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November  2003, 3(4): 619-642. doi: 10.3934/dcdsb.2003.3.619

Positive feedback control of Rayleigh-Bénard convection

B. A. Wagner 1, , Andrea L. Bertozzi 2, and  L. E. Howle 3,

1. 

Weierstrass-Institute for Applied Analysis and Stochastics, Mohrenstr. 39 D-10117 Berlin, Germany
2. 

Department of Mathematics, Duke University, Durham, NC 27708, Department of Mathematics, Univ. of California Los Angeles, Los Angeles, CA 90095, United States
3. 

Department of Mechanical Engineering, Center for Nonlinear and Complex Systems, Duke University, Durham, NC 27708, United States

Received  December 2002 Revised  May 2003 Published  August 2003

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: control, Galerkin projection, illposedness., Rayleigh-Bénard convection.
Mathematics Subject Classification: 35B32, 35B37, 76D05, 76D55, 93C2.
Citation: B. A. Wagner, Andrea L. Bertozzi, L. E. Howle. Positive feedback control of Rayleigh-Bénard convection. Discrete & Continuous Dynamical Systems - B, 2003, 3 (4) : 619-642. doi: 10.3934/dcdsb.2003.3.619
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