Modeling and Control of Physical Processes using Proper Orthogonal Decomposition

Proper orthogonal decomposition (POD) technique (or the Karhunan Lo eve procedure) has been used to obtain low dimensional dynamical models of many applications in engineering and science. In principle, the idea is to start with an ensemble of data, called snapshots, collected from an experiment or a numerical procedure of a physical system. The POD technique is then used to produce a set of basis functions which spans the snapshot collection. When these basis functions are used in a Galerkin procedure, they yield a-nite dimensional dynamical system with the smallest possible degrees of freedom. In this context, it is assumed that the physical system has a mathematical model, which may not be available for many physical and/or industrial applications. In this paper, we consider the steady state Rayleigh-B enard convection whose mathematical model is assumed to be unknown, but numerical data are available. The aim of the paper is to show that, using the obtained ensemble of data, POD can be used to model accurately the natural convection. Furthermore, this approach is very eecient in the sense that it uses the smallest possible number of parameters and thus is suited for process control. Particularly, we consider two boundary control problems: (a) tracking problem and (b) avoiding hot spot in a certain region of the domain.