Optimal control of a tumor growth model with hyperbolic relaxation of the chemical potential

In this paper, we study the optimal control of a phase field model for a tumor growth model of Cahn–Hilliard type in which the often assumed parabolic relaxation of the chemical potential is replaced by a hyperbolic one. Both the cases when the double-well potential governing the phase evolution is of either regular or logarithmic type are covered by the analysis. We show the Fréchet differentiability of the associated control-to-state operator in suitable Banach spaces and establish first-order necessary optimality conditions in terms of a variational inequality involving the adjoint state variables. The necessary optimality conditions are then used to derive sparsity results for the optimal controls.