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

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.