Stability of abstract thermo-elastic semigroups

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 corresponding thermoelastic system with the Fourier law. The proof of the main results uses, respectively, the methodology introduced by Haraux in [11] and generalized by Ammari and Tucsnak in [8], 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 a C0C0-semigroup, in a Hilbert space, by a polynomial estimation of the resolvent of its generator obtained by Borichev and Tomilov [9]. Illustrating examples are given.