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. Project supported by the National Natural Science Foundation of China (Grant Nos. 69674011, 19671054) and Science Foundation of Shanxi University.