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TECHNICAL PAPERS

A Reduction Method for the Boundary Control of the Heat Conduction Equation

[+] Author and Article Information
H. M. Park, O. Y. Kim

Department of Chemical Engineering, Sogang University, Seoul, Korea

J. Dyn. Sys., Meas., Control 122(3), 435-444 (Nov 18, 1998) (10 pages) doi:10.1115/1.1286365 History: Received November 18, 1998
Copyright © 2000 by ASME
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Figures

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The system and relevant boundary conditions
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Definition of shape functions. (a) Temporal shape functions; (b) spatial shape functions.
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Some dominant empirical eigenfunctions with large eigenvalues. (a) The first eigenfunction (λ1=0.741); (b) the second eigenfunction (λ2=0.156); (c) the third eigenfunction (λ3=4.969×10−2); (d) the fourth eigenfunction (λ4=2.127×10−2).
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Some typical empirical eigenfunctions with small eigenvalues. (a) The 11th eigenfunction (λ11=1.136×10−3); (b) the 12th eigenfunction (λ12=8.807×10−4); (c) the 13th eigenfunction (λ13=5.832×10−4); (d) the 14th eigenfunction (λ14=4.251×10−4).
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Relative errors of the low dimensional dynamic model with respect to the finite difference solution for various heat flux functions q(x,t)
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The space-time contours of the optimal control obtained either by the FDM-CG or by the KLG-CG
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The temporal profiles of the optimal control at x=0.5 and x=0.8 obtained either by the FDM-CG or by the KLG-CG. The original control, Eqs. (24a24b) is also displayed for comparison. (a) x=0.5; (b) x=0.8.
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Convergence of the KLG-CG for various initial approximations of q(x,t). (a) Profiles of various initial approximations at x=0.5; (b) converged profiles of optimal control q(x,t) at x=0.5; (c) profiles of various initial approximations at x=0.8; (d) converged profiles of optimal control q(x,t) at x=0.8.

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