About the stability to Timoshenko system with pointwise dissipation

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August 2022, 15(8): 2289-2303. doi: 10.3934/dcdss.2022078

About the stability to Timoshenko system with pointwise dissipation

Jaime E. Muñoz Rivera ^1,, and Maria Grazia Naso ^2,

1.	National Laboratory for Scientific Computation, Rua Getulio Vargas 333, Petrópolis, RJ, Brazil, DMAT - Universidad del Bío-Bío, Avenida Collao 1202, Casilla 5 - Concepción, Chile, IM - UFRJ, Cidade Universitária, Rio de Janeiro, Brazil
2.	DICATAM, Università degli Studi di Brescia, Via Valotti 9, 25133 Brescia, Italy

^* Corresponding author

In honor of Maurizio Grasselli's 60th Birthday

Received December 2021 Published August 2022 Early access March 2022

Fund Project: The first author is supported by Brazilian - CNPq grant 310249/2018-0 while the second one is partially supported by the Gruppo Nazionale per la Fisica Matematica of the Istituto Nazionale di Alta Matematica

In this paper we study the Timoshenko model over the interval $ (0, \ell) $ with pointwise dissipation at $ \xi\in (0, \ell) $. We prove that this dissipation produces exponential stability when $ \xi\in \mathbb{Q}\ell $ and $ \xi\ne \frac{n}{2m+1}\ell $, where $ n, m\in \mathbb{N} $ and $ n $, and $ 2m+1 $ are co-prime.

Keywords: Timoshenko system, boundary feedback, pointwise dissipation, asymptotic behavior, stability.

Mathematics Subject Classification: Primary: 35B35, 35Q74; Secondary: 74H40, 74K10, 93D15.

Citation: Jaime E. Muñoz Rivera, Maria Grazia Naso. About the stability to Timoshenko system with pointwise dissipation. Discrete and Continuous Dynamical Systems - S, 2022, 15 (8) : 2289-2303. doi: 10.3934/dcdss.2022078

References:

