Skip to main content
SpringerLink
Log in

Spectral theory for delay equations

  • Conference paper
Systems, Approximation, Singular Integral Operators, and Related Topics

Part of the book series: Operator Theory: Advances and Applications ((OT,volume 129))

Abstract

For dynamical systems governed by feedback laws, time delays arise naturally in the feedback loop to represent effects due to communication, transmission, transportation or inertia effects. The introduction of time delays in a system of differential equations results in an infinite dimensional state space. The solution operator associated with a differential delay equation is a nonself-adjoint operator defined on a Banach space. This implies that general abstract theorems cannot directly be applied. In this paper we discuss the spectral properties of autonomous and periodic differential delay equations, series expansions of solutions, completeness of eigenvectors and generalized eigenvectors, and solutions of delay equations that decay faster than any exponential.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. H. Amann, Gewöhnliche Differentialgleichungen, de Gruyter, Berlin, 1983.

    Google Scholar 

  2. H. T. Banks and A. Manitius, Projection series for retarded functional differential equations with applications to optimal control problems, J. Diff. Eqn., 18(1975), 296–332.

    Article  MathSciNet  MATH  Google Scholar 

  3. R. Gellman and K. L. Cooke, Differential-Difference Equations, Academic Press, New York, 1963.

    Google Scholar 

  4. R. Boas, Entire Functions, Academic Press, New York, 1954.

    MATH  Google Scholar 

  5. M. C. Delfour and A. Manitius, The structural operator F and its role in the theory of retarded systems I, J. Math. Anal. Appl., 73(1980), 466–490.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. C. Delfour and A. Manitius, The structural operator F and its role in the theory of retarded systems II, J. Math. Anal. Appl., 74(1980), 359–381.

    Article  MathSciNet  MATH  Google Scholar 

  7. O. Diekmann, S. A. van Gils, S. M. Verduyn Lunel, and H. O. Walther, Delay Equations: Functional-, Complex-, and Nonlinear Analysis, Springer-Verlag, New York, Applied Mathematical Sciences Vol. 110, 1995.

    MATH  Google Scholar 

  8. L. Glass and M. C. Mackey, Pathological conditions resulting from instabilities in physiological control systems, Ann. N.Y. Acad. Sci., 316(1979), 214–235.

    Article  Google Scholar 

  9. I. Gohberg, S. Goldberg, and M.A. Kaashoek, Classes of Linear Operators I, Birkhäuser Verlag, Basel, 1990.

    MATH  Google Scholar 

  10. I. Gohberg, M. A. Kaashoek, and S.M. Verduyn Lunel, New completeness theorems for compact operators and applications, In preparation.

    Google Scholar 

  11. I. C. Gohberg and E. I. Sigal, An operator generalization of the logarithmic residue theorem and the theorem of Rouché, Mat. Sb., 84(1971), 609–629. (Russian) (Math. USSR-Sb., 13(1971), 603–625.)

    Google Scholar 

  12. W. Hahn, On difference differential equations with periodic coefficients, J. Math. Anal. Appl., 3(1961), 70–101.

    Article  MathSciNet  MATH  Google Scholar 

  13. J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, Applied Mathematical Sciences Vol. 99, 1993.

    MATH  Google Scholar 

  14. J.K. Hale and S. M. Verduyn Lunel, Effects of small delays on stability and control, Operator Theory and Analysis, The M. A. Kaashoek Anniversary Volume (eds. H. Bart, I. Gohberg and A. C. M. Ran), Operator Theory: Advances and Applications, Vol. 122, pp. 275–301, Birkhäuser, 2001.

    Google Scholar 

  15. D. Henry, Small solutions of linear autonomous functional differential equations, J. Differential Eqns., 8(1970), 494–501.

    Article  MathSciNet  MATH  Google Scholar 

  16. E. Hille and R. Phillips, Functional Analysis and Semigroups, American Mathematical Society, Providence, RI, 1957.

    MATH  Google Scholar 

  17. M.A. Kaashoek and S.M. Verduyn Lunel, Characteristic matrices and spectral properties of evolutionary systems, Trans. Amer. Math. Soc., 334(1992), 479–517.

    Article  MathSciNet  MATH  Google Scholar 

  18. T. Kato, Perturbation Theory for Linear Operators (2nd edn.), Springer-Verlag, Berlin, 1976.

    Book  Google Scholar 

  19. B. Krauskopf and D. Lenstra, Fundamental Issues of Nonlinear Laser Dynamics, AIP Conference Proceedings, Volume 548, American Institute of Physics, Melville, New York, 2000.

