Pareto optimal structures producing resonances of minimal decay under L1L1-type constraints

Resonances optimization is studied under the constraint ‖B‖1≤m‖B‖1≤m on the nonnegative function B∈L1(0,ℓ)B∈L1(0,ℓ) representing the resonator structure. The problem is to design for a given frequency α∈Rα∈R a structure that generates a resonance ω   on the line α+iRα+iR with minimal possible decay rate |Imω|. We generalize the problem replacing B by a nonnegative measure, and show that optimal measures consist of finite number of point masses. This yields non-existence of optimizers for the problem over absolutely continuous measures. We derive restrictions on optimal masses and their positions. This reduces the original infinitely-dimensional problem to optimization over four real parameters. For low frequencies, we explicitly find optimizers. The technique is based on the two-parameter perturbation method and the notion of local boundary point, which is introduced as a generalization of local extrema to vector optimization problems. Special attention is paid to multiple and non-differentiable resonances.