  In this paper an N-pursuer vs. M-evader team conflict is studied. The
differential game of border defense is addressed and we focus on the game of
degree in the region of the state space where the pursuers are able to win.
This work extends classical differential game theory to simultaneously address
weapon assignments and multi-player pursuit-evasion scenarios. Saddle-point
strategies that provide guaranteed performance for each team regardless of the
actual strategies implemented by the opponent are devised. The players' optimal
strategies require the co-design of cooperative optimal assignments and optimal
guidance laws. A representative measure of performance is proposed and the
Value function of the game is obtained. It is shown that the Value function is
continuous, continuously differentiable, and that it satisfies the
Hamilton-Jacobi-Isaacs equation - the curse of dimensionality is overcome and
the optimal strategies are obtained. The cases of N=M and N>M are considered.
In the latter case, cooperative guidance strategies are also developed in order
for the pursuers to exploit their numerical advantage. This work provides a
foundation to formally analyze complex and high-dimensional conflicts between
teams of N pursuers and M evaders by means of differential game theory.
