On the local null controllability of a viscous Burgers' system in finite time

This paper is devoted to the local null controllability of the Burgers control system $y_t-y_{xx}+y{y}_x=u(t)$ on a bounded interval imposed by the zero Dirichlet boundary condition. It is known from the work of Marbach that this control system is not locally null controllable in small time. In this paper, we prove that the system is not locally null controllable in finite time as well. Our approach is inspired by the works of Coron, Koenig, and Nguyen, and Nguyen on the controllability of the KdV system and is different from the one of Marbach.