  In this paper we characterize the stabilization for some thermo-elastic type
system with Cattaneo law and we prove that the exponential or polynomial
stability of this system implies a polynomial stability of the correspond
thermoelastic system with the Fourier law. The proof of the main results uses,
respectively, the methodology introduced in Ammari and Tucsnak, where the
exponential stability for the closed loop problem is reduced to an
observability estimate for the corresponding uncontrolled system and a
characterization, of the polynomial stability for C0-semigroup in Hilbert space
by a polynomial estimation of the associated resolvante of the generator of
this semigroup, obtained by Borichev-Tomilov.
