A reduction method for the boundary control of the heat conduction equation

The Karhunen-Loeve Galerkin procedure (Park, H. M., and Cho, D. H., 1996, " Low Dimensional Modeling of Flow Reactors," Int J. Heat Mass Transf., 39, p p. 3311-3323) is a type of reduction method that can be used to solve linea r or nonlinear partial differential equations by reducing them to minimal s ets of algebraic of ordinary differential equations. In this work, the meth od is used in conjunction with a conjugate gradient technique to solve the boundary optimal control problems of the heat conduction equations. It is d emonstrated that the Karhunen-Loeve Galerkin procedure is well suited for t he problems of control of optimization, where one has to solve the governin g equations repeatedly but one can also estimate the approximate solution s pace based on the range of control variables. Choices of empirical eigenfun ctions to be employed in the Karhunen-Loeve Galerkin procedure and issues c oncerning the implementations of the method are discussed. Compared to the traditional methods, the Karhunen-Loeve Galerkin procedure is found to solv e the optimal control problems very efficiently without losing accuracy.