040. |
[6] |
R. Datko, J. Lagnese and M. P. Polis, An example on the effect of time delays in boundary feedback stabilization of wave equations, SIAM J. Control Optim., 24 (1986), 152-156.
doi: 10.1137/0324007. |
[7] |
V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms, Springer Series in Computational Mathematics, 5 Springer, Berlin, 1986.
doi: 10.1007/978-3-642-61623-5. |
[8] |
F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differential Equations, 1 (1985), 43-56. |
[9] |
V. Komornik, Rapid boundary stabilization of the wave equation, SIAM J. Control Optim., 29 (1991), 197-208.
doi: 10.1137/0329011. |
[10] |
V. Komornik, Exact Controllability and Stabilization. The Multiplier Method, RAM: Research in Applied Mathematics, 36, Masson, Paris, 1994. |
[11] |
V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., 69 (1990), 33-54. |
[12] |
J. Lagnese, Decay of solutions of wave equation in a bounded region with boundary dissipation, J. Differential Equations, 50 (1983), 163-182.
doi: 10.1016/0022-0396(83)90073-6. |
[13] |
J. Lagnese, Note on boundary stabilization of wave equations, SIAM J. Control and Optim., 26 (1988), 1250-1256.
doi: 10.1137/0326068. |
[14] |
I. Lasiecka and R. Triggiani, Uniform exponential energy decay of wave equations in a bounded region with $L_2(0,T;L_2(\Sigma))$-feedback control in the Dirichlet boundary conditions, J. Differential Equations, 66 (1987), 340-390.
doi: 10.1016/0022-0396(87)90025-8. |
[15] |
J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] Masson, Paris, 1988. |
[16] |
K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping, Z. angew. Math. Phys., 57 (2006), 419-432.
doi: 10.1007/s00033-005-0029-2. |
[17] |
Ö. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equations, IEEE Trans. Automat. Control., 40 (1995), 1626-1630.
doi: 10.1109/9.412634. |
[18] |
S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006), 1561-1585.
doi: 10.1137/060648891. |
[19] |
S. Nicaise and C. Pignotti, Exponential stability of second-order evolution equations with structural damping and dynamic boundary delay feedback, IMA J. Math. Control Inform., 28 (2011), 417-446.
doi: 10.1093/imamci/dnr012. |
[20] |
J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857.
doi: 10.2307/1999112. |
[21] |
G. Q. Xu, S. P. Yung and L. K. Li, Stabilization of wave systems with input delay in the boundary control, ESAIM: Control Optim. Calc. Var., 12 (2006), 770-785.
doi: 10.1051/cocv:2006021. |
[22] |
E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping, Comm. Partial Differential Equations, 15 (1990), 205-235.
doi: 10.1080/03605309908820684. |
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