# American Institute of Mathematical Sciences

September  2008, 10(4): 925-956. doi: 10.3934/dcdsb.2008.10.925

## Control for fast and stable Laminar-to-High-Reynolds-Numbers transfer in a 2D Navier-Stokes channel flow

 1 Escuela Superior de Ingenieros, Dpto. de Ingeniera Aeroespacial, Camino de los Descubrimientos s.n., 41092 Sevilla, Spain 2 Université d’Orléans, UFR Sciences, Fédération Denis Poisson Mathématiques, Laboratoire MAPMO, UMR 6628, Route de Chartres, BP 6759, 45067 Orléans Cedex 2 3 Labo. Jacques-Louis Lions, Univ. Pierre et Marie Curie and Institut Universitaire de France, B.C. 187, 4 place Jussieu, 75252 Paris Cedex 05, France

Received  February 2007 Revised  March 2008 Published  August 2008

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.
Citation: Rafael Vázquez, 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 - B, 2008, 10 (4) : 925-956. doi: 10.3934/dcdsb.2008.10.925
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