On the observability of abstract time-discrete linear parabolic equations, in "Rev

This article aims at analyzing the observability properties of time-discrete approximation schemes of abstract parabolic equations ż+Az = 0, where A is a self-adjoint positive definite operator with dense domain and compact resolvent. We analyze the observability properties of these diffusive systems for an observation operator B ∈ L(D(Aν), Y) with ν < 1/2. Assuming that the continuous system is observable, we prove uni-form observability results for suitable time-discretization schemes within the class of conveniently filtered data. We also propose a HUM type al-gorithm to compute discrete approximations of the exact controls. Our approach also applies to sequences of operators which are uniformly ob-servable. In particular, our results can be combined with the existing ones on the observability of space semi-discrete systems, yielding observability properties for fully discrete approximation schemes