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

Abstract : In this work, we are interested in the small time local null controllability for the viscous Burgers' equation $y_t - y_{xx} + y y_x = u(t)$ on the line segment $[0,1]$, 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 $[f_1,[f_1,f_0]]$ 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 weak $H^{-5/4}$ norm of the control $u$. The proof requires the careful derivation of an integral kernel operator and the estimation of residues by means of weakly singular integral operator estimates.
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https://hal.archives-ouvertes.fr/hal-01229493
Contributor : Frédéric Marbach <>
Submitted on : Monday, November 16, 2015 - 4:48:36 PM
Last modification on : Friday, May 21, 2021 - 3:35:08 AM
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Frédéric Marbach. An obstruction to small time local null controllability for a viscous Burgers' equation. Annales Scientifiques de l'École Normale Supérieure, Société mathématique de France, In press, 51 (5), pp.1129-1177. ⟨10.24033/asens.2373⟩. ⟨hal-01229493⟩

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