Stability analysis for a Rao-Nakra sandwich beam equation with time-varying weights and frictional dampings

Adel M. Al-Mahdi; Maher Noor; Mohammed M. Al-Gharabli; Baowei Feng; Abdelaziz Soufyane; Adel M. Al-Mahdi; Maher Noor; Mohammed M. Al-Gharabli; Baowei Feng; Abdelaziz Soufyane

doi:10.3934/math.2024615

AIMS Mathematics

2024, Volume 9, Issue 5: 12570-12587. doi: 10.3934/math.2024615

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Research article

Stability analysis for a Rao-Nakra sandwich beam equation with time-varying weights and frictional dampings

1.
Department of Mathematics, King Fahd University of Petroleum & Minerals, Dhahran 31261, Saudi Arabia
2.
The Interdisciplinary Research Center in Construction and Building Materials, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
3.
DCC-Math, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia
4.
Department of Mathematics, Southwestern University of Finance and Economics, Chengdu 611130, China
5.
Department of Mathematics and Statistics, University of Sharjah Sharjah, United Arab Emirates

Received: 16 January 2024 Revised: 18 March 2024 Accepted: 25 March 2024 Published: 01 April 2024
MSC : 35B40, 93D15, 93D20

In this paper, we studied the asymptotic behavior of solutions for a Rao-Nakra sandwich beam equation with time-varying weights and frictional damping terms acting complementarily in the domain. We studied the effect of the three damping on the asymptotic behavior of the energy function. Under nonrestrictive on the growth assumption on the frictional damping terms, we established exponential and general energy decay rates for this system by using the multiplier approach. The results generalized some earlier decay results on the Rao-Nakra sandwich beam equation.
- multilayer beam,
- multiplier method,
- stability
Citation: Adel M. Al-Mahdi, Maher Noor, Mohammed M. Al-Gharabli, Baowei Feng, Abdelaziz Soufyane. Stability analysis for a Rao-Nakra sandwich beam equation with time-varying weights and frictional dampings[J]. AIMS Mathematics, 2024, 9(5): 12570-12587. doi: 10.3934/math.2024615

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Abstract

In this paper, we studied the asymptotic behavior of solutions for a Rao-Nakra sandwich beam equation with time-varying weights and frictional damping terms acting complementarily in the domain. We studied the effect of the three damping on the asymptotic behavior of the energy function. Under nonrestrictive on the growth assumption on the frictional damping terms, we established exponential and general energy decay rates for this system by using the multiplier approach. The results generalized some earlier decay results on the Rao-Nakra sandwich beam equation.

