Lyapunov Fonksiyonları ile Fraksiyonel Diferansiyel Denklemlerin Kararlılığı

Öz

Bu çalışmada, fraksiyonel ve tamsayı mertebe içeren lineer olmayan otonom diferansiyel denklem sistemlerinin asimptotik kararlılığı araştırıldı. Lineer olmayan otonom fraksiyonel sistemlerin asimptotik kararlılığını göstermek için bazı yeterli şartlar elde edilerek Lyapunov’un ikinci metodu kullanıldı. Ayrıca elde edilen sonuçları pekiştirmek için iki örnek verildi.

Anahtar Kelimeler

• Asimptotik Kararlılık,Fraksiyonel Diferansiyel Denklemler,Lyapunov Fonksiyonları

On Stability of Fractional Differential Equations with Lyapunov Functions

Öz

We discuss the asymptotic stability of autonomous nonlinear fractional order systems, in which the state equations contain integer derivative and fractional order. We use the Lyapunov's second method to derive some sufficient conditions to ensure asymptotic stability of nonlinear fractional order differential equations. We also give two examples in order to consolidate the obtained results.

Anahtar Kelimeler

• Asymptotic Stability,Fractional Differential Equations,Lyapunov Functional

Kaynakça

1. [1] Kilbas A. A., Srivastava H. M., Trujillo J. J. Theory and Applications of Fractional Differential Equations, Elsevier Science B. V., Amsterdam, 2006.
2. [2] Podlubny I. Fractional Differential Equations, Academic Press, SanDiego, 1999.
3. [3] Miller K.S., Ross B. An Introduction to the Fractional Calculus and Differential Equations, John Wiley, New York, 1993.
4. [4] Lakshmikantham V., Vatsala A.S. Theory of fractional differential inequalities and applications. Commun. Appl. Anal., 11, 395-402, 2007.
5. [5] Lakshmikantham V., Vatsala A.S. Basic theory of fractional differential equations. Nonlinear Anal., 69, 2677-2682, 2008.
6. [6] Lakshmikantham V. Theorem of fractional functional differential equations. Nonlinear Anal., 69, 3337-3343, 2008.
7. [7] Lakshmikantham V., Vatsala A. S. Basic theory of fractional differential equations, Nonlinear Anal., 69, 2677-2682, 2008.
8. [9] Diethelm K. The Analysis of Fractional Differential Equations. Springer, Heidelberg, Dordrecht, London, New York, 2010.

9. [8] Lyapunov A.M. The general problem of the stability of motion. Translated from Edouard Davaux's French translation (1907) of the 1892 Russian original and edited by A. T. Fuller. Taylor Francis, Ltd., London, 1992.
10. [10] Trigeassou J.C., Maamri N., Sabatier J., Oustaloup A. A Lyapunov approach to the stability of fractional differential equations. Signal Process, 91(3), 437-445, 2011.
11. [11] Matignon D. Stability results for fractional differential equations with applications to control processing. Comput. Eng. Syst. Appl., 2, 963-968, 1996.
12. [12] Deng W., Li C., Lu J. Stability analysis of linear fractional differential system with multiple time delays. Nonlinear Dynam., 48, 409-416, 2007.
13. [13] Liao Z., Peng C., Li W., Wang Y. Robust stability analysis for a class of fractional order systems with uncertain parameters. J. Franklin Inst., 348, 1101-1113, 2011.
14. [14] Wang J., Lv L., Zhou Y. New concepts and results in stability of fractional differential equations. Commun. Nonlinear Sci. Numer. Simul., 17, 2530-2538, 2012.
15. [15] Choi S.K., Koo N. The monotonic preperty and stability of solution of fractional differential equations. Nonlinear Anal., 74, 6530-6536, 2011.
16. [16] Li Y., Chen Y., Pudlubny I. Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl., 59, 1810-1821, 2010.
17. [17] Matignon D. Stability results for fractional differential equations with applications to control processing. in:IMACS-SMCProceedings, Lille, p. 963-968, France, 1996.
18. [18] Aguila-Camacho N., Duarte-Mermoud M. A., Gallegos J. A. Lyapunov functions for fractional order systems. Commun Nonlinear Sci Numer Simulat, 19, 2951-2957, 2014.
19. [19] Zhou X.F., Hu L.G., Liu S., Jiang W. Stability criterion for a class of nonlinear fractional differential systems. Appl. Math. Lett., 28, 25-29, 2014.
20. [20] Li Y., Chen Y.Q., Podlubny I. Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica, 45(8), 1965-1969, 2009.
21. [21] Li C.P., Zhang F.R. A survey on the stability of fractional differential equations. Eur. Phys. J. Spec. Top., 193, 27-47, 2011.