On the observability of abstract time-discrete linear parabolic equations

This article aims at analyzing the observability properties of time-discrete approximation schemes of abstract parabolic equations. (z) over dot + 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 is an element of L(D(A(nu)), Y) with nu 1/2. Assuming that the continuous system is observable, we prove uniform observability results for suitable time-discretization schemes within the class of conveniently filtered data. We also propose a HUM type algorithm to compute discrete approximations of the exact controls. Our approach also applies to sequences of operators which are uniformly observable. 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.