[1]	F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 643-669. doi: 10.1007/s00030-007-5033-0.
[2]	K. Ammari, Uniform stabilization of beams by means of a pointwise feedback, Differential Integral Equations, 14 (2001), 1153-1167.
[3]	F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations, 194 (2003), 82-115. doi: 10.1016/S0022-0396(03)00185-2.
[4]	C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.
[5]	M. M. Cavalcanti, W. J. Corrêa, V. N. Domingos Cavalcanti, M. A. Jorge Silva and J. P. Zanchetta, Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping, Z. Angew. Math. Phys., 72 (2021), Paper No. 191, 20 pp. doi: 10.1007/s00033-021-01622-7.
[6]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, F. A. Falcão Nascimento, I. Lasiecka and J. H. Rodrigues, Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Z. Angew. Math. Phys., 65 (2014), 1189-1206. doi: 10.1007/s00033-013-0380-7.
[7]	V. Danese, F. Dell'Oro and V. Pata, Stability analysis of abstract systems of Timoshenko type, J. Evol. Equ., 16 (2016), 587-615. doi: 10.1007/s00028-015-0314-2.
[8]	A. Djebabla and N.-E. Tatar, Stabilization of the Timoshenko beam by thermal effect, Mediterr. J. Math., 7 (2010), 373-385. doi: 10.1007/s00009-010-0058-8.
[9]	K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000.
[10]	H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221-251. doi: 10.1007/s00205-009-0220-2.
[11]	M. Grasselli, V. Pata and G. Prouse, Longtime behavior of a viscoelastic Timoshenko beam. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 337-348. doi: 10.3934/dcds.2004.10.337.
[12]	A. Y. Khapalov, Exponential decay for the one-dimensional wave equation with internal pointwise damping, Math. Methods Appl. Sci., 20 (1997), 1171-1183. doi: 10.1002/(SICI)1099-1476(19970925)20:14<1171::AID-MMA907>3.0.CO;2-B.
[13]	J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429. doi: 10.1137/0325078.
[14]	A. Labuschagne, N. F. J. van Rensburg and A. J. van der Merwe, Comparison of linear beam theories, Math. Comput. Modelling, 49 (2009), 20-30. doi: 10.1016/j.mcm.2008.06.006.
[15]	K. S. Liu, Energy decay problems in the design of a point stabilizer for coupled string vibrating systems, SIAM J. Control Optim., 26 (1988), 1348-1356. doi: 10.1137/0326076.
[16]	S. A. Messaoudi and A. Fareh, Energy decay in a Timoshenko-type system of thermoelasticity of type Ⅲ with different wave-propagation speeds, Arab. J. Math. (Springer), 2 (2013), 199-207. doi: 10.1007/s40065-012-0061-y.
[17]	J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems–-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278. doi: 10.1016/S0022-247X(02)00436-5.
[18]	A. F. Neves, H. de S. Ribeiro and O. Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal., 67 (1986), 320-344. doi: 10.1016/0022-1236(86)90029-7.
[19]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[20]	J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.
[21]	C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett., 18 (2005), 535-541. doi: 10.1016/j.aml.2004.03.017.
[22]	B. Said-Houari and A. Kasimov, Decay property of Timoshenko system in thermoelasticity, Math. Methods Appl. Sci., 35 (2012), 314-333. doi: 10.1002/mma.1569.
[23]	A. Soufyane, Exponential stability of the linearized nonuniform Timoshenko beam, Nonlinear Anal. Real World Appl., 10 (2009), 1016-1020. doi: 10.1016/j.nonrwa.2007.11.019.
[24]	H. Timoshenko, Vibration Problems in Engineering, D van Nostrand Company, Inc., New-York, 1937.
[25]	R. W. Traill-Nash and A. R. Collar, The effects of shear flexibility and rotatory inertia on the bending vibrations of beams, Quart. J. Mech. Appl. Math., 6 (1953), 186-222. doi: 10.1093/qjmam/6.2.186.
[26]	Q.-X. Yan, S.-H. Hou and D.-X. Feng, Asymptotic behavior of Timoshenko beam with dissipative boundary feedback, J. Math. Anal. Appl., 269 (2002), 556-577. doi: 10.1016/S0022-247X(02)00036-7.

show all references

References:

