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access icon free Stability of fractional-order nonlinear systems by Lyapunov direct method

In this study, by using a characterisation of functions having a fractional derivative, the authors propose a rigorous fractional Lyapunov function candidate method to analyse the stability of fractional-order nonlinear systems. First, they prove an inequality concerning the fractional derivatives of convex Lyapunov functions without the assumption of the existence of the derivative of pseudo-states. Second, they establish fractional Lyapunov functions to fractional-order systems without the assumption of the global existence of solutions. Their theorems fill the gaps and strengthen results in some existing papers.

References

    1. 1)
      • 22. Zhou, X.F., Hu, L.G., Jiang, W.: ‘Stability criterion for a class of nonlinear fractional differential systems’, Appl. Math. Lett., 2014, 28, pp. 2529.
    2. 2)
      • 14. 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).
    3. 3)
      • 19. Aghababa, M.P.: ‘Stabilization of a class of fractional-order chaotic systems using a non-smooth control methodology’, Nonlinear Dyn., 2017, 89, (2), pp. 13571370.
    4. 4)
      • 15. Vainikko, G.: ‘Which functions are fractionally differentiable?’, J. Anal. Appl., 2016, 35, pp. 465487.
    5. 5)
      • 20. Ding, D., Qi, D., Wang, Q.: ‘Nonlinear Mittag–Leffler stabilisation of commensurate fractional order nonlinear systems’, IET Control Theory Appl., 2014, 9, (5), pp. 681690.
    6. 6)
      • 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),.
    7. 7)
      • 18. Baleanu, D., Mustafa, O.: ‘On the global existence of solutions to a class of fractional differential equations’, Comput. Math. Appl., 2010, 59, pp. 18351841.
    8. 8)
      • 21. Shen, J., Lam, J.: ‘Non-existence of finite-time stable equilibria in fractional-order nonlinear systems’, Automatica, 2014, 50, pp. 547551.
    9. 9)
      • 6. Li, Y., Chen, Y., Podlubny, I.: ‘Mittag–Leffler stability of fractional order nonlinear dynamic systems’, Automatica, 2009, 45, pp. 19651969.
    10. 10)
      • 5. 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.
    11. 11)
      • 16. Rockafellar, R.T.: ‘Convex analysis’ (Princeton University Press, Princeton, New Jersey, 1972).
    12. 12)
      • 9. Chen, W., Dai, H., Song, Y., et al: ‘Convex Lyapunov functions for stability analysis of fractional order systems’, IET Control Theory Appl., 2017, 11, (5), pp. 10701074.
    13. 13)
      • 3. Samko, S.G., Kilbas, A.A., Marichev, O.I.: ‘Fractional integrals and derivatives: theory and applications’ (Gordon and Breach Science Publishers, Switzerland, 1993).
    14. 14)
      • 13. Lakshmikantham, V., Leela, S., Devi, J.: ‘Theory of fractional dynamic systems’ (Cambridge Scientific Publishers Ltd, England, 2009).
    15. 15)
      • 11. Liu, S., Jiang, W., Li, X., et al: ‘Lyapunov stability analysis of fractional nonlinear systems’, Appl. Math. Lett., 2016, 51, pp. 1319.
    16. 16)
      • 4. 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.
    17. 17)
      • 17. Heinonen, J. (2005). ‘Lectures on Lipschitz analysis’. Technical Report, University of Jyväskylä.
    18. 18)
      • 12. Fernadez-Anaya, G., Nava-Antonio, G., Jamous-Galante, J., et al: ‘Lyapunov functions for a class of nonlinear systems using Caputo derivative’, Commun. Nonlinear Sci. Numer. Simul., 2017, 43, pp. 9199.
    19. 19)
      • 8. Duarte-Mermoud, M.A., Aguila-Camacho, N., Gallegos, J.A.: ‘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.
    20. 20)
      • 2. Oldham, K., Spanier, J.: ‘The fractional calculus’ (Academic Press, New York, 1974).
    21. 21)
      • 10. Yunquan, Y., Chunfang, M.: ‘Mittag-Leffler stability of fractional order Lorenz and Lorenz family systems’, Nonlinear Dyn., 2016, 83, (3), pp. 12371246.
    22. 22)
      • 7. Aguila-Camacho, N., Duarte-Mermoud, M.A., Gallegos, J.A.: ‘Lyapunov functions for fractional order systems’, Commun. Nonlinear Sci. Numer. Simul., 2014, 19, (9), pp. 29512957.
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