In this paper we study linear conservative systems of finitedimensioncoupled with an infinite dimensional system of diffusive type. Computing the time-derivative of anappropriate energy functional along the solutions helps us toprove the well-posedness of the systemand a stability property.But in order to prove asymptotic stability we need to applya sufficient spectral condition. We also illustrate the sharpness of thiscondition by exhibiting some systems for which we do not have the asymptoticproperty.
