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

TL;DR

This paper demonstrates a second order obstruction to small time local null controllability for the viscous Burgers' equation, despite infinite information propagation speed, using a quadratic expansion and weak norm estimates.

Contribution

It introduces a novel second order obstruction to controllability for the viscous Burgers' equation, beyond classical Lie bracket conditions.

Findings

Obstruction involves the weak H^{-5/4} norm of control

Classical Lie bracket conditions fail to detect this obstruction

Proof uses integral kernel operator and weakly singular integral estimates

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.