Computational Economics

, Volume 31, Issue 2, pp 181–206 | Cite as

Numerical Solution of Optimal Control Problems with Constant Control Delays

  • Ulrich Brandt-Pollmann
  • Ralph Winkler
  • Sebastian Sager
  • Ulf Moslener
  • Johannes P. Schlöder


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.


Delayed differential equations Delayed optimal control Numerical optimization 

JEL Classification

C63 C61 


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Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • Ulrich Brandt-Pollmann
    • 1
  • Ralph Winkler
    • 2
  • Sebastian Sager
    • 1
  • Ulf Moslener
    • 3
  • Johannes P. Schlöder
    • 1
  1. 1.Interdisciplinary Center for Scientific ComputingUniversity of HeidelbergHeidelbergGermany
  2. 2.CER-ETH – Center of Economic Research at ETH ZurichZurichSwitzerland
  3. 3.Centre for European Economic Research (ZEW)MannheimGermany

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