Control for fast and stable laminar-to-high-Reynolds-numbers transfer in a 2D Navier-Stokes channel flow

Abstract

We consider the problem of generating and tracking a trajectory between two arbitrary parabolic profiles of a periodic 2D channel flow, which is linearly unstable for high Reynolds numbers. Also known as the Poiseuille flow, this problem is frequently cited as a paradigm for transition to turbulence. Our procedure consists in generating an exact trajectory of the nonlinear system that approaches exponentially the objective profile. Using a backstepping method, we then design boundary control laws guaranteeing that the error between the state and the trajectory decays exponentially in L2 , H1 , and H2 norms. The result is first proved for the linearized Stokes equations, then shown to hold locally for the nonlinear Navier-Stokes system.