Boundary exact controllability for a porous elastic Timoshenko system

Manoel J. Santos; Carlos A. Raposo; Leonardo R. S. Rodrigues

Applications of Mathematics (2020)

  • Volume: 65, Issue: 4, page 343-354
  • ISSN: 0862-7940

Abstract

top
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.

How to cite

top

Santos, Manoel J., Raposo, Carlos A., and Rodrigues, Leonardo R. S.. "Boundary exact controllability for a porous elastic Timoshenko system." Applications of Mathematics 65.4 (2020): 343-354. <http://eudml.org/doc/297266>.

@article{Santos2020,
abstract = {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.},
author = {Santos, Manoel J., Raposo, Carlos A., Rodrigues, Leonardo R. S.},
journal = {Applications of Mathematics},
keywords = {boundary exact controllability; Timoshenko beam; porous elasticity},
language = {eng},
number = {4},
pages = {343-354},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Boundary exact controllability for a porous elastic Timoshenko system},
url = {http://eudml.org/doc/297266},
volume = {65},
year = {2020},
}

TY - JOUR
AU - Santos, Manoel J.
AU - Raposo, Carlos A.
AU - Rodrigues, Leonardo R. S.
TI - Boundary exact controllability for a porous elastic Timoshenko system
JO - Applications of Mathematics
PY - 2020
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 343
EP - 354
AB - 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.
LA - eng
KW - boundary exact controllability; Timoshenko beam; porous elasticity
UR - http://eudml.org/doc/297266
ER -

