Optimal decay rate of vibrating beam equations controlled by combined boundary feedback forces

The optimal decay rate problem is considered for boundary control system modeling by a flexible structure consisting of a Eular-Bernoulli beam. Controls are a bending moment in proportion to angular velocity and a shear force in proportion to velocity. A sensitivity asymptotic analysis of the system's eigenvalues and eigenfunctions is set up. It is proved that, for every 0<K 2<+∞ and 0<-K 1<+∞, all the generalized eigenfunctions of form a Riesz basis ofV×H, and the optimal exponential decay rate can be obtained from the spectrum of the system.