# Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains

Privat Y., Trélat E., Zuazua E. J EUR MATH SOC. Vol. 18, No. 5 (2016), pp. 1043–1111 DOI: 10.4171/JEMS/608

Abstract: We consider the wave and Schrödinger equations on a bounded open connected subset Ω of a Riemannian manifold, with Dirichlet, Neumann or Robin boundary conditions whenever its boundary is nonempty. We observe the restriction of the solutions to a measurable subset ω of Ω during a time interval [0,T] with T>0. It is well known that, if the pair (ω,T) satisfies the Geometric Control Condition (ω being an open set), then an observability inequality holds guaranteeing that the total energy of solutions can be estimated in terms of the energy localized in ω×(0,T).
We address the problem of the optimal location of the observation subset ω among all possible subsets of a given measure or volume fraction. A priori this problem can be modeled in terms of maximizing the observability constant, but from the practical point of view it appears more relevant to model it in terms of maximizing an average either over random initial data or over large time. This leads us to define a new notion of observability constant, either randomized, or asymptotic in time. In both cases we come up with a spectral functional that can be viewed as a measure of eigenfunction concentration. Roughly speaking, the subset ω has to be chosen so to maximize the minimal trace of the squares of all eigenfunctions. Considering the convexified formulation of the problem, we prove a no-gap result between the initial problem and its convexified version, under appropriate quantum ergodicity assumptions, and compute the optimal value. Our results reveal intimate relations between shape and domain optimization, and the theory of quantum chaos (more precisely, quantum ergodicity properties of the domain Ω).
We prove that in 1D a classical optimal set exists only for exceptional values of the volume fraction, and in general one expects relaxation to occur and therefore classical optimal sets not to exist. We then provide spectral approximations and present some numerical simulations that fully confirm the theoretical results in the paper and support our conjectures.
Finally, we provide several remedies to nonexistence of an optimal domain. We prove that when the spectral criterion is modified to consider a weighted one in which the high frequency components are penalized, the problem has then a unique classical solution determined by a finite number of low frequency modes. In particular the maximizing sequence built from spectral approximations is stationary.