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

Previous Article
Semigroup-theoretic approach to identification of linear diffusion coefficients
DCDS-S Home
This Issue
Next Article
A symmetry result for degenerate elliptic equations on the Wiener space with nonlinear boundary conditions and applications

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
Related Papers

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 & 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

[1]	Serge Nicaise, Julie Valein. Stabilization of the wave equation on 1-d networks with a delay term in the nodal feedbacks. Networks & Heterogeneous Media, 2007, 2 (3) : 425-479. doi: 10.3934/nhm.2007.2.425
[2]	Ferhat Mohamed, Hakem Ali. Energy decay of solutions for the wave equation with a time-varying delay term in the weakly nonlinear internal feedbacks. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 491-506. doi: 10.3934/dcdsb.2017024
[3]	Yanni Guo, Genqi Xu, Yansha Guo. Stabilization of the wave equation with interior input delay and mixed Neumann-Dirichlet boundary. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2491-2507. doi: 10.3934/dcdsb.2016057
[4]	Kim Dang Phung. Boundary stabilization for the wave equation in a bounded cylindrical domain. Discrete & Continuous Dynamical Systems - A, 2008, 20 (4) : 1057-1093. doi: 10.3934/dcds.2008.20.1057
[5]	Behzad Azmi, Karl Kunisch. Receding horizon control for the stabilization of the wave equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (2) : 449-484. doi: 10.3934/dcds.2018021
[6]	Serge Nicaise. Internal stabilization of a Mindlin-Timoshenko model by interior feedbacks. Mathematical Control & Related Fields, 2011, 1 (3) : 331-352. doi: 10.3934/mcrf.2011.1.331
[7]	Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693
[8]	Martin Gugat, Günter Leugering, Ke Wang. Neumann boundary feedback stabilization for a nonlinear wave equation: A strict $H^2$-lyapunov function. Mathematical Control & Related Fields, 2017, 7 (3) : 419-448. doi: 10.3934/mcrf.2017015
[9]	Mohammad Akil, Ali Wehbe. Stabilization of multidimensional wave equation with locally boundary fractional dissipation law under geometric conditions. Mathematical Control & Related Fields, 2019, 9 (1) : 97-116. doi: 10.3934/mcrf.2019005
[10]	Mokhtar Kirane, Belkacem Said-Houari, Mohamed Naim Anwar. Stability result for the Timoshenko system with a time-varying delay term in the internal feedbacks. Communications on Pure & Applied Analysis, 2011, 10 (2) : 667-686. doi: 10.3934/cpaa.2011.10.667
[11]	Yaru Xie, Genqi Xu. Exponential stability of 1-d wave equation with the boundary time delay based on the interior control. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 557-579. doi: 10.3934/dcdss.2017028
[12]	Guo Lin, Haiyan Wang. Traveling wave solutions of a reaction-diffusion equation with state-dependent delay. Communications on Pure & Applied Analysis, 2016, 15 (2) : 319-334. doi: 10.3934/cpaa.2016.15.319
[13]	Andrei Fursikov. Stabilization of the simplest normal parabolic equation. Communications on Pure & Applied Analysis, 2014, 13 (5) : 1815-1854. doi: 10.3934/cpaa.2014.13.1815
[14]	Gilbert Peralta, Karl Kunisch. Interface stabilization of a parabolic-hyperbolic pde system with delay in the interaction. Discrete & Continuous Dynamical Systems - A, 2018, 38 (6) : 3055-3083. doi: 10.3934/dcds.2018133
[15]	Abdelkarim Kelleche, Nasser-Eddine Tatar. Existence and stabilization of a Kirchhoff moving string with a delay in the boundary or in the internal feedback. Evolution Equations & Control Theory, 2018, 7 (4) : 599-616. doi: 10.3934/eect.2018029
[16]	Qingwen Hu, Huan Zhang. Stabilization of turning processes using spindle feedback with state-dependent delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (10) : 4329-4360. doi: 10.3934/dcdsb.2018167
[17]	Alberto Bressan, Fabio S. Priuli. Nearly optimal patchy feedbacks. Discrete & Continuous Dynamical Systems - A, 2008, 21 (3) : 687-701. doi: 10.3934/dcds.2008.21.687
[18]	Lassaad Aloui, Moez Khenissi. Boundary stabilization of the wave and Schrödinger equations in exterior domains. Discrete & Continuous Dynamical Systems - A, 2010, 27 (3) : 919-934. doi: 10.3934/dcds.2010.27.919
[19]	Tai-Chia Lin. Vortices for the nonlinear wave equation. Discrete & Continuous Dynamical Systems - A, 1999, 5 (2) : 391-398. doi: 10.3934/dcds.1999.5.391
[20]	V. Pata, Sergey Zelik. A remark on the damped wave equation. Communications on Pure & Applied Analysis, 2006, 5 (3) : 611-616. doi: 10.3934/cpaa.2006.5.611

2018 Impact Factor: 0.545

American Institute of Mathematical Sciences