An Inverse Problem for a Viscoelastic Timoshenko Beam Model

Abstract

We consider the Timoshenko model for a viscoelastic beam. This model consists in a system of two coupled Volterra integrodifferential equations describing the evolution of the mean displacement $w$ and of the mean angle of rotation $\varphi$. The damping mechanism is characterized by two time-dependent memory kernels, $a$ and $b$, which are a priori unknown. Provided that $(w, \varphi)$ solves a suitable initial and boundary value problem for the evolution system, the inverse problem of determining a and b fçom supplementary information is analyzed. A result of existence and uniqueness on a given bounded time interval is proved. In addition, Lipschitz continuous dependence of the solution $(w, \varphi,a,b)$ on the data is shown.