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In this article, we study the boundary null-controllability properties of theone-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 controlfunction is acting either on the density or velocity component at one end ofthe interval. We first prove that the linearized system, with a Dirichletboundary control on the density component and homogeneous Dirichlet boundaryconditions 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 isbased on solving a mixed parabolic-hyperbolic moments problem and to do so, weperform a spectral analysis for the associated adjoint operator which is themain involved part of this work. As a corollary, we also prove that the systemis 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 boundarypoints w.r.t. time, when a control is applied on the velocity part through aDirichlet condition, we can only able to prove that the system isnull-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 ofthe associated adjoint operator are observable for higher frequencies whereasfor the lower frequencies it is hard to conclude anything. Aparabolic-hyperbolic joint Ingham-type inequality which we prove in thisarticle, leads to an observability inequality in the space $\mathcal{H}^*$ andthe controllability result follows. The significant point is that the momentsmethod does not yield a better space for the null-controllability when acontrol acts on the velocity part.

Boundary null-controllability of 1d linearized compressible Navier-Stokes system by one control force