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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

Rafael Vázquez 1, , Emmanuel Trélat 2, and  Jean-Michel Coron 3,

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.
Keywords: controllability, stabilization., Navier-Stokes equations, Poiseuille flows.
Mathematics Subject Classification: Primary: 93B05, 93C20; Secondary: 35B3.
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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