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 PDF BibTeX XML Cite \textit{Y. Li} et al., Comput. Math. Appl. 59, No. 5, 1810--1821 (2010; Zbl 1189.34015) Full Text: DOI OpenURL 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 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.