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.

Brandt-Pollmann, Ulrich, Ralph Winkler, Sebastian Sager, Ulf Moslener and Johannes Schlöder (2008), Numerical Solution of Optimal Control Problems with Constant Control Delays, Computational Economics 31, 181-206.


Brandt-Pollmann, Ulrich
Winkler, Ralph
Sager, Sebastian
Moslener, Ulf
Schlöder, Johannes


delayed differential equations, delayed optimal control, numerical optimization