Journal of Applied Mathematics

Uniform Stability of a Class of Fractional-Order Nonautonomous Systems with Multiple Time Delays

Tao Zou, Jianfeng Qu, Yi Chai, Maoyun Guo, and Congcong Liu

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In mathematics, to a large extent, control theory addresses the stability of solutions of differential equations, which can describe the behavior of dynamic systems. In this paper, a class of fractional-order nonautonomous systems with multiple time delays modeled by differential equations is considered. A sufficient condition is established for the existence and uniqueness of solutions for such systems involving Caputo fractional derivative, and the uniform stability of solution is studied. At last, two examples are given to demonstrate the applicability of our results.

Article information

Source
J. Appl. Math., Volume 2014 (2014), Article ID 808293, 8 pages.

Dates
First available in Project Euclid: 2 March 2015

Permanent link to this document
https://projecteuclid.org/euclid.jam/1425305525

Digital Object Identifier
doi:10.1155/2014/808293

Mathematical Reviews number (MathSciNet)
MR3170449

Zentralblatt MATH identifier
07010758

Citation

Zou, Tao; Qu, Jianfeng; Chai, Yi; Guo, Maoyun; Liu, Congcong. Uniform Stability of a Class of Fractional-Order Nonautonomous Systems with Multiple Time Delays. J. Appl. Math. 2014 (2014), Article ID 808293, 8 pages. doi:10.1155/2014/808293. https://projecteuclid.org/euclid.jam/1425305525


