Optimal control of a model for brain lactate kinetics

The aim of this paper is to first study a Cahn-Hilliard model for brain lactate kinetics with a control function. This control allows for optimal treatment administered to ill patients suffering from glioma, in order to reduce their brain lactate concentrations, and thereby to slow down the tumor growth. We establish the well-posedness of the problem and the continuity of the control-to-state mapping, the existence of a minimizer of the objective functional, and its Fréchet differentiability in suitable Banach spaces with respect to the control and with respect to time. Moreover, we derive the first-order necessary conditions that an optimal control has to satisfy. In the second part of the paper, we illustrate our theoretical results with numerical simulations using MRI data from the University Hospital of Poitiers.