Remarks on global controllability for the shallow-water system with two control forces

TL;DR

This paper investigates the controllability of a one-dimensional compressible Navier-Stokes system with boundary controls, demonstrating that exact global controllability cannot be achieved in finite time under certain conditions.

Contribution

It extends the understanding of controllability for fluid dynamics equations by showing limitations in controlling the system globally with boundary forces.

Findings

Exact global controllability does not hold for any finite time.

Controllability results are influenced by initial conditions and boundary control placement.

The study uses irrotational data and effective velocity concepts to derive these results.

Abstract

In this paper we deal with the compressible Navier-Stokes equations with a friction term in one dimension on an interval. We study the exact controllability properties of this equation with general initial condition when the boundary control is acting at both endpoints of the interval. Inspired by the work of Guerrero and Imanuvilov in \cite{GI} on the viscous Burger equation, we prove by choosing irrotational data and using the notion of effective velocity developed in \cite{cpde,cras} that the exact global controllability result does not hold for any time $T$.