Talk by Illia Karabash

The optical engineering fabrication and numerical experiments for high-Q cavities led to a series of new analytical and computational problems related to optimization of resonances. The talk is devoted to the problem how to design an open resonator that has a dissipation eigenvalue as close as possible to the real line under certain fabrication constraints. It is planned to explain in the 1-dimensional case of TEM-modes in layered structures the rigorous analytical background for such problems and, in particular, why the Pareto optimization settings of (doi.org/10.1016/j.jde.2014.04.002) are natural. Then we concentrate on the optimal control reformulation developed jointly with Herbert Koch and Ievgen Verbytskyi (doi.org/10.1016/j.matpur.2020.02.005), as well as on resulting Hamilton-Jacobi-Bellman PDEs and extremal synthesis. If time allows us, the eigenvalue optimization for 3-dimensional dissipative Maxwell systems, as well as its connections with (non-)unique continuation and ho mogenization, will be also described following a recent joint paper with Matthias Eller (doi.org/10.1007/s00526-024-02908-0)