Stabilization of Vibrating Beam with a Tip Mass Controlled by Combined Feedback Forces

AbstractA flexible structure consisting of a Euler–Bernoulli beam with a tip mass is considered. To stabilize this system we use a boundary control laws: −uxxx(1,t)+mutt(1,t)=−αut(1,t)+βuxxxt(1,t) and uxx(1,t)=−γuxt(1,t). A sensitivity asymptotic analysis of the system's eigenvalues and eigenfunctions is set up. We prove that all of the generalized eigenfunctions of (2.9) form a Riesz basis of H. By a new method, we prove that the operator A generates a C0 contraction semigroup T(t), t≥0. Furthermore T(t), t≥0, is uniformly exponentially stable and the optimal exponential decay rate can be obtained from the spectrum of the system