Unified Optimality Criteria for the Target-Attacker-Defender Game

The game consists of two teams, team A (Attacker) and team T/D (Target and Defender). The Attacker aims to minimise the separation between itself and the Target at time $ t_f $ tf; and team T/D aims to maximise the separation distance. Previous works in the literature identified three distinct optimality criteria to characterise the state-feedback Nash equilibrium. We introduce a new unifying paradigm given by the Critical Escape Trajectories Theorem, which simultaneously characterises the Target's escape set and the value function of the game in all regions, highlighting a deep underlying geometric connection between the differential game of kind and the differential game of degree. Leveraging previous results in the literature, the Critical Escape Trajectories Theorem is proved to be equivalent and is utilised to develop an efficient algorithm for the computation of the state-feedback Nash equilibrium.