Some Controllability Results for Linearized Compressible Navier-Stokes System with Maxwell's Law

TL;DR

This paper investigates the controllability of a linearized compressible Navier-Stokes system with Maxwell's law, revealing conditions under which the system is controllable or not, and providing a comprehensive analysis of its controllability properties.

Contribution

It establishes the null and approximate controllability results for the linearized system, identifying the precise conditions for controllability with localized or global controls.

Findings

System not null controllable with localized controls in density and stress

System is null controllable with global controls in density or stress

System is approximately controllable at large time with localized controls

Abstract

In this article, we study the control aspects of the one-dimensional compressible Navier-Stokes equations with Maxwell's law linearized around a constant steady state with zero velocity. We consider the linearized system with Dirichlet boundary conditions and with interior controls. We prove that the system is not null controllable at any time using localized controls in density and stress equations and even everywhere control in the velocity equation. However, we show that the system is null controllable at any time if the control acting in the density or stress equation is everywhere in the domain. This is the best possible null controllability result obtained for this system. Next, we show that the system is approximately controllable at large time using localized controls. Thus, our results give a complete understanding of the system in this direction.