that the weak observation properties derived in [@Phung09] for the waves in general geometric setting (not necessarily satisfying the geometric control conditions) can be used to recover control properties for the heat operator. Let us also point out that these transmutation techniques can also be used to derive numerical schemes to compute approximate controls for the heat equations [@MunchZuazua2010].

We also emphasize that Theorem \[Thm-Main\] states observability results for the time-discrete schemes observed through an observation operator ${{\mathcal{B}}}$ provided the continuous model is observable through ${{\mathcal{B}}}$. Hence the first step is to check the observability property for the continuous model, and these observability properties have to be checked in each situation. When the continuous model stands for a wave equation and ${{\mathcal{B}}}$ is a distributed observation operator or a boundary observation, the necessary and sufficient condition for observability is the so-called geometric control condition, see [@Bardos; @BurqGerard]. For what concerns plate or Schrödinger’s equations, the geometry still plays a role, but due to the infinite speed of propagation the situation is more intricate: for these models, observability holds when the geometric control condition is satisfied (see e.g. [@Miller2004] where this result is derived using transmutation techniques), but other less restrictive 