June  2022, 15(6): 1317-1338. doi: 10.3934/dcdss.2022098

Eventual differentiability of coupled wave equations with local Kelvin-Voigt damping

1. 

LR Analyse non-linéaire et géométrie, LR21ES08, Department of Mathematics, Faculty of Sciences of Tunis, University of Tunis El-Manar, 2092 El Manar Ⅱ, Tunisia

2. 

LR Analysis and Control of PDEs, LR22ES03, Department of Mathematics, Faculty of Sciences of Monastir, University of Monastir, Tunisia

* Corresponding author: Ahmed Bchatnia

Received  August 2021 Revised  April 2022 Published  June 2022 Early access  April 2022

In this work, we consider a coupled wave equations with partially and locally distributed Kelvin-Voigt damping, where only one equation is dissipative.

Under the assumption that the damping coefficient changes smoothly near the interface of the damped and undamped regions, we investigate the effectiveness of the indirect control, and we prove that the associated semigroup is eventually differential.

Citation: Ahmed Bchatnia, Nadia Souayeh. Eventual differentiability of coupled wave equations with local Kelvin-Voigt damping. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1317-1338. doi: 10.3934/dcdss.2022098
References:
[1]

F. AlabauP. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150.  doi: 10.1007/s00028-002-8083-0.

[2]

F. Ammar-Khodja and A. Bader, Stability of systems of one dimensional wave equations by internal or boundary control force, SIAM J. Control Optim., 39 (2001), 1833-1851.  doi: 10.1137/S0363012900366613.

[3]

K. AmmariF. Hassine and L. Robbiano, Stabilization for the wave equation with singular Kelvin-Voigt damping, Archive for Rational Mechanics and analysis, 236 (2020), 577-601.  doi: 10.1007/s00205-019-01476-4.

[4]

K. AmmariZ. Liu and F. Shel, Stability of the wave equations on a tree with local Kelvin-Voigt damping, Semigroup Forum, 100 (2020), 364-382.  doi: 10.1007/s00233-019-10064-7.

[5]

A. Bchatnia and N. Souayeh, Indirect stability of coupled wave equations with local Kelvin-Voigt damping, preprint.

[6]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[7]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.

[8]

F. Hassine and N. Souayeh, Stability for coupled waves with locally disturbed Kelvin-Voigt damping, Semigroup Forum, 102 (2021), 134-159.  doi: 10.1007/s00233-020-10142-1.

[9]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert space, Ann. Differential Equations, 1 (1985), 43-56. 

[10]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[11]

K. LiuZ. Liu and Q. Ahang, Eventual differentiability of a string, with local Kelvin-voigt damping, ESAIM: control and Calculus of Variations, 23 (2017), 443-454.  doi: 10.1051/cocv/2015055.

[12]

K. Liu and B. Rao, Exponential stability for the wave equation with local Kelvin-Voigt damping, Z. Angew. Math. Phys., 57 (2006), 419-432.  doi: 10.1007/s00033-005-0029-2.

[13]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.

[14]

Z. Liu and Q. Zhang, Stability of a string with local Kelvin-Voigt damping and non-smooth coefficient at interface, SIAM J. Control Optim., 54 (2016), 1859-1871.  doi: 10.1137/15M1049385.

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[16]

J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 248 (1984), 847-857.  doi: 10.2307/1999112.

[17]

M. Renardy, On localized Kelvin-Voigt damping, Z. Angew Math. Mech., 84 (2004), 280-283.  doi: 10.1002/zamm.200310100.

[18]

D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, Math. Anal. Appl., 173 (1993), 339-358.  doi: 10.1006/jmaa.1993.1071.

[19]

L. Tebou, A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping, C. R. Math. Acad. Sci. Paris, 350 (2012), 603-608.  doi: 10.1016/j.crma.2012.06.005.

show all references

References:
[1]

F. AlabauP. Cannarsa and V. Komornik, Indirect internal stabilization of weakly coupled evolution equations, J. Evol. Equ., 2 (2002), 127-150.  doi: 10.1007/s00028-002-8083-0.

[2]

F. Ammar-Khodja and A. Bader, Stability of systems of one dimensional wave equations by internal or boundary control force, SIAM J. Control Optim., 39 (2001), 1833-1851.  doi: 10.1137/S0363012900366613.

[3]

K. AmmariF. Hassine and L. Robbiano, Stabilization for the wave equation with singular Kelvin-Voigt damping, Archive for Rational Mechanics and analysis, 236 (2020), 577-601.  doi: 10.1007/s00205-019-01476-4.

[4]

K. AmmariZ. Liu and F. Shel, Stability of the wave equations on a tree with local Kelvin-Voigt damping, Semigroup Forum, 100 (2020), 364-382.  doi: 10.1007/s00233-019-10064-7.

[5]

A. Bchatnia and N. Souayeh, Indirect stability of coupled wave equations with local Kelvin-Voigt damping, preprint.

[6]

A. Borichev and Y. Tomilov, Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347 (2010), 455-478.  doi: 10.1007/s00208-009-0439-0.

[7]

K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, 194. Springer-Verlag, New York, 2000.

[8]

F. Hassine and N. Souayeh, Stability for coupled waves with locally disturbed Kelvin-Voigt damping, Semigroup Forum, 102 (2021), 134-159.  doi: 10.1007/s00233-020-10142-1.

[9]

F. L. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert space, Ann. Differential Equations, 1 (1985), 43-56. 

[10]

T. Kato, Perturbation Theory for Linear Operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995.

[11]

K. LiuZ. Liu and Q. Ahang, Eventual differentiability of a string, with local Kelvin-voigt damping, ESAIM: control and Calculus of Variations, 23 (2017), 443-454.  doi: 10.1051/cocv/2015055.

[12]

K. Liu and B. Rao, Exponential stability for the wave equation with local Kelvin-Voigt damping, Z. Angew. Math. Phys., 57 (2006), 419-432.  doi: 10.1007/s00033-005-0029-2.

[13]

Z. Liu and B. Rao, Characterization of polynomial decay rate for the solution of linear evolution equation, Z. Angew. Math. Phys., 56 (2005), 630-644.  doi: 10.1007/s00033-004-3073-4.

[14]

Z. Liu and Q. Zhang, Stability of a string with local Kelvin-Voigt damping and non-smooth coefficient at interface, SIAM J. Control Optim., 54 (2016), 1859-1871.  doi: 10.1137/15M1049385.

[15]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44. Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[16]

J. Prüss, On the spectrum of $C_0$-semigroups, Trans. Amer. Math. Soc., 248 (1984), 847-857.  doi: 10.2307/1999112.

[17]

M. Renardy, On localized Kelvin-Voigt damping, Z. Angew Math. Mech., 84 (2004), 280-283.  doi: 10.1002/zamm.200310100.

[18]

D. L. Russell, A general framework for the study of indirect damping mechanisms in elastic systems, Math. Anal. Appl., 173 (1993), 339-358.  doi: 10.1006/jmaa.1993.1071.

[19]

L. Tebou, A constructive method for the stabilization of the wave equation with localized Kelvin-Voigt damping, C. R. Math. Acad. Sci. Paris, 350 (2012), 603-608.  doi: 10.1016/j.crma.2012.06.005.

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