zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Mittag-Leffler stability of fractional order nonlinear dynamic systems. (English) Zbl 1185.93062
Summary: We propose the definition of Mittag-Leffler stability and introduce the fractional Lyapunov direct method. Fractional comparison principle is introduced and the application of Riemann-Liouville fractional order systems is extended by using Caputo fractional order systems. Two illustrative examples are provided to illustrate the proposed stability notion.

MSC:
93C30Control systems governed by other functional relations
34A08Fractional differential equations
93C10Nonlinear control systems
93C30Control systems governed by other functional relations
93D05Lyapunov and other classical stabilities of control systems
WorldCat.org
Full Text: DOI
References:
[1] Bagley, R. L.; Torvik, P. J.: Fractional calculus -- A different approach to the analysis of viscoelastically damped structures, AIAA journal 21, No. 5, 741-748 (1983) · Zbl 0514.73048 · doi:10.2514/3.8142
[2] Bagley, R. L.; Torvik, P. J.: A theoretical basis for the application of fractional calculus to viscoelasticity, Journal of theology 27, No. 3, 201-210 (1983) · Zbl 0515.76012 · doi:10.1122/1.549724
[3] Chen, Y. Q. (2006). Ubiquitous fractional order controls? In Proceedings of the second IFAC workshop on fractional differentiation and its applications
[4] Chen, Y. Q.; Moore, K. 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
[5] Chua, L. O.: Memristor -- the missing circuit element, IEEE transactions on circuit theory 18, No. 5, 507-519 (1971)
[6] Cohen, I.; Golding, I.; Ron, I. G.; Ben-Jacob, E.: Bio-uiddynamics of lubricating bacteria, Mathematical methods in the applied sciences 24, 1429-1468 (2001) · Zbl 1097.76618 · doi:10.1002/mma.190
[7] Khalil, H. K.: Nonlinear systems, (2002) · Zbl 1003.34002
[8] Li, Y., Chen, Y. Q., Podlubny, I., & Cao, Y. (2008). Mittag-leffler stability of fractional order nonlinear dynamic systems. In Proceedings of the 3rd IFAC workshop on fractional differentiation and its applications · Zbl 1185.93062
[9] Li, C.; Deng, W.: Remarks on fractional derivatives, Applied mathematics and computation 187, No. 2, 777-784 (2007) · Zbl 1125.26009 · doi:10.1016/j.amc.2006.08.163
[10] Miller, K. S.; Samko, S. G.: Completely monotonic functions, Integral transforms and and special functions 12, No. 4, 389-402 (2001) · Zbl 1035.26012 · doi:10.1080/10652460108819360
[11] Momani, S.; Hadid, S.: 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
[12] Podlubny, I.: 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
[13] Podlubny, I.: Fractional differential equations, (1999) · Zbl 0924.34008
[14] Podlubny, I.: Geometric and physical interpretation of fractional integration and fractional differentiation, Fractional calculus and applied analysis 5, No. 4, 367-386 (2002) · Zbl 1042.26003
[15] Podlubny, I.; Heymans, N.: Physical interpretation of initial conditions for fractional differential equations with Riemann-Liouville fractional derivatives, Rheologica acta 45, No. 5, 765-772 (2006)
[16] Sabatier, J. (2008). On stability of fractional order systems. In Plenary lecture VIII on 3rd IFAC workshop on fractional differentiation and its applications
[17] 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
[18] Tarasov, V. E.: Fractional derivative as fractional power of derivative, International journal of mathematics 18, No. 3, 281-299 (2007) · Zbl 1119.26011 · doi:10.1142/S0129167X07004102
[19] Xu, M.; Tan, W.: 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
[20] Zhang, L. -G.; Li, J. -M.; Chen, G. -P.: Extension of Lyapunov second method by fractional calculus, Pure and applied mathematics 21, No. 3 (2005) · Zbl 1118.33001