[1]	F. Alabau-Boussouira, Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, NoDEA Nonlinear Differential Equations Appl., 14 (2007), 643-669. doi: 10.1007/s00030-007-5033-0.
[2]	K. Ammari, Uniform stabilization of beams by means of a pointwise feedback, Differential Integral Equations, 14 (2001), 1153-1167.
[3]	F. Ammar-Khodja, A. Benabdallah, J. E. Muñoz Rivera and R. Racke, Energy decay for Timoshenko systems of memory type, J. Differential Equations, 194 (2003), 82-115. doi: 10.1016/S0022-0396(03)00185-2.
[4]	C. J. K. Batty and T. Duyckaerts, Non-uniform stability for bounded semi-groups on Banach spaces, J. Evol. Equ., 8 (2008), 765-780. doi: 10.1007/s00028-008-0424-1.
[5]	M. M. Cavalcanti, W. J. Corrêa, V. N. Domingos Cavalcanti, M. A. Jorge Silva and J. P. Zanchetta, Uniform stability for a semilinear non-homogeneous Timoshenko system with localized nonlinear damping, Z. Angew. Math. Phys., 72 (2021), Paper No. 191, 20 pp. doi: 10.1007/s00033-021-01622-7.
[6]	M. M. Cavalcanti, V. N. Domingos Cavalcanti, F. A. Falcão Nascimento, I. Lasiecka and J. H. Rodrigues, Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Z. Angew. Math. Phys., 65 (2014), 1189-1206. doi: 10.1007/s00033-013-0380-7.
[7]	V. Danese, F. Dell'Oro and V. Pata, Stability analysis of abstract systems of Timoshenko type, J. Evol. Equ., 16 (2016), 587-615. doi: 10.1007/s00028-015-0314-2.
[8]	A. Djebabla and N.-E. Tatar, Stabilization of the Timoshenko beam by thermal effect, Mediterr. J. Math., 7 (2010), 373-385. doi: 10.1007/s00009-010-0058-8.
[9]	K.-J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Graduate Texts in Mathematics, vol. 194, Springer-Verlag, New York, 2000.
[10]	H. D. Fernández Sare and R. Racke, On the stability of damped Timoshenko systems: Cattaneo versus Fourier law, Arch. Ration. Mech. Anal., 194 (2009), 221-251. doi: 10.1007/s00205-009-0220-2.
[11]	M. Grasselli, V. Pata and G. Prouse, Longtime behavior of a viscoelastic Timoshenko beam. Partial differential equations and applications, Discrete Contin. Dyn. Syst., 10 (2004), 337-348. doi: 10.3934/dcds.2004.10.337.
[12]	A. Y. Khapalov, Exponential decay for the one-dimensional wave equation with internal pointwise damping, Math. Methods Appl. Sci., 20 (1997), 1171-1183. doi: 10.1002/(SICI)1099-1476(19970925)20:14<1171::AID-MMA907>3.0.CO;2-B.
[13]	J. U. Kim and Y. Renardy, Boundary control of the Timoshenko beam, SIAM J. Control Optim., 25 (1987), 1417-1429. doi: 10.1137/0325078.
[14]	A. Labuschagne, N. F. J. van Rensburg and A. J. van der Merwe, Comparison of linear beam theories, Math. Comput. Modelling, 49 (2009), 20-30. doi: 10.1016/j.mcm.2008.06.006.
[15]	K. S. Liu, Energy decay problems in the design of a point stabilizer for coupled string vibrating systems, SIAM J. Control Optim., 26 (1988), 1348-1356. doi: 10.1137/0326076.
[16]	S. A. Messaoudi and A. Fareh, Energy decay in a Timoshenko-type system of thermoelasticity of type Ⅲ with different wave-propagation speeds, Arab. J. Math. (Springer), 2 (2013), 199-207. doi: 10.1007/s40065-012-0061-y.
[17]	J. E. Muñoz Rivera and R. Racke, Mildly dissipative nonlinear Timoshenko systems–-global existence and exponential stability, J. Math. Anal. Appl., 276 (2002), 248-278. doi: 10.1016/S0022-247X(02)00436-5.
[18]	A. F. Neves, H. de S. Ribeiro and O. Lopes, On the spectrum of evolution operators generated by hyperbolic systems, J. Funct. Anal., 67 (1986), 320-344. doi: 10.1016/0022-1236(86)90029-7.
[19]	A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[20]	J. Prüss, On the spectrum of $C_{0}$-semigroups, Trans. Amer. Math. Soc., 284 (1984), 847-857. doi: 10.2307/1999112.
[21]	C. A. Raposo, J. Ferreira, M. L. Santos and N. N. O. Castro, Exponential stability for the Timoshenko system with two weak dampings, Appl. Math. Lett., 18 (2005), 535-541. doi: 10.1016/j.aml.2004.03.017.
[22]	B. Said-Houari and A. Kasimov, Decay property of Timoshenko system in thermoelasticity, Math. Methods Appl. Sci., 35 (2012), 314-333. doi: 10.1002/mma.1569.
[23]	A. Soufyane, Exponential stability of the linearized nonuniform Timoshenko beam, Nonlinear Anal. Real World Appl., 10 (2009), 1016-1020. doi: 10.1016/j.nonrwa.2007.11.019.
[24]	H. Timoshenko, Vibration Problems in Engineering, D van Nostrand Company, Inc., New-York, 1937.
[25]	R. W. Traill-Nash and A. R. Collar, The effects of shear flexibility and rotatory inertia on the bending vibrations of beams, Quart. J. Mech. Appl. Math., 6 (1953), 186-222. doi: 10.1093/qjmam/6.2.186.
[26]	Q.-X. Yan, S.-H. Hou and D.-X. Feng, Asymptotic behavior of Timoshenko beam with dissipative boundary feedback, J. Math. Anal. Appl., 269 (2002), 556-577. doi: 10.1016/S0022-247X(02)00036-7.

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