Taming Heterogeneity and Complexity of Embedded Control

Fast Tracking of Poiseuille Trajectories in Navier Stokes 2D Channel Flow

 

Rafael Vázquez¹, Emmanuel Trélat² and Jean-Michel Coron³

¹ Department of Mechanical and Aerospace Eng. Univ. of California San Diego, La Jolla, 92037 CA, USA Email:[e-mail]

² Univ. Paris-Sud, Lab. Math. UMR 8628, 91405 Orsay Cedex, France [e-mail]

³ Institut Universitaire de France and Univ. Paris-Sud, Lab. Math. UMR 8628, 91405 Orsay Cedex, France [e-mail]

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.

KEYWORDS: Distributed parameter systems, Partial differential equations, Tracking systems, Fourier analysis, Nonlinear control systems

1. Introduction

One of the few situations in which analytic expressions for solutions of the stationary flow field are available is the channel flow problem. Also known as the ...