American Institute of Mathematical Sciences

Advanced
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 and Continuous Dynamical Systems - B, 2003, 3 (4) : 619-642. doi: 10.3934/dcdsb.2003.3.619
[1]

Tian Ma, Shouhong Wang. Attractor bifurcation theory and its applications to Rayleigh-Bénard convection. Communications on Pure and Applied Analysis, 2003, 2 (4) : 591-599. doi: 10.3934/cpaa.2003.2.591
[2]

Tingyuan Deng. Three-dimensional sphere $S^3$-attractors in Rayleigh-Bénard convection. Discrete and Continuous Dynamical Systems - B, 2010, 13 (3) : 577-591. doi: 10.3934/dcdsb.2010.13.577
[3]

Toshiyuki Ogawa. Bifurcation analysis to Rayleigh-Bénard convection in finite box with up-down symmetry. Communications on Pure and Applied Analysis, 2006, 5 (2) : 383-393. doi: 10.3934/cpaa.2006.5.383
[4]

Jungho Park. Dynamic bifurcation theory of Rayleigh-Bénard convection with infinite Prandtl number. Discrete and Continuous Dynamical Systems - B, 2006, 6 (3) : 591-604. doi: 10.3934/dcdsb.2006.6.591
[5]

Jhean E. Pérez-López, Diego A. Rueda-Gómez, Élder J. Villamizar-Roa. On the Rayleigh-Bénard-Marangoni problem: Theoretical and numerical analysis. Journal of Computational Dynamics, 2020, 7 (1) : 159-181. doi: 10.3934/jcd.2020006
[6]

Björn Birnir, Nils Svanstedt. Existence theory and strong attractors for the Rayleigh-Bénard problem with a large aspect ratio. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 53-74. doi: 10.3934/dcds.2004.10.53
[7]

Marco Cabral, Ricardo Rosa, Roger Temam. Existence and dimension of the attractor for the Bénard problem on channel-like domains. Discrete and Continuous Dynamical Systems, 2004, 10 (1&2) : 89-116. doi: 10.3934/dcds.2004.10.89
[8]

O. V. Kapustyan, V. S. Melnik, José Valero. A weak attractor and properties of solutions for the three-dimensional Bénard problem. Discrete and Continuous Dynamical Systems, 2007, 18 (2&3) : 449-481. doi: 10.3934/dcds.2007.18.449
[9]

Petr Knobloch. Error estimates for a nonlinear local projection stabilization of transient convection--diffusion--reaction equations. Discrete and Continuous Dynamical Systems - S, 2015, 8 (5) : 901-911. doi: 10.3934/dcdss.2015.8.901
[10]

Roland Pulch. Stability preservation in Galerkin-type projection-based model order reduction. Numerical Algebra, Control and Optimization, 2019, 9 (1) : 23-44. doi: 10.3934/naco.2019003
[11]

ShinJa Jeong, Mi-Young Kim. Computational aspects of the multiscale discontinuous Galerkin method for convection-diffusion-reaction problems. Electronic Research Archive, 2021, 29 (2) : 1991-2006. doi: 10.3934/era.2020101
[12]

Andrzej Just, Zdzislaw Stempień. Optimal control problem for a viscoelastic beam and its galerkin approximation. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 263-274. doi: 10.3934/dcdsb.2018018
[13]

Maya Bassam, Denis Mercier, Ali Wehbe. Optimal energy decay rate of Rayleigh beam equation with only one boundary control force. Evolution Equations and Control Theory, 2015, 4 (1) : 21-38. doi: 10.3934/eect.2015.4.21
[14]

Zhen-Zhen Tao, Bing Sun. Galerkin spectral method for elliptic optimal control problem with $L^2$-norm control constraint. Discrete and Continuous Dynamical Systems - B, 2022, 27 (8) : 4121-4141. doi: 10.3934/dcdsb.2021220
[15]

Gildas Besançon, Didier Georges, Zohra Benayache. Towards nonlinear delay-based control for convection-like distributed systems: The example of water flow control in open channel systems. Networks and Heterogeneous Media, 2009, 4 (2) : 211-221. doi: 10.3934/nhm.2009.4.211
[16]

Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete and Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473
[17]

Mats Gyllenberg, Yan Ping. The generalized Liénard systems. Discrete and Continuous Dynamical Systems, 2002, 8 (4) : 1043-1057. doi: 10.3934/dcds.2002.8.1043
[18]

Hung-Wen Kuo. The initial layer for Rayleigh problem. Discrete and Continuous Dynamical Systems - B, 2011, 15 (1) : 137-170. doi: 10.3934/dcdsb.2011.15.137
[19]

Na Li, Maoan Han, Valery G. Romanovski. Cyclicity of some Liénard Systems. Communications on Pure and Applied Analysis, 2015, 14 (6) : 2127-2150. doi: 10.3934/cpaa.2015.14.2127
[20]

Jaume Llibre, Claudia Valls. On the analytic integrability of the Liénard analytic differential systems. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 557-573. doi: 10.3934/dcdsb.2016.21.557

2021 Impact Factor: 1.497

Tools

Metrics

  • PDF downloads (61)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy