Second-order sufficient conditions in the sparse optimal control of a phase field tumor growth model with logarithmic potential

Abstract

This paper treats a distributed optimal control problem for a tumor growth model of viscous Cahn--Hilliard type. The evolution  of the tumor fraction is governed by a thermodynamic force induced by a double-well potential of logarithmic type. The cost functional contains a nondifferentiable term like the $L^1$--norm in order to enhance the occurrence of sparsity effects in the optimal controls, i.e., of subdomains of the space-time cylinder where the controls vanish.  In the context of cancer therapies, sparsity is very important in order that the patient is not exposed to  unnecessary intensive medical treatment. In this work, we focus on the derivation of second-order sufficient optimality conditions for the optimal control problem. While in previous works on the system under investigation such conditions  have been established for the case without sparsity, the case with sparsity has not been treated before.       2020 Mathematics Subject Classification 35K57, 37N25, 49J50, 49J52, 49K20, 49K40.