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

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.