Optimal Control and Parameters Identification for the Cahn-Hilliard Equations Modeling Tumor Growth

This paper is dedicated to the setting and analysis of an optimal control problem for a two-phase system composed of two non-linearly coupled Chan-Hilliard-type equations. The model describes the evolution of a tumor cell fraction and a nutrient-rich extracellular water volume fraction. The main objective of this paper is the identification of the system's physical parameters, such as the viscosities and the proliferation rate, in addition to the controllability of the system's unknowns. For this purpose, we introduce an adequate cost function to be optimized by analyzing a linearized system, deriving the adjoint system, and defining the optimality condition. Eventually, we provide a numerical simulation example illustrating the theoretical results. Finally, numerical simulations of a tumor growing in two and three dimensions are carried out in order to illustrate the evolution of such a clinical situation and to possibly suggest different treatment strategies.