在欧几里得空间中保护目标集不受单个攻击
Guarding a Target Set from a Single Attacker in the Euclidean Space
摘要：
研究了$n$ -维欧几里得空间中的两人目标防御博弈，其中攻击者试图进入一个封闭的凸目标集，而防御者则努力提前捕获攻击者.我们提供了一个完整的和普遍的微分对策为基础的解决方案，不仅包括最近的工作与类似的问题，其目标集有简单的，低维的几何形状，但也可以解决的问题，涉及非平凡的几何形状的高维目标集。博弈的价值函数是以半解析形式导出的，其中包含一个凸优化问题。当后一个问题具有封闭形式的解时，其中一个值函数用于解析构造障碍曲面，该障碍曲面将游戏的状态空间划分为获胜的玩家集合。对于障碍曲面没有解析表达式但目标集边界光滑的情况，得到了目标边界与障碍曲面投影之间的双射映射。利用Hamilton-Jacobi-Isaacs方程，证明了无论最优化问题是否有封闭解，所提出的最优状态反馈策略总是构成博弈的唯一鞍点.我们通过数值模拟来说明我们的解决方案。
英文摘要：
This paper addresses a two-player target defense game in the $n$-dimensional Euclidean space where an attacker attempts to enter a closed convex target set while a defender strives to capture the attacker beforehand. We provide a complete and universal differential game-based solution which not only encompasses recent work associated with similar problems whose target sets have simple, low-dimensional geometric shapes, but can also address problems that involve nontrivial geometric shapes of high-dimensional target sets. The value functions of the game are derived in a semi-analytical form that includes a convex optimization problem. When the latter problem has a closed-form solution, one of the value functions is used to analytically construct the barrier surface that divides the state space of the game into the winning sets of players. For the case where the barrier surface has no analytical expression but the target set has a smooth boundary, the bijective map between the target boundary and the projection of the barrier surface is obtained. By using Hamilton-Jacobi-Isaacs equation, we verify that the proposed optimal state feedback strategies always constitute the game's unique saddle point whether or not the optimization problem has a closed-form solution. We illustrate our solutions via numerical simulations.
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