Higher Order Stability Properties of a 2D Navier Stokes System with an Explicit Boundary Controller

In a previous work, we presented formulae for boundary control laws which stabilized the parabolic profile of an infinite channel flow, linearly unstable for high Reynolds number. Also know as the Poiseuille flow, this problem is frequently cited as a paradigm for transition to turbulence, whose stabilization for arbitrary Reynolds number, without using discretization, had so far been an open problem. L₂ stability was proved for the closed loop system. In this work, we extend the stability result to exponential stability in the H₁ and H ₂ norms, and we state and prove some properties of the stabilizing controller, guaranteeing that the control law is well behaved