Some Controllability Results for Linearized Compressible Navier-Stokes System

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 $T\mathrm{.}$ 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.