论文标题
在希尔伯特空间中的集合控制分析和设计的凸线几何方法
A Convex-Geometric Approach to Ensemble Control Analysis and Design in a Hilbert Space
论文作者
论文摘要
在本文中，我们在希尔伯特空间环境中使用凸几何方法来应对集合控制分析和设计的长期挑战。具体而言，我们将线性集合系统的控制作为希尔伯特空间中的凸的可行性问题，可以通过迭代加权投影来解决。这种非平凡的几何解释不仅可以实现系统的设计原理，用于构建可行，最佳和受约束的集合控制信号，而且还使得对集合可及性和可控性的数值检查成为可能。此外，我们将这种几何方法纳入了迭代框架中，并说明了其为转向双线性集合系统提供可行控制的能力。我们对线性和双线性集合的控制进行了各种数值实验，以验证理论发展并证明所提出的凸数几何方法的适用性。
In this paper, we tackle the long-standing challenges of ensemble control analysis and design using a convex-geometric approach in a Hilbert space setting. Specifically, we formulate the control of linear ensemble systems as a convex feasibility problem in a Hilbert space, which can be solved by iterative weighted projections. Such a non-trivial geometric interpretation not only enables a systematic design principle for constructing feasible, optimal, and constrained ensemble control signals, but also makes it possible for numerical examination of ensemble reachability and controllability. Furthermore, we incorporate this geometric approach into an iterative framework and illustrate its capability to derive feasible controls for steering bilinear ensemble systems. We conduct various numerical experiments on the control of linear and bilinear ensembles to validate the theoretical developments and demonstrate the applicability of the proposed convex-geometric approach.
