  It is well-known that electromagnetic dispersive structures such as
metamaterials can be modelled by generalized Drude-Lorentz models. The present
paper is the first of two articles dedicated to dissipative generalized
Drude-Lorentz open structures. 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 an approach based on frequency dependent
Lyapounov estimates, we show that this decay is polynomial in time. These
results extend to an unbounded structure the ones obtained for bounded media in
[18] via a quite different method based on the notion of cumulated past history
and semi-group theory. A great advantage of the approach developed here is to
be less abstract and directly connected to the physics of the system via energy
balances.
