Fast global null controllability for a viscous Burgers' equation despite the presence of a boundary layer

TL;DR

This paper demonstrates that the viscous Burgers' equation can be globally null controlled in small time despite boundary layer challenges, by leveraging hyperbolic-parabolic properties and advanced analytical techniques.

Contribution

It introduces a novel control strategy combining Cole-Hopf transform and Fourier series to achieve small time global null controllability despite boundary layer effects.

Findings

Successful control within small time frame

Effective boundary layer management using analytical methods

Extension of controllability results to viscous Burgers' equation

Abstract

In this work, we are interested in the small time global null controllability for the viscous Burgers' equation y_t - y_xx + y y_x = u(t) on the line segment [0,1]. The second-hand side is a scalar control playing a role similar to that of a pressure. We set y(t,1) = 0 and restrict ourselves to using only two controls (namely the interior one u(t) and the boundary one y(t,0)). In this setting, we show that small time global null controllability still holds by taking advantage of both hyperbolic and parabolic behaviors of our system. We use the Cole-Hopf transform and Fourier series to derive precise estimates for the creation and the dissipation of a boundary layer.