On the observability of abstract time-discrete linear parabolic equations

Abstract

This article aims at analyzing the observability properties of time-discrete approximation schemes of abstract parabolic equationsż + Az = 0, where A is a selfadjoint 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(A ν ), Y ) with ν < 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. Mathematics Subject Classification (2000)35K05 · 93B05 · 93B07 · 93B40 · 93C55

Uniform controllability properties for space/time-discretized parabolic equations

“…A filtering of the high frequencies is required (in the spectral representation of the continuous Laplace operator) and the convergence results obtained may seem not optimal: a 1 2 convergence order is found for a first-order scheme, namely the implicit Euler scheme. More interesting is the result of [4], where the authors prove that any controllable parabolic equation, be it discrete or continuous in space, is null controllable after time discretization upon the application of an appropriate filtering of the high-frequencies (in the spectral representation of the continuous or discrete Laplace operator). In [4] there is however no study of the convergence of the control function as the time step goes to zero.…”

“…More interesting is the result of [4], where the authors prove that any controllable parabolic equation, be it discrete or continuous in space, is null controllable after time discretization upon the application of an appropriate filtering of the high-frequencies (in the spectral representation of the continuous or discrete Laplace operator). In [4] there is however no study of the convergence of the control function as the time step goes to zero.…”

Carleman estimates for time-discrete parabolic equations and applications to controllability

“…In [7], the authors prove in a quite general framework that any controllable parabolic equation (even if discretized in the space variable) is null controllable after time discretization by an appropriate filtering of the high-frequencies. In [5], the authors study in a general setting the null-controllability of fullydiscrete approximations for parabolic equations and its convergence rate towards the semi-discrete (in space) case.…”

Some inverse and control problems for fluids

“…The work [8] extends the results in [50] to the fully discrete situation and proves the convergence towards a semi-discrete control, as the time step ∆t tends to zero. Let us also mention [20], where the authors prove that any controllable parabolic equation, be it discrete or continuous in space, is null-controllable after time discretization through the application of an appropriate filtering of the high frequencies. As far as we know, a strong convergence result in the framework of dual method is still missing.…”