The Capture-the-Flag Differential Game

The two-phase capture-the-flag differential game considers a Defender and its opponent, an Attacker. In Phase I the Attacker aims at capturing the flag while the Defender tries to prevent this outcome by intercepting the Attacker. Phase II is conditioned by the outcome of Phase I. If the Attacker is able to actually capture the flag at the end of Phase I, then Phase II takes place where the Attacker attempts to reach the safe zone while the Defender strives to intercept the Attacker before the latter can reach the safe haven. The saddle-point state-feedback strategies for each player and for each phase of the game are determined in this paper. Furthermore, for each phase, the Value function is obtained and it is shown to be continuous and continuously differentiable, and to satisfy the Hamilton-Jacobi-Isaacs equation.