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. 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 ω 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 ω 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 Ω, 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 Ω 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. * IRMAR, ENS Cachan Bretagne, Univ. Rennes 1, CNRS, UEB, av. Robert Schuman,

In this article, we consider parabolic equations on a bounded open connected subset Ω of IR n . We model and investigate the problem of optimal shape and location of the observation domain having a prescribed measure. This problem is motivated by the question of knowing how to shape and place sensors in some domain in order to maximize the quality of the observation: for instance, what is the optimal location and shape of a thermometer?We show that it is relevant to consider a spectral optimal design problem corresponding to an average of the classical observability inequality over random initial data, where the unknown ranges over the set of all possible measurable subsets of Ω of fixed measure. We prove that, under appropriate sufficient spectral assumptions, this optimal design problem has a unique solution, depending only on a finite number of modes, and that the optimal domain is semi-analytic and thus has a finite number of connected components. This result is in strong contrast with hyperbolic conservative equations (wave and Schrödinger) studied in [56] for which relaxation does occur.We also provide examples of applications to anomalous diffusion or to the Stokes equations. In the case where the underlying operator is any positive (possible fractional) power of the negative of the Dirichlet-Laplacian, we show that, surprisingly enough, the complexity of the optimal domain may strongly depend on both the geometry of the domain and on the positive power.The results are illustrated with several numerical simulations.

In this paper, we consider the homogeneous one-dimensional wave equation on [0, π] with Dirichlet boundary conditions, and observe its solutions on a subset ω of [0, π]. Let L ∈ (0, 1). We investigate the problem of maximizing the observability constant, or its asymptotic average in time, over all possible subsets ω of [0, π] of Lebesgue measure Lπ. We solve this problem by means of Fourier series considerations, give the precise optimal value and prove that there does not exist any optimal set except for L = 1/2. When L = 1/2 we prove the existence of solutions of a convexified minimization problem, proving a no gap result. We then provide and solve a modal approximation of this problem, show the oscillatory character of the optimal sets, the so called spillover phenomenon, which explains the lack of existence of classical solutions for the original problem.