Convection-Enabled Boundary Control of a 2D Channel Flow

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.