In this paper, the state reconstruction of the wave equation with general viscosity and non-collocated observation and control are given from both the theoretical aspect and the numerical aspect. The forward-backward observers-based algorithm is utilized to solve this problem. The formula for calculating the initial value is derived. Moreover, the iterative sequence is also built for any given guess value and it is showed that it strongly converges to initial value. However, because the exponential stabilities of the error systems between the original systems and the forward∖backward observers play important roles in the involved algorithm, the exponential stability of some related system is firstly discussed. Furthermore, combining the theory of port-Hamiltonian system and the finite difference, the semi-discretization scheme of the finite difference with order reduction is given and the uniform exponential stability of the semi-discretization system is verified by the method paralleling to the continuous system. Finally, the convergence analysis of the finite difference scheme is given and the convergence of the solution of the mixed finite element scheme induced by the finite difference scheme with order reduction to the solution of the continuous counterpart is also presented.

本文从理论和数值两个方面给出了具有一般黏度和非并置观测与控制的波动方程的状态重构。利用基于前后观察者的算法来解决该问题。导出用于计算初始值的公式。此外，还为任何给定的猜测值建立了迭代序列，并表明该迭代序列强烈收敛于初始值。但是，由于原始系统与前后观察者之间误差系统的指数稳定性在所涉及的算法中起着重要作用，因此首先讨论了一些相关系统的指数稳定性。此外，结合港口哈密顿系统理论和有限差分，给出了具有阶次降阶的有限差分的半离散化方案，并通过与连续系统并行的方法验证了半离散化系统的均匀指数稳定性。最后，给出了有限差分格式的收敛性分析，并给出了由有限差分格式引起的混合有限元格式解的收敛性。