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The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems

Author

Listed:
  • Hanaa Zitane

    (MACS Laboratory, Department of Mathematics, Faculty of Sciences, University of Moulay Ismail, Meknes 11201, Morocco
    The authors contributed equally to this work.)

  • Ali Boutoulout

    (MACS Laboratory, Department of Mathematics, Faculty of Sciences, University of Moulay Ismail, Meknes 11201, Morocco
    The authors contributed equally to this work.)

  • Delfim F. M. Torres

    (Center for Research and Development in Mathematics and Applications (CIDMA), Department of Mathematics, University of Aveiro, 3810-193 Aveiro, Portugal
    The authors contributed equally to this work.)

Abstract

We investigate the stability and stabilization concepts for infinite dimensional time fractional differential linear systems in Hilbert spaces with Caputo derivatives. Firstly, based on a family of operators generated by strongly continuous semigroups and on a probability density function, we provide sufficient and necessary conditions for the exponential stability of the considered class of systems. Then, by assuming that the system dynamics are symmetric and uniformly elliptical and by using the properties of the Mittag–Leffler function, we provide sufficient conditions that ensure strong stability. Finally, we characterize an explicit feedback control that guarantees the strong stabilization of a controlled Caputo time fractional linear system through a decomposition approach. Some examples are presented that illustrate the effectiveness of our results.

Suggested Citation

  • Hanaa Zitane & Ali Boutoulout & Delfim F. M. Torres, 2020. "The Stability and Stabilization of Infinite Dimensional Caputo-Time Fractional Differential Linear Systems," Mathematics, MDPI, vol. 8(3), pages 1-14, March.
    • Handle: RePEc:gam:jmathe:v:8:y:2020:i:3:p:353-:d:329040
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    References listed on IDEAS

    as
    1. Fengrong Zhang & Changpin Li & YangQuan Chen, 2011. "Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative," International Journal of Differential Equations, Hindawi, vol. 2011, pages 1-12, August.
    2. Mohammed M. Matar & Esmail S. Abu Skhail, 2018. "On Stability of Nonautonomous Perturbed Semilinear Fractional Differential Systems of Order," Journal of Mathematics, Hindawi, vol. 2018, pages 1-10, August.
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