Export citation

References

  • D. D. Demir, N. Bildik, and B. G. Sinir, “Application of fractional calculus in the dynamics of beams,” Boundary Value Problems, vol. 2012, article 135, 2012.
  • M. A. García-González, M. Fernández-Chimeno, L. Capdevila, E. Parrado, and J. Ramos-Castro, “An application of fractional differintegration to heart rate variability time series,” Computer Methods and Programs in Biomedicine, vol. 111, no. 1, pp. 33–40, 2013.
  • Z. Wang and X. Ma, “Application of fractional-order calculus approach to signal processing,” in Proceedings of the 6th IEEE Joint International Information Technology and Artificial Intelligence Conference (ITAIC '11), vol. 1, pp. 220–222, IEEE, 2011.
  • I. S. Jesus and J. A. T. Machado, “Application of fractional order concepts in the study of electrical potential,” in Dynamics, Games and Science. II, pp. 467–470, Springer, Heidelberg, Germany, 2011.
  • D. A. Benson, M. M. Meerschaert, and J. Revielle, “Fractional calculus in hydrologic modeling: a numerical perspective,” Advances in Water Resources, vol. 51, pp. 479–497, 2012.
  • I. B. Bapna and N. Mathur, “Application of fractional calculus in statistics,” International Journal of Contemporary Mathematical Sciences, vol. 7, no. 17–20, pp. 849–856, 2012.
  • H. Zhang, J. Cao, and W. Jiang, “General solution of linear fractional neutral differential difference equations,” Discrete Dynamics in Nature and Society, vol. 2013, Article ID 489521, 7 pages, 2013.
  • C. Song, T. Zhu, and J. Cao, “Existence of solutions for fractional-order neutral differential inclusions with impulsive and nonlocal conditions,” Discrete Dynamics in Nature and Society, vol. 2012, Article ID 363562, 14 pages, 2012.
  • J. A. Tenreiro Machado, “Time-delay and fractional derivatives,” Advances in Difference Equations, vol. 2011, Article ID 934094, 2011.
  • Z. Jiao and Y. Zhong, “Robust stability for fractional-order systems with structured and unstructured uncertainties,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3258–3266, 2012.
  • J. Lu, Y. Chen, and W. Chen, “Robust asymptotical stability of fractionalorder linear systems with structured perturbations,” Computers & Mathematics with Applications, vol. 66, no. 5, pp. 873–882, 2013.
  • H.-S. Ahn and Y. Chen, “Necessary and sufficient stability condition of fractional-order interval linear systems,” Automatica, vol. 44, no. 11, pp. 2985–2988, 2008.
  • Y. Li, Y. Chen, and I. Podlubny, “Mittag-Leffler stability of fractional order nonlinear dynamic systems,” Automatica, vol. 45, no. 8, pp. 1965–1969, 2009.
  • Z. Jiao and Y. Q. Chen, “Stability of fractional-order linear time-invariant systems with multiple noncommensurate orders,” Computers & Mathematics with Applications, vol. 64, no. 10, pp. 3053–3058, 2012.
  • N. Tan, Ö. Faruk Özgüven, and M. Mine Özyetkin, “Robust stability analysis of fractional order interval polynomials,” ISA Transactions, vol. 48, no. 2, pp. 166–172, 2009.
  • A. M. A. El-Sayed and Sh. A. Abd El-Salam, “On the stability of a fractional-order differential equation with nonlocal initial condition,” Electronic Journal of Qualitative Theory of Differential Equations, vol. 29, pp. 1–8, 2008.
  • M. P. Lazarević and A. M. Spasić, “Finite-time stability analysis of fractional order time-delay systems: Gronwall's approach,” Mathematical and Computer Modelling, vol. 49, no. 3-4, pp. 475–481, 2009.
  • C. Zeng, Y. Chen, and Q. Yang, “Robust controllability of interval fractional order linear time invariant stochastic systems,” in Proceedings of the IEEE 51st Annual Conference on Decision and Control (CDC '12), pp. 4047–4050, IEEE, 2012.
  • H. Shao, “New delay-dependent stability criteria for systems with interval delay,” Automatica, vol. 45, no. 3, pp. 744–749, 2009.
  • X. Meng, J. Lam, B. Du, and H. Gao, “A delay-partitioning approach to the stability analysis of discrete-time systems,” Automatica, vol. 46, no. 3, pp. 610–614, 2010.
  • C. W. Chen, “Stability analysis and robustness design of nonlinear systems: an nn-based approach,” Applied Soft Computing, vol. 11, pp. 2735–2742, 2011.
  • F. Qiu, B. Cui, and Y. Ji, “Further results on robust stability of neutral system with mixed time-varying delays and nonlinear perturbations,” Nonlinear Analysis: Real World Applications, vol. 11, no. 2, pp. 895–906, 2010.
  • K. Akbari Moornani and M. Haeri, “On robust stability of LTI fractional-order delay systems of retarded and neutral type,” Automatica, vol. 46, no. 2, pp. 362–368, 2010.
  • S. Kumar and N. Sukavanam, “Approximate controllability of fractional order semilinear systems with bounded delay,” Journal of Differential Equations, vol. 252, no. 11, pp. 6163–6174, 2012.
  • Z. Wang, X. Huang, and G. Shi, “Analysis of nonlinear dynamics and chaos in a fractional order financial system with time delay,” Computers & Mathematics with Applications, vol. 62, no. 3, pp. 1531–1539, 2011.
  • A. M. El-Sayed and F. M. Gaafar, “Stability of a nonlinear non-autonomous fractional order systems with different delays and non-local conditions,” Advances in Difference Equations, vol. 2011, article 47, 2011.
  • A. M. El-Sayed, F. M. Gaafar, and E. M. Hamadalla, “Stability for a nonlocal nonlocal non-autonomous system of fractional order differential equations with delays,” Electronic Journal of Differential Equations, vol. 2010, pp. 1–10, 2010.
  • C. Song, “Stability analysis of fractional order hopfield neural networks with delays,” American Journal of Engineering and Technology Research, vol. 11, no. 12.
  • H. Zhang, J. Cao, and W. Jiang, “Controllability criteria for linear fractional differential systems with state delay and impulses,” Journal of Applied Mathematics, vol. 2013, Article ID 146010, 9 pages, 2013.
  • J. Shen and J. Cao, “Necessary and sufficient conditions for consensus of delayed fractional-order systems,” Asian Journal of Control, vol. 14, no. 6, pp. 1690–1697, 2012.
  • S. G. Samko, A. A. Kilbas, and O. I. Marichev, Fractional Integrals and Derivatives, Gordon and Breach Science Publishers, New York, NY, USA, 1993.
  • I. Podlubny, Fractional Differential Equations, vol. 198 of Mathematics in Science and Engineering, Academic Press, San Diego, Calif, USA, 1999.
  • C. Corduneanu, Principles of Differential and Integral Equations, Chelsea Publishing, New York, NY, USA, 2nd edition, 1977. \endinput