Journal of Optimization Theory and Applications

, Volume 102, Issue 2, pp 345–371 | Cite as

Control of the Burgers Equation by a Reduced-Order Approach Using Proper Orthogonal Decomposition

  • K. Kunisch
  • S. Volkwein


Proper orthogonal decomposition (POD) is a method to derive reduced-order models for dynamical systems. In this paper, POD is utilized to solve open-loop and closed-loop optimal control problems for the Burgers equation. The relative simplicity of the equation allows comparison of POD-based algorithms with numerical results obtained from finite-element discretization of the optimality system. For closed-loop control, suboptimal state feedback strategies are presented.

Open-loop optimal control closed-loop optimal control Burgers equation proper orthogonal decomposition 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Choi, H., Hinze, M., and Kunisch, K., Suboptimal Control of Backward-Facing-Step-Flow, Preprint 571/97, Technical University of Technology, Berlin, Germany, 1997.Google Scholar
  2. 2.
    Choi, H., Temam, R., Moin, P., and Kim, J., Feedback Control for Unsteady Flow and Its Application to the Stochastic Burgers Equation, Journal of Fluid Mechanics, Vol. 253, pp. 509–543, 1993.Google Scholar
  3. 3.
    Ito, K., and Ravindran, S. S., A Reduced-Basis Method for Control Problems Governed by PDEs, Control and Estimation of Distributed Parameter Systems, International Series of Numerical Mathematics, Vol. 126, pp. 153–168, 1998.Google Scholar
  4. 4.
    Broomhead, D. S., and King, G. P., Extracting Qualitative Dynamics from Experimental Data, Physica, Vol. 20D, pp. 217–236, 1986.Google Scholar
  5. 5.
    Berkooz, G., Holmes, P., and Lumley, J. L., Turbulence, Coherent Structures, Dynamical Systems, and Symmetry, Cambridge Monographs on Mechanics, Cambridge University Press, Cambridge, England, 1996.Google Scholar
  6. 6.
    Burgers, J. M., Application of a Model System to Illustrate Some Points of the Statistical Theory of Free Turbulence, Proc. Acad. Sci. Amsterdam, Vol. 43, pp. 2–12, 1940.Google Scholar
  7. 7.
    Lighthill, M. J., Viscosity Effects in Sound Waves of Finite Amplitude, Surveys in Mechanics, pp. 250–351, 1956.Google Scholar
  8. 8.
    Chambers, D. H., Adrian, R. J., Moin, P., Stewart, D. S., and Sung, H. J., Karhunen-Loéve Expansion of the Burgers Model of Turbulence, Physics of Fluids, Vol. 31, pp. 2573–2582, 1988.Google Scholar
  9. 9.
    Tang, K. Y., Graham, W. R., and Peraire, J., Active Flow Control Using a Reduced-Order Model and Optimum Control, Technical Report, Computational Aerospace Sciences Laboratory, Department of Aeronautics and Astronautics, MIT, 1996.Google Scholar
  10. 10.
    Ly, H. V., and Tran, H. T., Proper Orthogonal Decomposition for Flow Calculations and Optimal Control in a Horizontal CVD Reactor, Preprint CRSC-TR98–12, Center for Research in Scientific Computation, North Carolina State University, 1998.Google Scholar
  11. 11.
    Noble, B., Applied Linear Algebra, Prentice-Hall, Englewood Cliffs, New Jersey, 1969.Google Scholar
  12. 12.
    Aubry, N., Lian, W. Y., and Titi, E. S., Preserving Symmetries in the Proper Orthogonal Decomposition, SIAM Journal on Scientific Computing, Vol. 14, pp. 483–505, 1993.Google Scholar
  13. 13.
    Sirovich, L., Turbulence and the Dynamics of Coherent Structures, Parts 1–3, Quarterly of Applied Mathematics, Vol. 45, pp. 561–590, 1987.Google Scholar
  14. 14.
    Golub, G. H., and Van Loan, C. F., Matrix Computations, Johns Hopkins University Press, Baltimore, Maryland, 1989.Google Scholar
  15. 15.
    Volkwein, S., Mesh Independence of an Augmented Lagrangian-SQP Method in Hilbert Spaces and Control Problems for the Burgers Equation, PhD Thesis, Department of Mathematics, University of Technology, Berlin, Germany, 1997.Google Scholar
  16. 16.
    Ito, K., and Kunisch, K., Augmented Lagrangian-SQP Methods in Hilbert Spaces and Application to Control in the Coefficient Problems, SIAM Journal on Optimization, Vol. 6, pp. 96–125, 1996.Google Scholar
  17. 17.
    Ito, K., and Kunisch, K., Optimal Control, Encyclopedia of Electrical and Electronics Engineering, John Wiley, New York, New York, Vol. 15, pp. 364–379, 1999.Google Scholar
  18. 18.
    Prager, W., Numerical Computation of the Optimal Feedback Law for Nonlinear Infinite-Time Horizon Control Problems, Technical Report, Karl-Franzens Universität, Graz, Austria, 1996.Google Scholar

Copyright information

© Plenum Publishing Corporation 1999

Authors and Affiliations

  • K. Kunisch
    • 1
  • S. Volkwein
    • 2
  1. 1.Institut für MathematikKarl-Franzens-Universität GrazGrazAustria
  2. 2.Institut für MathematikKarl-Franzens-Universität GrazGrazAustria

Personalised recommendations