半无界退化抛物型方程的源项反演问题

Inverse Source Problem for Semi-infinite Degenerate Parabolic Equations

摘要

采用人工边界方法和最优控制理论讨论半无界退化抛物型方程的正问题及其在附加条件下的源项反演问题.首先,针对正问题,利用人工边界方法将半无界区域问题转化为有界区域上的等价问题,并建立右边界满足的精确边界条件,再利用能量估计证明弱解的存在唯一性;其次,基于最优控制理论将原反问题转化为一个最优控制问题,证明最优解的存在性及其满足的必要条件;最后,利用必要条件证明最优解的稳定性,从而为该类方程的正反问题理论分析与数值求解提供理论支撑.

Abstract

In this paper, the artificial boundary method and the optimal control theory are adopted to investigate the forward problem of the semi-infinite degenerate parabolic equation and the inverse problem of determining the source term under additional conditions. First, for the forward problem, the artificial boundary method is used to transform the problem defined on a semi-infinite domain into an equivalent problem on a bounded domain, and exact boundary conditions at the right boundary are established. Then, the existence and uniqueness of the weak solution are demonstrated through energy estimates. Second, based on the optimal control theory, the original inverse problem is transformed into an optimal control problem, and the existence of an optimal solution and the necessary conditions it satisfies are proved. Finally, the stability of the optimal solution is proven by using the necessary conditions. This work provides both theoretical analysis and numerical solution support for the forward and inverse problems of such equations.

关键词

半无界 / 退化抛物型方程 / 最优控制 / 必要条件 / 适定性

Key words

semi-infinite / degenerate parabolic equation / optimal control / necessary condition / well-posedness

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基金资助

国家自然科学基金(61663018)

国家自然科学基金(11961042)

甘肃省自然科学基金(25JRRA163)

甘肃省自然科学基金(25JRRA952)

甘肃省基础研究创新群体项目(25JRRA805)