Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback

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June 2016, 9(3): 791-813. doi: 10.3934/dcdss.2016029

Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback

Serge Nicaise ^1, and Cristina Pignotti ^2,

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

Received March 2015 Revised July 2015 Published April 2016

Abstract
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We study the stabilization problem for the wave equation with localized Kelvin--Voigt damping and mixed boundary condition with time delay. By using a frequency domain approach we show that, under an appropriate condition between the internal damping and the boundary feedback, an exponential stability result holds. In this sense, this extends the result of [19] where, in a more general setting, the case of distributed structural damping is considered.

Keywords: delay feedbacks, Wave equation, stabilization..

Mathematics Subject Classification: Primary: 35L05; Secondary: 93D1.

Citation: Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete and Continuous Dynamical Systems - S, 2016, 9 (3) : 791-813. doi: 10.3934/dcdss.2016029

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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