不定常Kohn-Sham模型下的最优控制问题研究
Investigation of optimal control problems governed by a time-dependent Kohn-Sham model
摘要：
量子物理和化学中的许多应用模型都需要控制多电子系统以实现所需的目标构型。这个具有挑战性的任务似乎可能在不定常密度泛函理论（TDDFT）的框架，允许描述这些系统，同时避免了高维多粒子薛定谔方程。为此，考虑到不同的目标和双线性控制机制，Kohn-Sham TDDFT模型的最优控制问题的理论和数值解进行了研究。讨论了Kohn-Sham TDDFT最优性系统的最优控制解的存在性及其作为解的特征。为了验证这种控制框架，时间分裂离散的最优性系统和非线性共轭梯度法。数值实验的结果表明所提出的控制方法的计算能力。
英文摘要：
Many application models in quantum physics and chemistry require to control multi-electron systems to achieve a desired target configuration. This challenging task appears possible in the framework of time-dependent density functional theory (TDDFT) that allows to describe these systems while avoiding the high dimensionality resulting from the multi-particle Schrödinger equation. For this purpose, the theory and numerical solution of optimal control problems governed by a Kohn-Sham TDDFT model are investigated, considering different objectives and a bilinear control mechanism. Existence of optimal control solutions and their characterization as solutions to Kohn-Sham TDDFT optimality systems are discussed. To validate this control framework, a time-splitting discretization of the optimality systems and a nonlinear conjugate gradient scheme are implemented. Results of numerical experiments demonstrate the computational capability of the proposed control approach.
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