基于有限元外微积分的希尔伯特复空间最优控制
Optimal Control in Hilbert Complex Spaces with Finite Element Exterior Calculus
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中文总结 AI 辅助
该研究开发了基于有限元外微积分的希尔伯特复空间PDE约束最优控制框架，建立了相关理论结果，通过数值实验验证了其在不同区域的控制效果，并扩展到时变麦克斯韦控制场景。
中文摘要 AI 辅助
我们开发了一套适用于希尔伯特复空间上PDE约束最优控制的框架，以及通过有限元外微积分（FEEC）实现的保结构离散化，重点关注非平凡拓扑产生具有物理意义的全局模式的问题。我们将经过规范固定的局部状态与代表全局环流或通量的调和分量分离开，并引入有限维拓扑执行器，其可达模式由区域拓扑和执行器秩共同决定。我们建立了适定性、弱最优性条件、可达性结果以及保持调和结构的FEEC误差估计。在可缩和多连通区域上的数值实验展示了环流与腔室通量控制、秩依赖可达性、与网格无关的优化以及拓扑诱导的奇异收敛。我们进一步将该框架扩展到时变麦克斯韦控制，其中调和电环流和磁通量成为表现出明确守恒性和可达性性质的动力学状态。
英文摘要
We develop a framework for PDE-constrained optimal control on Hilbert complexes and its structure-preserving discretization by finite element exterior calculus (FEEC), with emphasis on problems where nontrivial topology creates physically meaningful global modes. We separate the gauge-fixed local state from harmonic components representing global circulation or flux, and introduce finite-dimensional topological actuators whose reachable modes are determined jointly by domain topology and actuator rank. We establish well-posedness, weak optimality conditions, reachability results, and FEEC error estimates preserving the harmonic structure. Numerical experiments on contractible and multiply connected domains demonstrate circulation and cavity- flux control, rank-dependent reachability, mesh-independent optimization, and topology-induced singular convergence. We further extend the framework to time-dependent Maxwell control, where harmonic electric circulation and magnetic flux become dynamical states exhibiting explicit conservation and reachability properties.
