Optimal control problems with time delays: Two case studies in biomedicine

Laurenz Göllmann; Helmut Maurer; Laurenz Göllmann; Helmut Maurer

doi:10.3934/mbe.2018051

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 5: 1137-1154. doi: 10.3934/mbe.2018051

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Optimal control problems with time delays: Two case studies in biomedicine

Laurenz Göllmann ¹,
Helmut Maurer ^{2
,
,}

1.
Münster University of Applied Sciences, Department of Mechanical Engineering, Stegerwaldstr. 39, 48565 Steinfurt, Germany
2.
University of Münster, Applied Mathematics: Institute of Analysis and Numerics, Einsteinstr. 62, 49143 Münster, Germany

Received: 30 April 2017 Accepted: 18 March 2018 Published: 01 October 2018
MSC : Primary: 49K15, 49M37; Secondary: 92C50

There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. The first case study is concerned with the delay differential model in [21] describing the tumour-immune response to a chemo-immuno-therapy. Assuming $ L^1$-type objectives, which are linear in control, we obtain optimal controls of bang-bang type. In the second case study, we introduce a control variable in the delay differential model of Hepatitis B virus infection developed in [7]. For $ L^1$-type objectives we obtain extremal controls of bang-bang type.
- Optimal control,
- time delays,
- discretization methods,
- chemo-immuno-therapy of cancer,
- control of Hepatitis B infection
Citation: Laurenz Göllmann, Helmut Maurer. Optimal control problems with time delays: Two case studies in biomedicine[J]. Mathematical Biosciences and Engineering, 2018, 15(5): 1137-1154. doi: 10.3934/mbe.2018051

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Abstract

There exists an extensive literature on delay differential models in biology and biomedicine, but only a few papers study such models in the framework of optimal control theory. In this paper, we consider optimal control problems with multiple time delays in state and control variables and present two applications in biomedicine. After discussing the necessary optimality conditions for delayed optimal control problems with control-state constraints, we propose discretization methods by which the delayed optimal control problem is transformed into a large-scale nonlinear programming problem. The first case study is concerned with the delay differential model in [21] describing the tumour-immune response to a chemo-immuno-therapy. Assuming $ L^1$-type objectives, which are linear in control, we obtain optimal controls of bang-bang type. In the second case study, we introduce a control variable in the delay differential model of Hepatitis B virus infection developed in [7]. For $ L^1$-type objectives we obtain extremal controls of bang-bang type.

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