Pointwise feedback stabilization of an Euler-Bernoulli beam in observations with time delay^★

¹ College of Science, North China University of Technology, Beijing 100144, P.R. China.
² School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, P.R. China.

^** Corresponding author: kyy@amss.ac.cn

Received: 26 February 2016
Accepted: 12 December 2017

Abstract

This paper considers a one-dimensional Euler-Bernoulli beam equation where two collocated actuators/sensors are presented at the internal point with pointwise feedback shear force and angle velocity at the arbitrary position ξ in the bounded domain (0,1). The boundary x = 0 is simply supported and at the other boundary x = 1 there is a shear hinge end. Both of the observation signals are subjected to a given time delay τ ( >0). Well-posedness of the open-loop system is shown to illustrate availability of the observer. An observer is then designed to estimate the state at the time interval when the observation is available, while a predictor is designed to predict the state at the time interval when the observation is not available. Pointwise output feedback controllers are introduced to guarantee the closed-loop system to be exponentially stable for the smooth initial values when ξ ∈ (0, 1) is a rational number satisfying ξ ≠ 2l∕(2m − 1) for any integers l, m. Simulation results demonstrate that the proposed feedback design effectively stabilizes the performance of the pointwise control system with time delay.