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This paper considers a subspace guarding game in high-dimensional space whichconsists of a play subspace and a target subspace. Two faster defenderscooperate to protect the target subspace by capturing an attacker which strivesto enter the target subspace from the play subspace without being captured. Aclosed-form solution is provided from the perspectives of kind and degree.Contributions of the work include the use of the attack subspace (AS) method toconstruct the barrier, by which the game winner can be perfectly predictedbefore the game starts. In addition to this inclusion, with the prioriinformation about the game result, a critical payoff function is designed whenthe defenders can win the game. Then, the optimal strategy for each player isexplicitly reformulated as a saddle-point equilibrium. Finally, we apply thesetheoretical results to a half-space guarding game in three-dimensional space.Since the whole achieved developments are analytical, they require a littlememory without computational burden and allow for real-time updates, beyond thecapacity of traditional Hamilton-Jacobi-Isaacs method. It is worth noting thatthis is the first time in the current work to consider the target guardinggames for arbitrary high-dimensional space, and in a fully analytical form.

Guarding a Subspace in High-Dimensional Space with Two Defenders and One Attacker