100 research outputs found
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
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 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 -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
Recommended from our members
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
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
First and Second Order Optimality Conditions for the Control of Fokker-Planck Equations
In this article we study an optimal control problem subject to the
Fokker-Planck equation The control variable is time-dependent and
possibly multidimensional, and the function depends on the space variable
and the control. The cost functional is of tracking type and includes a
quadratic regularization term on the control. For this problem, we prove
existence of optimal controls and first order necessary conditions. Main
emphasis is placed on second order necessary and sufficient 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
Optimal sparse boundary control for a semilinear parabolic equation with mixed control-state constraints
A problem of sparse optimal boundary control for a semilinear parabolic partial differential equation is considered, where pointwise bounds on the control and mixed pointwise control-state constraints are given. A standard quadratic objective functional is to be minimized that includes a Tikhonov regularization term and the L1-norm of the control accounting for the sparsity. Applying a recent linearization theorem, we derive first-order necessary optimality conditions in terms of a variational inequality under linearized mixed control state constraints. Based on this preliminary result, a Lagrange multiplier rule with bounded and measurable multipliers is derived and sparsity results on the optimal control are demonstrated.The first author was partially supported by the Spanish Ministerio de Economía y Competitividad under project MTM2017-83185-P. The second author was supported by the collaborative research center SFB 910, TU Berlin, project B6
- …