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Differential Equations with State-Dependent Delays

In: Online Optimization of Large Scale Systems

Author

Listed:
  • Eberhard P. Hofer

    (Universität Ulm, Abteilung Meß-, Regel- und Mikrotechnik)

  • Bernd Tibken

    (Universität—Gesamthochschule Wuppertal, Fachgruppe für Automatisierungstechnik und Technische Kybernetik (atk) Bergische)

  • Frank Lehn

    (Universität Ulm, Abteilung Meß-, Regel- und Mikrotechnik)

Abstract

In this article an introduction to the wide field of retarded or delay differential equations with state-dependent delays is given. Hereby, the most important features of this type of differential equations which are profoundly different from ordinary differential equations are discussed. The presence of discontinuities in higher derivatives of the solution of delay differential equations require suitable integration methods. Thus, an efficient numerical code for the integration of delay differential equations is presented. This code is based on a fourth order RK one step algorithm. Furthermore, the presence of discontinuities may lead to severe problems when parameters are to be estimated using higher order optimization techniques. These problems are pointed out and a systematic way for the analysis of the smoothness of a least squares objective function is presented.

Suggested Citation

  • Eberhard P. Hofer & Bernd Tibken & Frank Lehn, 2001. "Differential Equations with State-Dependent Delays," Springer Books, in: Martin Grötschel & Sven O. Krumke & Jörg Rambau (ed.), Online Optimization of Large Scale Systems, pages 413-432, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-04331-8_22
    DOI: 10.1007/978-3-662-04331-8_22
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