| Title:
|
Boundary exact controllability for a porous elastic Timoshenko system (English) |
| Author:
|
Santos, Manoel J. |
| Author:
|
Raposo, Carlos A. |
| Author:
|
Rodrigues, Leonardo R. S. |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
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0862-7940 (print) |
| ISSN:
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1572-9109 (online) |
| Volume:
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65 |
| Issue:
|
4 |
| Year:
|
2020 |
| Pages:
|
343-354 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
In this paper, we consider a one-dimensional system governed by two partial differential equations. Such a system models phenomena in engineering, such as vibrations in beams or deformation of elastic bodies with porosity. By using the HUM method, we prove that the system is boundary exactly controllable in the usual energy space. We will also determine the minimum time allowed by the method for the controllability to occur. (English) |
| Keyword:
|
boundary exact controllability |
| Keyword:
|
Timoshenko beam |
| Keyword:
|
porous elasticity |
| MSC:
|
93B05 |
| MSC:
|
93C20 |
| idZBL:
|
07250666 |
| idMR:
|
MR4134138 |
| DOI:
|
10.21136/AM.2020.0133-19 |
| . |
| Date available:
|
2020-09-07T09:43:57Z |
| Last updated:
|
2022-09-01 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/148336 |
| . |
| Reference:
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[1] Akil, M., Chitour, Y., Ghader, M., Wehbe, A.: Stability and exact controllability of a Timoshenko system with only one fractional damping on the boundary.(to appear) in Asymptotic Anal. 10.3233/ASY-191574 |
| Reference:
|
[2] Júnior, D. S. Almeida, Ramos, A. J. A., Santos, M. L.: Observability inequality for the \hbox{finite}-difference semi-discretization of the 1-d coupled wave equations.Adv. Comput. Math. 41 (2015), 105-130. Zbl 1314.65119, MR 3311475, 10.1007/s10444-014-9351-6 |
| Reference:
|
[3] Júnior, D. S. Almeida, Santos, M. L., Rivera, J. E. Muñoz: Stability to weakly dissipative Timoshenko systems.Math. Methods Appl. Sci. 36 (2013), 1965-1976. Zbl 1273.74072, MR 3091687, 10.1002/mma.2741 |
| Reference:
|
[4] Aouadi, M., Campo, M., Copetti, M. I. M., Fernández, J. R.: Existence, stability and numerical results for a Timoshenko beam with thermodiffusion effects.Z. Angew. Math. Phys. 70 (2019), Article ID 117, 26 pages. Zbl 1418.74018, MR 3982962, 10.1007/s00033-019-1161-8 |
| Reference:
|
[5] Araruna, F. D., Zuazua, E.: Controllability of the Kirchhoff system for beams as a limit of the Mindlin-Timoshenko system.SIAM J. Control Optim. 47 (2008), 1909-1938. Zbl 1170.74351, MR 2421335, 10.1137/060659934 |
| Reference:
|
[6] Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Nascimento, F. A. Falcão, Lasiecka, I., Rodrigues, J. H.: 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. Zbl 1316.35034, MR 3279525, 10.1007/s00033-013-0380-7 |
| Reference:
|
[7] Cowin, S. C., Nunziato, J. W.: Linear elastic materials with voids.J. Elasticity 13 (1983), 125-147. Zbl 0523.73008, 10.1007/BF00041230 |
| Reference:
|
[8] Dell'Oro, F., Pata, V.: On the stability of Timoshenko systems with Gurtin-Pipkin thermal law.J. Differ. Equations 257 (2014), 523-548. Zbl 1288.35074, MR 3200380, 10.1016/j.jde.2014.04.009 |
| Reference:
|
[9] Dridi, H., Djebabla, A.: On the stabilization of linear porous elastic materials by microtemperature effect and porous damping.Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 66 (2020), 13-25. Zbl 07194183, MR 4091442, 10.1007/s11565-019-00333-2 |
| Reference:
|
[10] Infante, J. A., Zuazua, E.: Boundary observability for the space semi-discretizations of the 1-d wave equation.M2AN, Math. Model. Numer. Anal. 33 (1999), 407-438. Zbl 0947.65101, MR 1700042, 10.1051/m2an:1999123 |
| Reference:
|
[11] Lagnese, J. E., Leugering, G., Schmidt, E. J. P. G.: Control of planar networks of Timoshenko beams.SIAM J. Control Optim. 31 (1993), 780-811. Zbl 0775.93107, MR 1214764, 10.1137/0331035 |
| Reference:
|
[12] Lagnese, J. E., Lions, J.-L.: Modelling Analysis and Control of Thin Plates.Recherches en Mathématiques Appliquées 6. Masson, Paris (1988). Zbl 0662.73039, MR 0953313 |
| Reference:
|
