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Control of the Burgers equation by a reduced-order approach using proper orthogonal decomposition. (English) Zbl 0949.93039

Summary: 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.

MSC:

93C20 Control/observation systems governed by partial differential equations
93B40 Computational methods in systems theory (MSC2010)
Full Text: DOI

References:

[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. · Zbl 0939.76027
[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. · Zbl 0810.76012 · doi:10.1017/S0022112093001880
[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. · Zbl 0908.93025 · doi:10.1007/978-3-0348-8849-3_12
[4] Broomhead, D. S., and King, G. P., Extracting Qualitative Dynamics from Experimental Data, Physica, Vol. 20D, pp. 217–236, 1986. · Zbl 0603.58040
[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. · Zbl 0890.76001
[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. · Zbl 0061.45710
[7] Lighthill, M. J., Viscosity Effects in Sound Waves of Finite Amplitude, Surveys in Mechanics, pp. 250–351, 1956.
[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. · doi:10.1063/1.866535
[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.
[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. · Zbl 1146.76631
[11] Noble, B., Applied Linear Algebra, Prentice-Hall, Englewood Cliffs, New Jersey, 1969. · Zbl 0203.33201
[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. · Zbl 0774.65084 · doi:10.1137/0914030
[13] Sirovich, L., Turbulence and the Dynamics of Coherent Structures, Parts 1–3, Quarterly of Applied Mathematics, Vol. 45, pp. 561–590, 1987. · Zbl 0676.76047
[14] Golub, G. H., and Van Loan, C. F., Matrix Computations, Johns Hopkins University Press, Baltimore, Maryland, 1989. · Zbl 0733.65016
[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.
[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. · Zbl 0846.65026 · doi:10.1137/0806007
[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.
[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. · Zbl 0956.65050
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