Semigroup model and stability of the structurally damped Timoshenko beam with boundary inputs

Abstract

This paper is concerned with the linear partial differential equation describing the motion of the Timoshenko beam with viscous internal dampings and with a certain type of boundary inputs. The equation is formulated as an evolution equation of the form dz( t)/dt = — Az( i) in a Hilbert space, and then it is proved for the linear operator — A to generate an analytic semigroup. Moreover, the stability of the boundary control system is examined. It is shown that in some cases the present type of boundary inputs degrades the degree of stability of the system. This is a remarkable contrast to the results of previous research which states that the same type of boundary inputs are effctive in stabilizing the Timoshenko beam without any structural damping.