A Riesz basis approach to exponential stability in thermoelasticity of type III

Using a Riesz basis approach, we investigate, in this paper, the exponential stability for a one-dimensional linear thermoelasticity of type III with Dirichlet-Dirichlet boundary conditions. A detailed spectral analysis gives that the spectrum of the system contains two parts: the point and continuous spectrum. It is shown that, by asymptotic analysis, there are three classes of eigenvalues: one is along the negative real axis approaching to — ∞, the second is approaching to a vertical line which parallels to the imagine axis, and the third class is distributed around the continuous spectrum which is an accumulation point of the last classes of eigenvalues. Moreover, it is pointed out that there is a sequence of generalized eigenfunctions, which forms a Riesz basis for the energy state space. Finally, the spectrum-determined growth condition holds true and the exponential stability of the system is then established.