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Mittag-Leffler stability theorem for fractional nonlinear systems with delay. (English) Zbl 1195.34013
Summary: Fractional calculus started to play an important role for analysis of the evolution of the nonlinear dynamical systems which are important in various branches of science and engineering. In this line of taught, we study the stability of fractional order nonlinear time-delay systems for Caputo’s derivative, and we prove two theorems for Mittag-Leffler stability of the fractional nonlinear time delay systems.

MSC:
34A08Fractional differential equations
26A33Fractional derivatives and integrals (real functions)
WorldCat.org
Full Text: DOI EuDML
References:
[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, 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 Applications, Gordon and Breach Science, 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, Redding, Conn, USA, 2006.
[6] J. A. T. Machado, A. M. Galhano, A. M. Oliveira, and J. K. Tar, “Optimal approximation of fractional derivatives through discrete-time fractions using genetic algorithms,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 3, pp. 482-490, 2010. · doi:10.1016/j.cnsns.2009.04.030
[7] 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
[8] 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
[9] P. J. Torvik and R. L. Bagley, “On the appearance of the fractional derivative in the behavior of real materials,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 294-298, 1984. · Zbl 1203.74022 · doi:10.1115/1.3167615
[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. E75-A, 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] B. S. Razumikhin, “On stability of systems with a delay,” Prikladnoi Matematiki i Mekhaniki, vol. 20, pp. 500-512, 1956.
[17] N. N. Krasovskiĭ, “On the application of the second method of Lyapunov for equations with time retardations,” Prikladnaja Matematika i Mehanika, vol. 20, pp. 315-327, 1956.
[18] 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
[19] 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
[20] 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
[21] S. Momani and S. Hadid, “Lyapunov stability solutions of fractional integrodifferential equations,” International Journal of Mathematics and Mathematical Sciences, vol. 47, no. 45-48, pp. 2503-2507, 2004. · Zbl 1074.45006 · doi:10.1155/S0161171204312366 · eudml:51757
[22] J. Sabatier, “On stability of fractional order systems,” in Proceedings of the Plenary Lecture VIII on 3rd IFAC Workshop on Fractional Differentiation and Its Applications, Ankara, Turkey, 2008.
[23] Y. Li, Y. 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 · doi:10.1016/j.camwa.2009.08.019
[24] J. R. Sadati, D. Baleanu, R. Ghaderi, N. A. Ranjbar, T. Abdeljawad (Maraaba), and F. Jarad, “Razumikhin stability theorem for fractional systems with delay,” Abstract and Applied Analysis, vol. 2010, Article ID 124812, 9 pages, 2010. · Zbl 1197.34157 · doi:10.1155/2010/124812 · eudml:232582
[25] V. L. Kharitonov and D. Hinrichsen, “Exponential estimates for time delay systems,” Systems & Control Letters, vol. 53, no. 5, pp. 395-405, 2004. · Zbl 1157.34355 · doi:10.1016/j.sysconle.2004.05.016