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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. Raouf Boucekkine & Omar Licandro & Luis A. Puch & Fernando del Rio, . "Vintage capital and the dynamics of the AK model," Working Papers 2000-01, FEDEA.
  2. Fabrice COLLARD & Omar LICANDRO & Luis A. PUCH, 2004. "The short-run dynamics of optimal growth models with delays," Economics Working Papers ECO2004/04, European University Institute.
  3. Patrick Asea & Paul J. Zak, 1997. "Time-to-Build and Cycles," UCLA Economics Working Papers 767, UCLA Department of Economics.
  4. Peeters, Marga, 1996. "Investment gestation lags: The difference between time-to-build and delivery lags," MPRA Paper 28549, University Library of Munich, Germany.
  5. BOUCEKKINE, RAOUF & de la CROIX, David & LICANDRO, Omar, 2004. "Modelling vintage structures with DDEs : principles and applications," CORE Discussion Papers 2004047, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  6. El-Hodiri, Mohamed A & Loehman, Edna & Whinston, Andrew B, 1972. "An Optimal Growth Model with Time Lags," Econometrica, Econometric Society, vol. 40(6), pages 1137-46, November.
  7. Boucekkine, R. & Germain, M. & Licandro, O., . "Replacement echoes in the vintage capital growth model," CORE Discussion Papers RP -1275, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  8. Boucekkine, Raouf & Licandro, Omar & Paul, Christopher, 1997. "Differential-difference equations in economics: On the numerical solution of vintage capital growth models," Journal of Economic Dynamics and Control, Elsevier, vol. 21(2-3), pages 347-362.
  9. Raouf Boucekkine & David de la Croix & Omar Licandro, 2006. "Vintage Capital," Economics Working Papers ECO2006/8, European University Institute.
  10. Boucekkine, Raouf & de la Croix, David & Licandro, Omar, 2002. "Vintage Human Capital, Demographic Trends, and Endogenous Growth," Journal of Economic Theory, Elsevier, vol. 104(2), pages 340-375, June.
  11. Winkler, Ralph & Brandt-Pollmann, Ulrich & Moslener, Ulf & Schlöder, Johannes, 2005. "On the Transition from Instantaneous to Time-Lagged Capital Accumilation: The Case of Leontief Type Production Functions," ZEW Discussion Papers 05-30, ZEW - Zentrum für Europäische Wirtschaftsforschung / Center for European Economic Research.
  12. BOUCEKKINE, Raouf & GERMAIN, Marc & LICANDRO, Omar & MAGNUS, Alphonse, . "Creative destruction, investment volatility, and the average age of capital," CORE Discussion Papers RP -1376, Université catholique de Louvain, Center for Operations Research and Econometrics (CORE).
  13. Raouf Boucekkine & Marc Germain & Omar Licandro & Alphonse Magnus, . "Numerical solution by iterative methods of a class of vintage capital models," Working Papers 98-20, FEDEA.
  14. Feichtinger, Gustav & Hartl, Richard F. & Kort, Peter M. & Veliov, Vladimir M., 2006. "Anticipation effects of technological progress on capital accumulation: a vintage capital approach," Journal of Economic Theory, Elsevier, vol. 126(1), pages 143-164, January.
  15. Boyer, M., 1976. "A Habit Forming Optimal Grouwth Model," Cahiers de recherche 7612, Universite de Montreal, Departement de sciences economiques.
  16. Kydland, Finn E & Prescott, Edward C, 1982. "Time to Build and Aggregate Fluctuations," Econometrica, Econometric Society, vol. 50(6), pages 1345-70, November.
  17. de la Croix, David & Licandro, Omar, 1999. "Life expectancy and endogenous growth," Economics Letters, Elsevier, vol. 65(2), pages 255-263, November.
  18. Christopher D. Carroll & Jody Overland & David N. Weil, 1995. "Saving and growth with habit formation," Finance and Economics Discussion Series 95-42, Board of Governors of the Federal Reserve System (U.S.).
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Citations

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Cited by:
  1. Cyril Bourgeois & Pierre-Alain Jayet, 2010. "Revisited water-oriented relationships between a set of farmers and an aquifer: accounting for lag effect," Working Papers 2010/06, INRA, Economie Publique.
  2. Edward J. Balistreri & Russell H. Hillberry & Thomas F. Rutherford, 2008. "Structural Estimation and Solution of International Trade Models with Heterogeneous Firms," Department of Economics - Working Papers Series 1056, The University of Melbourne.
  3. Ralph Winkler, 2008. "Optimal control of pollutants with delayed stock accumulation," CER-ETH Economics working paper series 08/91, CER-ETH - Center of Economic Research (CER-ETH) at ETH Zurich.

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