Null controllability of the Burgers system with distributed controls

This paper is devoted to present some positive and negative controllability results for the viscous Burgers equation. More precisely, in the context of null controllability with distributed controls, we present sharp estimates of the minimal time of controllability T(r)T(r) of initial data of L2L2-norm less or equal to r. In particular, we see that (global) null controllability does not hold in general (unless the control is exerted everywhere). The same results apply to similar boundary controllability systems with one boundary control.