  We deal with the 3D Navier-Stokes equation in a smooth simply connected
bounded domain, with controls on a non-empty open part of the boundary and a
Navier slip-with-friction boundary condition on the remaining, uncontrolled,
part of the boundary. We extend the small-time global exact controllability
result in [J. M. Coron, F. Marbach and F. Sueur, Small-time global exact
controllability of the Navier-Stokes equation with Navier slip-with-friction
boundary conditions, J. Eur. Math. Soc., 22 (2020),1625-1673.] from Leray weak
solutions to the case of smooth solutions.
  Our strategy relies on a refinement of the method of well-prepared
dissipation of the viscous boundary layers which appear near the uncontrolled
part of the boundary, which allows to handle the multi-scale features in a
finer topology.
  As a byproduct of our analysis we also obtain a small-time global approximate
Lagrangian controllability result, extending to the case of the Navier-Stokes
equations the recent results [O. Glass and T. Horsin, Approximate Lagrangian
controllability for the 2-D Euler equation. Application to the control of the
shape of vortex patches, J. Math. Pures Appl. (9), 93 (2010), 61-90], [O. Glass
and T. Horsin, Prescribing the motion of a set of particles in a
three-dimensional perfect fluid, SIAM J. Control Optim., 50 (2012), 2726-2742],
[T. Horsin and O. Kavian, Lagrangian controllability of inviscid incompressible
fluids: a constructive approach, ESAIM Control Optim. Calc. Var., 23 (2017),
1179-1200] in the case of the Euler equations and the result [O. Glass and T.
Horsin, Lagrangian controllability at low Reynolds number, ESAIM Control Optim.
Calc. Var., 22 (2016), 1040-1053] in the case of the steady Stokes equations.
