Issues in the modeling and boundary control of 2D heat flow: POD based modeling

Abstract

Low Dimensional (LD) modeling of systems governed by Partial Differential Equations (PDE) has been studied several times in the past. Various types of boundary excitations have been considered. This paper demonstrates how the external stimuli is made explicit in an autonomous set of ODES and how the excitations along nonpoint subdomains of the boundaries are handled. Dirichlet type boundary excitations are considered and 2D heat equation has been chosen as the test bed. Linearity of the system makes it a good choice for investigating the stability and performance issues. Proper Orthogonal Decomposition (POD) is used in the modeling stage and it is shown that the developed model reconstructs the essential dynamics of the solution of the PDE successfully. The contributions of the paper are on the effects of the number of modes on the model performance, spectral dependence of LD models to the initial and boundary conditions and the prime importance of a fundamental assumption.