  In this article, we study the boundary null-controllability properties of the
one-dimensional linearized (around $(Q_0,V_0)$ with constants $Q_0>0, V_0>0$)
compressible Navier-Stokes equations in the interval $(0,1)$ when a control
function is acting either on the density or velocity component at one end of
the interval. We first prove that the linearized system, with a Dirichlet
boundary control on the density component and homogeneous Dirichlet boundary
conditions on the velocity component, is null-controllable in $H^s_{per}(0,1)
\times L^2(0,1)$ for any $s > 1/2$ provided the time $T > 1$, where
$H^s_{per}(0,1)$ denotes the Sobolev space of periodic functions. The proof is
based on solving a mixed parabolic-hyperbolic moments problem and to do so, we
perform a spectral analysis for the associated adjoint operator which is the
main involved part of this work. As a corollary, we also prove that the system
is approximately controllable in $L^2(0,1) \times L^2(0,1)$ when $T>1$.
  On the other hand, assuming that the density is equal on the two boundary
points w.r.t. time, when a control is applied on the velocity part through a
Dirichlet condition, we can only able to prove that the system is
null-controllable in a strict subspace of finite codimension
$\mathcal{H}\subset H^s_{per}(0,1) \times L^2(0,1)$ for $s>1/2$ when $T>1$.
More precisely, in this case we are able to show that all the eigenfunctions of
the associated adjoint operator are observable for higher frequencies whereas
for the lower frequencies it is hard to conclude anything. A
parabolic-hyperbolic joint Ingham-type inequality which we prove in this
article, leads to an observability inequality in the space $\mathcal{H}^*$ and
the controllability result follows. The significant point is that the moments
method does not yield a better space for the null-controllability when a
control acts on the velocity part.
