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On the Effect of Small Delays to the Stability of Feedback Systems

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Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics

Part of the book series: Progress in Nonlinear Differential Equations and Their Applications ((PNLDE,volume 55))

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Abstract

The problem considered in this paper is the following: Assume that some asymptotic properties are known for the solutions of the equation

$$ \left( {{\rm{DE}}} \right)_{\rm{0}} \left\{ \begin{array}{l} u\prime {\rm{ = }}\left( {B + C} \right)u\left( t \right), t \ge {\rm{0,}} \\ + {\rm{initial conditions}}, \\ \end{array} \right. $$

where C ∈ L(X)X is a Banach space and (B,D(B)) is the generator of a strongly continuous semigroup in X.

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References

  1. Amann, H., Operator-valued Fourier multipliers,vector-valued Besov spaces, and applications, Math. Nachr. 186 (1997), 5–56.

    Article  MathSciNet  MATH  Google Scholar 

  2. Bátkai, A., Hyperbolicity of linear partial differential equations with delay, to appear in Int. Eq. Oper. Th.

    Google Scholar 

  3. Bátkai, A., Fasanga, E., Shvidkoy, R., Hyperbolicity of delay equations via Fourier multipliers, to appear in Acta. Sci. Math. (Szeged)

    Google Scholar 

  4. Bátkai, A., Piazzera, S., Semigroups and linear partial differential equations with delay, J. Math. Anal. Appl. 264 (2001), 1–20.

    Article  MathSciNet  MATH  Google Scholar 

  5. Bátkai, A., Piazzera, S., Damped wave equations with delay, Fields Institute Communications 29 (2001), 51–61.

    Google Scholar 

  6. Clark, S., Latushkin, Y., Montgomery-Smith, S, Randolph, T., Stability radius and internal versus external stability in Banach spaces: an evolution semigroup approach, SIAM J. Control. Optim., 38 (2000), 1757–1793 (electronic).

    Article  MathSciNet  MATH  Google Scholar 

  7. Datko, R., Is boundary control a realistic approach to the stabilization of vibrating elastic systems?, in: Ferreyra, Guillermo (ed.), “Evolution Equations”, Marcel Dekker, 133–140 (1994).

    Google Scholar 

  8. Datko, R. Two questions concerning the boundary control of elastic systems, J. Diff. Eq. 92 (1991), 27–44.

    Article  MathSciNet  MATH  Google Scholar 

  9. Datko, R. Not all feedback stabilized systems are robust with respect to small time delays in their feedback, SIAM J. Control and Optimization 26 (1988), 697–713.

    Article  MathSciNet  MATH  Google Scholar 

  10. Datko, R., Lagnese, J., Polis, M. P., An example on the effect of time delays in boundary feedback of wave equations,SIAM J. Control and Optimization 24 (1986), 152–156.

    Article  MathSciNet  MATH  Google Scholar 

  11. Datko, R., You, Y.C., Some second order vibrating systems cannot tolerate small time delays in their damping, J. Optim. Th. Appl. 70 (1991), 521–537.

    Article  MathSciNet  MATH  Google Scholar 

  12. Engel, K.-J., Nagel R., “One-parameter Semigroups for Linear Evolution Equations”, Springer-Verlag, Graduate Texts in Mathematics 194, 1999.

    Google Scholar 

  13. Györi, I., Pituk, M., Stability criteria for linear delay differential equations, Diff. Int. Eq. 10 (1997), 841–852.

    MATH  Google Scholar 

  14. Györi, I., Hartung, F., Turi, J., Preservation of stability in delay equations under delay perturbations, J. Math. Anal. Appl. 220 (1998), 290–313.

    Article  MathSciNet  MATH  Google Scholar 

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

    Google Scholar 

  16. Hale, J. K., Verduyn Lunel, S. M., Effects of time delays on the dynamics of feedback systems, in: Fiedler, Gröger, Sprekels (eds.), “EQUADIFF’99, International Conference on Differential Equations, Berlin 1999”, World Scientific, 257–266 (2000).

    Google Scholar 

  17. Hieber, M., Operator valued Fourier multipliers, “Topics in Nonlinear Analysis, The Herbert Amann Anniversary Volume” (J. Escher, G. Simonett, eds.), Birkhäuser, 1999, pp. 363–380.

    Chapter  Google Scholar 

  18. Hieber, M., A characterization of the growth bound of a semigroup via Fourier multipliers, “Evolution Equations and Their Applications in Physical and Life Sciences” (G. Lumer, L. Weis, eds.), Lecture Notes in Pure and Applied Mathematics Series 215, Marcel Dekker, 2001, pp. 121–124.

    Google Scholar 

  19. Latushkin, Y., Räbiger, F., Fourier multipliers in stability and control theory, Preprint, 2000.

    Google Scholar 

  20. Latushkin, Y., Shvidkoy, R., Hyperbolicity of semigroups and Fourier multipliers, in: “Systems, approximation, singular integral operators, and related topics. IWOTA 2000”, Birkhauser-Verlag, 2001

    Google Scholar 

  21. Maniar, L., Voigt, J., Linear delay equations in the LT context, Preprint, 2000.

    Google Scholar 

  22. Nagel, R. (ed.), “One-parameter Semigroups of Positive Operators”, Springer-Verlag, Lecture Notes Math. 1184, 1986.

    Google Scholar 

  23. Nagel, R., Piazzera, S., On the regularity properties of perturbed semigroups, Rend. Circ. Mat. Palermo (2) Suppl. 56 (1998), 99–110.

    MathSciNet  Google Scholar 

  24. Rebarber, R., Townly, S., Robustness with respect to delays for exponential stability of distributed parameter systems,SIAM J. Control and Optimization 37 (1998), 230–244.

    Article  Google Scholar 

  25. Weis, L., The stability of positive semigroups in LP spaces, Proc. Amer. Math. Soc. 123 (1995), 3089–3094.

    MathSciNet  MATH  Google Scholar 

  26. Wu, J., “Theory and Applications of Partial Functional Differential Equations”, Springer-Verlag, Appl. Math. Sci. 119, 1996.

    Book  Google Scholar 

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Author information

Author notes

  1. András Bátkai

    Present address: Institut für Analysis und Technische Mathematik TU-Wien, Wiedner Hauptstraße 8-10, A-1040, Vienna, Austria

Authors and Affiliations

  1. Department of Applied Analysis, Pf. 120, H-1518, Budapest, Hungary

    András Bátkai & Bálint Farkas

Authors
  1. András Bátkai
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  2. Bálint Farkas
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Editor information

Editors and Affiliations

  1. Department of Mathematics, University of Trento, via Sommarive 14, 38050, Povo, Trento, Italy

    Mimmo Iannelli

  2. Institute of Mathematics and Informatics, University of Mons-Hainaut, 20, place du Parc, 7000, Mons, Belgium

    Günter Lumer

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Bátkai, A., Farkas, B. (2003). On the Effect of Small Delays to the Stability of Feedback Systems. In: Iannelli, M., Lumer, G. (eds) Evolution Equations: Applications to Physics, Industry, Life Sciences and Economics. Progress in Nonlinear Differential Equations and Their Applications, vol 55. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-8085-5_5

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  • DOI: https://doi.org/10.1007/978-3-0348-8085-5_5

  • Publisher Name: Birkhäuser, Basel

  • Print ISBN: 978-3-0348-9433-3

  • Online ISBN: 978-3-0348-8085-5

  • eBook Packages: Springer Book Archive

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