Spectral properties of non-homogeneous Timoshenko beam and its rest to rest controllability

The controllability of a slowly rotating non-homogeneous beam clamped to a disc is considered. It is assumed that at the beginning the beam remains at the position of rest and it is supposed to rotate by the given angle and stop. The movement is governed by the system of two differential equations with non-constant coefficients: mass density, flexural rigidity and shear stiffness. To solve the problem of controllability, the spectrum of the operator generating the dynamics of the model is studied. Then the problem of controllability is reduced to the moment problem that is, in turn, solved with the use of the asymptotics of the spectrum.