Browse > Article
http://dx.doi.org/10.7858/eamj.2018.046

REDUCED-ORDER BASED DISTRIBUTED FEEDBACK CONTROL OF THE BENJAMIN-BONA-MAHONY-BURGERS EQUATION  

Jia, Li-Jiao (Department of Mathematics, Yanbian University)
Nam, Yun (Department of Mathematics, Ajou University)
Piao, Guang-Ri (Department of Mathematics, Yanbian University)
Publication Information
East Asian mathematical journal / v.34, no.5, 2018 , pp. 661-681 More about this Journal
Abstract
In this paper, we discuss a reduced-order modeling for the Benjamin-Bona-Mahony-Burgers (BBMB) equation and its application to a distributed feedback control problem through the centroidal Voronoi tessellation (CVT). Spatial distcritization to the BBMB equation is based on the finite element method (FEM) using B-spline functions. To determine the basis elements for the approximating subspaces, we elucidate the CVT approaches to reduced-order bases with snapshots. For the purpose of comparison, a brief review of the proper orthogonal decomposition (POD) is provided and some numerical experiments implemented including full-order approximation, CVT based model, and POD based model. In the end, we apply CVT reduced-order modeling technique to a feedback control problem for the BBMB equation.
Keywords
BBMB equation; reduced-order modeling; CVT; POD; B-spline finite element method; feedback control; linear quadratic regulator;
Citations & Related Records
연도 인용수 순위
  • Reference
1 J. L. Bona, and L. H. Luo, Asymptotic decomposition of nonlinear, dispersive wave equation with dissipation, Physica D 152-153 (2001), 363-383.   DOI
2 N. Aubry, W. Lian, and E. Titi, Preserving symmetries in the proper orthgonal decomposition, SIAM J. Sci. Comput. 14 (1993), 483-505.   DOI
3 G. Berkooz, P. Holmes, and J. Lumley, The proper orthogonal decomposition in the analysis of turbulent flows, Ann. Rev. Fluid Mech. 25 (1993), 539-575.   DOI
4 G. Berkooz, and E. Titi, Galerkin projections and the proper orthognal decomposition for equivariant equations, Phys. Lett. A 174 (1993), 94-102.   DOI
5 E. Christensen, M. Brons, and J. Sorensen, Evaluation of proper orthogonal decomposition-based decomposition techniques applied to parameter-dependent nonturbulent flows, SIAM J. Sci. Comput. 21 (2000), 1419-1434.
6 A. Deane, I. Kevrekidis, G. Karniadakis, and S. Orszag, Low-dimensional models for compex geometry flows: application to grooved channels and circular cylinders, Phys. Fluids A 3 (1991), 2337-2354.
7 M. Graham, and I. Kevrekidis, Pattern analysis and model reduciton: some alternative approaches to the Karhunen-Loeve decomposition, Comput. Chem. Engrg. 20 (1996), 495-506.   DOI
8 M. Krstic, On global stabilization of Burgers equaiton by boundary control, Systems Control Lett. 37 (1999), 123-141.   DOI
9 K. Kunisch, and S. Volkwein, Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition, J. Optim. Theory Appl. 102 (1999), 345-371.   DOI
10 H.V. Ly, and H.T. Tran, Modeling and control of physical processes using proper orthogonal decomposition, Comput. Math. Appl. 33 (2001), 223-236.
11 T. B. Benjamin, J. L. Bona, and J. J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philos. Trans. Roy. Soc. London 272 (1972), no. 1220, 47-78.   DOI
12 D. J. Korteweg, and G. de Vries, On the change of form of long waves advancing in a rectangular canal and on a new type of long stationary waves, Philos. Mag. 39 (1895), 422-443.   DOI
13 J. Lumley, Stochastic Tools in Turbulence, Academic, New York, (1971)
14 S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation, SIAM J. Control. Optim. 39 (2000), 1677-1696.
15 H. Park, and Y. Jang, Control of Burgers equation by means of mode reduction, Int. J. Engrg. Sci. 38 (2000), 785-805.   DOI
16 H. Park, and J. Lee, Solution of an inverse heat transfer problem by means of empirical reduction of modes, Z. Angew. Math. Phys. 51 (2000), 17-38.   DOI
17 H. Park, and W. Lee, An efficient method of solving the Navier-Stokes equations for flow control, Int. J. Numer. Mech. Engrg. 41 (1998), 1133-1151.   DOI
18 S. Ravindran, Proper orthogonal decomposition in optimal control of fluids, Int. J. Numer. Meth. Fluids 34 (2000), 425-448.   DOI
19 S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD, SIAM J. Sci. Comput. 15 (2000), 457-478.   DOI
20 S. Volkwein, Optimal control of a phase field model using the proper orthogonal decomposition, ZAMM 81 (2001), 83-97.   DOI
21 S. Volkwein, Proper orthogonal decomposition and singular value decomposition, Spezialforschungsbereich F003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht Nr. 153, Graz, (1999)
22 G.-R. Piao, and H.-C. Lee, Internal feedback control of the Benjamin-Bona-Mahony-Burgers equation. J. Korean Soc. Ind. Appl. Math. 18 (2014), 269-277.
