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Distributed Feedback Control of the Benjamin-Bona-Mahony-Burgers Equation by a Reduced-Order Model

Published online by Cambridge University Press:  06 March 2015

Guang-Ri Piao  and
Hyung-Chun Lee
Guang-Ri Piao
Affiliation:
Department of Mathematics, Yanbian University, Yanji 133002, China
Hyung-Chun Lee*
Affiliation:
Department of Mathematics, Ajou University, Suwon, 443-749, South Korea
*
*Corresponding author. Email addresses: grpiao@ybu.edu.en (G.-R. Piao), hclee@ajou.ac.kr (H.-C. Lee)
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Abstract

A reduced-order model for distributed feedback control of the Benjamin-Bona-Mahony-Burgers (BBMB) equation is discussed. To retain more information in our model, we first calculate the functional gain in the full-order case, and then invoke the proper orthogonal decomposition (POD) method to design a low-order controller and thereby reduce the order of the model. Numerical experiments demonstrate that a solution of the reduced-order model performs well in comparison with a solution for the full-order description.

Keywords

49J2076D0549B22B-spline finite element methodlinear quadratic regulatorfeedback controlreduced-order modelBBMB equation
Type
Research Article
Information
East Asian Journal on Applied Mathematics , Volume 5 , Issue 1 , February 2015 , pp. 61 - 74
Copyright
Copyright © Global-Science Press 2015 

