Global exponential stabilisation for the Burgers equation with localised control

Abstract

We consider the 1D viscous Burgers equation with a control localised in a finite interval. It is proved that, for any $\epsilon >0$, one can find a time $T$ of order $\log {\epsilon }^{-1}$ such that any initial state can be steered to the $\epsilon$-neighbourhood of a given trajectory at time $T$. This property combined with an earlier result on local exact controllability shows that the Burgers equation is globally exactly controllable to trajectories in a finite time. We also prove that the approximate controllability to arbitrary targets does not hold even if we allow infinite time of control.

19 pages