Quasi-normal-eigenvalue optimization is studied under constraints b 1 (x) ≤ B(x) ≤ b 2 (x) on structure functions B of 2-side open optical or mechanical resonators. We prove existence of various optimizers and provide an example when different structures generate the same optimal quasi-(normal-)eigenvalue. To show that quasi-eigenvalues locally optimal in various senses are in the spectrum Σ nl of the bang-bang eigenproblem yis the indicator function of C + , we obtain a variational characterization of Σ nl in terms of quasi-eigenvalue perturbations. To address the minimization of the decay rate | Im ω|, we study the bang-bang equation and explain how it excludes an unknown optimal B from the optimization process. Computing one of minimal decay structures for 1-side open settings, we show that it resembles gradually size-modulated 1-D stack cavities introduced recently in Optical Engineering. In 2-side open symmetric settings, our example has an additional centered defect. Nonexistence of global decay rate minimizers is discussed. MSC-classes: 49R05,

We consider a class of Markov processes with resettings, where at random times, the Markov processes are restarted from a predetermined point or a region. These processes are frequently applied in physics, chemistry, biology, economics, and in population dynamics. In this paper we establish the local large deviation principle (LLDP) for the Wiener processes with random resettings, where the resettings occur at the arrival time of a Poisson process. Here, at each resetting time, a new resetting point is selected at random, according to a conditional distribution.

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