In collaboration with Maxence Cassier (Aix Marseille Univ, CNRS, Centrale Marseille, Institut Fresnel) and Luis Alejandro Rosas Martinez (POEMS, CNRS, INRIA, ENSTA Paris).
It is well-known that electromagnetic dispersive structures such as metamaterials obey mathematical models whose construction, based on fundamental physical such as causality and passivity, emphasizes the role of Herglotz functions. Among these models an important class is provided by generalized Drude-Lorentz models, see e.g. [1]. In this work, we are interested in dissipative Drude-Lorentz open structures and we wish to quantify the loss in such media in terms of the long time decay rate of the electromagnetic energy for the corresponding Cauchy problem. By using two different approaches, one based on (frequency dependent) Lyapounov estimates and the other on modal analysis, we show that this decay is polynomial in time. These results generalize a part the ones obtained for bounded media in [2] via a quite different method based on the notion of cumulated past history and semi-group theory. A great advantage of the approaches developed here is to be directly connected to the physics of the system via energy balances or modes behavior.[-]