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

Sakine Esmaili () and Mohammad Reza Eslahchi ()
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Sakine Esmaili: Tarbiat Modares University
Mohammad Reza Eslahchi: Tarbiat Modares University

Journal of Optimization Theory and Applications, 2017, vol. 173, issue 3, No 15, 1013-1041

Abstract: 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.

Keywords: Optimal control; Ekeland variational principle; Multicellular tumor spheroid model; Parabolic–hyperbolic equation; Free boundary problem; 49J20; 49J15; 49K20; 49K15 (search for similar items in EconPapers)
Date: 2017
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DOI: 10.1007/s10957-016-1037-4

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