Optimal observability of the multi-dimensional wave and Schrödinger equations in quantum ergodic domains
Résumé
We consider the wave and Schrödinger equations on a bounded open connected subset $\Omega$ 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 $\omega$ of $\Omega$ during a time interval $[0, T]$ with $T>0$. It is well known that, if the pair $(\omega,T)$ satisfies the Geometric Control Condition ($\omega$ 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 $\omega \times (0, T)$. We address the problem of the optimal location of the observation subset $\omega$ among all possible subsets of a given measure or volume fraction. We solve it in two different situations. First, when a specific choice of the initial data is given and therefore we deal with a particular solution, we show that the problem always admits at least one solution that can be regular or of fractal type depending on the regularity of the initial data. This first problem of finding the optimal $\omega$ for each initial datum is a mathematical benchmark but, in view of applications, it is important to define a relevant criterion, not depending on the initial conditions and to choose the observation set in an uniform way, independent of the data and solutions under consideration. Through spectral decompositions, this leads to a second problem which consists of maximizing a spectral functional that can be viewed as a measure of eigenfunction concentration. Roughly speaking, the subset $\omega$ has to be chosen so to maximize the minimal trace of the squares of all eigenfunctions. This spectral criterion can be obtained and interpreted in two ways: on the one hand, it corresponds to a time asymptotic observability constant as the observation time interval tends to infinity, and on the other hand, to a randomized version of the deterministic observability inequality. We also consider 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 on $\Omega$, and compute the optimal value. We also give several examples in which a classical optimal set exists, although, as it happens in 1D, generically with respect to the manifold $\Omega$ and the volume fraction, 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. Our results highlight precise connections between optimal observability issues and quantum ergodic properties of the domain under consideration.
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