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In this article, we study boundary null controllability properties of thelinearized compressible Navier-Stokes equations in the interval $(0,2\pi)$ forboth barotropic and non-barotropic fluids using only one boundary control. Weconsider all the possible cases of the act of control for both systems(density, velocity and temperature). These controls are acting on the boundaryand are given as the difference of the values at $x=0$ and $x=2\pi$. In thissetup, using a boundary control acting in density, we first prove nullcontrollability of both the barotropic and non-barotropic systems at large timein the spaces $(\dot{L}^2(0,2\pi))^2$ and $(\dot{L}^2(0,2\pi))^3$ respectively(where the dot represents functions with mean value zero). When the control isacting in the velocity component, we prove null controllability at large timein the spaces $\dot{H}^1_{\text{per}}(0,2\pi)\times\dot{L}^2(0,2\pi)$ and$\dot{H}^1_{\text{per}}(0,2\pi)\times(\dot{L}^2(0,2\pi))^2$ respectively.Further, in both cases, we prove that these null controllability results aresharp with respect to the regularity of the initial states in velocity/temperature case, and time in the density case. Finally, for both barotropicand non-barotropic fluids, we prove that, under some assumptions, the systemcannot be approximately controllable at any time, whether there is a controlacting in density, velocity or temperature.

Null controllability of one-dimensional barotropic and non-barotropic linearized compressible Navier-Stokes system using one boundary control