Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

STABILITY OF FRACTIONAL-ORDER NONLINEAR SYSTEMS DEPENDING ON A PARAMETER

  • Ben Makhlouf, Abdellatif (Department of Mathematics University of Sfax Faculty of Sciences of Sfax) ;
  • Hammami, Mohamed Ali (Department of Mathematics University of Sfax Faculty of Sciences of Sfax) ;
  • Sioud, Khaled (Department of Mathematics Taibah University Faculty of Sciences at Yanbu)
  • Received : 2016.07.03
  • Accepted : 2016.11.29
  • Published : 2017.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we present a practical Mittag Leffler stability for fractional-order nonlinear systems depending on a parameter. A sufficient condition on practical Mittag Leffler stability is given by using a Lyapunov function. In addition, we study the problem of stability and stabilization for some classes of fractional-order systems.

Keywords

References

  1. B. Ben Hamed, Z. Haj Salem, and M. A. Hammami, Stability of nonlinear time-varying perturbed differential equations, Nonlinear Dynam. 73 (2013), no. 3, 1353-1365. https://doi.org/10.1007/s11071-013-0868-x
  2. A. Ben Makhlouf and M. A. Hammami, The convergence relation between ordinary and delay-integro-differential equations, Int. J. Dyn. Syst. Differ. Equ. 5 (2015), no. 3, 236-247. https://doi.org/10.1504/IJDSDE.2015.071004
  3. S. Burov and E. Barkai, Fractional Langevin equation: overdamped, underdamped, and critical behaviors, Phys. Rev. (3) 78 (2008), no. 3, 031112, 18 pp.
  4. L. Chen, Y. He, Y. Chai, and R. Wu, New results on stability and stabilization of a class of nonlinear fractional-order systems, Nonlinear Dynam. 75 (2014), no. 4, 633-641. https://doi.org/10.1007/s11071-013-1091-5
  5. L. Chen, Y. He, R. Wu, and L. Yin, Robust finite time stability of fractional-order linear delayed systems with nonlinear perturbations, Int. J. Control Autom. 12 (2014), no. 3, 697-702. https://doi.org/10.1007/s12555-013-0436-7
  6. L. Chen, Y. He, R. Wu, and L. Yin, New result on finite-time stability of fractional-order nonlinear delayed systems, J. Comput. Nonlin. Dyn. 10 (2015), 064504. https://doi.org/10.1115/1.4029784
  7. L. Chen, Y. He, R. Wu, and L. Yin, Finite-time stability criteria for a class of fractional-order neural networks with delay, Neural Comput. Appl. 27 (2016), 549-556. https://doi.org/10.1007/s00521-015-1876-1
  8. S. K. Choi, B. Kang, and N. Koo, Stability for Caputo fractional differential systems, Abstr. Appl. Anal. (2014), Article ID 610547, 6 pages.
  9. M. Corless and G. Leitmann, Continuous state feedback guaranteeing uniform ultimate boundedness for uncertain dynamic systems, IEEE Trans. Automat. Control 26 (1981), no. 5, 1139-1144. https://doi.org/10.1109/TAC.1981.1102785
  10. S. Das and P. Gupta, A mathematical model on fractional Lotka-Volterra equations, J. Theoret. Biol. 277 (2011), 1-6. https://doi.org/10.1016/j.jtbi.2011.01.034
  11. D. Ding, D. Qi, J. Peng, and Q.Wang, Asymptotic pseudo-state stabilization of commensurate fractional-order nonlinear systems with additive disturbance, Nonlinear Dynam. 81 (2015), no. 1-2, 667-677. https://doi.org/10.1007/s11071-015-2018-0
  12. D. Ding, D. Qi, and Q. Wang, Non-linear Mittag-Leffler stabilisation of commensurate fractional-order non-linear systems, IET Control Theory Appl. 9 (2015), no. 5, 681-690. https://doi.org/10.1049/iet-cta.2014.0642
