On the stabilization of laminated beams with delay

Kassimu Mpungu; Tijani A. Apalara; Mukhiddin Muminov

Applications of Mathematics (2021)

  • Volume: 66, Issue: 5, page 789-812
  • ISSN: 0862-7940

Abstract

top
Of concern in this paper is the laminated beam system with frictional damping and an internal constant delay term in the transverse displacement. Under suitable assumptions on the weight of the delay, we establish that the system's energy decays exponentially in the case of equal wave speeds of propagation, and polynomially in the case of non-equal wave speeds.

How to cite

top

Mpungu, Kassimu, A. Apalara, Tijani, and Muminov, Mukhiddin. "On the stabilization of laminated beams with delay." Applications of Mathematics 66.5 (2021): 789-812. <http://eudml.org/doc/297718>.

@article{Mpungu2021,
abstract = {Of concern in this paper is the laminated beam system with frictional damping and an internal constant delay term in the transverse displacement. Under suitable assumptions on the weight of the delay, we establish that the system's energy decays exponentially in the case of equal wave speeds of propagation, and polynomially in the case of non-equal wave speeds.},
author = {Mpungu, Kassimu, A. Apalara, Tijani, Muminov, Mukhiddin},
journal = {Applications of Mathematics},
keywords = {laminated beam; interfacial slip; delay; exponential and polynomial decay},
language = {eng},
number = {5},
pages = {789-812},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {On the stabilization of laminated beams with delay},
url = {http://eudml.org/doc/297718},
volume = {66},
year = {2021},
}

TY - JOUR
AU - Mpungu, Kassimu
AU - A. Apalara, Tijani
AU - Muminov, Mukhiddin
TI - On the stabilization of laminated beams with delay
JO - Applications of Mathematics
PY - 2021
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 66
IS - 5
SP - 789
EP - 812
AB - Of concern in this paper is the laminated beam system with frictional damping and an internal constant delay term in the transverse displacement. Under suitable assumptions on the weight of the delay, we establish that the system's energy decays exponentially in the case of equal wave speeds of propagation, and polynomially in the case of non-equal wave speeds.
LA - eng
KW - laminated beam; interfacial slip; delay; exponential and polynomial decay
UR - http://eudml.org/doc/297718
ER -

