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

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.