23 G.-R. Piao, and H.-C. Lee, Distributed Feedback Control of the Benjamin-Bona-Mahony-Burgers Equation by a Reduced-Order Model, East Asian Journal on Applied Mathematics 5 (2015), 61-74.   DOI
24 H.-C. Lee, and G.-R. Piao, Boundary feedback control of the Burgers equations by a reduced-order approach using centroidal Voronoi tessellations. J. Sci. Comput. 43 (2010), 369-387.   DOI
25 J. Burkardt, M. Gunzburger, and H.-C. Lee, POD and CVT-based reduced-order modeling of Navier-Stokes flows. Comput. Methods Appl. Mech. Engrg. 196 (2006), 337-355.   DOI
26 J. Burkardt, M. Gunzburger, and H.-C. Lee, Centroidal Voronoi tessellation-based reduced-order modeling of complex systems, SIAM J. Sci. Comput. 28 (2006), 459-484.
27 H.-C. Lee, S.-W. Lee, and G.-R. Piao, Reduced-order modeling of Burgers equations based on centroidal Voronoi tessellation. Int. J. Numer. Anal. Model. 4 (2007), 559-583.
28 P.M. Prenter, Splines and variational methods, John Wiley and Sons, New York, (1975)
29 Q. Du, V. Faber, and M. Gunzburger, Centroidal Voronoi tessellations : applications and algorithms, SIAM Review 41 (1999), 637-676.   DOI
30 L. Ju, Q. Du, and M. Gunzburger, Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations, J. Parallel Comput. 28 (2002), 1477-1500.   DOI
31 S. Lloyd, Least squares quantization in PCM, IEEE Trans. Infor. Theory, 28 (1982), 129-137.   DOI
32 J. MacQueen, Some methods for classification and analysis of multivariate observations, Proc. Fifth Berkeley Symposium on Mathematical Statistics and Probability, vol. 1, University of California, pp. 281-297, (1967)
33 G. Golub, and C. van Loan, Matrix computations, Johns Hopkins University, Baltimore, (1996)
34 C. T. Chen, Linear System Theory and Design, Holt, Rinehart and Winston, New York, NY, (1984)
35 K. Kunisch, and S. Volkwein, Control of burger's equation by a reduced order approach using proper orthogonal decomposition, J. Optim. Theory Appl. 102 no. 2, (1999), 345-371.   DOI
36 G.-R. Piao, H.-C. Lee, and J.-Y. Lee, Distributed feedback control of the Burgers equation by a reduced-order approach using weighted centroidal Voronoi tessellation, J. KSIAM 13 (2009), 293-305.
37 H.-C. Lee, and G.-R. Piao, Boundary feedback control of the Burgers equations by a reduced-order approach using centroidal Voronoi tessellations. J. Sci. Comput. 43 (2010), 369-387.   DOI
38 J. A. Burns, and S. Kang, A control problem for Burgers' equation with bounded input/oqtput, ICASE Report 90-45, 1990, NASA Langley research Center, Hampton, VA
39 J. L. Bona, W. G. Pritchard, and L. R. Scott, An evaluation of a model equation for water waves, Phil.Trans. R.Soc.London A 302 (1981), 457-510.   DOI
40 R. S. Johnson, A nonlinear equation incorporating damping and dispersion, J.Fluid Mech. 42 (1970), 49-60.   DOI
41 J. A. Burns, and S. Kang, A control problem for Burgers' equation with bounded input/oqtput, Nonlinear Dynamics, 2 (1991), 235-262.   DOI
42 D. L. Russell, and B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans. Amer. Math. Soc. 348 (1996), 3643-3672.   DOI
43 H. Grad, and P. N. Hu, Unified shock profile in a plasma, Phys, Fluids 10 (1967), 2596-2602.   DOI
44 A. Hasan, B. Foss, and O. M. Aamo, Boundary control of long waves in nonlinear dispersive systems, in: proc. of 1st Australian Control Conference, Melbourne, 2011.
45 A. Balogh, M, Krstic, Boundary control of the Korteweg-de Vries-Burgers equation: further results on stabilization and well-posedness with numerical demonstration, IEEE Transactions on Automatic Control 455 (2000), 1739-1745.
46 A. Balogh, M, Krstic, Burgers equation with nonlinear boundary feedback: $H^1$ stability, well-posedness and simulation, Math. Probl. Eng. 6 (2000), 189-200.   DOI
47 M. Krstic, On global stabilization of Burgers equation by boundary control, Systems Control Lett. 37 (1999), 123-141.   DOI
48 C. Laurent, L. Rosier, and B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Comm. Partial Differential Equations 35 (2010), 707-744.   DOI