References

top
  1. Akil, M., Chitour, Y., Ghader, M., Wehbe, A., 10.3233/ASY-191574, (to appear) in Asymptotic Anal. DOI10.3233/ASY-191574
  2. Júnior, D. S. Almeida, Ramos, A. J. A., Santos, M. L., 10.1007/s10444-014-9351-6, Adv. Comput. Math. 41 (2015), 105-130. (2015) Zbl1314.65119MR3311475DOI10.1007/s10444-014-9351-6
  3. Júnior, D. S. Almeida, Santos, M. L., Rivera, J. E. Muñoz, 10.1002/mma.2741, Math. Methods Appl. Sci. 36 (2013), 1965-1976. (2013) Zbl1273.74072MR3091687DOI10.1002/mma.2741
  4. Aouadi, M., Campo, M., Copetti, M. I. M., Fernández, J. R., 10.1007/s00033-019-1161-8, Z. Angew. Math. Phys. 70 (2019), Article ID 117, 26 pages. (2019) Zbl1418.74018MR3982962DOI10.1007/s00033-019-1161-8
  5. Araruna, F. D., Zuazua, E., 10.1137/060659934, SIAM J. Control Optim. 47 (2008), 1909-1938. (2008) Zbl1170.74351MR2421335DOI10.1137/060659934
  6. Cavalcanti, M. M., Cavalcanti, V. N. Domingos, Nascimento, F. A. Falcão, Lasiecka, I., Rodrigues, J. H., 10.1007/s00033-013-0380-7, Z. Angew. Math. Phys. 65 (2014), 1189-1206. (2014) Zbl1316.35034MR3279525DOI10.1007/s00033-013-0380-7
  7. Cowin, S. C., Nunziato, J. W., 10.1007/BF00041230, J. Elasticity 13 (1983), 125-147. (1983) Zbl0523.73008DOI10.1007/BF00041230
  8. Dell'Oro, F., Pata, V., 10.1016/j.jde.2014.04.009, J. Differ. Equations 257 (2014), 523-548. (2014) Zbl1288.35074MR3200380DOI10.1016/j.jde.2014.04.009
  9. Dridi, H., Djebabla, A., 10.1007/s11565-019-00333-2, Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 66 (2020), 13-25. (2020) Zbl07194183MR4091442DOI10.1007/s11565-019-00333-2
  10. Infante, J. A., Zuazua, E., 10.1051/m2an:1999123, M2AN, Math. Model. Numer. Anal. 33 (1999), 407-438. (1999) Zbl0947.65101MR1700042DOI10.1051/m2an:1999123
  11. Lagnese, J. E., Leugering, G., Schmidt, E. J. P. G., 10.1137/0331035, SIAM J. Control Optim. 31 (1993), 780-811. (1993) Zbl0775.93107MR1214764DOI10.1137/0331035
  12. Lagnese, J. E., Lions, J.-L., Modelling Analysis and Control of Thin Plates, Recherches en Mathématiques Appliquées 6. Masson, Paris (1988). (1988) Zbl0662.73039MR0953313
  13. Lions, J.-L., Contrôlabilité Exacte, Perturbations et Stabilisation de Systèmes Distribués. Tome 2. Perturbations, Recherches en Mathématiques Appliquées 9. Masson, Paris (1988), French. (1988) Zbl0653.93003MR0963060
  14. Lions, J.-L., 10.1137/1030001, SIAM Rev. 30 (1988), 1-68. (1988) Zbl0644.49028MR0931277DOI10.1137/1030001
  15. Lions, J.-L., Magenes, E., 10.1007/978-3-642-65393-3, Die Grundlehren der mathematischen Wissenschaften 183. Springer, Berlin (1973). (1973) Zbl0251.35001MR0350179DOI10.1007/978-3-642-65393-3
  16. Magaña, A., Quintanilla, R., 10.1016/j.ijsolstr.2005.06.077, Int. J. Solids Struct. 43 (2006), 3414-3427. (2006) Zbl1121.74361MR2221521DOI10.1016/j.ijsolstr.2005.06.077
  17. Medeiros, L. A., Exact controllability for a Timoshenko model of vibrations of beams, Adv. Math. Sci. Appl. 2 (1993), 47-61. (1993) Zbl0860.93014MR1239248
  18. Medeiros, L. A., Miranda, M. M., Lourêdo, A. T., Introduction Exact Control Theory: Method HUM, EDUEPB, Campina Grande (2013). (2013) 
  19. Mercier, D., Régnier, V., 10.3934/eect.2019021, Evol. Equ. Control Theory 8 (2019), 423. (2019) Zbl1428.35578MR3959450DOI10.3934/eect.2019021
  20. Rivera, J. E. Muñoz, Quintanilla, R., 10.1016/j.jmaa.2007.06.005, J. Math. Anal. Appl. 338 (2008), 1296-1309. (2008) Zbl1131.74019MR2386497DOI10.1016/j.jmaa.2007.06.005
  21. Pazy, A., 10.1007/978-1-4612-5561-1, Applied Mathematical Sciences 44. Springer, New York (1983). (1983) Zbl0516.47023MR0710486DOI10.1007/978-1-4612-5561-1
  22. Ramos, A. J. A., Júnior, D. S. Almeida, Freitas, M. M., Santos, M. J. dos, Santos, A. R., 10.1007/s11565-019-00334-1, Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 66 (2020), 113-134. (2020) Zbl07194190MR4091443DOI10.1007/s11565-019-00334-1
  23. Raposo, C. A., Apalara, T. A., Ribeiro, J. O., 10.1016/j.jmaa.2018.06.017, J. Math. Anal. Appl. 466 (2018), 819-834. (2018) Zbl1394.35265MR3818147DOI10.1016/j.jmaa.2018.06.017
  24. Raposo, C. A., Bastos, W. D., Santos, M. L., 10.1590/s0101-82052007000200003, Comput. Appl. Math. 26 (2007), 215-234. (2007) Zbl1182.35216MR2337709DOI10.1590/s0101-82052007000200003
  25. Said-Houari, B., Laskri, Y., 10.1016/j.amc.2010.08.021, Appl. Math. Comput. 217 (2010), 2857-2869. (2010) Zbl1342.74086MR2733729DOI10.1016/j.amc.2010.08.021
  26. Santos, M. L., Júnior, D. S. Almeida, 10.1007/s00033-016-0622-6, Z. Angew. Math. Phys. 67 (2016), Article ID 63, 18 pages. (2016) Zbl1351.35217MR3494484DOI10.1007/s00033-016-0622-6
  27. Shubov, M. A., 10.1093/imamci/17.4.375, IMA J. Math. Control Inf. 17 (2000), 375-395. (2000) Zbl0991.93016MR1797350DOI10.1093/imamci/17.4.375
  28. Soufyane, A., 10.1016/S0764-4442(99)80244-4, C. R. Acad. Sci., Paris, Sér. I, Math. 328 (1999), 731-734 French. (1999) Zbl0943.74042MR1680836DOI10.1016/S0764-4442(99)80244-4
  29. Timoshenko, S. P., 10.1080/14786442108636264, Lond. Edinb. Dubl. Phil. Mag., Ser. VI. 41 (1921), 744-746. (1921) DOI10.1080/14786442108636264
  30. Zhang, C., Hu, X., 10.1215/kjm/1250281029, J. Math. Kyoto Univ. 47 (2007), 643-655. (2007) Zbl1141.74036MR2402520DOI10.1215/kjm/1250281029
  31. Zuazua, E., Exact controllability for the semilinear wave equation, J. Math. Pures Appl., IX. Sér. 69 (1990), 1-31. (1990) MR1054122
  32. Zuazua, E., 10.1137/S0036144503432862, SIAM Rev. 47 (2005), 197-243. (2005) Zbl1077.65095MR2179896DOI10.1137/S0036144503432862

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.