Stable Poiseuille flow transfer for a Navier-Stokes system

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 Poisseuille flow, this problem is frequently cited as a paradigm for transition to turbulence. Our approach consists in generating an exact trajectory of the nonlinear system that approaches exponentially the objective profile. A boundary control law guarantees then that the error between the state and the trajectory decays exponentially in the L2 norm. The result is first proved for the linearized Stokes equations, then shown to hold for the nonlinear Navier-Stokes system