The stabilization of tree-shape networks of beams with time delays in the boundary controls

In this paper we consider the dynamical stability of a tree-shape networks of Timoshenko beams with time-delay terms in the boundary controls. The time-delay feedback controllers at the exterior vertices are designed to derive the beams back to its equilibrium position. We first get the wellposedness of the closed loop system. Under certain conditions, we show that this system is asymptotically stable. By spectral analysis, we also prove that there is a sequence of the generalized eigenvectors of the system operator that forms a Riesz basis with parentheses. Hence the spectrum determined growth condition holds. Finally, for a special case, we discuss the exponential stability of this closed loop system and give a simulation to support our results.