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.

在本文中，我们之前的工作（Liu and Guo，2019）中开发的空间半离散化有限差分格式用于近似欧拉-伯努利光束控制系统的精确可观测性和可控性。通过离散能量乘法器技术证明了均匀可观测性不等式。离散控制的统一可控性和统一有界性也得到了发展。与现有文献相比，所提出的方法具有潜在的以下优点：（a）它消除了数值粘度项的引入以实现均匀性；(b) 可以处理任何类型的边界条件，而无需谱分析，谱分析仅限于一些特殊的边界条件；