  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.