References

top
  1. Abdallah, C., Dorato, P., Benitez-Read, J., Byrne, R., 10.23919/ACC.1993.4793475, American Control Conference (ACC) IEEE, Piscataway (1993), 3106-3107. (1993) DOI10.23919/ACC.1993.4793475
  2. Benhassi, E. M. Ait, Ammari, K., Boulite, S., Maniar, L., 10.1007/s00028-009-0004-z, J. Evol. Equ. 9 (2009), 103-121. (2009) Zbl1239.93092MR2501354DOI10.1007/s00028-009-0004-z
  3. Apalara, T. A., Well-posedness and exponential stability for a linear damped Timoshenko system with second sound and internal distributed delay, Electron. J. Differ. Equ. 2014 (2014), Article ID 254, 15 pages. (2014) Zbl1315.35031MR3291754
  4. Apalara, T. A., 10.1080/00036811.2014.1000314, Appl. Anal. 95 (2016), 187-202. (2016) Zbl1336.35053MR3426195DOI10.1080/00036811.2014.1000314
  5. Apalara, T. A., 10.1016/S0252-9602(16)30042-X, Acta Math. Sci., Ser. B, Engl. Ed. 36 (2016), 815-830. (2016) Zbl1363.35238MR3479257DOI10.1016/S0252-9602(16)30042-X
  6. Apalara, T. A., 10.1007/s00033-017-0784-x, Z. Angew. Math. Phys. 68 (2017), Article ID 41, 16 pages. (2017) Zbl1379.35182MR3615474DOI10.1007/s00033-017-0784-x
  7. Apalara, T. A., 10.1007/s10473-019-0604-9, Acta Math. Sci., Ser. B, Engl. Ed. 39 (2019), 1517-1524. (2019) MR4069832DOI10.1007/s10473-019-0604-9
  8. Apalara, T. A., Messaoudi, S. A., 10.1007/s00245-014-9266-0, Appl. Math. Optim. 71 (2015), 449-472. (2015) Zbl1326.35033MR3346709DOI10.1007/s00245-014-9266-0
  9. Brezis, H., 10.1007/978-0-387-70914-7, Universitext. Springer, New York (2011). (2011) Zbl1220.46002MR2759829DOI10.1007/978-0-387-70914-7
  10. Cao, X.-G., Liu, D.-Y., Xu, G.-Q., 10.1007/s10883-007-9022-8, J. Dyn. Control Syst. 13 (2007), 313-336. (2007) Zbl1129.93010MR2337280DOI10.1007/s10883-007-9022-8
  11. Chen, Z., Liu, W., Chen, D., 10.11650/tjm/181109, Taiwanese J. Math. 23 (2019), 1227-1252. (2019) Zbl1425.74254MR4012377DOI10.11650/tjm/181109
  12. Datko, R., Lagnese, J., Polis, M. P., 10.1137/0324007, SIAM J. Control Optim. 24 (1986), 152-156. (1986) Zbl0592.93047MR818942DOI10.1137/0324007
  13. Feng, B., 10.1002/mma.4655, Math. Methods Appl. Sci. 41 (2018), 1162-1174. (2018) Zbl1403.35163MR3762342DOI10.1002/mma.4655
  14. Fridman, E., Nicaise, S., Valein, J., 10.1137/090762105, SIAM J. Control Optim. 48 (2010), 5028-5052. (2010) Zbl1214.93081MR2735515DOI10.1137/090762105
  15. Hansen, S. W., Spies, R. D., 10.1006/jsvi.1996.0913, J. Sound Vib. 204 (1997), 183-202. (1997) DOI10.1006/jsvi.1996.0913
  16. Kirane, M., Said-Houari, B., 10.1007/s00033-011-0145-0, Z. Angew. Math. Phys. 62 (2011), 1065-1082. (2011) Zbl1242.35163MR2860945DOI10.1007/s00033-011-0145-0
  17. Komornik, V., Zuazua, E., A direct method for the boundary stabilization of the wave equation, J. Math. Pures Appl., IX. Sér. 69 (1990), 35-54. (1990) Zbl0636.93064MR1054123
  18. Lasiecka, I., 10.1080/00036819208840140, Appl. Anal. 47 (1992), 191-212. (1992) Zbl0788.35008MR1333954DOI10.1080/00036819208840140
  19. Liu, Z., Zheng, S., Semigroups Associated with Dissipative Systems, Chapman & Hall/CRC Research Notes in Mathematics 398. Chapman & Hall/CRC, Boca Raton (1999). (1999) Zbl0924.73003MR1681343
  20. Lo, A., Tatar, N.-E., Stabilization of laminated beams with interfacial slip, Electron. J. Differ. Equ. 2015 (2015), Article ID 129, 14 pages. (2015) Zbl1321.35231MR3358501
  21. Lo, A., Tatar, N.-E., 10.1007/s12346-015-0147-y, Qual. Theory Dyn. Syst. 15 (2016), 517-540. (2016) Zbl1383.74056MR3563434DOI10.1007/s12346-015-0147-y
  22. Mustafa, M. I., 10.1016/j.jmaa.2011.05.066, J. Math. Anal. Appl. 383 (2011), 490-498. (2011) Zbl1222.35034MR2812399DOI10.1016/j.jmaa.2011.05.066
  23. Mustafa, M. I., 10.1063/1.5017923, J. Math. Phys. 59 (2018), Article ID 051508, 9 pages. (2018) Zbl1391.74124MR3806429DOI10.1063/1.5017923
  24. Mustafa, M. I., 10.1016/j.jmaa.2018.06.016, J. Math. Anal. Appl. 466 (2018), 619-641. (2018) Zbl06897084MR3818134DOI10.1016/j.jmaa.2018.06.016
  25. Nicaise, S., Pignotti, C., 10.1137/060648891, SIAM J. Control Optim. 45 (2006), 1561-1585. (2006) Zbl1180.35095MR2272156DOI10.1137/060648891
  26. Nicaise, S., Pignotti, C., Interior feedback stabilization of wave equations with time dependent delay, Electron. J. Differ. Equ. 2011 (2011), Article ID 41, 20 pages. (2011) Zbl1215.35098MR2788660
  27. Nicaise, S., Pignotti, C., Valein, J., 10.3934/dcdss.2011.4.693, Discrete Contin. Dyn. Syst., Ser. S 4 (2011), 693-722. (2011) Zbl1215.35030MR2746429DOI10.3934/dcdss.2011.4.693
  28. Nicaise, S., Valein, J., Fridman, E., 10.3934/dcdss.2009.2.559, Discrete Contin. Dyn. Syst., Ser. S 2 (2009), 559-581. (2009) Zbl1171.93029MR2525768DOI10.3934/dcdss.2009.2.559
  29. Pignotti, C., 10.1016/j.sysconle.2011.09.016, Syst. Control Lett. 61 (2012), 92-97. (2012) Zbl1250.93103MR2878692DOI10.1016/j.sysconle.2011.09.016
  30. Raposo, C. A., 10.1016/j.aml.2015.10.005, Appl. Math. Lett. 53 (2016), 85-91. (2016) Zbl1332.35042MR3426559DOI10.1016/j.aml.2015.10.005
  31. 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
  32. Tatar, N.-E., 10.1186/s13661-015-0432-3, Bound. Value Probl. 2015 (2015), Article ID 169, 11 pages. (2015) Zbl1338.35037MR3411322DOI10.1186/s13661-015-0432-3
  33. Wang, J.-M., Xu, G.-Q., Yung, S.-P., 10.1137/040610003, SIAM J. Control Optim. 44 (2005), 1575-1597. (2005) Zbl1132.93021MR2193496DOI10.1137/040610003
  34. Xu, G. Q., Yung, S. P., Li, L. K., 10.1051/cocv:2006021, ESAIM, Control Optim. Calc. Var. 12 (2006), 770-785. (2006) Zbl1105.35016MR2266817DOI10.1051/cocv:2006021
  35. Zhang, Q., Ran, M., Xu, D., 10.1080/00036811.2016.1197914, Appl. Anal. 96 (2017), 1867-1884. (2017) Zbl1373.65061MR3663769DOI10.1080/00036811.2016.1197914
  36. Zuazua, E., 10.1137/0328025, SIAM J. Control Optim. 28 (1990), 466-477. (1990) Zbl0695.93090MR1040470DOI10.1137/0328025

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.