Advertisement

Vietnam Journal of Mathematics

, Volume 46, Issue 3, pp 665–680 | Cite as

Asymptotic Stability of Linear Fractional Systems with Constant Coefficients and Small Time-Dependent Perturbations

  • Nguyen D. CongEmail author
  • Thai S. Doan
  • Hoang T. Tuan
Article

Abstract

Our aim in this paper is to investigate the asymptotic behavior of solutions of the perturbed linear fractional differential system. We show that if the original linear autonomous system is asymptotically stable and then under the action of small (either linear or nonlinear) nonautonomous perturbations, the trivial solution of the perturbed system is also asymptotically stable.

Keywords

Fractional differential equations Linear systems Stability Asymptotic stability 

Mathematics Subject Classification (2010)

34Dxx 34A30 26A33 

Notes

Acknowledgements

The authors are grateful to the referees for reading this paper and his/her comments and suggestions which helped to improve its content.

Funding Information

This research of the authors is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under Grant Number 101.03-2017.01. The authors are grateful to the referees for reading this paper and his/her comments and suggestions which helped to improve its content.

References

  1. 1.
    Adrianova, L.Ya: Introduction to Linear Systems of Differential Equations. Translations of Mathematical Monographs, vol. 46. American Mathematical Society, Providence (1995)CrossRefGoogle Scholar
  2. 2.
    Băleanu, D., Mustafa, O.G.: On the global existence of solutions to a class of fractional differential equations. Comput. Math. Appl. 59, 1835–1841 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Bonilla, B., Rivero, M., Trujillo, J.J.: On systems of linear fractional differential equations with constant coefficients. Appl. Math. Comput. 187, 68–78 (2007)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Chen, F., Nieto, J.J., Zhou, Y.: Global attractivity for nonlinear fractional differential equations. Nonlinear Anal. RWA 13, 287–298 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Coddington, E.A., Levinson, N.: Theory of Ordinary Differential Equations. McCrow-Hill, New York (1955)zbMATHGoogle Scholar
  6. 6.
    Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: On stable manifolds for planar fractional differential equations. Appl. Math. Comput. 226, 157–168 (2014)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Cong, N.D., Doan, T.S., Tuan, H.T.: On fractional Lyapunov exponent for solutions of linear fractional differential equations. Fract. Calc. Appl. Anal. 17, 285–306 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Cong, N.D., Doan, T.S., Tuan, H.T., Siegmund, S.: Structure of the fractional Lyapunov spectrum for linear fractional differential equations. Adv. Dyn. Syst. Appl. 9, 149–159 (2014)MathSciNetGoogle Scholar
  9. 9.
    Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: Linearized asymptotic stability for fractional differential equations. Electron. J. Qual. Theory Differ. Equ. 39, 1–13 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: On stable manifolds for fractional differential equations in high-dimensional spaces. Nonlinear Dyn. 86, 1885–1894 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Cong, N.D., Doan, T.S., Siegmund, S., Tuan, H.T.: Erratum to: On stable manifolds for fractional differential equations in high-dimensional spaces. Nonlinear Dyn. 86, 1895–1895 (2016)CrossRefzbMATHGoogle Scholar
  12. 12.
    Deng, W.H., Li, C.P., Lü, J.H.: Stability analysis of linear fractional differential systems with multiple time delays. Nonlinear Dyn. 48, 409–416 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Diethelm, K.: The Analysis of Fractional Differential Equations: an Application-Oriented Exposition Using Differential Operators of Caputo Type. Lecture Notes in Mathematics, vol. 2004. Springer-Verlag, Berlin Heidelberg (2010)CrossRefzbMATHGoogle Scholar
  14. 14.
    Gorenflo, R., Kilbas, A.A., Mainardi, F., Rogosin, S.V.: Mittag-Leffler Functions, Related Topics and Applications: Theory and Applications. Springer Monographs in Mathematics. Springer-Verlag, Berlin Heidelberg (2014)CrossRefzbMATHGoogle Scholar
  15. 15.
    Graef, J.R., Grace, S.R., Tunç, E.: Asymptotic behavior of solutions of nonlinear fractional differential equations with Caputo-type Hadamard derivatives. Fract. Calc. Appl. Anal. 20, 71–87 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kilbas, A.A., Srivastava, H.M., Trujillo, J.J.: Theory and Applications of Fractional Differential Equations. North-Holland Mathematics Studies, vol. 204. Elsevier Science B.V., Amsterdam (2006)Google Scholar
  17. 17.
    Lancaster, P., Tismenetsky, M.: The Theory of Matrices, 2nd edn with Applications. Academic Press, San Diego (1985)zbMATHGoogle Scholar
  18. 18.
    Losada, J., Nieto, J.J., Puorhadi, E.: On the attractivity of solutions for a class of multi-term fractional functional differential equations. J. Comput. Appl. Math. 312, 2–12 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Li, C.P., Zhang, F.R.: A survey on the stability of fractional differential equations. Eur. Phys. J. Special Top. 193, 27–47 (2011)CrossRefGoogle Scholar
  20. 20.
    Matignon, D.: Stability results for fractional differential equations with applications to control processing. In: Computational Engineering in Systems Applications, vol. 2, pp 963–968, Lille (1996)Google Scholar
  21. 21.
    Podlubny, I.: Fractional Differential Equations: an Introduction to Fractional Derivatives, Fractional Differential Equations, to Methods of Their Solution and Some of Their Applications. Mathematics in Science and Engineering, vol. 198. Academic Press, Inc, San Diego (1999)zbMATHGoogle Scholar
  22. 22.
    Qian, D., Li, C., Agarwal, R.P., Wong, P.J.Y.: Stability analysis of fractional differential systems with Riemann–Liouville derivative. Math. Comput. Model. 52, 862–874 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Sabatier, J., Moze, M., Farges, C.: LMI stability conditions for fractional order systems. Comput. Math. Appl. 59, 1594–1609 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Tisdell, C.C.: On the application of sequential and fixed-point method to fractional differential equations of arbitrary order. J. Integral Equ. Appl. 24, 283–319 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    Tuan, H.T.: On some special properties of Mittag-Leffler functions. arxiv:http://arXiv.org/abs/1708.02277 (2017)
  26. 26.
    Wen, X. -J., Wu, Z. -M., Lu, J. -G.: Stability analysis of a class of nonlinear fractional–order systems. IEEE Trans. Circ. Syst. II Express Briefs 55, 1178–1182 (2008)Google Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Nature Singapore Pte Ltd. 2018

Authors and Affiliations

  • Nguyen D. Cong
    • 1
    Email author
  • Thai S. Doan
    • 1
  • Hoang T. Tuan
    • 1
  1. 1.Institute of MathematicsVietnam Academy of Science and TechnologyHanoiVietnam

Personalised recommendations