An efficient computational method of boundary optimal control problems for the 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 practical as well as mathematically rigorous. Our eventual purpose is to extend this technique to the control problems of viscous fluid flows. The present method is based on the Karhunen-Loeve decomposition which is a technique of obtaining empirical eigenfunctions from the experimental or numerical data of 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 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. The present algorithm is well suited for the problems 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 solving boundary control problems of Burgers equation employing the lowdimensional model obtained by the Karhunen-Loeve Galerkin procedure is found to yield accurate results in a very efficient way, when the minimization of the objective function is performed using a conjugate gradient method.