  13. M. A. Duarte-Mermoud, N. Aguila-Camacho, J. A. Gallegos, and R. Castro-Linares, Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems, Commun. Nonlinear Sci. Numer. Simul. 22 (2015), no. 1-3, 650-659. https://doi.org/10.1016/j.cnsns.2014.10.008
  14. B. Ghanmi, Stability of impulsive systems depending on a parameter, Mathematical Methods in the Applied Sciences, DOI: 10.1002/mma.3717.(2015)
  15. M. A. Hammami, On the Stability of Nonlinear Control Systems with Uncertainty, J. Dynam. Control Systems 7 (2001), no. 2, 171-179. https://doi.org/10.1023/A:1013099004015
  16. R. Hilfer, Applications of Fractional Calculus in Physics, World Scientific, Singapore, 2000.
  17. A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Application of Fractional Differential Equations, Elsevier, New York, 2015.
  18. A. Leung, X. Li, Y. Chu, and X. Rao, Synchronization of fractional-order chaotic systems using unidirectional adaptive full-state linear error feedback coupling, Nonlinear Dynam. 82 (2015), no. 1-2, 185-199. https://doi.org/10.1007/s11071-015-2148-4
  19. Y. Li, Y. Q. Chen, and I. Podlubny, Mittag-Leffler stability of fractional order nonlinear dynamic systems, Automatica J. IFAC 45 (2009), no. 8, 1965-1969. https://doi.org/10.1016/j.automatica.2009.04.003
  20. Y. Lin, D. Sontag, and Y. Wang, A smooth converse Lyapunov theorem for Robust Stability, SIAM J. Control Optim. 34 (1996), no. 1, 124-160. https://doi.org/10.1137/S0363012993259981
  21. S. Liu, W. Jiang, X. Li, and X. Zhou, Lyapunov stability analysis of fractional nonlinear systems, Appl. Math. Lett. 51 (2016), 13-19. https://doi.org/10.1016/j.aml.2015.06.018
  22. M. S. Mahmoud, Improved robust stability and feedback stabilization criteria for time-delay systems, IMA J. Math. Control Inform. 26 (2009), no. 4, 451-466. https://doi.org/10.1093/imamci/dnp024
  23. D. Matignon, Stability result on fractional differential equations with applications to control processing, Proceedings of IMACS-SMC, 963-968, 1996.
  24. K. Miller and S. Samko, A note on the complete monotonicity of the generalized Mittag-Leffler function, Real Anal. Exchange 23 (1999), no. 2, 753-755.
  25. O. Naifar, A. Ben Makhlouf, and M. A. Hammami, Comments on Lyapunov stability theorem about fractional system without and with delay, Commun. Nonlinear Sci. Numer. Simul. 30 (2016), no. 1-3, 360-361. https://doi.org/10.1016/j.cnsns.2015.06.027
  26. I. Podlubny, Fractional Differential Equations, Academic Press, San Diego California, 1999.
  27. D. Qian, C. Li, R. P. Agarwal, and P. Wong, Stability analysis of fractional differential system with Riemann-Liouville derivative, Math. Comput. Modelling 52 (2010), no. 5-6, 862-874. https://doi.org/10.1016/j.mcm.2010.05.016
  28. Z. Qu, Exponential stability of a class of nonlinear dynamical systems with uncertainties, Systems Control Lett. 18 (1992), 301-307. https://doi.org/10.1016/0167-6911(92)90060-6
  29. Y. Ryabov and A. Puzenko, Damped oscillation in view of the fractional oscillator equation, Phys. Rev. 66 (2002), 184-201.
  30. Y. Wei, Y. Chen, S. Liang, and Y. Wang, A novel algorithm on adaptive backstepping control of fractional order systems, Math. Comput. Modelling 165 (2015), 395-402.
  31. H. Wu and K. Mizukami, Exponential stability of a class of nonlinear dynamical systems with uncertainties, Systems and Control Letters 21 (2016), 307-313.