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

Abstract

This paper deals with the approximation and homogenization of thermoelastic wave model. First, we study the homogenization problem of a weakly coupled thermoelastic wave model with rapidly varying coefficients, using a semigroup approach, two-scale convergence method and some variational techniques. We show that the limit semigroup can be obtained by using a weak version of the Trotter Kato convergence Theorem. Secondly, we consider the approximation of two thermoelastic wave model, one with exponential decay and the other one with polynomial decay. the numerical experiments indicate that the two discrete systems show different behavior of the spectra. Moreover, their discrete energies inherit the same behavior of the continuous ones. Finally we show numerically how the smoothness of data can impact the rate of decay of the energy associated the weakly coupled thermoelastic wave model.