Dynamical Analysis of an Axially Vibrating Nanorod

The aim of this paper is to study dynamical properties of an axially vibrating uniform nanorod. This analysis is performed by describing the model as an infinite-dimensional state-space system, with bounded control operator. In the absence of control forces, the system with homogeneous boundary conditions generates a strongly continuous semigroup. It is proved that the associated eigenfunctions form a Riesz basis for the energy space. It is shown that axial vibration of nanorod is stable phenomena but not exponentially stable. A necessary and sufficient condition for the approximate controllability is given.