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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

 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

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.
Citation: Serge Nicaise, Cristina Pignotti. Stability of the wave equation with localized Kelvin-Voigt damping and boundary delay feedback. Discrete & 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,, , ().   Google Scholar [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.  doi: 10.3233/ASY-151317.  Google Scholar [3] G. Chen, Control and stabilization for the wave equation in a bounded domain I,, SIAM J. Control Optim., 17 (1979), 66.  doi: 10.1137/0317007.  Google Scholar [4] G. Chen, Control and stabilization for the wave equation in a bounded domain II,, SIAM J. Control Optim., 19 (1981), 114.  doi: 10.1137/0319009.  Google Scholar [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.  doi: 10.1137/0326040.  Google Scholar [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.  doi: 10.1137/0324007.  Google Scholar [7] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms,, Springer Series in Computational Mathematics, 5 (1986).  doi: 10.1007/978-3-642-61623-5.  Google Scholar [8] F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar [9] V. Komornik, Rapid boundary stabilization of the wave equation,, SIAM J. Control Optim., 29 (1991), 197.  doi: 10.1137/0329011.  Google Scholar [10] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, RAM: Research in Applied Mathematics, 36 (1994).   Google Scholar [11] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl., 69 (1990), 33.   Google Scholar [12] J. Lagnese, Decay of solutions of wave equation in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163.  doi: 10.1016/0022-0396(83)90073-6.  Google Scholar [13] J. Lagnese, Note on boundary stabilization of wave equations,, SIAM J. Control and Optim., 26 (1988), 1250.  doi: 10.1137/0326068.  Google Scholar [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.  doi: 10.1016/0022-0396(87)90025-8.  Google Scholar [15] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] Masson, (1988).   Google Scholar [16] K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping,, Z. angew. Math. Phys., 57 (2006), 419.  doi: 10.1007/s00033-005-0029-2.  Google Scholar [17] Ö. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equations,, IEEE Trans. Automat. Control., 40 (1995), 1626.  doi: 10.1109/9.412634.  Google Scholar [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.  doi: 10.1137/060648891.  Google Scholar [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.  doi: 10.1093/imamci/dnr012.  Google Scholar [20] J. Prüss, On the spectrum of $C_{0}$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar [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.  doi: 10.1051/cocv:2006021.  Google Scholar [22] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial Differential Equations, 15 (1990), 205.  doi: 10.1080/03605309908820684.  Google Scholar

show all references

##### References:
 [1] K. Ammari and S. Gerbi, Interior feedback stabilization of wave equations with dynamic boundary delay,, , ().   Google Scholar [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.  doi: 10.3233/ASY-151317.  Google Scholar [3] G. Chen, Control and stabilization for the wave equation in a bounded domain I,, SIAM J. Control Optim., 17 (1979), 66.  doi: 10.1137/0317007.  Google Scholar [4] G. Chen, Control and stabilization for the wave equation in a bounded domain II,, SIAM J. Control Optim., 19 (1981), 114.  doi: 10.1137/0319009.  Google Scholar [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.  doi: 10.1137/0326040.  Google Scholar [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.  doi: 10.1137/0324007.  Google Scholar [7] V. Girault and P. A. Raviart, Finite Element Methods for Navier-Stokes Equations. Theory and Algorithms,, Springer Series in Computational Mathematics, 5 (1986).  doi: 10.1007/978-3-642-61623-5.  Google Scholar [8] F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces,, Ann. Differential Equations, 1 (1985), 43.   Google Scholar [9] V. Komornik, Rapid boundary stabilization of the wave equation,, SIAM J. Control Optim., 29 (1991), 197.  doi: 10.1137/0329011.  Google Scholar [10] V. Komornik, Exact Controllability and Stabilization. The Multiplier Method,, RAM: Research in Applied Mathematics, 36 (1994).   Google Scholar [11] V. Komornik and E. Zuazua, A direct method for the boundary stabilization of the wave equation,, J. Math. Pures Appl., 69 (1990), 33.   Google Scholar [12] J. Lagnese, Decay of solutions of wave equation in a bounded region with boundary dissipation,, J. Differential Equations, 50 (1983), 163.  doi: 10.1016/0022-0396(83)90073-6.  Google Scholar [13] J. Lagnese, Note on boundary stabilization of wave equations,, SIAM J. Control and Optim., 26 (1988), 1250.  doi: 10.1137/0326068.  Google Scholar [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.  doi: 10.1016/0022-0396(87)90025-8.  Google Scholar [15] J. L. Lions, Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 1,, Recherches en Mathématiques Appliquées [Research in Applied Mathematics] Masson, (1988).   Google Scholar [16] K. Liu and B. Rao, Exponential stability for the wave equations with local Kelvin-Voigt damping,, Z. angew. Math. Phys., 57 (2006), 419.  doi: 10.1007/s00033-005-0029-2.  Google Scholar [17] Ö. Mörgul, On the stabilization and stability robustness against small delays of some damped wave equations,, IEEE Trans. Automat. Control., 40 (1995), 1626.  doi: 10.1109/9.412634.  Google Scholar [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.  doi: 10.1137/060648891.  Google Scholar [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.  doi: 10.1093/imamci/dnr012.  Google Scholar [20] J. Prüss, On the spectrum of $C_{0}$-semigroups,, Trans. Amer. Math. Soc., 284 (1984), 847.  doi: 10.2307/1999112.  Google Scholar [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.  doi: 10.1051/cocv:2006021.  Google Scholar [22] E. Zuazua, Exponential decay for the semilinear wave equation with locally distributed damping,, Comm. Partial Differential Equations, 15 (1990), 205.  doi: 10.1080/03605309908820684.  Google Scholar
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