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On the asymptotic behaviour of two coupled strings through a fractional joint damper. (English) Zbl 1447.93277

Summary: In this paper, we investigate the large time behavior of one dimensional coupled vibrating systems with fractional control applied at the coupled point. We prove well-posedness by using the semigroup theory. Also we establish an optimal decay result by frequency domain method and Borichev-Tomilov theorem.

MSC:

93D15 Stabilization of systems by feedback
35B40 Asymptotic behavior of solutions to PDEs
93C80 Frequency-response methods in control theory
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[1] Achouri, Z.; Amroun, N.; Benaissa, A., The Euler-Bernoulli beam equation with boundary dissipation of fractional derivative type, Math. Methods Appl. Sci., 40, 11, 3837-3854 (2017) · Zbl 1366.93484 · doi:10.1002/mma.4267
[2] Arendt, W.; Batty, CJK, Tauberian theorems and stability of one-parameter semigroups, Trans. Am. Math. Soc., 306, 837-852 (1988) · Zbl 0652.47022 · doi:10.1090/S0002-9947-1988-0933321-3
[3] Bagley, RL; Torvik, PJ, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheology., 27, 201-210 (1983) · Zbl 0515.76012 · doi:10.1122/1.549724
[4] Bagley, RL; Torvik, PJ, A different approach to the analysis of viscoelastically damped structures, AIAA J., 21, 741-748 (1983) · Zbl 0514.73048 · doi:10.2514/3.8142
[5] Bagley, RL; Torvik, PJ, On the appearance of the fractional derivative in the behavior of real material, J. Appl. Mech., 51, 294-298 (1983) · Zbl 1203.74022
[6] Blanc, E.; Chiavassa, G.; Lombard, B., Biot-JKD model: simulation of 1D transient poroelastic waves with fractional derivatives, J. Comput. Phys., 237, 1-20 (2013) · Zbl 1286.76141 · doi:10.1016/j.jcp.2012.12.003
[7] Borichev, A.; Tomilov, Y., Optimal polynomial decay of functions and operator semigroups, Math. Ann., 347, 2, 455-478 (2010) · Zbl 1185.47044 · doi:10.1007/s00208-009-0439-0
[8] Choi, JU; Maccamy, RC, Fractional order Volterra equations with applications to elasticity, J. Math. Anal. Appl., 139, 448-464 (1989) · Zbl 0674.45007 · doi:10.1016/0022-247X(89)90120-0
[9] Chen, G.; Coleman, M.; West, HH, Pointwise stabilization in the middle of the span for second order systems nonuniform and uniform exponential decay of solutions, SIAM J. Appl. Math., 47, 751-780 (1987) · Zbl 0641.93047 · doi:10.1137/0147052
[10] Dautray, R.; Lions, JL, Analyse mathématique et calcul numérique pour les sciences et les techniques (1984), Masson: Tome I, Masson · Zbl 0749.35003
[11] Hardy, GH; Wright, EM, An Introduction to the Theory of Numbers (2008), Oxford: Oxford University Press, Oxford
[12] Huang, F., Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differ. Equ., 1, 43-55 (1985) · Zbl 0593.34048
[13] Liu, K-S, Energy decay problems in the design of a point stabilizer for coupled string vibrating systems, SIAM J. Control Optim., 26, 1348-1356 (1988) · Zbl 0662.93054 · doi:10.1137/0326076
[14] Luo, ZH; Guo, BZ; Morgul, O., Stability and Stabilization of Infinite Dimensional Systems with Applications (1999), London: Springer, London · Zbl 0922.93001
[15] Lyubich Yu, I.; Vu, QP, Asymptotic stability of linear differential equations in Banach spaces, Stud. Math., 88, 1, 37-42 (1988) · Zbl 0639.34050 · doi:10.4064/sm-88-1-37-42
[16] Mainardi, F.; Bonetti, E., The applications of real order derivatives in linear viscoelasticity, Rheol. Acta, 26, 64-67 (1988)
[17] Mbodje, B., Wave energy decay under fractional derivative controls, IMA J. Math. Control Inf., 23, 237-257 (2006) · Zbl 1095.93015 · doi:10.1093/imamci/dni056
[18] Mbodje, B.; Montseny, G., Boundary fractional derivative control of the wave equation, IEEE Trans. Autom. Control, 40, 368-382 (1995) · Zbl 0820.93034 · doi:10.1109/9.341815
[19] Podlubny, I., Fractional Differential Equations, Mathematics in Science and Engineering (1999), Cambridge: Academic Press, Cambridge · Zbl 0918.34010
[20] Prüss, J., On the spectrum of \(C_0\)-semigroups, Trans. Am. Math. Soc., 284, 2, 847-857 (1984) · Zbl 0572.47030 · doi:10.2307/1999112
[21] Rzepnicki, L.; Schnaubelt, R., Polynomial stability for a system of coupled strings, Bull. Lond. Math. Soc., 50, 6, 1117-1136 (2018) · Zbl 1406.35185 · doi:10.1112/blms.12212
[22] Scott, WT, Approximation to real irrationals by certain classes of rational fractions, Bull. Am. Math. Soc., 46, 124-129 (1940) · JFM 66.0183.02 · doi:10.1090/S0002-9904-1940-07152-6
[23] Samko, SG; Kilbas, AA; Marichev, OI, Fractional integrals and derivatives (1993), Amsterdam: Gordon and Breach, Amsterdam · Zbl 0818.26003
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