Asymptotic behavior of non-uniform Timoshenko beam acting on shear force with feedback controller

The spectrum and asymptotic behavior of a non-uniform Timoshenko beam with second sound is considered in this paper. In particular, we consider a new thermoelastic coupling on shear force. The velocity of the beam at one end is observed. A feedback controller is designed to stabilize the beam. Firstly, we prove the well-posedness of the considered problem and that the operator has compact resolvent and generates a C-0-semigroup in an appropriate space. We also show that there is a sequence of the generalized eigenvectors of the operator that forms a Riesz basis. Then based on the distribution of the spectrum, they prove that the corresponding C-0-semigroup decays exponentially independently of the wave speeds of propagation. However, it no longer decays exponentially if there is no feedback controller. As application, we consider the uniform problem to improve some already existing results and to find the best choice of the feedback ensuring the exponential stability. (C) 2017 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim