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

Mehdi Badra; Takéo Takahashi

ESAIM: Control, Optimisation and Calculus of Variations (2014)

  • Volume: 20, Issue: 3, page 924-956
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Badra, Mehdi, and Takahashi, Takéo. "On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems." ESAIM: Control, Optimisation and Calculus of Variations 20.3 (2014): 924-956. <http://eudml.org/doc/272838>.

@article{Badra2014,
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.},
author = {Badra, Mehdi, Takahashi, Takéo},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {approximate controllability; stabilizability; parabolic equation; finite dimensional control; coupled−Stokes and mhd system; coupled-Stokes and MHD system},
language = {eng},
number = {3},
pages = {924-956},
publisher = {EDP-Sciences},
title = {On the Fattorini criterion for approximate controllability and stabilizability of parabolic systems},
url = {http://eudml.org/doc/272838},
volume = {20},
year = {2014},
}

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
PB - EDP-Sciences
VL - 20
IS - 3
SP - 924
EP - 956
AB - 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.
LA - eng
KW - approximate controllability; stabilizability; parabolic equation; finite dimensional control; coupled−Stokes and mhd system; coupled-Stokes and MHD system
UR - http://eudml.org/doc/272838
ER -

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