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Asymptotic stability for \(2 \times 2\) fractional nabla difference systems

  • Jagan Mohan Jonnalagadda
Article
  • 27 Downloads

Abstract

In this article explicit criteria for asymptotic stability of linear two-dimensional fractional nabla difference systems with constant co-efficients, are presented. The main result gives a necessary and sufficient condition for asymptotic stability of the considered system in terms of trace and determinant of the matrix associated with the system. An example is provided to illustrate the applicability of established results.

Keywords

Fractional order Backward (nabla) difference Trace Determinant Eigenvalue Asymptotic stability 

Mathematics Subject Classification

34A08 39A06 39A30 

References

  1. 1.
    Kilbas AA, Srivastava HM, Trujillo JJ (2006) Theory and applications of fractional differential equations. Elsevier, AmsterdamzbMATHGoogle Scholar
  2. 2.
    Podlubny I (1999) Fractional differential equations. Academic Press, San DiegozbMATHGoogle Scholar
  3. 3.
    Matignon D (1996) Stability results for fractional differential equations with applications to control processing. In: Computational engineering in systems and application multiconference, IMACS, IEEE—SMC, Lille, vol 2, pp 963–968Google Scholar
  4. 4.
    Li CP, Zhang FR (2011) A survey on the stability of fractional differential equations. Eur Phys J Spec Top 193(1):27–47CrossRefGoogle Scholar
  5. 5.
    Petráš I (2009) Stability of fractional-order systems with rational orders: a survey. Fract Calc Appl Anal 12(3):269–298MathSciNetzbMATHGoogle Scholar
  6. 6.
    Zhou Y (2014) Basic theory of fractional differential equations. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  7. 7.
    Abu-Saris R, Al-Mdallal Q (2013) On the asymptotic stability of linear system of fractional-order difference equations. Fract Calc Appl Anal 16(3):613–629MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Čermák J, Győri I, Nechvátal L (2015) On explicit stability conditions for a linear fractional difference system. Fract Calc Appl Anal 18(3):651–672MathSciNetzbMATHGoogle Scholar
  9. 9.
    Mozyrska D, Wyrwas M (2017) Explicit criteria for stability of fractional \(h\)-difference two-dimensional systems. Int J Dynam Control 5:4–9MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Čermák J, Győri I, Nechvátal L (2013) Stability regions for linear fractional difference systems and their discretizations. Appl Math Comput 219:7012–7022MathSciNetzbMATHGoogle Scholar
  11. 11.
    Abdeljawad T (2011) On Riemann and Caputo fractional differences. Comput Math Appl 62:1602–1611MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Abdeljawad T, Atici FM (2012) On the definitions of nabla fractional operators. Abstr Appl Anal, Article ID 406757Google Scholar
  13. 13.
    Anastassiou GA (2010) nabla discrete fractional calculus and inequalities. Math Comput Model 51:562–571MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Atici FM, Eloe PW (2009) Discrete fractional calculus with the nabla operator. Electron J Qual Theory Differ Equat, Special Edition I, No 13Google Scholar
  15. 15.
    Bohner M, Peterson A (2001) Dynamic equations on time scales: an introduction with applications. Birkhäuser, BostonCrossRefzbMATHGoogle Scholar
  16. 16.
    Elaydi S (2005) An introduction to difference equations, 3rd edn. Springer, New YorkzbMATHGoogle Scholar
  17. 17.
    Goodrich C, Peterson AC (2015) Discrete fractional calculus. Springer, BaselCrossRefzbMATHGoogle Scholar
  18. 18.
    Kelley WG, Peterson AC (1991) Difference equations: an introduction with applications, 2nd edn. Academic Press, CambridgezbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsBirla Institute of Technology and Science PilaniHyderabadIndia

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