Global exponential stabilisation for the Burgers equation with localised control
Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 613-632.

We consider the 1D viscous Burgers equation with a control localised in a finite interval. It is proved that, for any ε>0, one can find a time T of order logε -1 such that any initial state can be steered to the ε-neighbourhood of a given trajectory at time T. This property combined with an earlier result on local exact controllability shows that the Burgers equation is globally exactly controllable to trajectories in a finite time that does not depend on the initial conditions.

Nous considérons l’équation de Burgers visqueuse 1D avec un contrôle localisé dans un intervalle fini. Nous montrons que, pour tout ε>0, on peut trouver un temps T d’ordre logε -1 tel que tout état initial peut être amené dans un ε-voisinage d’une trajectoire donnée au temps T. Cette propriété, jointe à un résultat précédent de contrôle local exact, montre que l’équation de Burgers est globalement exactement contrôlable vers les trajectoires en un temps fini qui ne dépend pas des conditions initiales.

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DOI: 10.5802/jep.53
Classification: 35L65, 35Q93, 93C20
Keywords: Burgers equation, exponential stabilisation, localised control, Harnack inequality
Armen Shirikyan 1

1 Département de mathématiques, Université de Cergy–Pontoise, CNRS UMR 8088 2 avenue Adolphe Chauvin, 95302 Cergy–Pontoise, France and
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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Armen Shirikyan. Global exponential stabilisation for the Burgers equation with localised control. Journal de l’École polytechnique — Mathématiques, Volume 4 (2017), pp. 613-632. doi : 10.5802/jep.53. https://jep.centre-mersenne.org/articles/10.5802/jep.53/

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