Reduced-order based distributed feedback control of the Benjamin-Bona-Mahony-Burgers equation. (English) Zbl 07121111
Summary: 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 regulatorReferences:
| [1] | 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. · Zbl 0229.35013 · doi:10.1098/rsta.1972.0032 |
| [2] | 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. · JFM 26.0881.02 · doi:10.1080/14786449508620739 |
| [3] | 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. |
| [4] | 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. · Zbl 0990.93049 |
| [5] | A. Balogh, M, Krstic, Burgers equation with nonlinear boundary feedback: \(H^1\) stability, well-posedness and simulation, Math. Probl. Eng. 6 (2000), 189-200. · Zbl 1058.93050 · doi:10.1155/S1024123X00001320 |
| [6] | M. Krstic, On global stabilization of Burgers equation by boundary control, Systems Control Lett. 37 (1999), 123-141. · Zbl 1074.93552 · doi:10.1016/S0167-6911(99)00013-4 |
| [7] | 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. · Zbl 0862.93035 · doi:10.1090/S0002-9947-96-01672-8 |
| [8] | 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. · Zbl 1213.93015 · doi:10.1080/03605300903585336 |
| [9] | J. L. Bona, and L. H. Luo, Asymptotic decomposition of nonlinear, dispersive wave equation with dissipation, Physica D 152-153 (2001), 363-383. · Zbl 0982.35018 · doi:10.1016/S0167-2789(01)00180-4 |
| [10] | N. Aubry, W. Lian, and E. Titi, Preserving symmetries in the proper orthgonal decomposition, SIAM J. Sci. Comput. 14 (1993), 483-505. · Zbl 0774.65084 · doi:10.1137/0914030 |
| [11] | 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:10.1146/annurev.fl.25.010193.002543 |
| [12] | G. Berkooz, and E. Titi, Galerkin projections and the proper orthognal decomposition for equivariant equations, Phys. Lett. A 174 (1993), 94-102. · doi:10.1016/0375-9601(93)90549-F |
| [13] | 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. · Zbl 0959.35018 |
| [14] | 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. · Zbl 0746.76021 |
| [15] | 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:10.1016/0098-1354(95)00040-2 |
| [16] | M. Krstic, On global stabilization of Burgers equaiton by boundary control, Systems Control Lett. 37 (1999), 123-141. · Zbl 1074.93552 · doi:10.1016/S0167-6911(99)00013-4 |
| [17] | 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. · Zbl 0949.93039 · doi:10.1023/A:1021732508059 |
| [18] | H.V. Ly, and H.T. Tran, Modeling and control of physical processes using proper orthogonal decomposition, Comput. Math. Appl. 33 (2001), 223-236. · Zbl 0966.93018 |
| [19] | J. Lumley, Stochastic Tools in Turbulence, Academic, New York, (1971) · Zbl 1230.76004 |
| [20] | S. Micu, On the controllability of the linearized Benjamin-Bona-Mahony equation, SIAM J. Control. Optim. 39 (2000), 1677-1696. · Zbl 1007.93035 |
| [21] | H. Park, and Y. Jang, Control of Burgers equation by means of mode reduction, Int. J. Engrg. Sci. 38 (2000), 785-805. · Zbl 1210.49032 · doi:10.1016/S0020-7225(99)00044-0 |
| [22] | 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. · Zbl 0963.65101 · doi:10.1007/PL00001505 |
| [23] | 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. · Zbl 0912.76061 · doi:10.1002/(SICI)1097-0207(19980330)41:6<1133::AID-NME329>3.0.CO;2-Y |
| [24] | S. Ravindran, Proper orthogonal decomposition in optimal control of fluids, Int. J. Numer. Meth. Fluids 34 (2000), 425-448. · Zbl 1005.76020 · doi:10.1002/1097-0363(20001115)34:5<425::AID-FLD67>3.0.CO;2-W |
| [25] | S. Ravindran, Reduced-order adaptive controllers for fluid flows using POD, SIAM J. Sci. Comput. 15 (2000), 457-478. · Zbl 1048.76016 · doi:10.1023/A:1011184714898 |
| [26] | S. Volkwein, Optimal control of a phase field model using the proper orthogonal decomposition, ZAMM 81 (2001), 83-97. · Zbl 1007.49019 · doi:10.1002/1521-4001(200102)81:2<83::AID-ZAMM83>3.0.CO;2-R |
| [27] | S. Volkwein, Proper orthogonal decomposition and singular value decomposition, Spezialforschungsbereich F003 Optimierung und Kontrolle, Projektbereich Kontinuierliche Optimierung und Kontrolle, Bericht Nr. 153, Graz, (1999) |
| [28] | 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. · Zbl 1331.93063 |
| [29] | 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. · Zbl 1320.93045 · doi:10.4208/eajam.210214.061214a |
| [30] | 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. · Zbl 1120.76323 · doi:10.1016/j.cma.2006.04.004 |
| [31] | 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. · Zbl 1111.65084 |
| [32] | 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. · Zbl 1154.35303 |
| [33] | 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. · Zbl 1203.93088 · doi:10.1007/s10915-009-9310-4 |
| [34] | P.M. Prenter, Splines and variational methods, John Wiley and Sons, New York, (1975) · Zbl 0344.65044 |
| [35] | Q. Du, V. Faber, and M. Gunzburger, Centroidal Voronoi tessellations : applications and algorithms, SIAM Review 41 (1999), 637-676. · Zbl 0983.65021 · doi:10.1137/S0036144599352836 |
| [36] | L. Ju, Q. Du, and M. Gunzburger, Probabilistic methods for centroidal Voronoi tessellations and their parallel implementations, J. Parallel Comput. 28 (2002), 1477-1500. · Zbl 1014.68202 · doi:10.1016/S0167-8191(02)00151-5 |
| [37] | S. Lloyd, Least squares quantization in PCM, IEEE Trans. Infor. Theory, 28 (1982), 129-137. · Zbl 0504.94015 · doi:10.1109/TIT.1982.1056489 |
| [38] | 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) · Zbl 0214.46201 |
| [39] | G. Golub, and C. van Loan, Matrix computations, Johns Hopkins University, Baltimore, (1996) · Zbl 0865.65009 |
| [40] | C. T. Chen, Linear System Theory and Design, Holt, Rinehart and Winston, New York, NY, (1984) |
| [41] | 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. |
| [42] | 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. · Zbl 1203.93088 · doi:10.1007/s10915-009-9310-4 |
| [43] | 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 |
| [44] | 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. · Zbl 0949.93039 · doi:10.1023/A:1021732508059 |
| [45] | 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. · Zbl 0497.76023 · doi:10.1098/rsta.1981.0178 |
| [46] | H. Grad, and P. N. Hu, Unified shock profile in a plasma, Phys, Fluids 10 (1967), 2596-2602. · doi:10.1063/1.1762081 |
| [47] | R. S. Johnson, A nonlinear equation incorporating damping and dispersion, J.Fluid Mech. 42 (1970), 49-60. · Zbl 0213.54904 · doi:10.1017/S0022112070001064 |
| [48] | J. A. Burns, and S. Kang, A control problem for Burgers’ equation with bounded input/oqtput, Nonlinear Dynamics, 2 (1991), 235-262. · doi:10.1007/BF00045296 |
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