Control for Fast and Stable Laminar-to-High-Reynolds-Numbers Transfer in a 2D Navier-Stokes Channel Flow

Rafael Vazquez (1) , Emmanuel Trélat (2) , Jean-Michel Coron (3)

1 University of California
2 MAPMO - Mathématiques - Analyse, Probabilités, Modélisation - Orléans
3 LM-Orsay - Laboratoire de Mathématiques d'Orsay

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 $L^2$, $H^1$, and $H^2$ norms. The result is first proved for the linearized Stokes equations, then shown to hold locally for the nonlinear Navier-Stokes system.

Mots clés: control; Navier-Stokes equations; Poiseuille flow

Domaines: Optimisation et contrôle [math.OC]; Equations aux dérivées partielles [math.AP]

Rafael Vazquez, Emmanuel Trélat, Jean-Michel Coron. Control for Fast and Stable Laminar-to-High-Reynolds-Numbers Transfer in a 2D Navier-Stokes Channel Flow. Discrete and Continuous Dynamical Systems - Series B, 2008, 10 (4), pp.925--956. ⟨hal-00131057v1⟩