Uniform Polynomial Decay and Approximation in Control of a Family of Abstract Thermoelastic Models

In this paper, we consider the approximation of abstract thermoelastic models. It is by now well known that approximated systems are not in general uniformly exponentially or polynomially stable with respect to the discretization parameter, even if the continuous system has this property. Our goal in this paper is to study the uniform exponential/polynomial stability of a sequence of a system of weakly coupled thermoelastic models. We prove that when 0 <= beta <1/2, the total energy of solutions is not uniformly exponentially stable, but it decays uniformly polynomially to zero. Finally, the results are applied to space semi-discretizations of thermoelastic beam equation in a bounded interval with homogeneous Dirichlet boundary conditions. We consider finite element, spectral element and finite difference semi-discretizations. Finally, we illustrate the mathematical results with several numerical experiments.