On two optimal control problems for magnetic fields

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)Two optimal control problems for instationary magnetization processes are considered in 3D spatial domains that include electrically conducting and non-conducting regions. The magnetic fields are generated by induction coils. In the first model, the induction coil is considered as part of the conducting region and the electrical current is taken as control. In the second, the coil is viewed as part of the non-conducting region and the electrical voltage is the control. Here, an integro-differential equation accounts for the magnetic induction law that couples the given electrical voltage with the induced electrical current in the induction coil. We derive first-order necessary optimality conditions for the optimal controls of both problems. Based on them, numerical methods of gradient type are applied. Moreover, we report on the application of model reduction by POD that lead to tremendous savings. Numerical tests are presented for academic 3D geometries but also for a real-world application

Optimal control of linear bottleneck problems

International audienceThe regularity of Lagrange multipliers for state-constrained optimal control problems belongs to the basic questions of control theory. Here, we investigate bottleneck problems arising from optimal control problems for PDEs with certain mixed control-state inequality constraints. We show how to obtain Lagrange multipliers in Lp-spaces for linear problems and give an application to linear parabolic optimal control problems

Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth

In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [12]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the $L^1-$norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions

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 $L^1$-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 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 $L^1$-norm leading to sparsity of optimal controls. For such cases, we derive second-order sufficient conditions for locally optimal controls

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

Sparse optimal control of a phase field system with singular potentials arising in the modeling of tumor growth

In this paper, we study an optimal control problem for a nonlinear system of reaction-diffusion equations that constitutes a simplified and relaxed version of a thermodynamically consistent phase field model for tumor growth originally introduced in [13]. The model takes the effect of chemotaxis into account but neglects velocity contributions. The unknown quantities of the governing state equations are the chemical potential, the (normalized) tumor fraction, and the nutrient extra-cellular water concentration. The equation governing the evolution of the tumor fraction is dominated by the variational derivative of a double-well potential which may be of singular (e.g., logarithmic) type. In contrast to the recent paper [10] on the same system, we consider in this paper sparsity effects, which means that the cost functional contains a nondifferentiable (but convex) contribution like the L1-norm. For such problems, we derive first-order necessary optimality conditions and conditions for directional sparsity, both with respect to space and time, where the latter case is of particular interest for practical medical applications in which the control variables are given by the administration of cytotoxic drugs or by the supply of nutrients. In addition to these results, we prove that the corresponding control-to-state operator is twice continuously differentiable between suitable Banach spaces, using the implicit function theorem. This result, which complements and sharpens a differentiability result derived in [10], constitutes a prerequisite for a future derivation of second-order sufficient optimality conditions

State-constrained semilinear elliptic optimization problems with unrestricted sparse controls

In this paper, we consider optimal control problems associated with semilinear elliptic equation equations, where the states are subject to pointwise constraints but there are no explicit constraints on the controls. A term is included in the cost functional promoting the sparsity of the optimal control. We prove existence of optimal controls and derive first and second order optimality conditions. In addition, we establish some regularity results for the optimal controls and the associated adjoint states and Lagrange multipliers.The first author was partially supported by Spanish Ministerio de Economía, Industria y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P