The Complete Differential Game of Active Target Defense

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.