Justification of the geometric solution of a target defense game with faster defenders and a convex target area using the HJI equation

A multi-defender single-invader target defense game is a differential game where the invader intends to enter a target area protected by a group of defenders, while the defenders intend to capture the invader before it enters. This game has been extensively studied and a geometric solution exists. However, this solution has only been justified under special cases. The main contribution of this paper is to prove that the geometric solution satisfies the HJI equation under the general condition. Specifically, the target area is not required to take a peculiar shape, such as circles, lines, etc. In addition, the defenders are allowed to move freely in the two-dimensional plane, the capture range of the defenders is non-zero, and the number of defenders is not restricted. This generalized formulation imposes an important challenge on an essential step of the proof, computing the derivatives of the value function. This challenge is resolved in this paper therefore the proof can be accomplished. The significance of studying the geometric solution is that it provides a state feedback control law using an adequate amount of computation. The target defense game is inherently challenging to be solved with numerical methods, because it is highly nonlinear and suffers from the curse of dimensionality. The proof presented in this paper provides a solid theoretic foundation for the geometric solution, so the difficulties raised by the numerical methods can be circumvented.