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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.
@article{COCV_2014__20_3_924_0,
author = {Badra, Mehdi and Takahashi, Tak\'eo},
title = {On the {Fattorini} criterion for approximate controllability and stabilizability of parabolic systems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {924--956},
publisher = {EDP-Sciences},
volume = {20},
number = {3},
year = {2014},
doi = {10.1051/cocv/2014002},
mrnumber = {3264229},
zbl = {1292.93022},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/cocv/2014002/}
}
TY - JOUR AU - Badra, Mehdi AU - Takahashi, Takéo TI - On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2014 SP - 924 EP - 956 VL - 20 IS - 3 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/cocv/2014002/ DO - 10.1051/cocv/2014002 LA - en ID - COCV_2014__20_3_924_0 ER -
%0 Journal Article %A Badra, Mehdi %A Takahashi, Takéo %T On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems %J ESAIM: Control, Optimisation and Calculus of Variations %D 2014 %P 924-956 %V 20 %N 3 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/cocv/2014002/ %R 10.1051/cocv/2014002 %G en %F COCV_2014__20_3_924_0
Badra, Mehdi; Takahashi, Takéo. On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 20 (2014) no. 3, pp. 924-956. doi: 10.1051/cocv/2014002
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