It is well known since the pionnering work of Bardos-Lebeau-Rauch 92’ that the observability and control of linear waves on bounded domains of the euclidian space $\mathbb{R}^n$ are governed by the so-called geometric control condition (GCC). It is a microlocal condition, i.e. a property in the cotangent bundle, linking the set $\omega$ on which the control is acting and the generalized bicharacteristics of the wave operator. The aim of this work is to investigate geometries where GCC is not not satisfied. More precisely, given an open set $\omega$, and without assuming GCC, we aim to determine which subsets of data in the energy space can be observed from $\omega$. 

Regional and microlocal results are obtained, as well as corresponding control results, under microlocal conditions much weaker than GCC. 

In doing this, we have to manage in crucial way the central question of unique continuation property.

This talk comes from a joint work with Sylvain Ervedoza ( Univ. Bordeaux ) and Enrique Zuazua ( Univ. Erlangen ) .