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

In this article, we study the boundary null-controllability of the one-dimensional linearized compressible Navier-Stokes equations in the interval $(0,1)$ with a Dirichlet boundary control only on the density component and homogeneous Dirichlet boundary conditions on the velocity component. We prove that the linearized system is null-controllable in $\dot{H}^s_{per}(0,1) \times L^2(0,1)$ for any $s > 1/2$ provided the time $T > 1$, where $\dot{H}^s_{per}(0,1)$ denotes the Sobolev space of periodic functions with zero mean value. As it is well-known, the time restriction is arising due to the presence of a transport equation in the system. The proof for the null-controllability is based on solving a mixed parabolic-hyperbolic moments problem. To apply this, we perform a spectral analysis of the associated adjoint operator which is the main involved part of this work.