退化抛物型方程零阶项系数和初值的反演问题  

Inverse Problem of Zero Order Coefficient and Initial Value of Degenerate Parabolic Equation

作　　者：任如霞 REN Ru-xia(School of Mathematics and Physics,Lanzhou Jiaotong University,Lanzhou 730070,China)

机构地区：[1]兰州交通大学数理学院,甘肃兰州730070

出　　处：《滨州学院学报》2020年第6期44-52,共9页Journal of Binzhou University

基　　金：国家自然科学基金项目(11461039,61663018);兰州交通大学“百名青年优秀人才培养计划”;甘肃省自然科学基金资助项目(18JR3RA122)。

摘　　要：研究了最优控制问题极小元的局部适定性。该优化问题涉及同时重构退化抛物型方程初始值和零阶项系数的反问题。与一般的优化问题不同,这里构造的代价函数是一个二元函数,包含两个自变量和两个独立的正则化参数,并推导出极小元必须满足的必要条件。特别地,由于代价函数中两个未知系数的状态不同,单参数优化问题的共轭理论不能应用于该问题。通过假设终端时间T相对较小,得到了关于极小元的L2估计,由此可推导出极小元的唯一性和稳定性。The local well-posedness of the minimizer of optimal control problems is studied in this paper.The optimization problem involves the inverse problem of simultaneously reconstructing the initial value and the zero order coefficient in a degenerate parabolic equation.Different from general optimization problems,the cost function constructed in the paper is a binary function containing two independent variables and two independent regularization parameters.The necessary conditions that minimizer must be satisfied is deduced.In particular,because of the different states of the two unknown coefficients in the cost function,the conjugate theory widely applied in the single-parameter optimization problem cannot be applied in this problem.By assuming that the terminal time T is relatively small,an L2 estimate of the minimizer is obtained,from which the uniqueness and stability of the minimizer can be derived immediately.

关 键 词：反问题 退化抛物型方程 最优控制 二元函数 稳定性 唯一性 

分 类 号：O175.26[理学—数学]