Optimal control problems with time delays: Two case studies in biomedicine
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 t...
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AIMS Press
2018-09-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2018051 |
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| author | Laurenz Göllmann Helmut Maurer |
| author_facet | Laurenz Göllmann Helmut Maurer |
| author_sort | Laurenz Göllmann |
| collection | DOAJ |
| description | 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. |
| format | Article |
| id | doaj-art-51c57eeaa97e40dc98764388fc05f7ff |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2018-09-01 |
| publisher | AIMS Press |
| record_format | Article |
| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-51c57eeaa97e40dc98764388fc05f7ff2025-01-24T02:41:02ZengAIMS PressMathematical Biosciences and Engineering1551-00182018-09-011551137115410.3934/mbe.2018051Optimal control problems with time delays: Two case studies in biomedicineLaurenz Göllmann0Helmut Maurer1Münster University of Applied Sciences, Department of Mechanical Engineering, Stegerwaldstr. 39, 48565 Steinfurt, GermanyUniversity of Münster, Applied Mathematics: Institute of Analysis and Numerics, Einsteinstr. 62, 49143 Münster, GermanyThere 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.https://www.aimspress.com/article/doi/10.3934/mbe.2018051optimal controltime delaysdiscretization methodschemo-immuno-therapy of cancercontrol of hepatitis b infection |
| spellingShingle | Laurenz Göllmann Helmut Maurer Optimal control problems with time delays: Two case studies in biomedicine Mathematical Biosciences and Engineering optimal control time delays discretization methods chemo-immuno-therapy of cancer control of hepatitis b infection |
| title | Optimal control problems with time delays: Two case studies in biomedicine |
| title_full | Optimal control problems with time delays: Two case studies in biomedicine |
| title_fullStr | Optimal control problems with time delays: Two case studies in biomedicine |
| title_full_unstemmed | Optimal control problems with time delays: Two case studies in biomedicine |
| title_short | Optimal control problems with time delays: Two case studies in biomedicine |
| title_sort | optimal control problems with time delays two case studies in biomedicine |
| topic | optimal control time delays discretization methods chemo-immuno-therapy of cancer control of hepatitis b infection |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2018051 |
| work_keys_str_mv | AT laurenzgollmann optimalcontrolproblemswithtimedelaystwocasestudiesinbiomedicine AT helmutmaurer optimalcontrolproblemswithtimedelaystwocasestudiesinbiomedicine |