Stabilisation of a viscous conservation law by a one-dimensional external force

TL;DR

This paper demonstrates that a viscous conservation law with external forcing can be exponentially stabilized and globally controlled using a localized one-dimensional control in a bounded domain.

Contribution

It introduces a method to stabilize and control a viscous conservation law with external force using a localized control, extending previous results to this specific setting.

Findings

Exponential stabilization of solutions achieved

Global approximate controllability to trajectories established

Control implemented via a localized one-dimensional input

Abstract

We study a damped scalar conservation law driven by the sum of a fixed external force and a localised one-dimensional control. The problem is considered in a bounded domain and is supplemented with the Dirichlet boundary condition. It is proved that any solution of the uncontrolled equation can be exponentially stabilised. As a consequence, we obtain the global approximate controllability to trajectories by a one-dimensional localised control.