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.

多防御者单入侵者目标防御博弈是入侵者意图进入由一群防御者保护的目标区域，而防御者意图在入侵者进入之前将其捕获的差分博弈。这个游戏已经被广泛研究并且存在一个几何解决方案。然而，这种解决方案仅在特殊情况下才被证明是合理的。本文的主要贡献是证明了几何解在一般条件下满足HJI方程。具体来说，目标区域不需要采用圆形、直线等奇特的形状。此外，允许防守者在二维平面内自由移动，防守者的捕获范围不为零，并且防御者的数量不受限制。这种广义的表述对证明的一个基本步骤提出了一个重要的挑战，即计算价值函数的导数。本文解决了这一挑战，因此可以完成证明。研究几何解的意义在于它使用足够的计算量提供了状态反馈控制律。目标防御博弈本身就很难用数值方法来解决，因为它是高度非线性的，并且受到维数灾难的影响。本文给出的证明为几何求解提供了坚实的理论基础，因此可以规避数值方法带来的困难。研究几何解的意义在于它使用足够的计算量提供了状态反馈控制律。目标防御博弈本身就很难用数值方法来解决，因为它是高度非线性的，并且受到维数灾难的影响。本文给出的证明为几何求解提供了坚实的理论基础，因此可以规避数值方法带来的困难。研究几何解的意义在于它使用足够的计算量提供了状态反馈控制律。目标防御博弈本身就很难用数值方法来解决，因为它是高度非线性的，并且受到维数灾难的影响。本文给出的证明为几何求解提供了坚实的理论基础，因此可以规避数值方法带来的困难。