Local exact controllability to constant trajectories for Navier-Stokes-Korteweg system

This talk is devoted to the study of controllability properties for a compressible Navier–Stokes–Korteweg system. I will present a result on local exact controllability to a constant equilibrium state, when the control is localized on the boundary of an open subset of $\R^d$, with $d\in\{1,2,3\}. The controllability result holds for arbitrarily small positive times and does not require any geometric condition on the control region. These properties contrast with control results for the compressible Navier–Stokes equations. The main idea is to exploit the parabolic effects generated by capillarity terms. In this talk, I will explain how these effects can be exploited within the framework of control theory.