  We consider the incompressible Navier-Stokes equations in a two-dimensional
channel. The tangential and normal velocities are assumed to be periodic in the
streamwise (horizontal) direction. Moreover, we consider no-slip boundary
conditions on the tangential velocity at the top and bottom walls of the
channel, and normal velocity actuation at the top and bottom walls. For an
arbitrarily large Reynolds number, we design the boundary control inputs to
achieve global exponential stabilization, in the L2 sense, of a chosen
parabolic Poiseuille profile. Moreover, we design the control inputs such that
they have zero mean, but non-zero cubic mean. The zero-mean property is to
ensure that the conservation of mass constraint is verified. The non-zero cubic
mean property is the key to exploiting the stabilizing effect of nonlinear
convection and achieving global stabilization independently of the size of the
Reynolds number. This paper is not only the first work where a closed-form
feedback law is proposed for global stabilization of parabolic Poiseuille
profiles for arbitrary Reynolds number but is also the first generalization of
the Cardano-Lyapunov formula, designed initially to stabilize scalar-valued
convective PDEs, to a vector-valued convective PDE with a divergence-free
constraint on the state.
