Séminaire de GéométrieOn the asymptotic of observability constants for wave like equations
vendredi 15 juin 2018 14:00 - Tours - Salle 1180 (Bât E2)
In this talk, we investigate asymptotic properties of the observability constants for wave equations as the observation time T tends to $+\infty$. We first consider a simplified one-dimensional wave equation involving a Sturm-Liouville operator and provide upper and lower estimates of the time observability constant in terms of the observation set Lebesgue measure. More generally, given a wave equation on a manifold without boundary, and given an arbitrary observation subset, we prove that the time-asymptotic observability constant is the minimum of two quantities: the first is a purely spectral one, and contains information on the quantum ergodic properties of the manifold; the second is purely geometric and gives an account for rays propagating in the manifold. We discuss some applications to control theory and shape optimization.