References

[1]	J.-M. Wang, B.-Z. Guo, B. Chentouf, Boundary feedback stabilization of a three-layer sandwich beam: Riesz basis approach, ESAIM: COCV, 12 (2006), 12–34. https://doi.org/10.1051/cocv:2005030 doi: 10.1051/cocv:2005030
[2]	R. Rajaram, Exact boundary controllability results for a Rao–Nakra sandwich beam, Syst. Control Lett., 56 (2007), 558–567. https://doi.org/10.1016/j.sysconle.2007.03.007 doi: 10.1016/j.sysconle.2007.03.007
[3]	S. W. Hansen, O. Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with clamped boundary conditions, ESAIM: COCV, 17 (2011), 1101–1132. https://doi.org/10.1051/cocv/2010040 doi: 10.1051/cocv/2010040
[4]	S. W. Hansen, O. Y. Imanuvilov, Exact controllability of a multilayer Rao-Nakra plate with free boundary conditions, Math. Control Relat. Field., 1 (2011), 189–230. https://doi.org/10.3934/mcrf.2011.1.189 doi: 10.3934/mcrf.2011.1.189
[5]	A. O. Ozer, S. W. Hansen, Uniform stabilization of a multilayer Rao-Nakra sandwich beam, Evol. Equ. Control Theor., 2 (2013), 695–710. https://doi.org/10.3934/eect.2013.2.695 doi: 10.3934/eect.2013.2.695
[6]	A. O. Ozer, S. W. Hansen, Exact boundary controllability results for a multilayer Rao-Nakra sandwich beam, SIAM J. Control Optim., 52 (2014), 1314–1337. https://doi.org/10.1137/120892994 doi: 10.1137/120892994
[7]	Z. Y. Liu, B. P. Rao, Q. Zhang, Polynomial stability of the Rao-Nakra beam with a single internal viscous damping, J. Differ. Equations, 269 (2020), 6125–6162. https://doi.org/10.1016/j.jde.2020.04.030 doi: 10.1016/j.jde.2020.04.030
[8]	Y. Wang, Boundary feedback stabilization of a Rao-Nakra sandwich beam, J. Phys.: Conf. Ser., 1324 (2019), 012044. https://doi.org/10.1088/1742-6596/1324/1/012044 doi: 10.1088/1742-6596/1324/1/012044
[9]	A. A. Allen, S. W. Hansen, Analyticity and optimal damping for a multilayer Mead-Markus sandwich beam, Discrete Contin. Dyn. Syst. B, 14 (2010), 1279–1292. https://doi.org/10.3934/dcdsb.2010.14.1279 doi: 10.3934/dcdsb.2010.14.1279
[10]	A. A. Allen, S. W. Hansen, Analyticity of a multilayer Mead-Markus plate, Nonlinear Anal., 71 (2009), e1835–e1842. https://doi.org/10.1016/j.na.2009.02.063 doi: 10.1016/j.na.2009.02.063
[11]	R. H. Fabiano, S. W. Hansen, Modeling and analysis of a three-layer damped sandwich beam, Conference Publications, 2001 (2001), 143–155. https://doi.org/10.3934/proc.2001.2001.143 doi: 10.3934/proc.2001.2001.143
[12]	S. W. Hansen, R. Rajaram, Simultaneous boundary control of a Rao-Nakra sandwich beam, In: Proceedings of the 44th IEEE Conference on Decision and Control, IEEE, 2005, 3146–3151. https://doi.org/10.1109/CDC.2005.1582645
[13]	S. W. Hansen, R. Rajaram, Riesz basis property and related results for a Rao-Nakra sandwich beam, Conference Publications, 2005 (2005), 365–375. https://doi.org/10.3934/proc.2005.2005.365 doi: 10.3934/proc.2005.2005.365
[14]	A. Ozkan Ozer, S. W. Hansen, Exact controllability of a Rayleigh beam with a single boundary control, Math. Control Signals Syst., 23 (2011), 199–222. https://doi.org/10.1007/s00498-011-0069-4 doi: 10.1007/s00498-011-0069-4
[15]	C. Yang, J.-M. Wang, Exponential stability of an active constrained layer beam actuated by a voltage source without magnetic effects, J. Math. Anal. Appl., 448 (2017), 1204–1227. https://doi.org/10.1016/j.jmaa.2016.11.067 doi: 10.1016/j.jmaa.2016.11.067
[16]	I. Lasiecka, D. Tataru, Uniform boundary stabilization of semilinear wave equations with nonlinear boundary damping, Differ. Integral Equ., 6 (1993), 507–533. https://doi.org/10.57262/die/1370378427 doi: 10.57262/die/1370378427
[17]	F. Alabau-Boussouira, Convexity and weighted integral inequalities for energy decay rates of nonlinear dissipative hyperbolic systems, Appl. Math. Optim., 51 (2005), 61–105. https://doi.org/10.1007/s00245 doi: 10.1007/s00245
[18]	F. Alabau-Boussouira, P. Cannarsa, D. Sforza, Decay estimates for second order evolution equations with memory, J. Funct. Anal., 254 (2008), 1342–1372. https://doi.org/10.1016/j.jfa.2007.09.012 doi: 10.1016/j.jfa.2007.09.012
[19]	F. Alabau-Boussouira, P. Cannarsa, A general method for proving sharp energy decay rates for memory-dissipative evolution equations, C. R. Math., 347 (2009), 867–872. https://doi.org/10.1016/j.crma.2009.05.011 doi: 10.1016/j.crma.2009.05.011
[20]	P. Martinez, A new method to obtain decay rate estimates for dissipative systems, ESAIM: COCV, 4 (1999), 419–444. https://doi.org/10.1051/cocv:1999116 doi: 10.1051/cocv:1999116
[21]	A. Guesmia, Inégalités intégrales et applications à la stabilisation des systèmes distribués non dissipatifs, PhD thesis, Université de Metz, 2006.
[22]	V. I. Arnol'd, Mathematical methods of classical mechanics, New York: Springer, 1989. https://doi.org/10.1007/978-1-4757-2063-1

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