基于终端观测条件反演非线性抛物型方程的辐射系数

Inversion of Radiation Coefficient in Nonlinear ParabolicEquation Based on Terminal Observation Condition

研究了利用终端观测数据重构非线性热传导方程辐射系数的反演问题。与一般的抛物型方程不同的是,非线性热传导方程不仅具有非线性源项,而且边界也是非线性的。主要研究了一维逆问题的理论分析。对于多维情况,同样是适用的。基于最优控制框架,考虑到原问题的不适定性,首先将原问题转化为一个优化问题,在对原问题做估计的同时,证明了控制泛函极小值的存在性以及极值存在的必要条件。其次由于最优控制问题是非凸的,一般不存在全局唯一性,该解只在一些特殊条件下是局部唯一的。最后简单给出了解的稳定性分析。需要指出的是,一般情况下,参数识别是一个非线性反问题,即使其基本方程是一个线性方程。模型复杂性的增大,一方面使得该模型可以刻画更多的物理现象,另一方面也会给相应的分析带来更大的困难。因此,逆问题P的非线性程度,在某种意义上,要高于那些由线性方程控制的非线性程度,在复杂性方面也是如此。

This article studies the inversion problem of reconstructing the radiation coefficient of a nonlinear heat conduction equation using terminal observation data.Different from the general parabolic equations,the nonlinear heat conduction equation not only has a nonlinear source term,but also boundary is nonlinear.Based on the optimal control framework,the original problem is first transformed into an optimization problem,and the existence and necessary conditions of the control functional minimum value are proved.Since the optimal control problem is generally non convex and lacks global uniqueness,which is common in optimization control problems,this article proves that the solution is locally unique.Finally,the stability analysis of the solution is given.It should be pointed out that,in general,parameter identification is a nonlinear inverse problem,even if the basic equation is a linear equation.The increase of the complexity of the model,on the other hand,will bring greater difficulties to the corresponding analysis.Therefore,the nonlinear degree of the inverse problem P is,in a sense,higher than that of those controlled by linear equations,and in terms of complexity.