WIAS Preprint No. 3020, (2023)

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


Authors

  • Sprekels, Jürgen
    ORCID: 0009-0000-0618-8604
  • Tröltzsch, Fredi

2020 Mathematics Subject Classification

  • 35K20 35K57 37N25 49J20 49J50 49J52 49K20 49K40

Keywords

  • Optimal control, tumor growth models, logarithmic potentials, second-order sufficient optimality conditions, sparsity

DOI

10.20347/WIAS.PREPRINT.3020

Abstract

his 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 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. The results obtained in this paper also improve the known results on this phase field model for the case without sparsity.

Appeared in

Download Documents