Stabilization of abstract thermo-elastic semigroup

TL;DR

This paper investigates the stabilization properties of thermo-elastic systems, showing how stability under Cattaneo law relates to stability under Fourier law, with implications for control and damping strategies.

Contribution

It characterizes the stabilization of thermo-elastic systems with Cattaneo law and links their stability to that of systems with Fourier law, using advanced semigroup techniques.

Findings

Exponential stability under Cattaneo law implies polynomial stability under Fourier law.

Polynomial stability of the uncontrolled system can be characterized by resolvent estimates.

Methodology combines Ammari and Tucsnak's approach with Borichev-Tomilov's resolvent analysis.

Abstract

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.