In the Target–Attacker–Defender differential game, an Attacker missile strives to capture a Target aircraft. The Target tries to escape the Attacker and is aided by a Defender missile which aims at intercepting the Attacker, before the latter manages to close in on the Target. The conflict between these intelligent adversaries is naturally modeled as a zero-sum differential game. The Game of Degree when the Attacker is able to win the Target–Attacker–Defender differential game has not been fully solved, and it is addressed in this paper. Previous attempts at designing the players’ strategies have not been proven to be optimal in the differential game sense. In this paper, the optimal strategies of the Game of Degree in the Attacker’s winning region of the state space are synthesized. Also, the value function is obtained, and it is shown that it is continuously differentiable, and it is the solution of the Hamilton–Jacobi–Isaacs equation. The obtained state feedback strategies are compared to recent results addressing this differential game. It is shown that the correct solution of the Target–Attacker–Defender differential game that provides a semipermeable Barrier surface is synthesized and verified in this paper.

在Tar​​get–Attacker–Defender差分游戏中，攻击者导弹努力捕获目标飞机。目标试图逃脱攻击者，并得到防御者导弹的帮助，该导弹旨在拦截攻击者，然后后者设法将目标封闭。这些聪明的对手之间的冲突自然被建模为零和微分博弈。攻击者能够赢得目标-攻击者-防御者的差分游戏时的度数博弈尚未完全解决，本文对此进行了介绍。在差异游戏意义上，先前设计玩家策略的尝试尚未被证明是最佳的。本文综合了状态空间攻击者获胜区域的度数博弈优化策略。同样，获得了价值函数，结果表明，它是连续可微的，是汉密尔顿-雅各比-艾萨克斯方程的解。将获得的状态反馈策略与针对该差分博弈的最新结果进行比较。结果表明，本文合成并验证了提供半渗透性阻隔表面的Target–Attacker–Defender微分博弈的正确解。