Uniformly semidiscretized approximation for exact observability and controllability of one-dimensional Euler–Bernoulli beam

Abstract

In this paper, a spacial semidiscretized finite difference scheme developed in our previous work (Liu and Guo, 2019) is used to approximate exact observability and controllability for Euler–Bernoulli beam control system. The uniform observability inequality is proved by discrete energy multiplier technique. The uniform controllability and uniform boundedness of the discrete controls are also developed. Compared with the existing literature, the proposed approach has achieved potentially the following advantages: (a) It removes the introduction of the numerical viscosity term to achieve uniformity; (b) It can deal with any type of boundary conditions without help of the spectral analysis which is limited only for some special boundary conditions; (c) The proofs of the uniform observability and controllability are simplified significantly with the similar techniques in dealing with the continuous counterpart.