  In this article, we study the local exact controllability to a constant
trajectory for a compressible Navier-Stokes-Korteweg system on the torus in
dimension $ d\in\{1,2,3\}$ when the control acts on an open subset. To be more
precise, we obtain the local exact controllability to the constant state
$(\rho_{\star},0)$ for arbitrary small positive times and without any geometric
condition on the control region. In order to do so, we analyze the control
properties of the linearized equation, and present a detailed study of the
observability of the adjoint equations. In particular, we shall exhibit the
parabolic (possibly also dispersive) structure of these adjoint equations.
Based on that, we will be able to recover observability of the adjoint system
through Carleman estimates.
