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2008, Volume 10, Issue 4: 925-956. Doi: 10.3934/dcdsb.2008.10.925
This issue Previous Article On a Hamiltonian PDE arising in magma dynamics Next Article Very weak solutions for the magnetohydrodynamic type equations

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

  • 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

Received:  February 1, 2007
Revised:  March 1, 2008
Published online:  August 1, 2008
Abstract / Introduction Related Papers Cited by
    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.
    Mathematics Subject Classification: Primary: 93B05, 93C20; Secondary: 35B37.

    Citation:
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