Previous |  Up |  Next


stability; Caputo derivative; Lyapunov function; fractional differential equation
The stability of the zero solution of a nonlinear nonautonomous Caputo fractional differential equation is studied using Lyapunov-like functions. The novelty of this paper is based on the new definition of the derivative of a Lyapunov-like function along the given fractional equation. Comparison results using this definition for scalar fractional differential equations are presented. Several sufficient conditions for stability, uniform stability and asymptotic uniform stability, based on the new definition of the derivative of Lyapunov functions and the new comparison result, are established.
[1] Aguila-Camacho, N., Duarte-Mermoud, M. A., Gallegos, J. A.: Lyapunov functions for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 19 (2014), 2951-2957. DOI 10.1016/j.cnsns.2014.01.022 | MR 3182869
[2] Băleanu, D., Mustafa, O. G.: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59 (2010), 1835-1841. DOI 10.1016/j.camwa.2009.08.028 | MR 2595957 | Zbl 1189.34006
[3] Burton, T. A.: Fractional differential equations and Lyapunov functionals. Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 74 (2011), 5648-5662. DOI 10.1016/ | MR 2819307 | Zbl 1226.34004
[4] Das, S.: Functional Fractional Calculus. Springer, Berlin (2011). MR 2807926 | Zbl 1225.26007
[5] Diethelm, K.: The Analysis of Fractional Differential Equations. An Application-Oriented Exposition Using Differential Operators of Caputo Type. Springer, Berlin (2010). MR 2680847 | Zbl 1215.34001
[6] Duarte-Mermoud, M. A., Aguila-Camacho, N., Gallegos, J. A., Castro-Linares, R.: Using general quadratic Lyapunov functions to prove Lyapunov uniform stability for fractional order systems. Commun. Nonlinear Sci. Numer. Simul. 22 (2015), 650-659. DOI 10.1016/j.cnsns.2014.10.008 | MR 3282452
[7] Hu, J.-B., Lu, G.-P., Zhang, S.-B., Zhao, L.-D.: Lyapunov stability theorem about fractional system without and with delay. Commun. Nonlinear Sci. Numer. Simul. 20 (2015), 905-913. DOI 10.1016/j.cnsns.2014.05.013 | MR 3255642 | Zbl 1311.34125
[8] Lakshmikantham, V., Leela, S., Sambandham, M.: Lyapunov theory for fractional differential equations. Commun. Appl. Anal. 12 (2008), 365-376. MR 2494983 | Zbl 1191.34007
[9] Lakshmikantham, V., Leela, S., Devi, J. Vasundhara: Theory of Fractional Dynamic Systems. Cambridge Scientific Publishers, Cambridge (2009).
[10] Li, Y., Chen, Y., Podlubny, I.: Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59 (2010), 1810-1821. DOI 10.1016/j.camwa.2009.08.019 | MR 2595955 | Zbl 1189.34015
[11] Li, C., Qian, D., Chen, Y.: On Riemann-Liouville and Caputo derivatives. Discrete Dyn. Nat. Soc. 2011 (2011), Article ID 562494, 15 pages. MR 2782260 | Zbl 1213.26008
[12] Li, C. P., Zhang, F. R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Special Topics 193 (2011), 27-47. DOI 10.1140/epjst/e2011-01379-1
[13] Podlubny, I.: Fractional Differential Equations. An Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering 198 Academic Press, San Diego (1999). MR 1658022 | Zbl 0924.34008
[14] Samko, S. G., Kilbas, A. A., Marichev, O. I.: Fractional Integrals and Derivatives: Theory and Applications. Transl. from the Russian. Gordon and Breach, New York (1993). MR 1347689 | Zbl 0818.26003
[15] Trigeassou, J. C., Maamri, N., Sabatier, J., Oustaloup, A.: A Lyapunov approach to the stability of fractional differential equations. Signal Process. 91 (2011), 437-445. Zbl 1203.94059
[16] Devi, J. Vasundhara, McRae, F. A., Drici, Z.: Variational Lyapunov method for fractional differential equations. Comput. Math. Appl. 64 (2012), 2982-2989. DOI 10.1016/j.camwa.2012.01.070 | MR 2989328
Partner of
EuDML logo