Advertisement

Mathematical Notes

, Volume 88, Issue 5–6, pp 626–636 | Cite as

Robust stability of a class of positive quasi-polynomials in Banach spaces

  • Bui The AnhEmail author
  • Nguyen Khoa Son
Article
  • 51 Downloads

Abstract

In this paper, we study the stability radii of positive quasipolynomials associated with linear functional difference equations in infinite-dimensional spaces. It is shown that the positive, real and complex stability radii coincide. Moreover, explicit formulas are derived for these stability radii and illustrated by a simple example.

Keywords

stability radius parameter perturbation positive quasipolynomial Banach lattice Perron-Frobenius theorem 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    A. Fischer, “Stability radii of infinite-dimensional positive systems,” Math. Control Signals Systems 10(3), 223–236 (1997).CrossRefMathSciNetzbMATHGoogle Scholar
  2. 2.
    D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory, Vol. 1, in Modelling, State Space Analysis, Stability and Robustness, Texts Appl. Math. (Springer-Verlag, Berlin, 2005), Vol. 48.Google Scholar
  3. 3.
    B. T. Anh and N. K. Son, “Stability radii of positive higher order difference system in infinite-dimensional spaces,” Systems Control Lett. 57(10), 822–827 (2008).CrossRefMathSciNetzbMATHGoogle Scholar
  4. 4.
    B. T. Anh, N. K. Son, and D. D. X. Thanh, “Stability radii of delay difference systems under affine parameter perturbations in infinite-dimensional spaces,” Appl. Math. Comput. 202(2), 562–570 (2008).CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    B. T. Anh and N. K. Son, “Stability radii of positive linear systems under affine parameter perturbations in infinite-dimensional spaces,” Positivity 12(4), 677–690 (2008).CrossRefMathSciNetzbMATHGoogle Scholar
  6. 6.
    B. T. Anh, N. K. Son, and D. D. X. Thanh, “Robust stability of Metzler operator and delay equation in L p([−h, 0];X),” Vietnam J. Math. 34(3), 357–368 (2006).MathSciNetzbMATHGoogle Scholar
  7. 7.
    A. V. Bulatov and F. Daimond, “Real structural stability radius of infinite-dimensional linear systems: Its estimate,” Avtomat. i Telemekh., No. 5, 24–33 (2002) [Autom. Remote Control 63 (5), 713–722 (2002)].Google Scholar
  8. 8.
    N. A. Bobylev and A. V. Bulatov, “A bound on the real stability radius of continuous-time linear infinite-dimensional systems,” in Computational Mathematics and Modeling (Fizmatlit, Moscow, 2001), Vol. 1, pp. 77–86 [in Russian].Google Scholar
  9. 9.
    S. Clark, Yu. Latushkin, S. Montgomery-Smith, and T. Randolph, “Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach,” SIAM J. Control Optim. 38(6), 1757–1793 (2000).CrossRefMathSciNetzbMATHGoogle Scholar
  10. 10.
    D. Hinrichsen and A. J. Pritchard, “Robust stability of linear evolution operators on Banach spaces,” SIAM J. Control Optim. 32(6), 1503–1541 (1994).CrossRefMathSciNetzbMATHGoogle Scholar
  11. 11.
    D. Hinrichsen and A. J. Pritchard, “Stability radii of linear systems,” Systems Control Lett. 7(1), 1–10 (1986).CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    M. Adimy and K. Ezzinbi, “Local existence and linearized stability for partial functional-differential equations,” Dynam. Systems Appl. 7(3), 389–403 (1998).MathSciNetzbMATHGoogle Scholar
  13. 13.
    M. Adimy and K. Ezzinbi, “A class of linear partial neutral functional-differential equations with nondense domain,” J. Differential Equations 147(2), 285–332 (1998).CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    J. K. Hale, Functional Differential Equations, in Appl. Math. Sci. (Springer-Verlag, New York, 1971), Vol. 3.Google Scholar
  15. 15.
    J. K. Hale and S. M. Verduyn Lunel, “Strong stabilization of neutral functional differential equations,” IMA J.Math. Control Inform. 19(1–2), 5–23 (2002).CrossRefMathSciNetzbMATHGoogle Scholar
  16. 16.
    O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel and H.-O. Walther, Delay Equations. Functional-, Complex-, and Nonlinear Analysis, in Appl. Math. Sci. (Springer-Verlag, Berlin, 1995), Vol. 110.Google Scholar
  17. 17.
    W. Michiels and T. Vyhlídal, “An eigenvalue based approach for the stabilization of linear time-delay systems of neutral type,” Automatica J. IFAC 41(6), 991–998 (2005).CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    H. H. Schaefer, Banach Lattices and Positive Operators, in Grundlehren Math. Wiss. (Springer-Verlag, Berlin, 1974), Vol. 215.zbMATHGoogle Scholar
  19. 19.
    P. Meyer-Nieberg, Banach Lattices, in Universitext (Springer-Verlag, Berlin, 1991).Google Scholar
  20. 20.
    A. C. Zaanen, Introduction to Operator Theory in Riesz Spaces (Springer-Verlag, Berlin, 1977).Google Scholar

Copyright information

© Pleiades Publishing, Ltd. 2010

Authors and Affiliations

  1. 1.Pedagogical UniversityHo Chi Minh CityVietnam
  2. 2.Hanoi Institute of MathematicsHanoiVietnam

Personalised recommendations