Global exponential stabilisation for the Burgers equation with localised control

Shirikyan, Armen

Journal de l’École polytechnique — Mathématiques, Tome 4 (2017), pp. 613-632.

Nous considérons l’équation de Burgers visqueuse 1D avec un contrôle localisé dans un intervalle fini. Nous montrons que, pour tout $\epsilon >0$, on peut trouver un temps $T$ d’ordre $log{\epsilon }^{-1}$ tel que tout état initial peut être amené dans un $\epsilon$-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.

We consider the 1D viscous Burgers equation with a control localised in a finite interval. It is proved that, for any $\epsilon >0$, one can find a time $T$ of order $log{\epsilon }^{-1}$ such that any initial state can be steered to the $\epsilon$-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.

Classification : 35L65, 35Q93, 93C20
Keywords: Burgers equation, exponential stabilisation, localised control, Harnack inequality
Mot clés : Équation de Burgers, stabilisation exponentielle, contrôle localisé, inégalité de Harnack

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