Some Controllability Results for Linearized Compressible Navier-Stokes System

TL;DR

This paper investigates the controllability of linearized compressible Navier-Stokes systems in one and two dimensions, showing limitations of control methods and conditions under which controllability is achievable.

Contribution

It provides new results on the null controllability of linearized compressible Navier-Stokes equations, highlighting when control is possible or impossible.

Findings

One-dimensional non-barotropic system controllable with full interior control.

Non-controllability of localized controls in certain cases.

Two-dimensional barotropic system not controllable by localized interior control.

Abstract

In this article, we study the null controllability of linearized compressible Navier-Stokes system in one and two dimension. We first study the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state. We prove that the linearized system around $(\bar \rho,0,\bar \theta)$, with $\bar \rho > 0,$ $ \bar \theta > 0$ is not null controllable by localized interior control or by boundary control. But the system is null controllable by interior controls acting everywhere in the velocity and temperature equation for regular initial condition. We also prove that the the one-dimensional compressible Navier-Stokes system for non-barotropic fluid linearized around a constant steady state $(\bar \rho,\bar v ,\bar \theta)$, with $\bar \rho > 0,$ $\bar v > 0,$ $\bar \theta > 0$ is not null controllable by localized interior control or by boundary control for small time $T.$ Next we consider two-dimensional compressible Navier-Stokes system for barotropic fluid linearized around a constant steady state $(\bar \rho, {\bf 0}).$ We prove that this system is also not null controllable by localized interior control.