标题： 关于正交区域上值迭代的一些几何行为：切换系统视角 显示英文标题

标题： On Some Geometric Behavior of Value Iteration on the Orthant: Switching System Perspective

摘要： 本文的主要目标是通过控制领域中的切换系统模型，为价值迭代提供更深入的见解。这些模型在价值迭代和切换系统理论之间建立了联系，并揭示了价值迭代在解决折扣马尔可夫决策问题时的额外几何行为。具体而言，本文的主要贡献有两个方面：1）我们提供了价值迭代的切换系统模型，并基于此模型为价值迭代的压缩性质提供了不同的证明；2）此外，从这些额外的见解出发，当初始迭代位于特殊区域时，证明了价值迭代的新几何行为。我们预计提出的视角可能有潜力成为一种有用的工具，在各种场景中应用。因此，这些方法的进一步发展可能是未来研究的一个有价值的途径。 显示英文摘要

摘要： In this paper, the primary goal is to offer additional insights into the value iteration through the lens of switching system models in the control community. These models establish a connection between value iteration and switching system theory and reveal additional geometric behaviors of value iteration in solving discounted Markov decision problems. Specifically, the main contributions of this paper are twofold: 1) We provide a switching system model of value iteration and, based on it, offer a different proof for the contraction property of the value iteration. 2) Furthermore, from the additional insights, new geometric behaviors of value iteration are proven when the initial iterate lies in a special region. We anticipate that the proposed perspectives might have the potential to be a useful tool, applicable in various settings. Therefore, further development of these methods could be a valuable avenue for future research.