同时重构非线性抛物方程的辐射系数和初始温度

Simultaneous reconstruction of the heat radiative coefficient and initial temperature for nonlinear parabolic equations

本文研究了同时重构抛物方程的辐射系数和初始温度的反问题.与一般的抛物方程不同的是,本文不仅具有非线性源项,而且边界条件也是非线性的.基于最优控制框架,首先将原问题转化为一个优化问题,证明了控制泛函极小值的必要条件.与以往的优化问题不同,本文构造的代价泛函是一个包含两个自变量和两个独立正则化参数的二元泛函.特别的,由于代价泛函中两个未知系数的状态不同,在单参数优化问题中广泛使用的共轭理论不能应用于我们的问题.

In this paper,the inverse problem of simultaneously reconstructing the radiation coefficient and the initial temperature of the parabolic equation is studied.Unlike the general parabolic equation,this paper not only has a nonlinear source term,but also the boundary conditions are nonlinear.Based on the optimal control framework,the original problem is first transformed into an optimization problem,and the necessary conditions for the minimum value of the control functional are proved.Being different from other ordinary optimization problems,the cost functional constructed in the paper is a binary functional which contains two independent variables and two independent regularization parameters.Particularly,since the status of the two unknown coefficients in the cost functional are different,the conjugate theory which is extensively used in single-parameter optimization problems cannot be applied for our problem.