[论文解读] Optimal Strategies for Guarding a Compact and Convex Target Area: A Differential Game Approach.

Yoonjae Lee, Efstathios Bakolas|arXiv (Cornell University)|Apr 1, 2021

Guidance and Control Systems参考文献 21被引用 4

一句话总结

本文提出了一个双人平面目标防御博弈的最优策略，其中追捕者防御一个紧致且凸的靶区，对抗逃避者。利用微分博弈理论和Isaacs方法，识别出一条分隔状态空间中捕获与逃脱子博弈的屏障曲面，推导出在一般凸靶区下连续可微的值函数，以及追捕者与逃避者双方的唯一鞍点状态反馈策略。

ABSTRACT

We revisit the two-player planar target-defense game posed in [1], a special class of pursuit-evasion games in which the pursuer (or defender) strives to defend a stationary target area from the evader (or intruder) who desires to reach it, if possible, or approach it as close as possible. In this paper, the target area is assumed to be a compact and convex set. Unlike classical two-player pursuit-evasion games, this game involves two subgames: a capture game and an escape game. In the capture game, where capture is assured, the evader attempts to minimize the distance between her final position and the target area whereas the pursuer tries to maximize the same distance. In the escape game, where capture is not guaranteed, the evader attempts to maximize the distance between herself and the pursuer at the moment that she reaches the target for the first time. Our solution approach is based on Isaacs classical method in differential games. We first identify the barrier surface that demarcates the state space of the game into two subspaces, each of which corresponds to the two aforementioned subgames, by means of geometric arguments. Thereafter, we derive the optimal strategies for the players in each subspace. We show that, as long as the target area is compact and convex, the value of the game in each subspace is always continuously differentiable, and the proposed optimal strategies correspond to the unique saddle-point state-feedback strategies for the players. We illustrate our proposed solutions by means of numerical simulations.

研究动机与目标

• 开发一种双人平面目标防御博弈的最优反馈策略，目标区域为紧致且凸的区域。
• 将博弈分析为两种不同的子博弈：捕获博弈（捕获是确定的）和逃脱博弈（捕获不确定）。
• 确定最优策略存在且唯一的条件，特别是与目标区域几何形状的关系。
• 证明值函数在整个状态空间上连续可微，确保策略转换的平滑性。

提出的方法

• 采用Isaacs经典的微分博弈方法，推导最优策略。
• 通过几何方法识别出屏障曲面，将状态空间划分为对应于捕获和逃脱子博弈的区域。
• 在每个子空间中分别推导值函数，确保其在屏障面上连续且可微。
• 将最优策略表述为追捕者与逃避者双方的鞍点状态反馈控制。
• 分析假设目标区域为紧致且凸，从而可将结果推广至一般形状。
• 利用数值仿真验证理论解，并展示策略行为。

实验结果

研究问题

• RQ1 当逃避者必然被捕获时，追捕者的最优策略是什么？其如何最小化与目标区域的最终距离？
• RQ2 当捕获不确定时，最优策略如何变化？逃避者在抵达目标时如何最大化与追捕者的距离？
• RQ3 目标区域的何种几何性质可确保存在一条平滑屏障曲面，将捕获与逃脱子博弈分隔？
• RQ4 尽管博弈具有分段结构，值函数如何在整个状态空间上保持连续可微？
• RQ5 在微分博弈框架下，何种条件可确保所推导策略构成唯一的鞍点均衡？

主要发现

• 分隔捕获与逃脱子博弈的屏障曲面具有几何确定性，其依赖于目标区域的紧致性和凸性。
• 值函数在整个状态空间上连续可微，确保最优策略之间的平滑过渡。
• 双方的最优策略均为由Hamilton-Jacobi-Isaacs方程导出的唯一鞍点状态反馈控制。
• 该解框架可普遍适用于任意紧致且凸的目标集合，不仅限于圆盘或多边形等特定形状。
• 数值仿真验证了理论预测，显示在不同初始条件下策略执行稳定且一致。
• 博弈结构确保追捕者的最优策略始终在捕获时刻最大化其与目标的最小距离，而逃避者在捕获子博弈中则最小化该距离。