Asymptotic observable symmetry for spacetime observability of waves II: Smooth bounded domains

Abstract

Continuing \cite{NWX} and Part I \cite{AOSI}, we further investigate symmetry for spacetime observability of wave equations. Building on asymptotic observable symmetry (AOS) introduced in \cite{AOSI}, we study symmetry under glancing propagation and caustics. We prove that, on every smooth bounded domain in ${\mathbb{R}}^2$ and for every positive observation time, there is a fixed measurable set satisfying the essential spacetime measurable geometric control condition. However, the observability inequality fails on this set.