Razumikhin stability theorem for fractional systems with delay. (English) Zbl 1197.34157

Summary: We study the stability of fractional-order nonlinear time-delay systems for Riemann-Liouville and Caputo derivatives, and we extend the Razumikhin stability theorem to fractional nonlinear time-delay systems.


34K37 Functional-differential equations with fractional derivatives
34K20 Stability theory of functional-differential equations
Full Text: DOI EuDML


[1] A. A. Kilbas, H. M. Srivastava, and J. J. Trujillo, Theory and Applications of Fractional Differential Equations, vol. 204 of North-Holland Mathematics Studies, Elsevier Science B.V., Amsterdam, The Netherlands, 2006. · Zbl 1155.35396 · doi:10.1134/S1064562406010029
[2] I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[3] S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives: Theory and Application, Gordon and Breach Science Publishers, Yverdon, Switzerland, 1993. · Zbl 0924.44003 · doi:10.1080/10652469308819017
[4] B. J. West, M. Bologna, and P. Grigolini, Physics of Fractal Operators, Institute for Nonlinear Science, Springer, New York, NY, USA, 2003. · Zbl 1088.37045 · doi:10.1137/S1111111102406038
[5] R. L. Magin, Fractional Calculus in Bioengineering, Begell House Publisher, Connecticut, Mass, USA, 2006.
[6] P. J. Torvik and R. L. Bagley, “On the appearance of the fractional derivative in the behavior of real materials,” Journal of Applied Mechanics, Transactions, vol. 51, no. 2, pp. 294-298, 1984. · Zbl 1203.74022 · doi:10.1115/1.3167615
[7] J.-P. Richard, “Time-delay systems: an overview of some recent advances and open problems,” Automatica, vol. 39, no. 10, pp. 1667-1694, 2003. · Zbl 1145.93302 · doi:10.1016/S0005-1098(03)00167-5
[8] J. Chen, D. M. Xu, and B. Shafai, “On sufficient conditions for stability independent of delay,” IEEE Transactions on Automatic Control, vol. 40, no. 9, pp. 1675-1680, 1995. · Zbl 0834.93045 · doi:10.1109/9.412644
[9] T. N. Lee and S. Dianat, “Stability of time-delay systems,” IEEE Transactions on Automatic Control, vol. 26, no. 4, pp. 951-953, 1981. · Zbl 0544.93052 · doi:10.1109/TAC.1981.1102755
[10] E. Weber, Linear Transient Analysis. Volume II, John Wiley & Sons, New York, NY, USA, 1956. · Zbl 0073.21801
[11] V. G. Jenson and G. V. Jeffreys, Mathematical Methods in Chemical Engineering, Academic Press, New York, NY, USA, 2nd edition, 1977. · Zbl 0413.00002
[12] M. Nakagava and K. Sorimachi, “Basic characteristics of a fractance device,” IEICE Transactions, Fundamentals, vol. 75, no. 12, pp. 1814-1818, 1992.
[13] P. Lanusse, A. Oustaloup, and B. Mathieu, “Third generation CRONE control,” in Proceedings of the IEEE International Conference on Systems, Man and Cybernetics, vol. 2, pp. 149-155, 1993.
[14] I. Podlubny, “Fractional-order systems and PI\lambda D\mu -controllers,” IEEE Transactions on Automatic Control, vol. 44, no. 1, pp. 208-214, 1999. · Zbl 1056.93542 · doi:10.1109/9.739144
[15] H.-F. Raynaud and A. Zergaïnoh, “State-space representation for fractional order controllers,” Automatica, vol. 36, no. 7, pp. 1017-1021, 2000. · Zbl 0964.93024 · doi:10.1016/S0005-1098(00)00011-X
[16] M. P. Lazarević, “Finite time stability analysis of PD\alpha fractional control of robotic time-delay systems,” Mechanics Research Communications, vol. 33, no. 2, pp. 269-279, 2006. · Zbl 1192.70008 · doi:10.1016/j.mechrescom.2005.08.010
[17] 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 · doi:10.1007/s11071-006-9094-0
[18] 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. · doi:10.1016/j.isatra.2008.10.003
[19] 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 · doi:10.1016/j.amc.2007.07.069
[20] S. Momani and S. Hadid, “Lyapunov stability solutions of fractional integrodifferential equations,” International Journal of Mathematics and Mathematical Sciences, no. 45-48, pp. 2503-2507, 2004. · Zbl 1074.45006 · doi:10.1155/S0161171204312366
[21] Y. Li, Y. Chen, and I. Podlubny, “Stability of fractional-order nonlinear dynamic systems: lyapunov direct method and generalized Mittag-Leffler stability,” Computers and Mathematics with Applications, vol. 59, no. 5, pp. 1810-1821, 2010. · Zbl 1189.34015 · doi:10.1016/j.camwa.2009.08.019
[22] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional-Differential Equations, vol. 99 of Applied Mathematical Sciences, Springer, New York, NY, USA, 1993. · Zbl 0787.34002
[23] G. Kequin, V. L. Kharitonov, and J. Chen, Stability of Time-Delay Systems, Birkhäuser, Basel, Switzerland, 2003.
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