Global exponential stabilisation for the Burgers equation with localised control

TL;DR

This paper demonstrates that the 1D viscous Burgers equation can be globally exponentially stabilized to a trajectory using localized control within finite time, but exact controllability to arbitrary targets is impossible even with infinite time.

Contribution

It proves global exponential stabilization and controllability results for the Burgers equation with localized control, extending previous local controllability findings.

Findings

Global exponential stabilization to trajectories in finite time.

Exact controllability to arbitrary targets is impossible, even with infinite time.

Control time scales logarithmically with the desired accuracy.

Abstract

We consider the 1D viscous Burgers equation with a control localised in a finite interval. It is proved that, for any $\varepsilon>0$, one can find a time $T$ of order $\log\varepsilon^{-1}$ such that any initial state can be steered to the $\varepsilon$-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.