Stability result of the Lamé system with a delay term in the internal fractional feedback

Abbes Benaissa; Soumia Gaouar

doi:10.2478/ausm-2021-0021

Abstract

In this article, we consider a Lamé system with a delay term in the internal fractional feedback. We show the existence and uniqueness of solutions by means of the semigroup theory under a certain condition between the weight of the delay term in the fractional feedback and the weight of the term without delay. Furthermore, we show the exponential stability by the classical theorem of Gearhart, Huang and Pruss.

References

[1] C. Abdallah, P. Dorato, J. Benitez-Read & R. Byrne, Delayed positive feedback can stabilize oscillatory system, ACC, San Francisco, (1993), 3106–3107.10.23919/ACC.1993.4793475
Search in Google Scholar Back to article
[2] W. Arendt & C. J. K. Batty, Tauberian theorems and stability of one-parameter semigroups, Trans. Am. Math. Soc., 306 (1988), 837–852.10.1090/S0002-9947-1988-0933321-3
Search in Google Scholar Back to article
[3] R. L. Bagley & P. J. Torvik, A theoretical basis for the application of fractional calculus to viscoelasticity, J. Rheology., 27 (1983), 201–210.10.1122/1.549724
Search in Google Scholar Back to article
[4] R. L. Bagley & P. J. Torvik, On the appearance of the fractional derivative in the behavior of real material, J. Appl. Mech., 51 (1983), 294–298.10.1115/1.3167615
Search in Google Scholar Back to article
[5] A. Benaissa, S. Gaouar, Asymptotic stability for the Lamé system with fractional boundary damping, Comput. Math. Appl., 77 (2019)-5, 1331–1346.10.1016/j.camwa.2018.11.011
Search in Google Scholar Back to article
[6] H. Brézis, Operateurs Maximaux Monotones et semi-groupes de contractions dans les espaces de Hilbert, Notas de Matemàtica (50), Universidade Federal do Rio de Janeiro and University of Rochester, North-Holland, Amsterdam, (1973).
Search in Google Scholar Back to article
[7] L. Gearhart, Spectral theory for contraction semigroups on Hilbert spaces, Trans. Amer. Math. Soc. 236 (1978), 385–394.10.1090/S0002-9947-1978-0461206-1
Search in Google Scholar Back to article
[8] H. Haddar, J. R Li and D. Matignon, Efficient solution of a wave equation with fractional-order dissipative terms, J. Comput. Appl. Math., 234 (2010)-6, 2003–2010.10.1016/j.cam.2009.08.051
Search in Google Scholar Back to article
[9] F. Huang, Characteristic conditions for exponential stability of linear dynamical systems in Hilbert spaces, Ann. Differ. Equ., 1 (1985), 43–55.
Search in Google Scholar Back to article
[10] I. Lyubich Yu & V. Q. Phóng, Asymptotic stability of linear differential equations in Banach spaces, Studia Mathematica, 88 (1988)-(1), 37–42.10.4064/sm-88-1-37-42
Search in Google Scholar Back to article
[11] F. Mainardi & E. Bonetti, The applications of real order derivatives in linear viscoelasticity, Rheol. Acta, 26 (1988), 64–67.10.1007/978-3-642-49337-9_11
Search in Google Scholar Back to article
[12] B. Mbodje, Wave energy decay under fractional derivative controls, IMA Journal of Mathematical Control and Information., 23 (2006), 237–257.10.1093/imamci/dni056
Search in Google Scholar Back to article
[13] S. Nicaise and C. Pignotti, Stability and instability results of the wave equation with a delay term in the boundary or internal feedbacks, SIAM J. Control Optim., 45 (2006)-5, 1561–1585.10.1137/060648891
Search in Google Scholar Back to article
[14] I. Podlubny, Fractional differential equations, Mathematics in Science and Engineering, 198 (1999), Academic Press.
Search in Google Scholar Back to article
[15] J. Prüss, On the spectrum of C₀-semigroups, Transactions of the American Mathematical Society. 284 (1984)-2, 847–857.10.1090/S0002-9947-1984-0743749-9
Search in Google Scholar Back to article
[16] S. G. Samko, A. A. Kilbas, O. I. Marichev, emphFractional integrals and derivatives, Amsterdam: Gordon and Breach (1993) [Engl. Trans. from the Russian (1987)].
Search in Google Scholar Back to article
[17] I. H. Suh & Z. Bien, Use of time delay action in the controller design, IEEE Trans. Autom. Control, 25 (1980) 600–603.10.1109/TAC.1980.1102347
Search in Google Scholar Back to article

Stability result of the Lamé system with a delay term in the internal fractional feedback

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