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Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. (English) Zbl 1189.34015
Summary: Stability of fractional-order nonlinear dynamic systems is studied using Lyapunov direct method with the introductions of Mittag-Leffler stability and generalized Mittag-Leffler stability notions. With the definitions of Mittag-Leffler stability and generalized Mittag-Leffler stability proposed, the decaying speed of the Lyapunov function can be more generally characterized which include the exponential stability and power-law stability as special cases. Finally, four worked out examples are provided to illustrate the concepts.

34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
34D20Stability of ODE
37C75Stability theory
Full Text: DOI
[1] Momani, Shaher; Hadid, Samir: Lyapunov stability solutions of fractional integrodifferential equations, International journal of mathematics and mathematical sciences 47, 2503-2507 (2004) · Zbl 1074.45006 · doi:10.1155/S0161171204312366
[2] Zhang, Long-Ge; Li, Jun-Min; Chen, Guo-Pei: Extension of Lyapunov second method by fractional calculus, Pure and applied mathematics 3, 1008-5513 (2005)
[3] Vasily E. Tarasov, Fractional stability, 2007. Available online: http://arxiv.org/abs/0711.2117v1 · Zbl 1119.26011
[4] YangQuan Chen, Ubiquitous fractional order controls? in: Proceedings of the Second IFAC Workshop on Fractional Differentiation and Its Applications, Porto, Portugal, 2006
[5] Jocelyn Sabatier, On stability of fractional order systems, in: Plenary Lecture VIII on 3rd IFAC Workshop on Fractional Differentiation and its Applications, Ankara, Turkey, 05--07 November, 2008
[6] Jocelyn Sabatier, Mathieu Merveillaut, Rachid Malti, Alain Oustaloup, On a representation of fractional order systems: Interests for the initial condition problem, Paper accepted to the next IFAC Fractional Derivative and its Applications Workshop, FDA 08, 5--7 November, Ankara, Turkey · Zbl 1221.34019
[7] Manuel D. Ortigueira, Fernando J. Coito, Initial conditions: What are we talking about? Paper accepted to the next IFAC Fractional Derivative and its Applications Workshop, FDA 08, 5--7 November, Ankara, Turkey · Zbl 1125.26011
[8] Yan Li, YangQuan Chen, Igor Podlubny, Yongcan Cao, Mittag-leffler stability of fractional order nonlinear dynamic systems, Paper accepted to the next IFAC Fractional Derivative and its Applications Workshop, FDA 08, 5--7 November, Ankara, Turkey · Zbl 1185.93062
[9] Chen, Yangquan; Moore, Kevin L.: Analytical stability bound for a class of delayed fractional order dynamic systems, Nonlinear dynamics 29, 191-200 (2002) · Zbl 1020.34064 · doi:10.1023/A:1016591006562
[10] Podlubny, Igor: Fractional-order systems and $PI{\lambda}$D${\mu}$-controllers, IEEE transactions on automatic control 44, No. 1, 208-214 (1999) · Zbl 1056.93542 · doi:10.1109/9.739144
[11] Podlubny, Igor: Fractional differential equations, (1999)
[12] Sabatier, J.; Agrawal, O. P.; Machado, J. A. Tenreiro: Advances in fractional calculus--theoretical developments and applications in physics and engineering, (2007) · Zbl 1116.00014
[13] Xu, Mingyu; Tan, Wenchang: Intermediate processes and critical phenomena: theory, method and progress of fractional operators and their applications to modern mechanics, Science in China: series G physics, mechanics and astronomy 49, No. 3, 257-272 (2006) · Zbl 1109.26005 · doi:10.1007/s11433-006-0257-2
[14] Sastry, Shankar; Bodson, Marc: Adapative control-stability convergence and robustness, (1989) · Zbl 0721.93046
[15] Appleby, John A. D.; Reynolds, David W.: On the non-exponential convergence of asymptotically stable solutions of linear scalar Volterra integro-differential equations, Journal of integral equations and applications 14, No. 2, 109-118 (2002) · Zbl 1041.45009 · doi:10.1216/jiea/1031328362
[16] Miller, Kenneth S.; Samko, Stefan G.: Completely monotonic functions, Integral transforms and special functions 12, No. 4, 389-402 (2001) · Zbl 1035.26012 · doi:10.1080/10652460108819360
[17] Khalil, Hassan K.: Nonlinear systems, (2002) · Zbl 1003.34002