发表机构
Institute of Mathematics, Technische Universität Ilmenau（尹尔瑙工业大学数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种针对无限维线性二次最优控制的Newton--Kleinman迭代，通过线性化Lur'e恒等式并采用预解式归一化处理无界控制，保证稳定性和单调收敛，并在有限维情形恢复经典迭代，适用于圆柱尾流和弹性问题。
AI 中文摘要
我们针对可能具有无界控制和观测算子的无限维系统的线性二次最优控制，发展了一种Newton--Kleinman迭代。在此设定下，经典的代数Riccati方程未必有意义，因此不能作为迭代的基础。我们转而直接在系统节点域上对Lur'e恒等式进行线性化。从任意稳定反馈出发，所得迭代是良定义的，保持稳定性，并产生单调递减的成本算子序列。基于预解式的归一化使得该构造适用于无界控制。在有限成本可检测性和有限维输入空间的条件下，迭代强收敛于最优值算子，且与固定的归一化参数无关。在有限维情形下，自然的零状态归一化恰好恢复经典的Newton--Kleinman迭代。该公式导致一种基于移位系统求解和输入空间上求逆的低秩数值实现。我们在线性化圆柱尾流（具有边界牵引控制和无界观测）以及一个具有边界牵引控制和边界位移观测的阻尼线性弹性问题上展示了该方法，在每个自适应加密空间网格上使用一次Newton--Kleinman更新。
英文摘要
We develop a Newton--Kleinman iteration for linear-quadratic optimal control of infinite-dimensional systems with possibly unbounded control and observation operators. In this setting, the classical algebraic Riccati equation need not be meaningful and therefore cannot serve as the basis of the iteration. We instead linearize a Lur'e identity formulated directly on the system-node domain. Starting from any stabilizing feedback, the resulting iteration is well defined, preserves stability, and yields a monotonically decreasing sequence of cost operators. A resolvent-based normalization makes the construction applicable to unbounded control. Under finite-cost detectability and for a finite-dimensional input space, the iterates converge strongly to the optimal value operator, independently of the fixed normalization parameter. In finite dimensions, the natural zero-state normalization recovers exactly the classical Newton--Kleinman iteration. The formulation leads to a low-rank numerical realization based on shifted system solves and inversions on the input space. We illustrate the method on a linearized cylinder wake with boundary traction control and unbounded observation, using one Newton--Kleinman update per adaptively refined spatial mesh, and on a damped linear elasticity problem with boundary traction control and boundary displacement observation.
