Global controllability of the micropolar fluids with Navier slip-with-friction and Dirichlet boundary conditions

In this paper, we study the global controllability towards the trajectories of the micropolar fluids in a smooth bounded 2D or 3D domain. The controls only act on a small part of the boundary which intersects all its connected components. On the remaining part of the boundary, the vector field obeys Navier slip-with-friction boundary conditions and the angular velocity obeys Dirichlet boundary conditions. Our strategy is to extend the original domain and thus transform the boundary control into a local internal control problem of the extended domain. Adapting the method introduced by Coron et al. (2020) [15], we establish a global approximate controllability result by using the Return Method, the well-prepared dissipation method and the analysis of appropriate asymptotic boundary layer expansions. We also obtain the local controllability to trajectories for the micropolar system by a new Carleman inequality and Kakutani's fixed point theorem.