Sparse Optimal Control of a Phase Field Tumor Model with Mechanical Effects

In this paper, we study an optimal control problem for a macroscopic mechanical tumor model based on the phase field approach. The model couples a Cahn--Hilliard-type equation to a system of linear elasticity and a reaction-diffusion equation for a nutrient concentration. By taking advantage of previous analytical well-posedness results established by the authors, we seek optimal controls in the form of a boundary nutrient supply as well as concentrations of cytotoxic and antiangiogenic drugs that minimize a cost functional involving mechanical stresses. Special attention is given to sparsity effects, where with the inclusion of convex nondifferentiable regularization terms to the cost functional, we can infer from the first-order optimality conditions that the optimal drug concentrations can vanish on certain time intervals.