LAUSR

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.