9 research outputs found
Second-order sufficient conditions in the sparse optimal control of a phase field tumor growth model with logarithmic potential
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 -norm in order to enhance the occurrence of sparsity effects in the
optimal control, 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.Comment: arXiv admin note: text overlap with arXiv:2303.16708,
arXiv:2104.0981
Second-order sufficient conditions in the sparse optimal control of a phase field tumor growth model with logarithmic potential
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
Second-order sufficient conditions for sparse optimal control of singular Allen--Cahn systems with dynamic boundary conditions
In this paper we study the optimal control of a parabolic initial-boundary
value problem of Allen--Cahn type with dynamic boundary conditions. Phase field
systems of this type govern the evolution of coupled diffuse phase transition
processes with nonconserved order parameters that occur in a container and on
its surface, respectively. It is assumed that the nonlinear function driving
the physical processes within the bulk and on the surface are double well
potentials of logarithmic type whose derivatives become singular at the
boundary of their respective domains of definition. For such systems, optimal
control problems have been studied in the past. We focus here on the situation
when the cost functional of the optimal control problem contains a
nondifferentiable term like the -norm leading to sparsity of optimal
controls. For such cases, we derive second-order sufficient conditions for
locally optimal controls