Stabilization of tree-shaped network of Timoshenko beams

In this paper we study stabilization problem of tree-shaped network of Timoshenko beams which consists of three beams. Suppose that the root of the network is clamped, at the interior node, the displacement are continuous, and the forces satisfy the transmission conditions. The feedback controllers at exterior vertices are applied to stabilize the system. We show that the closed loop system is asymptotically stable. By spectral analysis, we show that the spectrum of the system operator consists of all eigenvalues and distributes in a strip parallel to the imaginary axis, the generalized eigenfunctions of the system forms a Riesz basis with parentheses for the state space under some conditions. Finally, we prove that the closed loop system is stable exponentially.