DOI QR코드

DOI QR Code

INTERNAL FEEDBACK CONTROL OF THE BENJAMIN-BONA-MAHONY-BURGERS EQUATION

  • 투고 : 2014.02.27
  • 심사 : 2014.09.11
  • 발행 : 2014.09.25

초록

A numerical scheme is proposed to control the BBMB (Benjamin-Bona-Mahony-Burgers) equation, and the scheme consists of three steps. Firstly, BBMB equation is converted to a finite set of nonlinear ordinary differential equations by the quadratic B-spline finite element method in spatial. Secondly, the controller is designed based on the linear quadratic regulator (LQR) theory; Finally, the system of the closed loop compensator obtained on the basis of the previous two steps is solved by the backward Euler method. The controlled numerical solutions are obtained for various values of parameters and different initial conditions. Numerical simulations show that the scheme is efficient and feasible.

키워드

참고문헌

  1. T.B. Benjamin, J.L. Bona, J.J. Mahony, Model equations for long waves in nonlinear dispersive systems, Philosophical Transactions of the Royal Society of London 272(1220)(1972) 47-78. https://doi.org/10.1098/rsta.1972.0032
  2. D.J. Korteweg, 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, Philosophical Magazine 39(1895) 422-443. https://doi.org/10.1080/14786449508620739
  3. M.A. Raupp, Galerkin methods applied to the Benjamin-Bona-Mahony equation, Boletim da Sociedade Brazilian Mathematical 6(1)(1975) 65-77. https://doi.org/10.1007/BF02584873
  4. L.Wahlbin, Error estimates for a Galerkin mehtod for a class of model equations for long waves, Numerische Mathematik 23(4)(1975) 289-303.
  5. R.E. Ewing, Time-stepping Galerkin methods for nonlinear Sobolev partial differential equation, SIAM Journal on Numerical Analysis 15(5)(1978) 1125-1150. https://doi.org/10.1137/0715075
  6. D.N. Arnold, J. Douglas Jr.,and V. Thomee, Superconvergence of finite element approximation to the solution of a Sobolev equation in a single space variable, Mathematics of computation 36(153)(1981) 737-743.
  7. T. Ozis, A. Esen, S. Kutluay, Numerical solution of Burgers equation by quadratic B-spline finite element, Appl Math Comput 165(2005) 237-249. https://doi.org/10.1016/j.amc.2004.04.101
  8. E.N. Aksan, Quadratic B-spline finite element method for numerical solution of the Burgers equation, Appl Math Compu 174(2006) 884-896. https://doi.org/10.1016/j.amc.2005.05.020
  9. A. Hasan, B. Foss, O.M. Aamo, Boundary control of long waves in nonlinear dispersive systems. in: proc. of 1st Australian Control Conference, Melbourne, 2011.
  10. 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.
  11. A. Balogh, M, Krstic, Burgers equation with nonlinear boundary feedback: $H^1$ stability, well-posedness and simulation, Machematical Problems in Engineering 6(2000) 189-200. https://doi.org/10.1155/S1024123X00001320
  12. M. Krstic, On global stabilization of Burgers equation by boundary control, Systems and Control Letters 37(1999) 123-141. https://doi.org/10.1016/S0167-6911(99)00013-4
  13. D.L. Russell, B.-Y. Zhang, Exact controllability and stabilizability of the Korteweg-de Vries equation, Trans Amer Math Soc 348(1996) 3643-3672. https://doi.org/10.1090/S0002-9947-96-01672-8
  14. C. Laurent, L. Rosier, B.-Y. Zhang, Control and stabilization of the Korteweg-de Vries equation on a periodic domain, Communications in Partial Diferential Equations 35(2010) 707-744. https://doi.org/10.1080/03605300903585336
  15. J.L. Bona, L.H. Luo, Asymptotic decomposition of nonlinear, dispersive wave equation with dissipation, Physica D 152-153(2001) 363-383. https://doi.org/10.1016/S0167-2789(01)00180-4
  16. P.M. Prenter, Splines and variational methods, John Wiley and Sons, New York, 1975.
  17. C.T. Chen, Linear System Theory and Design, Holt, Rinehart and Winston, New York, NY, 1984.
  18. G.-R. Piao, H.-C. Lee, 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.
  19. H.-C. Lee, G.-R. Piao, Boundary feedback control of the Burgers equaitons by a reduced-order approach using centroidal Voronoi tessellations, J. Sci. Comput. 43(2010) 369-387. https://doi.org/10.1007/s10915-009-9310-4
  20. 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
  21. J.A. Burns and S. Kang, A control problem for Burgers' equation with bounded input/oqtput, Nonlinear Dynamics 2(1991) 235-262. https://doi.org/10.1007/BF00045296
  22. J.L. Bona, W.G. Pritchard, L.R. Scott, An evaluation of a model equation for water waves, Phil.Trans.R.Soc.London A 302(1981) 457-510. https://doi.org/10.1098/rsta.1981.0178
  23. H. Grad, P.N. Hu, Unified shock profile in a plasma, Phys, Fluids 10(1967) 2596-2602. https://doi.org/10.1063/1.1762081
  24. R.S. Johnson, A nonlinear equation incorporating damping and dispersion, J.Fluid Mech. 42(1970) 49-60. https://doi.org/10.1017/S0022112070001064