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Numerical solution of optimal control problems with constant control delays

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Author Info

  • Ulrich Brandt-Pollmann

    (Interdisciplinary Center for Scientific Computing, University of Heidelberg, Germany)

  • Ralph Winkler

    ()
    (Center of Economic Research (CER-ETH) at ETH Zürich)

  • Sebastian Sager

    (Interdisciplinary Center for Scientific Computing, University of Heidelberg, Germany)

  • Ulf Moslener

    (Centre for European Economic Research (ZEW), Mannheim, Germany)

  • Johannes P. Schlöder

    (Interdisciplinary Center for Scientific Computing, University of Heidelberg, Germany)

Abstract

We investigate a class of optimal control problems that exhibit constant exogenously given delays in the control in the equation of motion of the differential states. Therefore, we formulate an exemplary optimal control problem with one stock and one control variable and review some analytic properties of an optimal solution. However, analytical considerations are quite limited in case of delayed optimal control problems. In order to overcome these limits, we reformulate the problem and apply direct numerical methods to calculate approximate solutions that give a better understanding of this class of optimization problems. In particular, we present two possibilities to reformulate the delayed optimal control problem into an instantaneous optimal control problem and show how these can be solved numerically with a state-of-the-art direct method by applying Bock’s direct multiple shooting algorithm. We further demonstrate the strength of our approach by two economic examples.

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Bibliographic Info

Paper provided by CER-ETH - Center of Economic Research (CER-ETH) at ETH Zurich in its series CER-ETH Economics working paper series with number 06/59.

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Length: 26 pages
Date of creation: Oct 2006
Date of revision:
Handle: RePEc:eth:wpswif:06-59

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Keywords: delayed differential equations; delayed optimal control; numerical optimization; time-to-build;

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References

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  1. Boucekkine, Raouf & Licandro, Omar & Puch, Luis A. & del Rio, Fernando, 2005. "Vintage capital and the dynamics of the AK model," Journal of Economic Theory, Elsevier, Elsevier, vol. 120(1), pages 39-72, January.
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  5. Boyer, Marcel, 1978. "A Habit Forming Optimal Growth Model," International Economic Review, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, Department of Economics, University of Pennsylvania and Osaka University Institute of Social and Economic Research Association, vol. 19(3), pages 585-609, October.
  6. Raouf, BOUCEKKINE & David de la Croix & Omar Lidandro, 2004. "Modelling vintage structures with DDEs : principles and applications," Discussion Papers (IRES - Institut de Recherches Economiques et Sociales), Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES) 2004004, Université catholique de Louvain, Institut de Recherches Economiques et Sociales (IRES).
  7. El-Hodiri, Mohamed A & Loehman, Edna & Whinston, Andrew B, 1972. "An Optimal Growth Model with Time Lags," Econometrica, Econometric Society, Econometric Society, vol. 40(6), pages 1137-46, November.
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  11. Luis A. Puch & Fabrice Collard & Omar Licandro, 2004. "The short-run dynamics of optimal growth models with delays," Computing in Economics and Finance 2004, Society for Computational Economics 117, Society for Computational Economics.
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Citations

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Cited by:
  1. Edward J. Balistreri & Russell H. Hillberry & Thomas F. Rutherford, 2008. "Structural Estimation and Solution of International Trade Models with Heterogeneous Firms," CER-ETH Economics working paper series, CER-ETH - Center of Economic Research (CER-ETH) at ETH Zurich 08/89, CER-ETH - Center of Economic Research (CER-ETH) at ETH Zurich.
  2. Cyril Bourgeois & Pierre-Alain Jayet, 2010. "Revisited water-oriented relationships between a set of farmers and an aquifer: accounting for lag effect," Working Papers, INRA, Economie Publique 2010/06, INRA, Economie Publique.
  3. Ralph Winkler, 2008. "Optimal control of pollutants with delayed stock accumulation," CER-ETH Economics working paper series, CER-ETH - Center of Economic Research (CER-ETH) at ETH Zurich 08/91, CER-ETH - Center of Economic Research (CER-ETH) at ETH Zurich.

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