An obstruction to small-time  local null controllability  for a viscous Burgers' equation

Marbach, Frédéric

doi:10.24033/asens.2373

An obstruction to small-time local null controllability for a viscous Burgers' equation
[Obstruction à la contrôlabilité locale en temps petit pour une équation de Burgers visqueuse]

Marbach, Frédéric

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1129-1177.

Résumé
Abstract

Nous nous intéressons à la contrôlabilité locale en temps petit pour l'équation de Burgers visqueuse $y_{t} - y_{x x} + y y_{x} = u (t)$ , posée sur un segment, avec des conditions de Dirichlet nulles au bord. Le terme source au second membre est un contrôle scalaire qui joue un rôle similaire à celui d'une pression. Dans ce contexte, la condition de crochet de Lie nécessaire classique introduite par Sussmann ne permet pas de conclure. Cependant, en utilisant un développement à l'ordre deux du système étudié, nous mettons en lumière une obstruction de nature quadratique à la contrôlabilité locale en temps petit. Cette obstruction tient alors même que la vitesse de propagation de l'information dans cette équation de Burgers est infinie. Elle fait intervenir la norme $H^{- 5 / 4}$ du contrôle. La démonstration nécessite le calcul soigneux du noyau d'un opérateur intégral, ainsi que l'estimation d'opérateurs résiduels à l'aide de la théorie de régularité pour les opérateurs intégraux faiblement singuliers.

In this work, we are interested in the small-time local null controllability for the viscous Burgers' equation $y_{t} - y_{x x} + y y_{x} = u (t)$ on a line segment, with null boundary conditions. The second-hand side is a scalar control playing a role similar to that of a pressure. In this setting, the classical Lie bracket necessary condition introduced by Sussmann fails to conclude. However, using a quadratic expansion of our system, we exhibit a second order obstruction to small-time local null controllability. This obstruction holds although the information propagation speed is infinite for the Burgers equation. Our obstruction involves the $H^{- 5 / 4}$ norm of the control. The proof requires the careful derivation of an integral kernel operator and the estimation of residues by means of weakly singular integral operator estimates.

Publié le : 2018-11-12

DOI : 10.24033/asens.2373

Classification : 93B05, 93C20, 47G10.
Keywords: Burgers, controllability, quadratic obstruction, weakly singular integral operator.
Mot clés : Burgers, contrôlabilité, obstruction quadratique, opérateur intégral faiblement singulier.

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     author = {Marbach, Fr\'ed\'eric},
     title = {An obstruction to small-time  local null controllability  for a viscous {Burgers'} equation},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {1129--1177},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 51},
     number = {5},
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     mrnumber = {3942039},
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Marbach, Frédéric. An obstruction to small-time  local null controllability  for a viscous Burgers' equation. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 51 (2018) no. 5, pp. 1129-1177. doi : 10.24033/asens.2373. http://www.numdam.org/articles/10.24033/asens.2373/

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