如何理解对于变系数波方程控制为必要工具的黎曼几何

摘要

五十年来,变系数波方程的精确能控性一直是一个困难的问题,已经有许多文献把精确能控性转化为不可验证的假设,这些假设是不可验证的原因是能控性是全局性质,且经典的分析方法仅能很好地处理局部问题,而对全局问题不能和好地处理.微分几何估计是十几年前引入的,目的是给出变系数波方程精确能控的可验证条件,从此在振动和结构动力系统的建模和控制问题取得许多重要进展,这里通过简单地黎几何估计和一些主要的方法,说明为什么它是给出精确能控性可验证条件的一个必要工具.%The exact controllability of the wave equation with variable coefficients had been a difficult topic for almost fifty years.There were many papers which changed the controllability into some uncheckable assumptions.The reason that these assumptions are uncheckable is because that controllability is a global property and the classical analysis works well for local problems only and is insufficient to cope with global problems.The differential geometrical approach was introduced more than a decade ago where the original motivation was to give checkable conditions to the exact controllability of the wave equation with variable coefficients.Since then,many important advances in modeling and control in vibrational and structural dynamics have been made.Here we will briefly compare the Riemannian geometrical approach with some other main methods to show why it is a necessary tool to give checkable assumptions for control lability.