×

On the stability of some discrete fractional nonautonomous systems. (English) Zbl 1235.93206

Summary: Using the Lyapunov direct method, the stability of discrete nonautonomous systems within the frame of the Caputo fractional difference is studied. The conditions for uniform stability, uniform asymptotic stability, and uniform global stability are discussed.

MSC:

93D15 Stabilization of systems by feedback
93D05 Lyapunov and other classical stabilities (Lagrange, Poisson, \(L^p, l^p\), etc.) in control theory
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Dractional Differential Equations, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1206.26007
[2] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542
[3] G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Applications, Gordon and Breach Science, Yverdon, Switzerland, 1993. · Zbl 0924.44003
[4] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Springer-Verlag, New York, NY, USA, 2003. · Zbl 1225.62144
[5] R. L. Magin, Fractional Calculus in Bioengineering, Begell House, Connecticut, Conn, USA, 2006.
[6] R. L. Bagley and P. J. Torvik, “On the appearance of the fractional derivative in the behavior of real materials,” Journal of Applied Mechanics, Transactions ASME, vol. 51, no. 2, pp. 294-298, 1984. · Zbl 1203.74022
[7] K. S. Miller and B. Ross, “Fractional difference calculus,” in Proceedings of the International Symposium on Univalent Functions, Fractional Calculus and Their Applications, pp. 139-152, Nihon University, Koriyama, Japan, 1989. · Zbl 0693.39002
[8] F. M. Atici and P. W. Eloe, “A transform method in discrete fractional calculus,” International Journal of Difference Equations, vol. 2, no. 2, pp. 165-176, 2007.
[9] F. M. Atici and P. W. Eloe, “Initial value problems in discrete fractional calculus,” Proceedings of the American Mathematical Society, vol. 137, no. 3, pp. 981-989, 2009. · Zbl 1166.39005
[10] F. M. Atıcı and P. W. Eloe, “Discrete fractional calculus with the nabla operator,” Electronic Journal of Qualitative Theory of Differential Equations, no. 3, pp. 1-12, 2009. · Zbl 1189.39004
[11] T. Abdeljawad and D. Baleanu, “Fractional differences and integration by parts,” Journal of Computational Analysis and Applications, vol. 13, no. 3, pp. 574-582, 2011. · Zbl 1225.39008
[12] T. Abdeljawad, “On Riemann and Caputo fractional differences,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1602-1611, 2011. · Zbl 1228.26008
[13] J. Chen, D. M. Xu, and B. Shafai, “On sufficient conditions for stability independent of delay,” Institute of Electrical and Electronics Engineers. Transactions on Automatic Control, vol. 40, no. 9, pp. 1675-1680, 1995. · Zbl 0834.93045
[14] M. P. Lazarević, “Finite time stability analysis of fractional control of robotic time-delay systems,” Mechanics Research Communications, vol. 33, no. 2, pp. 269-279, 2006. · Zbl 1192.70008
[15] W. Deng, C. Li, and J. Lü, “Stability analysis of linear fractional differential system with multiple time delays,” Nonlinear Dynamics, vol. 48, no. 4, pp. 409-416, 2007. · Zbl 1185.34115
[16] F. Merrikh-Bayat and M. Karimi-Ghartemani, “An efficient numerical algorithm for stability testing of fractional-delay systems,” ISA Transactions, vol. 48, no. 1, pp. 32-37, 2009.
[17] X. Zhang, “Some results of linear fractional order time-delay system,” Applied Mathematics and Computation, vol. 197, no. 1, pp. 407-411, 2008. · Zbl 1138.34328
[18] S. Momani and S. Hadid, “Lyapunov stability solutions of fractional integrodifferential equations,” International Journal of Mathematics and Mathematical Sciences, no. 4548, pp. 2503-2507, 2004. · Zbl 1074.45006
[19] Y. Li, Y. Q. Chen, and I. Podlubny, “Stability of fractional-order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1810-1821, 2010. · Zbl 1189.34015
[20] R. P. Agarwal, Difference Equations and Inequalities, Theory, Methods, and Application, Marcel Dekker, New York, NY, USA, 2000. · Zbl 0952.39001
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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.