Pareto optimal structures producing resonances of minimal decay under $L^1$-type constraints

Optimization of resonances associated with 1-D wave equations in inhomogeneous media is studied under the constraint $\| B \|_1 <= m$ on the nonnegative function $B \in L^1 (0,\ell)$ that represents the medium's structure. From the Physics and Optimization points of view, it convenient to generalize the problem replacing $B$ by a nonnegative measure $d M$ and imposing on $d M$ the condition that its total mass is $<= m$. The problem is to design for a given frequency $\alpha \in R$ a medium that generates a resonance $\omega$ on the line $\alpha + i R$ with a minimal possible decay rate $| Im \omega |$. Such resonances are said to be of minimal decay and form a Pareto frontier. We show that corresponding optimal measures consist of finite number of point masses, and that this result yields non-existence of optimizers for the problem over the set of absolutely continuous measures $B(x) dx$. Then we derive restrictions on optimal point masses and their positions. These restrictions are strong enough to calculate optimal $d M$ if the optimal resonance $\omega$, the first point mass $m_1$, and one more geometric parameter are known. This reduces the original infinitely-dimensional problem to optimization over four real parameters. For small frequencies, we explicitly find the Pareto set and the corresponding optimal measures $d M$. The technique of the paper is based on the two-parameter perturbation method and the notion of local boundary point. The latter is introduced as a generalization of local extrema to vector optimization problems.