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Stabilization of abstract thermo-elastic semigroup

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.