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

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 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.
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Discrete and Continuous Dynamical Systems - Series B, American Institute of Mathematical Sciences, 2008, 10 (4), pp.925--956
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https://hal.archives-ouvertes.fr/hal-00131057
Contributeur : Emmanuel Trélat <>
Soumis le : dimanche 24 août 2008 - 10:19:05
Dernière modification le : mercredi 17 octobre 2018 - 19:50:49
Document(s) archivé(s) le : samedi 26 novembre 2016 - 00:56:20

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poiseuille.pdf
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• HAL Id : hal-00131057, version 2

### Citation

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, American Institute of Mathematical Sciences, 2008, 10 (4), pp.925--956. 〈hal-00131057v2〉

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