On the stabilization of a thermoelastic laminated beam system with microtemperature effects
DOI:
https://doi.org/10.24193/subbmath.2025.2.06Keywords:
Laminated beam, Microtemperatures, Exponential stability, Energy methodAbstract
The present article investigates a one dimensional thermoelastic laminated beam with microtemperature effects. Using the energy method we prove in the case of zero thermal conductivity that the unique dissipation due to the microtemperatures is strong enough to exponentially stabilize the system if and only if the wave speeds of the system are equal. Our result is new and improves previous results in the literature.References
[1] Apalara, T.A., Uniform stability of a laminated beam with structural damping and second sound, Z. Angew. Math. Phys., 68(2017), 1-16.
[2] Apalara, T.A., On the stability of a thermoelastic laminated beam, Acta Math. Sci., 39(2019), 1517-1524.
[3] Cabanillas, V.R., M endez, T.Q., Ramos, A.J.A., Laminated beams with thermoelasticity acting on the shear force, Math. Methods Appl. Sci., 46(2023), 1352-1374.
[4] 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.
[5] Chen, Z., Liu, W., Chen, D., General decay rates for a laminated beam with memory, Taiwanese J. Math., 23(2019), 1227-1252.
[6] Djellali, F., Stabilization of laminated beam with structural damping and a heat conduction of Gurtin-Pipkin's law, Appl. Anal., 102(2023), 4659-4677.
[7] Djellali, F., Well posedness and stability result for a thermoelastic laminated beam with structural damping, Ric. Mat., 73(2024), 2049-2073.
[8] Djellali, F., On the stabilization of a type III thermoelastic laminated beam with structural memory, SeMA J., 81(2024), 263{281.
[9] Djellali, F., Apalara, T.A., General decay for laminated beams with structural memory and modified thermoelasticity of type III, Ann. Univ. Ferrara Sez. VII Sci. Mat., 69(2023), 541-560.
[10] Djellali, F., Apalara, T.A., Sai a, O., New Exponential Stability Result for Thermoelastic Laminated Beams with Structural Damping and Second Sound, Acta Appl. Math., 184(2023), 12.
[11] Djellali, F., Cabanillas, V.R., Al-Mahdi, A.M., Exponential stabilization of laminated beams with Gurtin-Pipkin thermal law the case of equal speeds, J. Integral Equations Appl., 36(2024), 183-202.
[12] Djeradi, F.S., Yazid, F., Georgiev, S.G., Hajjej, Z., Zennir, K., On the time decay for a thermoelastic laminated beam with microtemperature e ects, nonlinear weight, and nonlinear time-varying delay, AIMS Math., 8(2023), 26096-26114.
[13] Feng, B., On a thermoelastic laminated Timoshenko beam: Well posedness and stability, Complexity, 2020(2020), 1-13.
[14] Hansen, S.W., Spies, R.D., Structural damping in laminated beams due to interfacial slip, J. Sound Vibration, 204(1997), 183-202.
[15] Khochemane, H.E., Exponential stability for a thermoelastic porous system with microtemperatures e ects, Acta Appl. Math., 173(2021), 1-14.
[16] Liu, W., Zhao, W., Stabilisation of a thermoelastic laminated beam with past history, Appl. Math. Optim., 80(2019), 103-133.
[17] Liu, W., Zhao, W., On the stability of a laminated beam with structural damping and Gurtin-Pipkin thermal law, Nonlinear Anal. Model. Control, 26(2021), 396-418.
[18] Liu, W., Zhao, W., Exponential and polynomial decay for a laminated beam with Fourier's law of heat conduction and possible absence of structural damping, Front. Math. China, 16(2021), 997-1021.
[19] Liu, Z., Zheng, S., Semigroups associated with dissipative systems, Boca, Raton, Chapman Hall/CRC, 1999.
[20] Lo, A., Tatar, N.E., Stabilization of laminated beams with interfacial slip, Electron. J. Differential Equations, 2015(2015), 1-14.
[21] Lo, A., Tatar, N.E., Uniform stability of a laminated beam with structural memory, Qual. Theory Dyn. Syst., 15(2016), 517-540.
[22] Lo, A., Tatar, N.E., Exponential stabilization of a structure with interfacial slip, Discrete Contin. Dyn. Syst., 36(2016), 6285-6306.
[23] M endez, T.Q., Cabanillas, V.R., Ramos, A.J.A., Stability results for a laminated thermoviscoelastic system with Fourier's law, Z. Angew. Math. Phys., 73(2022), 152.
[24] Meradji, S., Boudeliou, M., Djebabla, A., New stability number of the Timoshenko system with only microtemperature e ects and without thermal conductivity, Eur. J. Math. Comput. Appl., 12(2024), 94-109.
[25] Mukiawa, S.E., Apalara, T.A., Messaoudi, S., A general and optimal stability result for a laminated beam, J. Integral Equations Appl., 32(2020), 341-359.
[26] Mukiawa, S.E., Apalara, T.A., Messaoudi, S., A stability result for a memory-type laminated-thermoelastic system with Maxwell-Cattaneo heat conduction, J. Thermal Stresses, 43(2020), 1437-1466.
[27] Mustafa, M.I., Laminated Timoshenko beams with viscoelastic damping, J. Math. Anal. Appl., 466(2018), 619-641.
[28] Mustafa, M.I., On the stabilization of viscoelastic laminated beams with interfacial slip, Z. Angew. Math. Phys., 69(2018), 1-14.
[29] Nonato, C., Raposo, C., Feng, B., Exponential stability for a thermoelastic laminated beam with nonlinear weights and time-varying delay, Asymptot. Anal., 126(2022), 157-185.
[30] Pazy, A., Semigroups of linear operators and applications to partial differential equations, New York, Springer-Verlag, 1983.
[31] Saci, M., Djebabla, A., On the stability of linear porous elastic materials with microtemperatures effects, J. Thermal Stresses, 43(2020), 1300-1315.
[32] 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.