A sufficient condition for observability of waves by measurable subsets
Résumé
We consider the wave equation on a closed Riemannian manifold $(M,g)$. Given a measurable subset $\omega$ of $M$ and given $T>0$, we establish that, if the metric $g$ is of class $C^2$ and if $\omega$ is regular enough, if the average time over $[0,T]$ of geodesic rays crossing $\overline\omega$ is greater than $1/2$ then the Geometric Control Condition is satisfied and thus the wave equation is observable on $(\omega,T)$. Our proof relies on measuring the discrepancy of this average time in $\overline\omega$ with respect to $\mathring{\omega}$. We show that our assumptions are essentially sharp.
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