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Asymptotic stability of nonlinear fractional neutral singular systems

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Abstract

In this paper, fractional calculus has been introduced into neutral singular systems. The (asymptotical) stability and (generalized) Mittag-Leffler stability of nonlinear fractional neutral singular systems with Caputo and Riemann-Liouville derivatives are studied, respectively. Several sufficient conditions guaranteeing stability of such systems are established by using the Lyapunov direct method.

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References

  1. Podlubny, I.: Fractional Differential Equations. Academic Press, New York (1999)

    MATH  Google Scholar 

  2. Miller, K.S., Ross, B.: An Introduction to the Fractional Calculus and Fractional Differential Equation. Wiley, New York (1993)

    Google Scholar 

  3. Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Application of Fractional Differential Equations. Elsevier, Amsterdam (2006)

    Google Scholar 

  4. Li, Y., Chen, Y., Podlubny, I.: Mittag-Leffler stability of fractional order nonlinear dynamic systems. Automatica 45, 1965–1969 (2009)

    Article  MATH  MathSciNet  Google Scholar 

  5. Li, Y., Chen, Y., Podlubny, I.: Stability of fractional order nonlinear dynamic systems: Lyapunov direct method and generalized Mittag-Leffler stability. Comput. Math. Appl. 59, 1810–1821 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  6. Zhang, F., Li, C., Chen, Y.: Asymptotical stability of nonlinear fractional differential system with Caputo derivative. Int. J. Differ. Equ. 1–12 (2011)

  7. Baleanu, D., Sadati, S.J., Ghaderi, R., Ranjbar, A., Abdeljawad Maraaba, T., Jarad, F.: Razumikhin stability theorem for fractional systems with delay. Abstr. Appl. Anal. 23–31 (2010)

  8. Sadati, S.J., Baleanu, D., Ranjbar, A., Ghaderi, R., Abdeljawad Maraaba, T.: Mittag-Leffler stability theorem for fractional nonlinear systems with delay. Abstr. Appl. Anal. 25–31 (2010)

  9. Dai, L.: Singular Control Systems. Springer, New York (1989)

    Book  MATH  Google Scholar 

  10. Kunkel, P., Mehrmann, V.: Differential-Algebraic Equations: Analysis and Numerical Solution. EMS Publishing House, Zürich (2006)

    Book  Google Scholar 

  11. Campbell, S.L.: Singular linear systems of differential equations with delays. Appl. Anal. 11, 129–136 (1980)

    Article  MATH  Google Scholar 

  12. Jiang, W.: Eigenvalue and stability of singular differential delay systems. J. Math. Anal. Appl. 297, 305–316 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  13. Jiang, W., Song, W.: Controllability of singular systems with control delay. Automatica 37, 1873–1877 (2001)

    Article  MATH  Google Scholar 

  14. Lu, G., Ho, D.W.C.: Generalized quadratic stability for continuous-time singular systems with nonlinear perturbation. IEEE Trans. Autom. Control 51, 818–823 (2006)

    Article  MathSciNet  Google Scholar 

  15. Li, H., Li, H., Zhong, S.: Stability of neutral type descriptor system with mixed delays. Chaos Solitons Fractals 33, 1796–1800 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  16. Liu, S., Jiang, W.: Asymptotic stability of nonlinear descriptor systems with infinite delays. Ann. Differ. Equ. 26, 174–180 (2010)

    MATH  MathSciNet  Google Scholar 

  17. Sun, Y.J.: Stability criterion for a class of descriptor systems with discrete distributed time delays. Chaos Solitons Fractals 33, 986–993 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  18. Fridman, E.: Stability of linear descriptor systems with delay: a Lyapunov-based approach. J. Math. Anal. Appl. 273, 24–44 (2002)

    Article  MATH  MathSciNet  Google Scholar 

  19. Liu, S., Jiang, W.: Mittag-Leffler stability of nonlinear fractional neutral singular systems. Commun. Nonlinear Sci. Numer. Simul. 17, 3961–3966 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  20. Li, C., Deng, W.: Remarks on fractional derivatives. Appl. Math. Comput. 187, 777–784 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  21. Huang, L.: Stability Theory. Peking University Press, Beijing (1992)

    Google Scholar 

  22. Hale, J.K.: Theory of Functional Differential Equations. Springer, New York (1977)

    Book  MATH  Google Scholar 

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Acknowledgements

This work is supported by the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20093401120001), the Natural Science Foundation of Anhui Province(No. 11040606M12), the Natural Science Foundation of Anhui Education Bureau (No. KJ2010A035) and the 211 project of Anhui University (No. KJJQ1102).

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Authors and Affiliations

  1. School of Mathematics, Anhui University, Hefei, 230039, China

    Yanfen Lu, Ranchao Wu & Zhiquan Qin

Authors
  1. Yanfen Lu
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  2. Ranchao Wu
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  3. Zhiquan Qin
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Correspondence to Ranchao Wu.

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Lu, Y., Wu, R. & Qin, Z. Asymptotic stability of nonlinear fractional neutral singular systems. J. Appl. Math. Comput. 45, 351–364 (2014). https://doi.org/10.1007/s12190-013-0726-5

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  • DOI: https://doi.org/10.1007/s12190-013-0726-5

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