Some Optimization Problems in Electromagnetism

Venue: The event will take in presence only for the Phd student and part of the commission and online through the ZOOM platform. To get the access codes please contact the secretary office (phd.maths [at] unitn.it)
Time: 10:00

Gabriele Caselli - PhD in Mathematics, University of Trento

Abstract:
Electromagnetism and optimal control stand out as a topics that feature impactful applications in modern engineering, as well as challenging theoretical aspects of mathematical analysis. Within this context, a major role is played by the search of necessary and sufficient conditions characterizing optimal solutions, as they are functional to numerical algorithms aiming to approximate such solutions. During this seminar I will be discussing three standalone optimization problems that share the underlying framework of Maxwell-related PDEs. First, I will present an optimal control problem driven by a quasi-linear magneto-static obstacle problem featuring first-order state constraints. The non-linearity allows to suitably model electromagnetic waves in the presence of ferromagnetic materials, while the first-order obstacle is relevant for applications in the field of magnetic shielding. Existence theory and the derivation of an optimality system are addressed with an approximation technique based on a relaxation-penalization of the variational inequality. Second, I will analyze an eddy current problem controlled through a dipole type source, i.e. a Dirac mass with fixed position and variable intensity: well-posedness of the state equation through a fundamental solution approach and first order conditions are dealt with. To conclude, I will discuss the computation of the topological derivative for shape functionals constrained to low-frequency electromagnetic problems (closely related to the eddy current model), with respect to the inclusion/removal of conducting material; the results are obtained using a Lagrangian approach.

Supervisors:
Alberto Valli (University of Trento, Italy)
Irwin Yousept (University of Duisburg-Essen, Germany)