平面快慢动力系统中进入-退出现象的最优控制
On the optimal control of entry-exit phenomena in planar fast-slow dynamical systems
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中文总结 AI 辅助
本文针对平面快慢动力系统中的进入-退出现象，提出基于几何奇异摄动理论的最优控制方法，通过玩具模型解析推导并证明最优控制存在性，且算法有限步收敛。
中文摘要 AI 辅助
我们研究了一种几何奇异摄动理论（GSPT）方法，用于涉及平面快慢动力系统中进入-退出现象的最优控制问题。与以往工作不同，我们假设控制仅作用于慢动力学，并考虑最小时间最优控制问题（即，我们的目标是使到达某个目标的时间最小化）。我们将此方法应用于一个简化的快慢玩具模型，该模型以进入-退出过程的重复为特征，从而能够对控制动力学进行透明的解析探索，并使我们能够显式地推导出关键结果。首先，我们分别分析每个非自治的进入-退出现象，并推导出控制它们的最优方式。其次，通过人为生成一个稳定的极限环，我们确定在何种条件下能够到达目标。最后，在这些条件下，我们将所有进入-退出过程结合起来，并证明最优控制的存在性。我们的方法主要利用GSPT和动态规划的技术。实际上，我们表明我们的问题可以以一种类似于Bellman方程的方式表述，尽管并非其标准形式。基于这种类Bellman公式的数值模拟说明了我们的解析结果。特别地，最优控制通过一种在有限步内收敛的算法推导得出。最后，我们将这种新方法扩展到更复杂的平面快慢动力系统中。
英文摘要
We study a geometric singular perturbation theory (GSPT) approach to optimal control problems involving entry-exit phenomena in planar fast-slow dynamical systems. In contrast to previous work, we assume that the control acts only on the slow dynamics and we consider minimum-time optimal control problems (i.e., we aim at minimizing the time to reach a certain target). We apply this method to a simplified fast-slow toy model, characterized by a repetition of entry-exit processes, enabling a transparent analytical exploration of the control dynamics and allowing us to derive the key results explicitly. First, we analyze each non-autonomous entry-exit phenomenon separately, and we derive the optimal way to control them. Second, by artificially generating a stable limit cycle, we determine under which conditions it is possible to reach the target. Finally, under such conditions, we combine all entry-exit processes together and we demonstrate the existence of an optimal control. Our approach mainly exploits techniques of GSPT and dynamic programming. Indeed, we show that our problem can be formulated in a way resembling the Bellman equation, even though not in its standard setting. Numerical simulations, built on such Bellman-like formulation, illustrate our analytical results. In particular, the optimal control is derived with an algorithm that converges in a finite number of steps. Lastly, we extend this novel approach to more complex planar fast-slow dynamical systems.
发表机构
- Simon Fraser University（西蒙菲莎大学）
- Technical University of Munich（慕尼黑工业大学）
- Università degli Studi di Trento（特伦托大学）
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