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References

[1]Ali, A.H.A., Gardner, G.A. and Gardner, L.R.T., A collocation solution for Burgers equation using cubic B-spline finite elements, Comp. Meth. Appl. Mech. Engrg. 100, 325–337 (1992)Google Scholar
[2]Aubry, N., Lian, W. and Titi, E., Preserving symmetries in the proper orthogonal decomposition, SIAM J. Sci. Comp. 14, 483–505 (1993)Google Scholar
[3]Balogh, A. and Krstic, M, Boundary control of the Korteweg-de Vries-Burgers equation: further results on stabilisation and well-posedness, with numerical demonstration, IEEE Trans. Automatic Control 455, 1739–1745 (2000)Google Scholar
[4]Balogh, A. and Krstic, M., Burgers equation with nonlinear boundary feedback: H 1stability, well-posedness and simulation, Math. Problems in Engineering 6, 189–200 (2000)Google Scholar
[5]Benjamin, T.B., Bona, J.L. and Mahony, J.J., Model equations for long waves in nonlinear dispersive systems, Phil. Trans. Royal Soc. (London) 272 (1220), 47–78 (1972)Google Scholar
[6]Berkooz, G., Holmes, P. and Lumley, J., The proper orthogonal decomposition in the analysis of turbulent flows, Ann. Rev. Fluid Mech. 25, 539–575 (1993)Google Scholar
[7]Berkooz, G. and Titi, E., Galerkin projections and the proper orthogonal decomposition for equiv-ariant equations, Phys, Lett. A 174, 94–102 (1993)Google Scholar
[8]Bona, J.L. and Luo, L.H., Asymptotic decomposition of nonlinear, dispersive wave equation with dissipation, Physica D 152–153, 363–383 (2001)Google Scholar
[9]Burkardt, J., Gunzburger, M. and Lee, H.-C., POD and CVT-based reduced-order modeling of Navier-Stokes flows, Comp. Meth. Appl. Mech. Engrg. 196, 337–355 (2006)Google Scholar
[10]Burns, J.A. and Kang, S., A control problem for Burgers’ equation with bounded input/oqtput, ICASE Report 90–45, 1990, NASA Langley research Center, Hampton, VA; Nonlinear Dynamics 2, 235–262 (1991)Google Scholar
[11]Chen, C.T., Linear System Theory and Design, Holt, Rinehart and Winston, New York, 1984Google Scholar
[12]Christensen, E., Brons, M. and Sorensen, J., Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows, SIAM J. Sci. Comp. 21, 1419–1434 (2000)Google Scholar
[13]Deane, A., Kevrekidis, I., Karniadakis, G. and Orszag, S., Low-dimensional models for complex geometry flows: application to grooved channels and circular cylinders, Phys. Fluids A 3, 2337–2354 (1991)Google Scholar
[14]Golub, G. and van Loan, C., Matrix Computations, Johns Hopkins University, Baltimore, 1996Google Scholar
[15]Graham, M. and Kevrekidis, I., Pattern analysis and model reduction: some alternative approaches to the Karhunen-Lóeve decomposition, Comp. Chem. Engrg. 20, 495–506 (1996)Google Scholar
[16]Hasan, A., Foss, B. and Aamo, O.M., Boundary control of long waves in nonlinear dispersive systems, in Proc. 1st Australian Control Conference, Melbourne (2011)Google Scholar
[17]Korteweg, D.J. and de Vries, G., On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Phil. Magazine 39, 422–443 (1895)Google Scholar
[18]Krstic, M., On global stabilisation of Burgers equaiton by boundary control, Systems and Control. Letters. 37, 123–141 (1999)Google Scholar
[19]Kunisch, K. and Volkwein, S., Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl. 102, 345–371 (1999)Google Scholar
[20]Kutluay, S. and Ucar, Y., Numerical solutions of the coupled Burgers’ equation by the Galerkin quadratic B-spline finite element method, Math. Meth. Appl. Sci. 36, 2403–2415 (2013)Google Scholar
[21]Kutluay, S. and Ucar, Y., A quadratic B-spline Galerkin approach for solving a coupled KdV equation, Math. Model. Anal. 18, 103–121 (2013)Google Scholar
[22]Lee, H.-C. and Piao, G.-R., Boundary feedback control of the Burgers equations by a reduced-order approach using centroidal Voronoi tessellations, J. Sci. Comp. 43, 369–387 (2010)Google Scholar
[23]Lumley, J., Stochastic Tools in Turbulence, Academic Press, New York (1971)Google Scholar
[24]Ly, H.V. and Tran, H.T., Modeling and control of physical processes using proper orthogonal decomposition, Comp. Math. Appl. 33, 223–236 (2001)Google Scholar
[25]Micu, S., On the controllability of the linearised Benjamin-Bona-Mahony equation, SIAM J. Control. Optim. 39, 1677–1696 (2000)Google Scholar
[26]Omrani, K. and Ayadi, M., Finite difference discretisation of the Benjamin-Bona-Mahony-Burgers equation, Num. Meth. Partial Differential Equations 24, 239–248 (2008)Google Scholar
[27]Park, H. and Jang, Y., Control of Burgers equation by means of mode reduction, Int. J. Eng. Sci. 38, 785–805 (2000)Google Scholar
[28]Park, H. and Lee, J., Solution of an inverse heat transfer problem by means of empirical reduction of modes, Z. Angew. Math. Phys. 51, 17–38 (2000)CrossRefGoogle Scholar
[29]Park, H. and Lee, W., An efficient method of solving the Navier-Stokes equations for flow control, Int. J. Numer. Mech. Eng. 41, 1133–1151 (1998)Google Scholar
[30]Piao, G.-R., Lee, H.-C. and Lee, J.-Y., Distributed feedback control of the Burgers equation by a reduced-order approach using weighted centroidal Voronoi tessellation, J. KSIAM 13, 293–305 (2009)Google Scholar
[31]Prenter, P.M., Splines and Variational Methods, Wiley, New York (1975)Google Scholar
[32]Volkwein, S., Optimal control of a phase field model using the proper orthogonal decomposition, ZAMM 81, 83–97 (2001)Google Scholar
[33]Volkwein, S., Proper orthogonal decomposition and singular value decomposition, Spezial-forschungsbereich F003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht Nr. 153, Graz (1999)Google Scholar