A Reduction Method for the Boundary Control of the Heat Conduction Equation

The Karhunen–Loe`ve Galerkin procedure (Park, H. M., and Cho, D. H., 1996, “Low Dimensional Modeling of Flow Reactors,” Int. J. Heat Mass Transf., 39, pp. 3311–3323) is a type of reduction method that can be used to solve linear or nonlinear partial differential equations by reducing them to minimal sets of algebraic or ordinary differential equations. In this work, the method is used in conjunction with a conjugate gradient technique to solve the boundary optimal control problems of the heat conduction equations. It is demonstrated that the Karhunen–Loe`ve Galerkin procedure is well suited for the problems of control or optimization, where one has to solve the governing equations repeatedly but one can also estimate the approximate solution space based on the range of control variables. Choices of empirical eigenfunctions to be employed in the Karhunen–Loe`ve Galerkin procedure and issues concerning the implementations of the method are discussed. Compared to the traditional methods, the Karhunen–Loe`ve Galerkin procedure is found to solve the optimal control problems very efficiently without losing accuracy. [S0022-0434(00)00603-1]