An efficient computational method of boundary optimal control problems for the Burgers equation. (English) Zbl 0949.76024

From the summary: We suggest an efficient method of solving optimal boundary control problems for the Burgers equation. Our purpose is also to extend this technique to the control problems of fluid flows. The method is based on the Karhunen-Loève decomposition, which is a technique of obtaining empirical eigenfunctions from experimental or numerical data. Employing these empirical eigenfunctions as basis functions of a Galerkin procedure, one can a priori limit the function space considered to the smallest linear subspace that is sufficient to describe the observed phenomena, and consequently reduce the Burgers equation to a set of ordinary differential equations with a minimum degree of freedom. The resulting low-dimensional model of Burgers equation is shown to simulate the original system almost exactly.


76D55 Flow control and optimization for incompressible viscous fluids
49J20 Existence theories for optimal control problems involving partial differential equations
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[1] Burgers, J.M., A mathematical model illustrating the theory of turbulence, Adv. appl. mech., 1, 171-199, (1948)
[2] Hopf, E., The partial differential equation ur + uux = μuxx, Comm. pure appl. math., 3, 201-230, (1950)
[3] Cole, J.D., On the quasi-linear parabolic equation occuring in aerodynamics, Quart. appl. math., 9, 225-236, (1951) · Zbl 0043.09902
[4] Park, H.M.; Cho, D.H., Low dimensional modeling of flow reactors, Int. J. heat mass trans., 39, 3311-3323, (1996)
[5] Loève, M., Probability theory, (1955), Van Nostrand Princeton, NJ · Zbl 0108.14202
[6] Courant, R.; Hilbert, D., ()
[7] Ku, H.C.; Taylor, T.D.; Hirsh, R.S., Pseudospectral methods for solution of the incompressible Navier-Stokes equations, Comput. fluids, 15, 195-214, (1987) · Zbl 0622.76028
[8] Fletcher, R.; Reeves, R.M., Function minimization by conjugate gradients, The computer J., 7, 149-154, (1964) · Zbl 0132.11701
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