About this Journal Submit a Manuscript Table of Contents
International Journal of Differential Equations
Volume 2011 (2011), Article ID 635165, 12 pages
http://dx.doi.org/10.1155/2011/635165
Research Article

Asymptotical Stability of Nonlinear Fractional Differential System with Caputo Derivative

1Department of Mathematics, Shanghai University, Shanghai 200444, China
2School of Mathematics and Computational Science, China University of Petroleum (East China), Qingdao 266555, China
3Electrical and Computer Engineering Department, Utah State University, Logan, UT 84322-4160, USA

Received 18 April 2011; Accepted 14 June 2011

Academic Editor: Fawang Liu

Copyright © 2011 Fengrong Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. R. C. Koeller, “Applications of fractional calculus to the theory of viscoelasticity,” Journal of Applied Mechanics, vol. 51, no. 2, pp. 299–307, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  2. J. T. Machado, V. Kiryakova, and F. Mainardi, “Recent history of fractional calculus,” Communications in Nonlinear Science and Numerical Simulation, vol. 16, no. 3, pp. 1140–1153, 2011. View at Publisher · View at Google Scholar
  3. D. Matignon, “Stability results for fractional differential equations with applications to control processing,” in Proceedings of the IMACS-SMC, vol. 2, pp. 963–968, 1996.
  4. W. H. Deng, C. P. Li, and Q. Guo, “Analysis of fractional differential equations with multi-orders,” Fractals, vol. 15, no. 2, pp. 173–182, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. W. H. Deng, C. P. Li, and J. H. Lü, “Stability analysis of linear fractional differential system with multiple time delays,” Nonlinear Dynamics, vol. 48, no. 4, pp. 409–416, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  6. M. Moze, J. Sabatier, and A. Oustaloup, “LMI characterization of fractional systems stability,” in Advances in Fractional Calculus, pp. 419–434, Springer, Dordrecht, The Netherlands, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  7. D. L. Qian and C. P. Li, “Stability analysis of the fractional differential systems with Miller-Ross sequential derivative,” in Proceedings of the 8th World Congress on Intelligent Control and Automation, pp. 213–219, Jinan, China, 2010.
  8. D. L. Qian, C. P. Li, R. P. Agarwal, and P. J. Y. Wong, “Stability analysis of fractional differential system with Riemann-Liouville derivative,” Mathematical and Computer Modelling, vol. 52, no. 5-6, pp. 862–874, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  9. A. G. Radwan, A. M. Soliman, A. S. Elwakil, and A. Sedeek, “On the stability of linear systems with fractional-order elements,” Chaos, Solitons & Fractals, vol. 40, no. 5, pp. 2317–2328, 2009.
  10. J. Sabatier, M. Moze, and C. Farges, “LMI stability conditions for fractional order systems,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1594–1609, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  11. M. S. Tavazoei and M. Haeri, “A note on the stability of fractional order systems,” Mathematics and Computers in Simulation, vol. 79, no. 5, pp. 1566–1576, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  12. V. Daftardar-Gejji and A. Babakhani, “Analysis of a system of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 293, no. 2, pp. 511–522, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  13. K. Diethelm and N. J. Ford, “Analysis of fractional differential equations,” Journal of Mathematical Analysis and Applications, vol. 265, no. 2, pp. 229–248, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  14. Y. Li, Y. Q. Chen, and I. Podlubny, “Mittag-Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  15. 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. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  16. Z. Denton and A. S. Vatsala, “Fractional integral inequalities and applications,” Computers & Mathematics with Applications, vol. 59, no. 3, pp. 1087–1094, 2010. View at Zentralblatt MATH
  17. T. C. Hu, D. L. Qian, and C. P. Li, “Comparison theorems for fractional differential equations,” Communication on Applied Mathematics and Computation, vol. 23, no. 1, pp. 97–103, 2009 (Chinese).
  18. C. P. Li and Z. G. Zhao, “Introduction to fractional integrability and differentiability,” European Physical Journal —Special Topics, vol. 193, no. 1, pp. 5–26, 2011. View at Publisher · View at Google Scholar
  19. L. Huang, Stability Theory, Peking University Press, Beijing, China, 1992.
  20. V. Lakshmikantham, S. Leela, and D. J. Vasundhara, Theory of Fractional Dynamic Systems, Cambridge Scientific Publishers, Cambridge UK, 2009.
  21. H. Pollard, “The completely monotonic character of the Mittag-Leffler function Eα(x),” Bulletin of the American Mathematical Society, vol. 54, pp. 1115–1116, 1948. View at Publisher · View at Google Scholar
  22. I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  23. K. S. Miller and S. G. Samko, “A note on the complete monotonicity of the generalized Mittag-Leffler function,” Real Analysis Exchange, vol. 23, no. 2, pp. 753–755, 1997.
  24. W. R. Schneider, “Completely monotone generalized Mittag-Leffler functions,” Expositiones Mathematicae, vol. 14, no. 1, pp. 3–16, 1996. View at Zentralblatt MATH