This paper considers pursuit-evasion differential games in the Euclidean plane where an evader is engaged by multiple pursuers and point capture is required. The players have simple motion (i.e., holonomic) in the manner of Isaacs, and the pursuers are faster than the evader. The attention of this paper is confined to the case where the pursuers have the same speed, and so the game’s parameter is that the evader/pursuers speed ratio is $0<\mu <1$. State feedback capture strategies and an evader strategy that yields a lower bound on his/her time-to-capture are devised using a geometric method. It is shown that, in group/swarm pursuit, when the players are in general position, capture is effected by one, two, or three critical pursuers, and this is irrespective of the size $N(>3)$ of the pursuit pack. Group pursuit devolves into pure pursuit by one of the pursuers or into a pincer movement pursuit by two or three pursuers who isochronously capture the evader. The critical pursuers are identified. However, these geometric method-based pursuit and evasion strategies are optimal only in a part of the state space where a strategic saddle point is obtained and the value of the differential game is established. As such, these strategies are suboptimal. To fully explore the differential game’s high-dimensional state space and get a better understanding of group pursuit, numerical experimentation is undertaken. The state space region where the geometric solution of the group pursuit differential game is the optimal solution becomes larger the smaller the speed ratio parameter is.

本文考虑了在欧几里得飞机上追逃逃避的差分博弈，其中多个追逃者参与了逃避逃避，并且需要点捕获。玩家以艾萨克（Isaacs）的方式进行简单的运动（即完整），并且追赶者比逃避者快。本文的注意力仅限于追赶者具有相同速度的情况，因此游戏的参数是逃避者/追赶者的速度比为$0<\mu <\mathrm{1个}$。使用几何方法设计了状态反馈捕获策略和在捕获时间上下限的逃避策略。结果表明，在小组/群追逐中，当玩家处于一般位置时，俘获是由一个，两个或三个关键性追逐者进行的，这与大小无关$ñ（>3）$追求包。团体追逐演变为一个追逐者的纯粹追逐，或由两个或三个同步捕捉逃避者的追逐者的钳子运动追逐。确定关键的追求者。但是，这些基于几何方法的追随和逃避策略仅在状态空间中获得战略鞍点并确定了微分博弈的价值的部分中是最佳的。因此，这些策略不是最佳的。为了充分探索微分博弈的高维状态空间并更好地理解群体追求，我们进行了数值实验。速度比参数越小，群追随对策的几何解是最优解的状态空间区域越大。