Regional and partial observability and control of waves

We establish sharp regional observability results for solutions of the wave equation in a bounded domain of $\Omega \subset \mathbb{R}^n$, in case where the geometric control condition is not satisfied. Assuming that the waves are observed on a non-empty open subset $\omega\subset \Omega$ and that the initial data are supported in another open subset $\mathscr{O} \subset\Omega$, we derive estimates for the energy of initial data localized in $\mathscr{O}$, in terms of the energy measured on the observation set $(0,T) \times \omega$. This holds under a suitable geometric condition relating the time horizon T and the subdomains $\omega$ and $\mathscr{O}$. By duality, we obtain new controllability results for the wave equation, ensuring that the projection of the solution onto $\mathscr{O}$ can be controlled by means of controls supported in $\omega$, with optimal spatial support. We also present several extensions of the main result, including the case of boundary observations, as well as a characterization of the observable fraction of the energy of the initial data from partial measurements on $(0,T) \times \omega$. Applications to wave control are discussed accordingly.