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Optimal Control for a Parabolic–Hyperbolic Free Boundary Problem Modeling the Growth of Tumor with Drug Application

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  • Sakine Esmaili

    (Tarbiat Modares University)

  • Mohammad Reza Eslahchi

    (Tarbiat Modares University)

Abstract

In this paper, we study two optimal control problems for a free boundary problem, which models tumor growth with drug application. This free boundary problem is a multicellular tumor spheroid model and includes five time-dependent partial differential equations. The tumor considered in this model consists of three kinds of cells: proliferative cells, quiescent cells and dead cells. Three different first-order hyperbolic equations are given, which describe the evolution of cells, and other two second-order parabolic equations describe the diffusions of nutrient (e.g., oxygen and glucose) and drug concentrations. Existence and uniqueness of optimal controls are also proved. We use tangent-normal cone techniques to obtain necessary conditions. Then, we employ the Ekeland variational principle to show that there exists unique optimal control for each optimal control problem.

Suggested Citation

  • Sakine Esmaili & Mohammad Reza Eslahchi, 2017. "Optimal Control for a Parabolic–Hyperbolic Free Boundary Problem Modeling the Growth of Tumor with Drug Application," Journal of Optimization Theory and Applications, Springer, vol. 173(3), pages 1013-1041, June.
    • Handle: RePEc:spr:joptap:v:173:y:2017:i:3:d:10.1007_s10957-016-1037-4
    DOI: 10.1007/s10957-016-1037-4
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    References listed on IDEAS

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    1. V. Barbu & M. Iannelli, 1999. "Optimal Control of Population Dynamics," Journal of Optimization Theory and Applications, Springer, vol. 102(1), pages 1-14, July.
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    Cited by:

    1. Nasresfahani, F. & Eslahchi, M.R., 2021. "Error analysis of finite difference/collocation method for the nonlinear coupled parabolic free boundary problem modeling plaque growth in the artery," Applied Mathematics and Computation, Elsevier, vol. 405(C).

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