Numerical Solution of Optimal Control Problems with Constant Control Delays

Computational Economics, Vol. 31, pp. 181-206, 2008

27 Pages Posted: 14 Jan 2008 Last revised: 23 Aug 2008

Ullrich Brandt-Pollmann

University of Heidelberg - Interdisciplinary Institute for Environmental Economics

Ralph Winkler

University of Bern - Department of Economics; University of Bern - Oeschger Centre for Climate Change Research

Sebastian Sager

University of Heidelberg - Interdisciplinary Center for Scientific Computing

Ulf Moslener

Frankfurt School of Finance & Management gemeinnützige GmbH

Johannes Schloeder

Interdisciplinary Center for Scientific Computing

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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.

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

JEL Classification: C63, C61

Suggested Citation

Brandt-Pollmann, Ullrich and Winkler, Ralph and Sager, Sebastian and Moslener, Ulf and Schloeder, Johannes, Numerical Solution of Optimal Control Problems with Constant Control Delays. Computational Economics, Vol. 31, pp. 181-206, 2008. Available at SSRN: https://ssrn.com/abstract=1082801

Ullrich Brandt-Pollmann (Contact Author)

University of Heidelberg - Interdisciplinary Institute for Environmental Economics ( email )

Grabengasse 1
Heidelberg
Germany

Ralph Winkler

University of Bern - Department of Economics ( email )

Schanzeneckstrasse 1
Bern, 3001
Switzerland
+41 31 631 4508 (Phone)

HOME PAGE: http://www.ralph-winkler.de

University of Bern - Oeschger Centre for Climate Change Research ( email )

Zähringerstrasse 25
Bern, 3012
Switzerland

Sebastian Sager

University of Heidelberg - Interdisciplinary Center for Scientific Computing ( email )

Im Neuenheimer Feld 368
Heidelberg, D-69120
Germany

Ulf Moslener

Frankfurt School of Finance & Management gemeinnützige GmbH ( email )

Sonnemannstraße 9-11
Frankfurt am Main, 60314
Germany

Johannes Schloeder

Interdisciplinary Center for Scientific Computing ( email )

Im Neuenheimer Feld 368
Heidelberg, D-69120
Germany

HOME PAGE: http://www.iwr.uni-heidelberg.de/

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