The aim of the paper is to reduce one spectral optimization problem, which involves the minimization of the decay rate $|\mathrm{Im}\kappa |$ of a resonance κ, to a collection of optimal control problems on the Riemann sphere $\hat{\mathbb{C}}$. This reduction allows us to apply methods of extremal synthesis to the structural optimization of layered optical cavities. We start from a dual problem of minimization of the resonator length and give several reformulations of this problem that involve Pareto optimization of the modulus $|\kappa |$ of a resonance, minimum-time control problems on $\hat{\mathbb{C}}$, and associated Hamilton-Jacobi-Bellman equations. Various types of controllability properties are studied in connection with the existence of optimizers and with the relationship between the Pareto optimal frontiers of minimal decay and minimal modulus. We give explicit examples of optimal resonances and describe qualitatively properties of the Pareto frontiers near them. A special representation of bang-bang controlled trajectories is combined with the analysis of extremals to obtain various bounds on optimal widths of layers. We propose a new method of computation of optimal symmetric resonators based on minimum-time control and compute with high accuracy several Pareto optimal frontiers and high-Q resonators.

本文的目的是减少一个频谱优化问题，该问题涉及衰减率的最小化 $|\mathrm{林}\kappa |$共振κ对黎曼球上最优控制问题的集合$\hat{\mathbb{C}}$。这种减少使我们能够将极值合成方法应用于分层光腔的结构优化。我们从使谐振器长度最小化的双重问题开始，并给出涉及该问题的若干重新公式化，其中涉及模数的帕累托优化。$|\kappa |$ 共振，最小时间控制问题 $\hat{\mathbb{C}}$，以及相关的Hamilton-Jacobi-Bellman方程。结合优化器的存在以及最小衰减和最小模量的帕累托最优边界之间的关系，研究了各种类型的可控制性。我们给出了最佳共振的显式示例，并定性描述了它们附近的帕累托边界的性质。爆炸控制轨迹的特殊表示与极值分析相结合，以获得层的最佳宽度上的各种边界。我们提出了一种基于最小时间控制的最优对称谐振器计算新方法，并以高精度计算了多个帕累托最优边界和高Q谐振器。