On Control-Oriented Modeling in Heat Transfer Based on a Projection Technique and the Method of Integrodifferential Relations

In this paper, projection techniques of an initial-boundary value problem formulated originally in PDEs to a system of ODEs of small dimension are presented. Properties of various quadratic relations derived from energy terms that arise in different problems of mathematical physics are studied and discussed. As an example, integrodifferential formulations of a 2-D heat transfer problem are considered. A feedforward control problem for heat transfer processes in a metal plate is reduced to a system of ordinary differential equations based on the finite element method and semi-discrete approximations with polynomial basis functions for which an analysis of the problem is performed. Finally, results of numerical simulations are presented and discussed.