Mittag-Leffler stability for a Timoshenko problem

Tatar, Nasser-eddine

International Journal of Applied Mathematics and Computer Science, Tome 31 (2021) no. 2, pp. 219-232.

Résumé

A Timoshenko system of a fractional order between zero and one is investigated here. Using a fractional version of resolvents, we establish an existence and uniqueness theorem in an appropriate space. Moreover, it is proved that lower order fractional terms (in the rotation component) are capable of stabilizing the system in a Mittag-Leffler fashion. Therefore, they deserve to be called damping terms. This is shown through the introduction of some new functionals and some fractional inequalities, and the establishment of some properties, involving fractional derivatives. In the case of different wave speeds of propagation we obtain convergence to zero.

Détail

Keywords: Caputo fractional derivative, Mittag-Leffler stability, multiplier technique, resolvent operator

Bibliographie

[1] [1] Alabau-Boussouira, F. (2007). Asymptotic behavior for Timoshenko beams subject to a single nonlinear feedback control, Nonlinear Differential Equations and Applications 14(5): 643–669.

[2] [2] Alikhanov, A. (2012). Boundary value problems for the diffusion equation of the variable order in differential and difference settings, Applied Mathematics and Computation 219(8): 3938–3946.

[3] [3] Almeida Júnior, D., Santos, M.L. and Muñoz Rivera, J. (2013). Stability to weakly dissipative Timoshenko systems, Mathematical Methods in the Applied Sciences 36(14): 1965–1976.

[4] [4] Ammar-Khodja, F., Kerbal, S. and Soufyane, A. (2007). Stabilization of the nonuniform Timoshenko beam, Journal of Mathematical Analysis and Applications 327(1): 525–538.

[5] [5] Anderson, J., Moradi, S. and Rafiq, T. (2018). Non-linear Langevin and fractional Fokker–Planck equations for anomalous diffusion by Lévy stable processes, Entropy 20(10): 1–12.

[6] [6] Atanackovic, T., Pilipovic, S. and Zorica, D. (2007). A diffusion wave equation with two fractional derivatives of different order, Journal of Physics A: Mathematical and Theoretical 40(20): 5319.

[7] [7] Bajlekova, E. (2001). Fractional Evolution Equations in Banach Spaces, PhD thesis, Eindhoren University of Technology, Eindhoven.

[8] [8] Beghin, L. and Orsingher, E. (2003). The telegraph process stopped at stable-distributed times and its connection with the fractional telegraph equation, Fractional Calculus and Applied Analysis 6(2): 187–204.

[9] [9] Cavalcanti, M., Cavalcanti, V., Nascimento, F., Lasiecka, I. and Rodrigues, J. (2014). Uniform decay rates for the energy of Timoshenko system with the arbitrary speeds of propagation and localized nonlinear damping, Zeitschrift für angewandte Mathematik und Physik 65(6): 1189–1206.

[10] [10] Chen, J., Liu, F. and Anh, V. (2008). Analytical solution for the time-fractional telegraph equation by the method of separating variables, Journal of Mathematical Analysis and Applications 338(2): 1364–1377.

[11] [11] Duarte-Mermoud, M., Aguila-Camacho, N., Gallegos, J. and Castro-Linares, R. (2015). Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Communications in Nonlinear Science and Numerical Simulation 22(1–3): 650–659.

[12] [12] Gallegos, J., Duarte-Mermoud, M., Aguila-Camacho, N. and Castro-Linares, R. (2015). On fractional extensions of Barbalat lemma, Systems Control Letters 84(1): 7–12.

[13] [13] Gorenflo, R., Kilbas, A., Mainardi, F. and Rogosin, S. (2014). Mittag-Leffler Functions, Related Topics and Applications, Vol. 2, Springer, Heidelberg.

[14] [14] Kaczorek, T. (2020). Global stability of nonlinear feedback systems with fractional positive linear parts, International Journal of Applied Mathematics and Computer Science 30(3): 493–499, DOI: 10.34768/amcs-2020-0036.

[15] [15] Klamka, J., Babiarz, A., Czornik, A. and Niezabitowski, M. (2020). Controllability and stability of semilinear fractional order systems, in P. Kulczycki et al. (Eds), Automatic Controls, Robotics, and Information Processing, Studies in Systems, Decision and Control, Vol. 296, Springer, Cham, pp. 267–290.

[16] [16] Li, Y., Chen, Y. and Podlubny, I. (2010). Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability, Computers Mathematics with Applications 59(5): 1810–1821.

[17] [17] Mainardi, F. (2010). Fractional Calculus and Waves in Linear Viscoelasticity: An Introduction to Mathematical Models, Imperial College Press, London.

[18] [18] Momani, S. (2005). Analytic and approximate solutions of the space-and time-fractional telegraph equations, Applied Mathematics and Computation 170(2): 1126–1134.

[19] [19] Mustafa, M. and Messaoudi, S. (2010). General energy decay rates for a weakly damped Timoshenko system, Journal of Dynamical and Control Systems 16(2): 211–226.

[20] [20] Orsingher, E. and Beghin, L. (2004). Time-fractional telegraph equations and telegraph processes with Brownian time, Probability Theory and Related Fields 128(1): 141–160.

[21] [21] Podlubny, I. (1999). Fractional Differential Equations, Academic Press, San Diego.

[22] [22] Prüss, J. (2013). Evolutionary Integral Equations and Applications, Birkhäuser, Basel.

[23] [23] Raposo, C., Ferreira, J., Santos, M. and Castro, N. (2005). Exponential stability for the Timoshenko system with two weak dampings, Applied Mathematics Letters 18(5): 535–541.

[24] [24] Rivera, J. and Racke, R. (2008). Timoshenko systems with indefinite damping, Journal of Mathematical Analysis and Applications 341(2): 1068–1083.

[25] [25] Rivera, M., Racke, J. and Null, R. (2002). Global stability for damped Timoshenko systems, Discrete Continuous Dynamical Systems 9(6): 1625–1639.

[26] [26] Sandev, T. and Tomovski, . (2019). Fractional Equations and Models: Theory and Applications, Developments in Mathematics, Springer Nature Switzerland AG, Cham.

[27] [27] Sklyar, G.M. and Szkibiel, G. (2013). Controlling a non-homogeneous Timoshenko beam with the aid of the torque, International Journal of Applied Mathematics and Computer Science 23(3): 587–598, DOI: 10.2478/amcs-2013-0044.

[28] [28] Soufyane, A. and Whebe, A. (2003). Uniform stabilization for the Timoshenko beam by a locally distributed damping, Electronic Journal of Differential Equations 2003(29): 1–14.