[13] Lions, J.-L.: Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2. Perturbations.Recherches en Mathématiques Appliquées 9. Masson, Paris (1988), French. Zbl 0653.93003, MR 0963060 |
| Reference:
|
[14] Lions, J.-L.: Exact controllability, stabilization and perturbations for distributed systems.SIAM Rev. 30 (1988), 1-68. Zbl 0644.49028, MR 0931277, 10.1137/1030001 |
| Reference:
|
[15] Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications. Volume III.Die Grundlehren der mathematischen Wissenschaften 183. Springer, Berlin (1973). Zbl 0251.35001, MR 0350179, 10.1007/978-3-642-65393-3 |
| Reference:
|
[16] Magaña, A., Quintanilla, R.: On the time decay of solutions in one-dimensional theories of porous materials.Int. J. Solids Struct. 43 (2006), 3414-3427. Zbl 1121.74361, MR 2221521, 10.1016/j.ijsolstr.2005.06.077 |
| Reference:
|
[17] Medeiros, L. A.: Exact controllability for a Timoshenko model of vibrations of beams.Adv. Math. Sci. Appl. 2 (1993), 47-61. Zbl 0860.93014, MR 1239248 |
| Reference:
|
[18] Medeiros, L. A., Miranda, M. M., Lourêdo, A. T.: Introduction Exact Control Theory: Method HUM.EDUEPB, Campina Grande (2013). |
| Reference:
|
[19] Mercier, D., Régnier, V.: Decay rate of the Timoshenko system with one boundary damping.Evol. Equ. Control Theory 8 (2019), 423. Zbl 1428.35578, MR 3959450, 10.3934/eect.2019021 |
| Reference:
|
[20] Rivera, J. E. Muñoz, Quintanilla, R.: On the polynomial decay in elastic solids with voids.J. Math. Anal. Appl. 338 (2008), 1296-1309. Zbl 1131.74019, MR 2386497, 10.1016/j.jmaa.2007.06.005 |
| Reference:
|
[21] Pazy, A.: Semigroups of Linear Operators and Applications to Partial Differential Equations.Applied Mathematical Sciences 44. Springer, New York (1983). Zbl 0516.47023, MR 0710486, 10.1007/978-1-4612-5561-1 |
| Reference:
|
[22] Ramos, A. J. A., Júnior, D. S. Almeida, Freitas, M. M., Santos, M. J. dos, Santos, A. R.: Exponential stabilization for porous elastic system with one boundary dissipation.Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 66 (2020), 113-134. Zbl 07194190, MR 4091443, 10.1007/s11565-019-00334-1 |
| Reference:
|
[23] Raposo, C. A., Apalara, T. A., Ribeiro, J. O.: Analyticity to transmission problem with delay in porous-elasticity.J. Math. Anal. Appl. 466 (2018), 819-834. Zbl 1394.35265, MR 3818147, 10.1016/j.jmaa.2018.06.017 |
| Reference:
|
[24] Raposo, C. A., Bastos, W. D., Santos, M. L.: A transmission problem for the Timoshenko system.Comput. Appl. Math. 26 (2007), 215-234. Zbl 1182.35216, MR 2337709, 10.1590/s0101-82052007000200003 |
| Reference:
|
[25] Said-Houari, B., Laskri, Y.: A stability result of a Timoshenko system with a delay term in the internal feedback.Appl. Math. Comput. 217 (2010), 2857-2869. Zbl 1342.74086, MR 2733729, 10.1016/j.amc.2010.08.021 |
| Reference:
|
[26] Santos, M. L., Júnior, D. S. Almeida: On porous-elastic system with localized damping.Z. Angew. Math. Phys. 67 (2016), Article ID 63, 18 pages. Zbl 1351.35217, MR 3494484, 10.1007/s00033-016-0622-6 |
| Reference:
|
[27] Shubov, M. A.: Exact controllability of damped Timoshenko beam.IMA J. Math. Control Inf. 17 (2000), 375-395. Zbl 0991.93016, MR 1797350, 10.1093/imamci/17.4.375 |
| Reference:
|
[28] Soufyane, A.: Stabilisation de la poutre de Timoshenko.C. R. Acad. Sci., Paris, Sér. I, Math. 328 (1999), 731-734 French. Zbl 0943.74042, MR 1680836, 10.1016/S0764-4442(99)80244-4 |
| Reference:
|
[29] Timoshenko, S. P.: On the correction for shear of the differential equation for transverse vibrations of prismatic bars.Lond. Edinb. Dubl. Phil. Mag., Ser. VI. 41 (1921), 744-746. 10.1080/14786442108636264 |
| Reference:
|
[30] Zhang, C., Hu, X.: Exact controllability of a Timoshenko beam with dynamical boundary.J. Math. Kyoto Univ. 47 (2007), 643-655. Zbl 1141.74036, MR 2402520, 10.1215/kjm/1250281029 |
| Reference:
|
[31] Zuazua, E.: Exact controllability for the semilinear wave equation.J. Math. Pures Appl., IX. Sér. 69 (1990), 1-31. Zbl 38.49017, MR 1054122 |
| Reference:
|
[32] Zuazua, E.: Propagation, observation, and control of waves approximated by finite difference methods.SIAM Rev. 47 (2005), 197-243. Zbl 1077.65095, MR 2179896, 10.1137/S0036144503432862 |
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