先稳定再优化：最优控制中作为预条件器的反馈变换
Stabilize-then-optimize: Feedback transformations as preconditioners in optimal control
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中文总结 AI 辅助
研究最优控制问题，核心方法是利用反馈变换重新表述问题，主要贡献是降低控制到状态映射的范数，改善条件数，通过多种方程示例及数值例子展示了该方法的有效性。
中文摘要 AI 辅助
许多最优控制的数值算法通过控制到状态映射消除状态，例如凝聚方法或最优性系统的预条件共轭梯度法。控制到状态映射的范数直接进入这些方法的收敛估计。本文表明，使用反馈变换可重新表述最优控制问题以降低（反馈后的）控制到状态映射的范数，从而显著改善相关条件数。通过抛物、双曲或椭圆等常微分方程和偏微分方程示例进行说明，并给出通过反馈改善解算子范数的构造方法，还通过数值示例展示了该方法的有效性。
英文摘要
Many numerical algorithms for optimal control leverage an elimination of the state via the control-to-state map such as condensed approaches or preconditioned conjugate gradient methods for the optimality system. As such, the norm of the control-to-state map directly enters the convergence estimates for these methods, e.g., via the condition number of the associated linear system. In this work we show that using feedback transformations one may reformulate the optimal control problem to decrease the norm of the (feedbacked) control-to-state map, leading to a drastic improvement of the involved condition numbers. We illustrate the abstract approach for ordinary and partial differential equations such as parabolic, hyperbolic or elliptic equations. For each of these problem classes we provide a constructive method to improve solution operator norms via feedbacks. Further, we showcase the efficacy of the method by means of various numerical examples with elliptic, parabolic and hyperbolic partial differential equations.
