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Numerical solution of optimal control problems with constant control delays

Author

Listed:
  • 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.

Suggested Citation

  • Ulrich Brandt-Pollmann & Ralph Winkler & Sebastian Sager & Ulf Moslener & Johannes P. Schlöder, 2006. "Numerical solution of optimal control problems with constant control delays," CER-ETH Economics working paper series 06/59, CER-ETH - Center of Economic Research (CER-ETH) at ETH Zurich.
    • Handle: RePEc:eth:wpswif:06-59
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    References listed on IDEAS

    as
    1. Raouf Boucekkine & David Croix & Omar Licandro, 2004. "MODELLING VINTAGE STRUCTURES WITH DDEs: PRINCIPLES AND APPLICATIONS," Mathematical Population Studies, Taylor & Francis Journals, vol. 11(3-4), pages 151-179.
    2. Jody Overland & Christopher D. Carroll & David N. Weil, 2000. "Saving and Growth with Habit Formation," American Economic Review, American Economic Association, vol. 90(3), pages 341-355, June.
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    4. Boucekkine, Raouf & Germain, Marc & Licandro, Omar, 1997. "Replacement Echoes in the Vintage Capital Growth Model," Journal of Economic Theory, Elsevier, vol. 74(2), pages 333-348, June.
    5. 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.
    6. Fabrice Collard & Omar Licandro & Luis A. Puch, 2008. "The short-run Dynamics of Optimal Growth Model with Delays," Annals of Economics and Statistics, GENES, issue 90, pages 127-143.
    7. Boucekkine, Raouf & Germain, Marc & Licandro, Omar & Magnus, Alphonse, 1998. "Creative Destruction, Investment Volatility, and the Average Age of Capital," Journal of Economic Growth, Springer, vol. 3(4), pages 361-384, December.
    8. Raouf Boucekkine & David de la Croix & Omar Licandro, 2006. "Vintage Capital," Economics Working Papers ECO2006/8, European University Institute.
    9. de la Croix, David & Licandro, Omar, 1999. "Life expectancy and endogenous growth," Economics Letters, Elsevier, vol. 65(2), pages 255-263, November.
    10. Boucekkine, Raouf & Germain, Marc & Licandro, Omar & Magnus, Alphonse, 2001. "Numerical solution by iterative methods of a class of vintage capital models," Journal of Economic Dynamics and Control, Elsevier, vol. 25(5), pages 655-669, May.
    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 - Leibniz Centre for European Economic Research.
    12. 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, vol. 120(1), pages 39-72, January.
    13. Kydland, Finn E & Prescott, Edward C, 1982. "Time to Build and Aggregate Fluctuations," Econometrica, Econometric Society, vol. 50(6), pages 1345-1370, November.
    14. 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.
    15. Benhabib, Jess & Rustichini, Aldo, 1991. "Vintage capital, investment, and growth," Journal of Economic Theory, Elsevier, vol. 55(2), pages 323-339, December.
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    17. 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.
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    19. Peeters, Marga, 1996. "Investment gestation lags: The difference between time-to-build and delivery lags," MPRA Paper 28549, University Library of Munich, Germany.
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    Citations

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    Cited by:

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    2. Balistreri, Edward J. & Hillberry, Russell H. & Rutherford, Thomas F., 2011. "Structural estimation and solution of international trade models with heterogeneous firms," Journal of International Economics, Elsevier, vol. 83(2), pages 95-108, March.
    3. Rădulescu, I.R. & Cândea, D. & Halanay, A., 2016. "Optimal control analysis of a leukemia model under imatinib treatment," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 121(C), pages 1-11.
    4. 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.
    5. Leon A. Petrosyan & David W.K. Yeung, 2020. "Cooperative Dynamic Games with Durable Controls: Theory and Application," Dynamic Games and Applications, Springer, vol. 10(4), pages 872-896, December.
    6. 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.
    7. David W. K. Yeung & Leon A. Petrosyan, 2019. "Cooperative Dynamic Games with Control Lags," Dynamic Games and Applications, Springer, vol. 9(2), pages 550-567, June.

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    More about this item

    Keywords

    delayed differential equations; delayed optimal control; numerical optimization; time-to-build;
    All these keywords.

    JEL classification:

    • C63 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Computational Techniques
    • C61 - Mathematical and Quantitative Methods - - Mathematical Methods; Programming Models; Mathematical and Simulation Modeling - - - Optimization Techniques; Programming Models; Dynamic Analysis

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