一类Maxwell方程最优控制问题的差分格式
Difference Schemes for a Class of Maxwell Equation Optimal Control Problems
DOI: 10.12677/ORF.2022.124152, PDF,    国家自然科学基金支持
作者: 梅 练, 罗贤兵:贵州大学数学与统计学院,贵州 贵阳
关键词: Maxwell方程最优控制FDTD格式四阶差分格式Maxwell’s Equation Optimal Control FDTD Scheme Four-Order Difference Scheme
摘要: 针对一类时域Maxwell方程最优控制问题,借助Lagrange乘子法和Helmholtz分解理论,推导出由状态方程、对偶状态方程、一阶最优性条件构成的最优性系统,分别利用时域有限差分(FDTD)格式和四阶差分格式离散最优性系统,得到最优性系统的离散格式。最后,建立数值算例分别求解基于两种差分格式的Maxwell方程最优控制问题。为了避免求解大型耦合代数方程,采用迭代法进行计算,数值结果验证了理论分析结论的正确性。
Abstract: Aiming at a class of time-domain Maxwell equation optimal control problems, an optimality system composed of equation of state, dual equation of state and first-order optimality condition is derived by using Lagransssge multiplier method and Helmholtz decomposition theory. The finite-difference time-domain (FDTD) scheme and fourth-order difference scheme are used to discretization the optimality system, respectively. The discrete scheme of the optimality system is obtained. Finally, numerical examples are established to solve the Maxwell equation optimal control problem based on the two difference schemes. In order to avoid solving large coupled algebraic equations, the iterative method is used for calculation, and the numerical results verify the correctness of the theoretical analysis.
文章引用:梅练, 罗贤兵. 一类Maxwell方程最优控制问题的差分格式[J]. 运筹与模糊学, 2022, 12(4): 1439-1451. https://doi.org/10.12677/ORF.2022.124152

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