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Global attractivity and asymptotic stability of mixed-order fractional systems

Global attractivity and asymptotic stability of mixed-order fractional systems

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This study investigates the asymptotic properties of mixed-order fractional systems. By using the variation of constants formula, properties of real Mittag-Leffler functions, and Banach fixed-point theorem, the authors first propose an explicit criterion guaranteeing global attractivity for a class of mixed-order linear fractional systems. The criterion is easy to check requiring the system's matrix to be strictly diagonally dominant (C1) and elements on its main diagonal to be negative (C2). The authors then show the asymptotic stability of the trivial solution to a non-linear mixed-order fractional system linearised along with its equilibrium point such that its linear part satisfies the conditions (C1) and (C2). Two numerical examples with simulations are given to illustrate the effectiveness of the results over existing ones in the literature.


    1. 1)
      • 16. Gorenflo, R., Mainardi, F.: ‘Fractional calculus: integral and differential equations of fractional order’, in Carpinteri, A., Mainardi, F. (Eds.): ‘Fractals and fractional calculus in continuum mechanics’ (Springer, Berlin, 1997), pp. 223276.
    2. 2)
      • 5. Gorenflo, R., Kilbas, A., Mainardi, F., et al: ‘Mittag-leffler functions, related topics and applications (Springer monographs in mathematics)’ (Springer-Verlag, Berlin, Heidelberg, 2014).
    3. 3)
      • 7. Badri, V., Sojoodi, M.: ‘Robust fixed-order dynamic output feedback controller design for fractional-order systems’, IET Control Theory Applic., 2018, 12, (9), pp. 12361243.
    4. 4)
      • 21. Petras, I.: ‘Stability of fractional-order systems with rational orders: a survey’, Fractional Calculus Appl. Anal., 2009, 12, (3), pp. 269298.
    5. 5)
      • 23. Diethelm, K., Siegmund, S., Tuan, H.T.: ‘Asymptotic behavior of solutions of linear multi-order fractional differential systems’, Fractional Calculus Appl. Anal., 2017, 20, (5), pp. 11651195.
    6. 6)
      • 3. Lakshmikantham, V., Leela, S., Devi, J.: ‘Theory of fractional dynamic systems’ (Cambridge Scientific Publishers Ltd., England, 2009).
    7. 7)
      • 13. Tuan, H.T., Trinh, H.: ‘Stability of fractional-order nonlinear systems by Lyapunov direct method’, IET Control Theory Applic., 2018, 12, (17), pp. 24172422.
    8. 8)
      • 11. He, B.B., Zhou, H.C., Chen, Y., et al: ‘Asymptotical stability of fractional order systems with time delay via an integral inequality’, IET Control Theory Applic., 2018, 12, (12), pp. 17481754.
    9. 9)
      • 8. Chen, W., Dai, H., Song, Y., et al: ‘Convex Lyapunov functions for stability analysis of fractional order systems’, IET Control Theory Applic., 2017, 11, (5), pp. 10701074.
    10. 10)
      • 2. Diethelm, K.: ‘The analysis of fractional differential equations. an application-oriented exposition using differential operators of Caputo type (Lecture Notes in Mathematics, 2004)’ (Springer-Verlag, Berlin, 2010).
    11. 11)
      • 22. Shen, J., Lam, J.: ‘Stability and performance analysis for positive fractional-order systems with time-varying delays’, IEEE Trans. Autom. Control, 2016, 61, (9), pp. 26762681.
    12. 12)
      • 9. Cong, N.D., Son, D.T., Siegmund, S., et al: ‘Linearized asymptotic stability for fractional differential equations’, Electron. J. Qual. Theory Diff. Equ., 2016, 39, pp. 113.
    13. 13)
      • 20. Deng, W., Li, C., Lu, J.: ‘Stability analysis of linear fractional differential system with multiple time delays’, Nonlinear Dyn., 2007, 48, (4), pp. 409416.
    14. 14)
      • 1. Bandyopadhyay, B., Kamal, S.: ‘Stabilization and control of fractional order systems: a sliding mode approach’ (Lecture Notes in Electrical Engineering 317) (Springer International Publishing, Switzerland, 2015).
    15. 15)
      • 17. Torvik, P.J., Bagley, R.L.: ‘On the appearance of the fractional derivative in the behavior of real materials’, J. Appl. Mech., 1984, 51, pp. 294298.
    16. 16)
      • 6. Aguila-Camacho, N., Duarte-Mermoud, M., Gallegos, J., et al: ‘Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems’, Commun. Nonlinear Sci. Numer. Simul., 2015, 22, (1-3), pp. 650659.
    17. 17)
      • 18. Brandibur, O., Kaslik, E.: ‘Stability of two-component incommensurate fractional-order systems and applications to the investigation of a FitzHugh-Nagumo neuronal model’, Math. Methods Appl. Sci., 2018, 41, (11), pp. 71827194.
    18. 18)
      • 12. Li, Y., Zhao, D., Chen, Y., et al: ‘Finite energy Lyapunov function candidate for fractional order general nonlinear systems’, Commun. Nonlinear Sci. Numer. Simul., 2019, 78, p. 104886.
    19. 19)
      • 10. Cong, N.D., Son, D.T., Siegmund, S., et al: ‘An instability theorem for nonlinear fractional differential systems’, Dis. Continuous Dyn. Syst., B, 2017, 22, (8), pp. 30793090.
    20. 20)
      • 25. Cong, N.D., Tuan, H.T.: ‘Generation of nonlocal fractional dynamical systems by fractional differential equations’, J. Integral Equ. Appl., 2017, 29, (4), pp. 585608.
    21. 21)
      • 24. Feckan, M., Wang, J.: ‘Mixed order fractional differential equations’, Mathematics, 2017, 5, (4), p. 61.
    22. 22)
      • 19. Škovránek, T., Podlubny, I., Petráš, I.: ‘Modelling of the national economies in state-space: a fractional calculus approach’, Econ. Model., 2012, 29, pp. 13221327.
    23. 23)
      • 15. Li, Y., Chen, Y., Podlubny, I.: ‘Mittag-Leffler stability of fractional order nonlinear dynamic systems’, Automatica, 2009, 45, pp. 19651969.
    24. 24)
      • 4. Podlubny, I.: ‘Fractional differential equations. an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications, (Mathematics in Science and Engineering 1998)’ (Academic Press Inc., CA, 1999).
    25. 25)
      • 14. Tuan, H.T., Trinh, H.: ‘A linearized stability theorem for nonlinear delay fractional differential equations’, IEEE Trans. Autom. Control, 2018, 63, (9), pp. 31803186.

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