Applications of Mathematics, Vol. 70, No. 2, pp. 149-168, 2025
Exponential stability for Timoshenko model with thermal effect
Luiz Gutemberg Rosário Miranda, Bruno Magalhães Alves
Received July 23, 2024. Published online May 13, 2025.
Abstract: We performe an exponential decay analysis for a Timoshenko-type system under the thermal effect by constructing the Lyapunov functional. More precisely, this thermal effect is acting as a mechanism for dissipating energy generated by the bending of the beam, acting only on the vertical displacement equation, different from other works already existing in the literature. Furthermore, we show the good placement of the problem using semigroup theory.
References: [1] D. S. Almeida Júnior, I. Elishakoff, A. J. A. Ramos, L. G. R. Miranda: The hypothesis of equal wave speeds for stabilization of Timoshenko beam is not necessary anymore: The time delay cases. IMA J. Appl. Math. 84 (2019), 763-796. DOI 10.1093/imamat/hxz014 | MR 3987834 | Zbl 1476.74067
[2] D. S. Almeida Júnior, M. L. Santos, J. E. Muñoz Rivera: Stability to weakly dissipative Timoshenko systems. Math. Methods Appl. Sci. 36 (2013), 1965-1976. DOI 10.1002/mma.2741 | MR 3091687 | Zbl 1273.74072
[3] D. S. Almeida Júnior, M. L. Santos, J. E. Muñoz Rivera: Stability to 1-D thermoelastic Timoshenko beam acting of shear force. Z. Angew. Math. Phys. 65 (2014), 1233-1249. DOI 10.1007/s00033-013-0387-0 | MR 3279528 | Zbl 1316.35044
[4] T. A. Apalara, S. A. Messaoudi, M. I. Mustafa: Energy decay in thermoelasticity type III with viscoelastic damping and delay term. Electron. J. Differ. Equ. 2012 (2012), Article ID 128, 15 pages. MR 2967193 | Zbl 1254.35144
[5] A. Borichev, Y. Tomilov: Optimal polynomial decay of functions and operator semigroups. Math. Ann. 347 (2010), 455-478. DOI 10.1007/s00208-009-0439-0 | MR 2606945 | Zbl 1185.47044
[6] P. S. Casas, R. Quintanilla: Exponential decay in one-dimensional porous-thermo-elasticity. Mech. Res. Commun. 32 (2005), 652-658. DOI 10.1016/j.mechrescom.2005.02.015 | MR 2158183 | Zbl 1192.74156
[7] L. Gearhart: Spectral theory for contraction semigroups on Hilbert space. Trans. Am. Math. Soc. 236 (1978), 385-394. DOI 10.1090/S0002-9947-1978-0461206-1 | MR 0461206 | Zbl 0326.47038
[8] F. Huang: Characteristic condition for exponential stability of linear dynamical systems in Hilbert spaces. Ann. Differ. Equations 1 (1985), 43-56. MR 0834231 | Zbl 0593.34048
[9] J. E. Muñoz Rivera, R. Racke: Mildly dissipative nonlinear Timoshenko systems - global existence and exponential stability. J. Math. Anal. Appl. 276 (2002), 248-278. DOI 10.1016/S0022-247X(02)00436-5 | MR 1944350 | Zbl 1106.35333
[10] J. Prüss: On the spectrum of $C_0$-semigroups. Trans. Am. Math. Soc. 284 (1984), 847-857. DOI 10.1090/S0002-9947-1984-0743749-9 | MR 0743749 | Zbl 0572.47030
[11] A. Soufyane: Stabilisation de la poutre de Timoshenko. C. R. Acad. Sci., Paris, Sér. I, Math. 328 (1999), 731-734. (In French.) DOI 10.1016/S0764-4442(99)80244-4 | MR 1680836 | Zbl 0943.74042
[12] S. P. Timoshenko: On the correction for shear of the differential equation for transverse vibrations of prismatic bars. Phil. Mag. (6) 41 (1921), 744-746. DOI 10.1080/14786442108636264
Affiliations: Luiz Gutemberg Rosário Miranda (corresponding author), Faculty of Mathematics, Federal University of Pará, Raimundo Santana Street, s/n, 68721-000, Salinópolis-Pa, Brazil, e-mail: luizmiranda@ufpa.br; Bruno Magalhães Alves, Faculty of Coastal and Oceanic Engineering, Federal University of Pará, Raimundo Santana Street, s/n, 68721-000, Salinópolis-Pa, Brazil, e-mail: bruno06br1@gmail.com
