Barrier Analysis of the Two-Deadline Game

This paper presents an in-depth study on the dead-line differential game, which is a fundamental type within qualitative differential games. Challenges exist in the research, primarily due to the complexity of the Retrograde Path Equations (RPEs) in the two-deadline game, often preventing the determination of the barrier's analytical form. To address this problem, we introduce a novel method using decomposition. We decompose the entire game into two subgames, apply Isaacs's approach to construct the barriers for each subgame individually, and then obtain the complete barrier of the entire game. In the two-dimensional space, this paper discusses the variations in the barriers and capture regions as the distance between the pursuer's position and the positions of the two deadlines changes. We have validated the methods and conclusions through a series of examples, demonstrating their effectiveness. This research offers new insights into the theoretical study of deadline games, providing more strategic choices for decision-makers in practical applications.