Stabilization and Riesz basis of a star-shaped network of Timoshenko beams

Abstract

In this paper we consider a star-shaped network of Timoshenko beams, with feedback acting on the common vertex of the star. Suppose that the exterior vertices of this network are clamped, and at the common vertex the displacement and rotational angles are continuous. For this network, feedback controllers are designed at the common vertex so as to move the beams back to their equilibrium position. We show that the operator determined by the closed loop system has a compact resolvent and generates a C 0 semigroup in an appropriate Hilbert space. Under certain conditions, we prove that the closed loop system is asymptotically stable. We also show that there is a sequence of the generalized eigenvectors of the system operator, which forms a Riesz basis with parentheses. Hence the spectrum determined growth condition holds.