Discrete and Continuous Dynamical Systems - Series B (DCDS-B)

Positive feedback control of Rayleigh-Bénard convection

Pages: 619 - 642, Volume 3, Issue 4, November 2003      doi:10.3934/dcdsb.2003.3.619

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B. A. Wagner - Weierstrass-Institute for Applied Analysis and Stochastics, Mohrenstr. 39 D-10117 Berlin, Germany (email)
Andrea L. Bertozzi - Department of Mathematics, Duke University, Durham, NC 27708, Department of Mathematics, Univ. of California Los Angeles, Los Angeles, CA 90095, United States (email)
L. E. Howle - Department of Mechanical Engineering, Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, United States (email)

Abstract: We consider the problem of active feedback control of Rayleigh-Bénard convection via shadowgraphic measurement. Our theoretical studies show, that when the feedback control is positive, i.e. is tuned to advance the onset of convection, there is a critical threshold beyond which the system becomes linearly ill-posed so that short-scale disturbances are greatly amplified. Experimental observation suggests that finite size effects become important and we develop a theory to explain these contributions. As an efficient modelling tool for studying the dynamics of such a controlled pattern forming system, we use a Galerkin approximation to derive a dimension reduced model.

Keywords:  Galerkin projection, control, Rayleigh-Bénard convection, illposedness.
Mathematics Subject Classification:  35B32, 35B37, 76D05, 76D55, 93C20.

Received: December 2002;      Revised: May 2003;      Available Online: August 2003.