分段仿射系统无限时间最优控制的值函数研究
On the Value Function of Infinite-Horizon Optimal Control of Piecewise Affine Systems
中文总结 AI 辅助
本文针对具有ℓ₁或ℓ∞阶段代价的PWA系统CITOC问题，通过显式例子证明其值函数片段数可无穷，并建立确保值函数为真PWA函数的充分条件，为相关学习控制方案提供支撑。
中文摘要 AI 辅助
本文研究具有ℓ₁或ℓ∞阶段代价的分段仿射（PWA）系统的约束无限时间最优控制（CITOC）问题的值函数结构。现有工作（如文献[1]）已证明该值函数在状态上为PWA函数，但未分析其是否为真PWA函数（即在紧集上具有有限个仿射片段）或片段数是否可无穷。我们通过一个显式例子证明后者确实可能发生，该例子还有助于建立严格且易验证的充分条件，确保所得值函数为真PWA函数。我们的理论发现补充了线性二次情形等已知结果，为近期针对PWA系统的基于学习的控制方案提供支撑，全文通过数值例子说明所提结果。
英文摘要
In this paper, we study the structure of the value function in constrained infinite-time optimal control (CITOC) problems of piecewise affine (PWA) systems, with $\ell_1$ or $\ell_\infty$ stage cost. Existing works, such as [1], establish that the resulting value function is PWA in the state. However, existing results do not analyze whether the value function is a proper PWA function, i.e., with a finite number of affine pieces over compact sets, or whether the number of pieces can be infinite. We show that the latter case is indeed possible by means of an explicit example, which is also instrumental in establishing rigorous and easily verifiable sufficient conditions that ensure that the resulting value function is a proper PWA function. Our theoretical findings complement well-known results, e.g., the linear-quadratic case, and serve as support for recent learning-based control schemes for PWA systems. Throughout the paper, the proposed results are illustrated by means of a numerical example.
