Journal of Industrial and Management Optimization (JIMO)

Theory and applications of optimal control problems with multiple time-delays

Pages: 413 - 441, Volume 10, Issue 2, April 2014      doi:10.3934/jimo.2014.10.413

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Laurenz Göllmann - Department of Mechanical Engineering, Münster University of Applied Sciences, Stegerwaldstrasse 39, 48565 Steinfurt, Germany (email)
Helmut Maurer - Institute of Computational and Applied Mathematics, Westfälische Wilhelms-Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany (email)

Abstract: In this paper we study optimal control problems with multiple time delays in control and state and mixed type control-state constraints. We derive necessary optimality conditions in the form of a Pontryagin type Minimum Principle. A discretization method is presented by which the delayed control problem is transformed into a nonlinear programming problem. It is shown that the associated Lagrange multipliers provide a consistent numerical approximation for the adjoint variables of the delayed optimal control problem. We illustrate the theory and numerical approach on an analytical example and an optimal control model from immunology.

Keywords:  Optimal control, multiple delays, maximum principle, discretization method, optimal control of innate immune response.
Mathematics Subject Classification:  Primary: 49J15; Secondary: 37N25, 37N40.

Received: November 2012;      Revised: July 2013;      Available Online: October 2013.