A Closed-Form Feedback Controller for Stabilization of Linearized Navier-Stokes Equations: The 2D Poisseuille Flow

We present a formula for a boundary control law which stabilizes the parabolic profile of an infinite channel flow, which is linearly unstable for high Reynolds numbers. Also known as the Poisseuille flow, this problem is frequently cited as a paradigm for transition to turbulence, whose stabilization for arbitrary Reynolds numbers, without using discretization, has so far been an open problem. Our result achieves exponential stability in the L²norm for the linearized Navier-Stokes equations, guaranteeing local stability for the fully nonlinear system. Explicit solutions are obtained for the closed loop system. This is the first time explicit formulae are produced for solutions of the Navier-Stokes equations. The result is presented for the 2D case for clarity of exposition. An extension to 3D is available and will be presented in a future publication.