Irwin Yousept

A regularization method for the numerical solution of elliptic boundary control problems with pointwise state constraints

Abstract A Lavrentiev type regularization technique for solving elliptic boundary control problems with pointwise state constraints is considered. The main concept behind this regularization is to look for controls in the range of the adjoint control-to-state mapping. After investigating the analysis of the method, a semismooth Newton method based on the optimality conditions is presented. The theoretical results are confirmed by numerical tests. Moreover, they are validated by comparing the regularization technique with standard numerical codes based on the discretize-then-optimize concept.

Optimal Control of Three-Dimensional State-Constrained Induction Heating Problems with Nonlocal Radiation Effects

The paper is concerned with a class of optimal heating problems in semiconductor single crystal growth processes. To model the heating process, time-harmonic Maxwell equations are considered in the system of the state. Due to the high temperatures characterizing crystal growth, it is necessary to include nonlocal radiation boundary conditions and a temperature-dependent heat conductivity in the description of the heat transfer process. The first goal of this paper is to prove existence and uniqueness of the state. The regularity analysis associated with the time-harmonic Maxwell equations is also studied. In the second part of the paper, existence and uniqueness of the solution of the corresponding linearized equation are shown. With this result at hand, the differentiability of the control-to-state operator is derived. Finally, based on the theoretical results, first order necessary optimality conditions for an associated optimal control problem are established.

Mesh-independence of semismooth Newton methods for Lavrentiev-regularized state constrained nonlinear optimal control problems

A class of nonlinear elliptic optimal control problems with mixed control-state constraints arising, e.g., in Lavrentiev-type regularized state constrained optimal control is considered. Based on its first order necessary optimality conditions, a semismooth Newton method is proposed and its fast local convergence in function space as well as a mesh-independence principle for appropriate discretizations are proved. The paper ends by a numerical verification of the theoretical results including a study of the algorithm in the case of vanishing Lavrentiev-parameter. The latter process is realized numerically by a combination of a nested iteration concept and an extrapolation technique for the state with respect to the Lavrentiev-parameter.

Optimal control of Maxwell's equations with regularized state constraints

This paper is devoted to an optimal control problem of Maxwell’s equations in the presence of pointwise state constraints. The control is given by a divergence-free three-dimensional vector function representing an applied current density. To cope with the divergence-free constraint on the control, we consider a vector potential ansatz. Due to the lack of regularity of the control-to-state mapping, existence of Lagrange multipliers cannot be guaranteed. We regularize the optimal control problem by penalizing the pointwise state constraints. Optimality conditions for the regularized problem can be derived straightforwardly. It also turns out that the solution of the regularized problem enjoys higher regularity which then allows us to establish its convergence towards the solution of the unregularized problem. The second part of the paper focuses on the numerical analysis of the regularized optimal control problem. Here the state and the control are discretized by Nédélec’s curl-conforming edge elements. Employing the higher regularity property of the optimal control, we establish an a priori error estimate for the discretization error in the

State-Constrained Optimal Control of Semilinear Elliptic Equations with Nonlocal Radiation Interface Conditions

\boldsymbol{H}(\bold{curl})

Regularization of state-constrained elliptic optimal control problems with nonlocal radiation interface conditions

-norm. The paper ends by numerical results including a numerical verification of our theoretical results.

Optimal Control of Quasilinear

We consider a control- and state-constrained optimal control problem governed by a semilinear elliptic equation with nonlocal interface conditions. These conditions occur during the modeling of diffuse-gray conductive-radiative heat transfer. The nonlocal radiation interface condition and the pointwise state constraints represent the particular features of this problem. To deal with the state constraints, continuity of the state is shown, which allows us to derive first-order necessary conditions. Afterwards, we establish second-order sufficient conditions that account for strongly active sets and ensure local optimality in an

\boldsymbol{H}(\mathbf{curl})

L^2

-Elliptic Partial Differential Equations in Magnetostatic Field Problems

-neighborhood.

Shape sensitivities for an inverse problem in magnetic induction tomography based on the eddy current model

Abstract A state-constrained optimal control problem with nonlocal radiation interface conditions arising from the modeling of crystal growth processes is considered. The problem is approximated by a Moreau-Yosida type regularization. Optimality conditions for the regularized problem are derived and the convergence of the regularized problems is shown. In the last part of the paper, some numerical results are presented.