    Chapter  Google Scholar 

  20. B. Ja. Levin, Distribution of Zeros of Entire Function, Amer. Math. Soc., Providence, 1972.

    Google Scholar 

  21. N. Levinson and C. McCalla, Completeness and independence of the exponential solutions of some functional differential equations, Studies in Appl. Math., 53(1974), 1–15.

    MathSciNet  Google Scholar 

  22. Yu. I. Lyubich and V. A. Tkachenko, Floquet’s theory for equations with retarded argument, Diff. Uravneniya, 5(1969), 648–656.

    MATH  Google Scholar 

  23. J. Mallet-Paret and S. M. Verduyn Lunel, Asymptotics of the Floquet multipliers for periodic delay equations with rational period, In preparation.

    Google Scholar 

  24. A. Manitius, Completeness and F-completeness of eigenfunctions associated with retarded functional differential equations, J. Differential Eqns., 35(1980), 1–29.

    Article  MathSciNet  MATH  Google Scholar 

  25. V. Matsaev and S. M. Verduyn Lunel, Spectral theory for periodic differential delay equations, In preparation.

    Google Scholar 

  26. D Salamon, Control and Observation of Neutral Systems, Research Notes on Mathematics 91, Pitman, London, 1984.

    Google Scholar 

  27. S. M. Verduyn Lunel, A sharp version of Henry’s theorem on small solutions, J. Differential Eqns., 62(1986), 266–274.

    Article  MATH  Google Scholar 

  28. S. M. Verduyn Lunel, Series expansions and small solutions for Volterra equations of convolution type, J. Differential Eqns., 85(1990), 17–53.

    Article  MATH  Google Scholar 

  29. S. M. Verduyn Lunel, The closure of the generalized eigenspace of a class of infinitesimal generators, Proc. Roy. Soc. Edinburgh Sect. A, 117(1991), 171–192.

    Article  MathSciNet  MATH  Google Scholar 

  30. S. M. Verduyn Lunel, Series expansions for functional differential equations, Integral Equations and Operator Theory, 22(1995), 93–123.

    Article  MathSciNet  MATH  Google Scholar 

  31. S. M. Verduyn Lunel, About completeness for a class of unbounded operators, J. Differential Eqns., 120(1995), 108–132.

    Article  MATH  Google Scholar 

  32. S. M. Verduyn Lunel, Inverse problems for nonself-adjoint evolutionary systems, Topics in Functional Differential and Difference Equations (eds. T. Faria and P. Freitas), Fields Institute Communications, Vol. 29, pp. 321–347, American Mathematical Society, Providence, 2001.

    Google Scholar 

  33. S. M. Verduyn Lunel and D. V. Yakubovich, A functional model approach to linear neutral functional differential equations, Integral Equations and Operator Theory, 27(1997), 347–378.

    Article  MathSciNet  MATH  Google Scholar 

  34. H.-O. Walther, Bifurcation from periodic solutions in functional differential equations, Math. Z., 182(1983), 269–289.

    Article  MathSciNet  MATH  Google Scholar 

  35. A. M. Zverkin, The completeness of a system of Floquet type solutions for equations with retardations, Differential Equations, 4(1968), 249–251.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Mathematisch Instituut, Universiteit Leiden, P.O. Box 9512, 2300 RA, Leiden, The Netherlands

    Sjoerd M. Verduyn Lunel

Authors
  1. Sjoerd M. Verduyn Lunel
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Département de Mathématiques, Université de Bordeaux I, 351, cours de la Libération, 33405, Talence, France

    Alexander A. Borichev  & Nikolai K. Nikolski  & 

Rights and permissions

Reprints and permissions

Copyright information

© 2001 Springer Basel AG

About this paper

Cite this paper

Lunel, S.M.V. (2001). Spectral theory for delay equations. In: Borichev, A.A., Nikolski, N.K. (eds) Systems, Approximation, Singular Integral Operators, and Related Topics. Operator Theory: Advances and Applications, vol 129. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8362-7_19

Download citation

  • DOI: https://doi.org/10.1007/978-3-0348-8362-7_19

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9534-7

  • Online ISBN: 978-3-0348-8362-7

  • eBook Packages: Springer Book Archive

Publish with us

Policies and ethics

Search

Navigation