| Title:
|
On the stabilization of laminated beams with delay (English) |
| Author:
|
Mpungu, Kassimu |
| Author:
|
A. Apalara, Tijani |
| Author:
|
Muminov, Mukhiddin |
| Language:
|
English |
| Journal:
|
Applications of Mathematics |
| ISSN:
|
0862-7940 (print) |
| ISSN:
|
1572-9109 (online) |
| Volume:
|
66 |
| Issue:
|
5 |
| Year:
|
2021 |
| Pages:
|
789-812 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
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. (English) |
| Keyword:
|
laminated beam |
| Keyword:
|
interfacial slip |
| Keyword:
|
delay |
| Keyword:
|
exponential and polynomial decay |
| MSC:
|
35B40 |
| MSC:
|
35L56 |
| MSC:
|
93D15 |
| MSC:
|
93D20 |
| MSC:
|
93D23 |
| idZBL:
|
07396178 |
| idMR:
|
MR4299885 |
| DOI:
|
10.21136/AM.2021.0056-20 |
| . |
| Date available:
|
2021-08-18T08:32:25Z |
| Last updated:
|
2023-11-06 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/149083 |
| . |
| Reference:
|
[1] Abdallah, C., Dorato, P., Benitez-Read, J., Byrne, R.: Delayed positive feedback can stabilize oscillatory systems.American Control Conference (ACC) IEEE, Piscataway (1993), 3106-3107. 10.23919/ACC.1993.4793475 |
| Reference:
|
[2] Benhassi, E. M. Ait, Ammari, K., Boulite, S., Maniar, L.: Feedback stabilization of a class of evolution equations with delay.J. Evol. Equ. 9 (2009), 103-121. Zbl 1239.93092, MR 2501354, 10.1007/s00028-009-0004-z |
| Reference:
|
[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. Zbl 1315.35031, MR 3291754 |
| Reference:
|
[4] Apalara, T. A.: Asymptotic behavior of weakly dissipative Timoshenko system with internal constant delay feedbacks.Appl. Anal. 95 (2016), 187-202. Zbl 1336.35053, MR 3426195, 10.1080/00036811.2014.1000314 |
| Reference:
|
[5] Apalara, T. A.: Uniform decay in weakly dissipative Timoshenko system with internal distributed delay feedbacks.Acta Math. Sci., Ser. B, Engl. Ed. 36 (2016), 815-830. Zbl 1363.35238, MR 3479257, 10.1016/S0252-9602(16)30042-X |
| Reference:
|
[6] Apalara, T. A.: Uniform stability of a laminated beam with structural damping and second sound.Z. Angew. Math. Phys. 68 (2017), Article ID 41, 16 pages. Zbl 1379.35182, MR 3615474, 10.1007/s00033-017-0784-x |
| Reference:
|
[7] Apalara, T. A.: On the stability of a thermoelastic laminated beam.Acta Math. Sci., Ser. B, Engl. Ed. 39 (2019), 1517-1524. MR 4069832, 10.1007/s10473-019-0604-9 |
| Reference:
|
[8] Apalara, T. A., Messaoudi, S. A.: An exponential stability result of a Timoshenko system with thermoelasticity with second sound and in the presence of delay.Appl. Math. Optim. 71 (2015), 449-472. Zbl 1326.35033, MR 3346709, 10.1007/s00245-014-9266-0 |
| Reference:
|
[9] Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations.Universitext. Springer, New York (2011). Zbl 1220.46002, MR 2759829, 10.1007/978-0-387-70914-7 |
| Reference:
|
[10] Cao, X.-G., Liu, D.-Y., Xu, G.-Q.: Easy test for stability of laminated beams with structural damping and boundary feedback controls.J. Dyn. Control Syst. 13 (2007), 313-336. Zbl 1129.93010, MR 2337280, 10.1007/s10883-007-9022-8 |
| Reference:
|
[11] Chen, Z., Liu, W., Chen, D.: General decay rates for a laminated beam with memory.Taiwanese J. Math. 23 (2019), 1227-1252. Zbl 1425.74254, MR 4012377, 10.11650/tjm/181109 |
| Reference:
|
[12] Datko, R., Lagnese, J., Polis, M. P.: An example on the effect of time delays in boundary feedback stabilization of wave equations.SIAM J. Control Optim. 24 (1986), 152-156. Zbl 0592.93047, MR 818942, 10.1137/0324007 |
| Reference:
|
[13] Feng, B.: Well-posedness and exponential decay for laminated Timoshenko beams with time delays and boundary feedbacks.Math. Methods Appl. Sci. 41 (2018), 1162-1174. Zbl 1403.35163, MR 3762342, 10.1002/mma.4655 |
| Reference:
|
[14] Fridman, E., Nicaise, S., Valein, J.: Stabilization of second order evolution equations with unbounded feedback with time-dependent delay.SIAM J. Control Optim. 48 (2010), 5028-5052. Zbl 1214.93081, MR 2735515, 10.1137/090762105 |
| Reference:
|
[15] Hansen, S. W., Spies, R. D.: Structural damping in laminated beams due to interfacial slip.J. Sound Vib. 204 (1997), 183-202. 10.1006/jsvi.1996.0913 |
| Reference:
|
