  As a first approach to the study of systems coupling finite and infinite
dimensional natures, this article addresses the stability of a system of
ordinary differential equations coupled with a classic heat equation using a
Lyapunov functional technique. Inspired from recent developments in the area of
time delay systems, a new methodology to study the stability of such a class of
distributed parameter systems is presented here. The idea is to use a
polynomial approximation of the infinite dimensional state of the heat equation
in order to build an enriched energy functional. A well known efficient
integral inequality (Bessel inequality) will allow to obtain stability
conditions expressed in terms of linear matrix inequalities. We will eventually
test our approach on academic examples in order to illustrate the efficiency of
our theoretical results.
