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The aim of the paper is to reduce one spectral optimization problem, whichinvolves the minimization of the decay rate $|\mathrm{Im} \, k |$ of aresonance $k$, to a collection of optimal control problems on the Riemannsphere $\widehat{\mathbb{C}}$. This reduction allows us to apply methods ofextremal synthesis to the structural optimization of layered optical cavities.We start from a dual problem of minimization of the resonator length and giveseveral reformulations of this problem that involve Pareto optimization of themodulus $|k|$ of a resonance, a minimum-time control problem on$\widehat{\mathbb{C}}$, and associated Hamilton-Jacobi-Bellman equations.Various types of controllability properties are studied in connection with theexistence of optimizers and with the relationship between the Pareto optimalfrontiers of minimal decay and minimal modulus. We give explicit examples ofoptimal resonances and describe qualitatively properties of the Paretofrontiers near them. A special representation of bang-bang controlledtrajectories is combined with the analysis of extremals to obtain variousbounds on optimal widths of layers. We propose a new method of computation ofoptimal symmetric resonators based on minimum-time control and compute withhigh accuracy several Pareto optimal frontiers and high-Q resonators.

Pareto optimization of resonances and minimum-time control