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Li, Yan; Chen, Yangquan; Podlubny, Igor

Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. (English) Zbl 1189.34015

Comput. Math. Appl. 59, No. 5, 1810-1821 (2010).
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.

Cited in 411 Documents

MSC:

34A08 Fractional ordinary differential equations
26A33 Fractional derivatives and integrals
34D20 Stability of solutions to ordinary differential equations
37C75 Stability theory for smooth dynamical systems

Keywords:

fractional-order dynamic system; nonautonomous system; fractional Lyapunov direct method; generalized Mittag-Leffler stability; fractional comparison principle
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\textit{Y. Li} et al., Comput. Math. Appl. 59, No. 5, 1810--1821 (2010; Zbl 1189.34015)
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References:

[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
[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), 03-0291-04, 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
[10] Podlubny, Igor, Fractional-order systems and \(\text{PI}^\lambda \text{D}^\mu\)-controllers, IEEE transactions on automatic control, 44, 1, 208-214, (1999) · Zbl 1056.93542
[11] Podlubny, Igor, Fractional differential equations, (1999), Academic Press · Zbl 0924.34008
[12] Sabatier, J.; Agrawal, O.P.; Machado, J.A.Tenreiro, Advances in fractional calculus – theoretical developments and applications in physics and engineering, (2007), Springer · 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, 3, 257-272, (2006) · Zbl 1109.26005
[14] Sastry, Shankar; Bodson, Marc, Adapative control-stability convergence and robustness, (1989), Prentice Hall · 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, 2, 109-118, (2002) · Zbl 1041.45009
[16] Miller, Kenneth S.; Samko, Stefan G., Completely monotonic functions, Integral transforms and special functions, 12, 4, 389-402, (2001) · Zbl 1035.26012
[17] Khalil, Hassan K., Nonlinear systems, (2002), Prentice Hall · Zbl 1003.34002
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