An efficient computational method of boundary optimal control problems forthe Burgers equation

The Burgers equation is a simple one-dimensional model of the Navier-Stokes equation. In the present paper, we suggest an efficient method of solving optimal boundary control problems of the Burgers equation, which is practic al as well as mathematically rigorous. Our eventual purpose is to extend th is technique to the control problems of viscous fluid flows. The present me thod is based on the Karhunen-Loeve decomposition which is a technique of o btaining empirical eigenfunctions from the experimental or numerical data o f a system. 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 ordina ry differential equations with a minimum degree of freedom. The resulting l ow-dimensional model of Burgers equation is shown to simulate the original system almost exactly. The present algorithm is well suited for the problem s of control or optimization, where one has to solve the governing equation repeatedly but one can also estimate the approximate solution space based on the range of control variables. The present method of serving boundary c ontrol problems of Burgers equation employing the low-dimensional model obt ained by the Karhunen-Loeve Galerkin procedure is found to yield accurate r esults in a very efficient way, when the minimization of the objective func tion is performed using a conjugate gradient method. (C) 1998 Elsevier Scie nce S.A. All rights reserved.