[16] Kirane, M., Said-Houari, B.: Existence and asymptotic stability of a viscoelastic wave equation with a delay.Z. Angew. Math. Phys. 62 (2011), 1065-1082. Zbl 1242.35163, MR 2860945, 10.1007/s00033-011-0145-0 |
| Reference:
|
[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. Zbl 0636.93064, MR 1054123 |
| Reference:
|
[18] Lasiecka, I.: Global uniform decay rates for the solutions to wave equation with nonlinear boundary conditions.Appl. Anal. 47 (1992), 191-212. Zbl 0788.35008, MR 1333954, 10.1080/00036819208840140 |
| Reference:
|
[19] Liu, Z., Zheng, S.: Semigroups Associated with Dissipative Systems.Chapman & Hall/CRC Research Notes in Mathematics 398. Chapman & Hall/CRC, Boca Raton (1999). Zbl 0924.73003, MR 1681343 |
| Reference:
|
[20] Lo, A., Tatar, N.-E.: Stabilization of laminated beams with interfacial slip.Electron. J. Differ. Equ. 2015 (2015), Article ID 129, 14 pages. Zbl 1321.35231, MR 3358501 |
| Reference:
|
[21] Lo, A., Tatar, N.-E.: Uniform stability of a laminated beam with structural memory.Qual. Theory Dyn. Syst. 15 (2016), 517-540. Zbl 1383.74056, MR 3563434, 10.1007/s12346-015-0147-y |
| Reference:
|
[22] Mustafa, M. I.: Uniform stability for thermoelastic systems with boundary time-varying delay.J. Math. Anal. Appl. 383 (2011), 490-498. Zbl 1222.35034, MR 2812399, 10.1016/j.jmaa.2011.05.066 |
| Reference:
|
[23] Mustafa, M. I.: Boundary control of laminated beams with interfacial slip.J. Math. Phys. 59 (2018), Article ID 051508, 9 pages. Zbl 1391.74124, MR 3806429, 10.1063/1.5017923 |
| Reference:
|
[24] Mustafa, M. I.: Laminated Timoshenko beams with viscoelastic damping.J. Math. Anal. Appl. 466 (2018), 619-641. Zbl 06897084, MR 3818134, 10.1016/j.jmaa.2018.06.016 |
| Reference:
|
[25] Nicaise, S., Pignotti, C.: Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks.SIAM J. Control Optim. 45 (2006), 1561-1585. Zbl 1180.35095, MR 2272156, 10.1137/060648891 |
| Reference:
|
[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. Zbl 1215.35098, MR 2788660 |
| Reference:
|
[27] Nicaise, S., Pignotti, C., Valein, J.: Exponential stability of the wave equation with boundary time-varying delay.Discrete Contin. Dyn. Syst., Ser. S 4 (2011), 693-722. Zbl 1215.35030, MR 2746429, 10.3934/dcdss.2011.4.693 |
| Reference:
|
[28] Nicaise, S., Valein, J., Fridman, E.: Stability of the heat and of the wave equations with boundary time-varying delays.Discrete Contin. Dyn. Syst., Ser. S 2 (2009), 559-581. Zbl 1171.93029, MR 2525768, 10.3934/dcdss.2009.2.559 |
| Reference:
|
[29] Pignotti, C.: A note on stabilization of locally damped wave equations with time delay.Syst. Control Lett. 61 (2012), 92-97. Zbl 1250.93103, MR 2878692, 10.1016/j.sysconle.2011.09.016 |
| Reference:
|
[30] Raposo, C. A.: Exponential stability for a structure with interfacial slip and frictional damping.Appl. Math. Lett. 53 (2016), 85-91. Zbl 1332.35042, MR 3426559, 10.1016/j.aml.2015.10.005 |
| Reference:
|
[31] 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:
|
[32] Tatar, N.-E.: Stabilization of a laminated beam with interfacial slip by boundary controls.Bound. Value Probl. 2015 (2015), Article ID 169, 11 pages. Zbl 1338.35037, MR 3411322, 10.1186/s13661-015-0432-3 |
| Reference:
|
[33] Wang, J.-M., Xu, G.-Q., Yung, S.-P.: Exponential stabilization of laminated beams with structural damping and boundary feedback controls.SIAM J. Control Optim. 44 (2005), 1575-1597. Zbl 1132.93021, MR 2193496, 10.1137/040610003 |
| Reference:
|
[34] Xu, G. Q., Yung, S. P., Li, L. K.: Stabilization of wave systems with input delay in the boundary control.ESAIM, Control Optim. Calc. Var. 12 (2006), 770-785. Zbl 1105.35016, MR 2266817, 10.1051/cocv:2006021 |
| Reference:
|
[35] Zhang, Q., Ran, M., Xu, D.: Analysis of the compact difference scheme for the semilinear fractional partial differential equation with time delay.Appl. Anal. 96 (2017), 1867-1884. Zbl 1373.65061, MR 3663769, 10.1080/00036811.2016.1197914 |
| Reference:
|
[36] Zuazua, E.: Uniform stabilization of the wave equation by nonlinear boundary feedback.SIAM J. Control Optim. 28 (1990), 466-477. Zbl 0695.93090, MR 1040470, 10.1137/0328025 |
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