Uniform controllability properties for space/time-discretized parabolic equations

Abstract

This article is concerned with the analysis of semi-discrete-in-space and fully-discrete approximations of the null controllability (and controllability to the trajectories) for parabolic equations. We propose an abstract setting for space discretizations that potentially encompasses various numerical methods and we study how the controllability problems depend on the discretization parameters. For time discretization we use θ -schemes with θ ∈ [ 1 2 , 1]. For the proofs of controllability we rely on the strategy introduced by Lebeau and Robbiano (Comm Partial Differ Equ 20:335-356, 1995) for the null-controllability of the heat equation, which is based on a spectral inequality. We obtain relaxed uniform observability estimates in both the semi-discrete and fully-discrete frameworks, and associated uniform controllability properties. For the practical computation of the control functions we follow J.-L. Lions' Hilbert Uniqueness Method strategy, exploiting the relaxed uniform The three authors were partially supported by l'Agence Nationale de la Recherche under grant ANR-07-JCJC-0139-01. The CNRS Pticrem project facilitated the writing of this article.

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.…”

A mixed formulation for the direct approximation of L 2-weighted controls for the linear heat equation

“…This fact and the hugeness of H has raised many authors to relax the controllability problem: precisely, the constraint (2). We mention the references [5,3,35] and notably [2,18,25] for some numerical realizations.…”

On the Numerical Controllability of the Two-Dimensional Heat, Stokes and Navier–Stokes Equations

“…Moreover, in (1), we assume that G ∈ L ∞ (Q T ); in (2) and (3), ν > 0. Let us first consider the system (1).…”

“…Let us recall the definitions of some usual spaces in the context of incompressible fluids: For any y 0 ∈ H, T > 0 and v ∈ L 2 (q T ), there exists exactly one solution (y, π) to the Stokes equations (2) and (since we are in the 2D case), also one solution (y, π) to the Navier-Stokes equations (3). In both cases…”

“…For instance, if we look for a minimal L 2 norm null control for (1), we are led by duality to an unconstrained extremal problem in a huge space that cannot be approximated efficiently with usual finite dimensional spaces; see however [2,1] for detailed comments on this issue.…”