Stability of a Timoshenko System with Localized Kelvin–Voigt Dissipation

We consider the Timoshenko beam with localized Kelvin–Voigt dissipation distributed over two components: one of them with constitutive law of the type C1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C^1$$\end{document}, and the other with discontinuous law. The third component is simply elastic, where the viscosity is not effective. Our main result is that the decay depends on the position of the components. We will show that the system is exponentially stable if and only if the component with discontinuous constitutive law is not in the center of the beam. When the discontinuous component is in the middle, the solution decays polynomially.