This paper deals with the stability analysis of a system of ordinary differential equations coupled to a vectorial transport equation. We develop here a new method to study the stability of this class of systems using linear matrix inequalities led by the choice of an appropriate Lyapunov functional. The main idea in our approach is to build a Lyapunov functional by enriching the basic energy of the coupled system under study with specific terms involving an approximation of the infinite dimensional state of the transport equation. To this end, we will exploit Legendre polynomials and their properties, and use a Bessel inequality to measure the contribution of our approximation. We will then give our exponential stability results and their proofs. Our approach will finally be tested on academic examples.