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We analyze the optimal control of an electromagnetic process in type-II superconductivity. The PDE-constrained optimization problem is to find an optimal applied current density, which steers the electromagnetic fields to the desired ones in the presence of a type-II superconductor. The governing PDE system for the electromagnetic fields consists of hyperbolic evolution Maxwell equations with a nonlinear and nonsmooth constitutive law for the electric field and the current density based on the Bean critical-state model. Through the use of the Maxwell theory, the semigroup theory, Helmholtz decomposition, and results on maximal monotone operators, we develop a mathematical theory including an existence analysis and first-order necessary optimality conditions for the nonsmooth PDE-constrained optimization problem.

Abstract. In this paper the shape derivative of an objective depending on the solution of an eddy current approximation of Maxwell's equations is obtained. Using a Lagrangian approach in the spirit of Delfour and Zolésio, the computation of the shape derivative of the solution of the state equation is bypassed. This theoretical result is applied to magnetic impedance tomography, which is an imaging modality aiming at the contactless mapping (identification) of the unknown electrical conductivities inside an object given measurements recorded by receiver coils.Key words. magnetic induction tomography, inverse problems, shape optimization, shape derivative AMS subject classifications. Primary 49N45,35Q61; Secondary 47J30, 49Q101. Introduction. Magnetic induction tomography (MIT) is a low-contrast resolution mapping modality for non-contacting measurement of electric properties of conducting materials which has been developed for industrial processes about twenty years ago and whose use is more recent in medical imaging; see [12,17,26,27,30]. The technology of MIT involves an oscillating magnetic field generated by a transmitter coil, which in turn induces an eddy current inside the conducting materials. The resulting magnetic field arising from the eddy current depends on the conductivity distribution in the region surrounded by the coil array and it is detected by a set of receiver coils. Compared to other imaging techniques such as electrical impedance tomography, the advantage of MIT is to avoid electrode-skin contact since the magnetic field can penetrate through the insulating barrier, making it a contactless and non-invasive technique. The goal of this paper is to study an inverse problem of a simplified MIT system governed by the time-harmonic eddy current equations. In fact, given measurements, which depend on the electric field E and the current density J and which are recorded at receiver coils surrounding the region of interest, we aim at reconstructing the conductivity (function) associated with objects contained in the region of interest.Eddy current equations play a significant role in many advanced technologies involving electromagnetic phenomena. These equations arise from Maxwell's equations by neglecting the presence of displacement currents. In a low frequency range, they provide a reasonable approximation of the full Maxwell equations; see [2]. For recent results on the mathematical and numerical analysis of the optimal control of eddy current equations, we refer to [24,35,[40][41][42]. The use of the eddy current model in MIT is justified by the small wavelength of its operating frequencies. Typically, the frequencies lie between 10 and 100 MHz, i.e. in the range of some micrometers, so that the wavelength is small compared to the size of the conductor.The inverse problem of MIT is ill-posed, in the sense that, besides other structural difficulties, there is no uniqueness and stability of the solution. In many practical applications, this may be exacerbated by the small number of available mea...

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 controlto-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-thenoptimize concept.

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 Lavrentievparameter. Mathematics Subject Classification (2000)35J60 · 49K20 · 49M05 · 65K10