超材料中Maxwell方程最优控制问题的建模与数值计算

The optimal control problems governed by Maxwell equations in metamaterials play a very important role in the design and applications of metamaterials. Usually, one can not find analytical solutions of the optimal control problems. Therefore, it is of great importance to find the reliable and efficient numerical methods. This project focus on the modelings and numerical calculations of the optimal control problems governed by Maxwell equations in metamaterials. We first establish the mathematical models with reasonable control variables, state variables and state equations. Then using theories of Helmholtz decomposition, weakly convergence, semigroup and so on, we present rigorous mathematical analyses including existence of optimal controls, regularity results and optimality conditions. Based on these results, we study finite element methods of the optimal control problems. Some approaches, such as discrete Helmholtz decomposition, discrete compactness property of elements and embedding results are employed to derive convergence analysis and error estimations. We aim to eventually propose a general framework of finite element method for solving the optimal control problems governed by Maxwell equations in metamaterials. At last, we try to establish the high order discrete schemes, explore superconvergence analysis and develop efficient algorithms. The innovative results of this project will enrich the connotation of optimal control of electromagnetic wave propagation in the metamaterials and finite element methods.

超材料中Maxwell方程最优控制问题的研究在超材料的设计和应用中发挥着重要作用。由于最优控制问题的精确解通常难以求出，从而寻找其可靠、高效的数值逼近方法具有非常重要的理论意义和应用价值。本项目拟就超材料中Maxwell方程最优控制问题的建模和数值计算展开研究。首先合理选择控制变量、状态变量和状态方程，建立最优控制数学模型；然后基于Helmholtz分解、弱收敛和半群等理论给出严格的数学分析(包括最优解的存在性、正则性、最优性条件等)；在此基础上发展其有限元方法，利用离散Helmholtz分解、单元的离散紧致性、嵌入理论等实现收敛性分析和误差估计，形成超材料中Maxwell方程最优控制问题有限元方法的一般格式；进一步探索建立高阶离散格式和超收敛格式，开发其高效算法。项目的创新性成果将丰富超材料中电磁波传播优化控制问题和有限元方法研究的内涵。

鉴于Maxwell方程最优控制问题在电磁材料设计和应用中的重要作用，本项目围绕电磁场最优控制问题的建模和数值计算，从优化问题和控制方程两个角度开展了数学理论及其有限元方法的深入研究。通过创新性思路和方法的探索，取得了一些有积极意义的成果：建立了超材料中Drude型Maxwell方程最优控制问题的数学模型，完成了数学理论分析并给出有限元离散格式。对一类最优控制问题和与最优控制问题密切相关的变分不等式等非线性问题提出了非协调有限元离散的收敛性分析新方法，形式了高精度分析新模式，为电磁场最优控制问题的高效有限元方法研究提供了新思路；对电磁场最优控制问题的几类控制方程构造了全离散有限元格式，得到了最优误差估计和超收敛结果，为发展新的电磁场最优控制模型并进行数值模拟奠定了基础。项目的创新性成果为实现电磁波传播的优化控制提供了理论支撑和数值方法，丰富了最优控制问题和有限元方法研究的内涵。