一类确定波动方程势函数的反问题

An Inverse Problem of Identifying Potential Function of Wave Equation

本文研究了一类利用终端观测数据重构波动方程势函数的反问题.这一研究是识别地下介质物理力学参数的重要方法,且在地球物理和工程技术等领域中有重要应用.该问题有两个主要困难:一、极值原理不再成立;二、终端观测值不仅包含位移,还包含了终端时刻的速度.首先基于优化方法,将原问题转化为最优控制问题,进而建立了最优解的存在性及所满足的必要条件.需特别注意,共轭方程中位移与速度的位置恰好相反,这一点与抛物情形是完全不同的.此外,还证明了最优解的局部唯一性和稳定性.这一结果是新而有趣的,为将来的数值模拟和工程应用打下坚实的理论基础.

This paper investigates an inverse problem of using terminal observations to reconstruct the potential function of the wave equation.The research is an important method to identify the physical and mechanical parameters of underground media,and has important applications in geophysics and engineering technology.There are two main difficulties in this problem:first,the extremum principle is no longer valid;second,the terminal observations include not only the displacement,but also the velocity at the terminal.Firstly,based on the optimal control framework,the problem is transformed into the optimal control problem,and then the existence and the necessary conditions of the optimal solution are established.It should be noted that the position of displacement and velocity in the adjoint equation is exactly the opposite,which is completely different from that in the parabolic case.In addition,the local uniqueness and st ability of the optimal solution are proved.This result is novel and interesting,and lays a solid theoretical foundation for future numerical simulation and engineering applications.