On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems

¹ LMAP, UMR CNRS 5142, UNIV PAU & PAYS ADOUR, 64013 Pau Cedex, France
mehdi.badra@univ-pau.fr
² Inria, 54600 Villers-lès-Nancy, France
takeo.takahashi@inria.fr
³ Université de Lorraine, IECN, UMR 7502, 54506 Vandoeuvre-lès-Nancy, France
⁴ CNRS, IECN, UMR 7502, 54506 Vandoeuvre-lès-Nancy, France

Received: 20 October 2012
Revised: 8 October 2013

Abstract

In this paper, we consider the well-known Fattorini’s criterion for approximate controllability of infinite dimensional linear systems of type y′ = Ay + Bu. We precise the result proved by Fattorini in [H.O. Fattorini, SIAM J. Control 4 (1966) 686–694.] for bounded input B, in the case where B can be unbounded or in the case of finite-dimensional controls. More precisely, we prove that if Fattorini’s criterion is satisfied and if the set of geometric multiplicities of A is bounded then approximate controllability can be achieved with finite dimensional controls. An important consequence of this result consists in using the Fattorini’s criterion to obtain the feedback stabilizability of linear and nonlinear parabolic systems with feedback controls in a finite dimensional space. In particular, for systems described by partial differential equations, such a criterion reduces to a unique continuation theorem for a stationary system. We illustrate such a method by tackling some coupled Navier−Stokes type equations (MHD system and micropolar fluid system) and we sketch a systematic procedure relying on Fattorini’s criterion for checking stabilizability of such nonlinear systems. In that case, the unique continuation theorems rely on local Carleman inequalities for stationary Stokes type systems.