A geometric approach to reach-avoid games with time limits

The differential games have been widely used to analyze the conflicts between intelligent agents. Motivated by the fact that the agents always have finite energy or time requirements, a novel reach-avoid game with time limits is investigated in this work. The attacker aims to reach the target region without being captured or reaching its time limit, while the defender strives to intercept the attacker or delay it. This game is beyond the scope of the classical Hamilton-Jacobi-Isaacs (HJI) approach. To make the problem possible to solve, we introduce the concept of reaching region and provide the optimal strategies of the players based on it. Using these strategies, we construct a hypersurface, called the barrier, in the game state space which partitions it into two parts that lead to different outcomes of the game. In this work, the complete analytical expressions of the barrier in all possible situations are provided. The game results can be obtained by substituting the initial states into the related expression and there is barely any computational burden. Compare to the existing works, the game with time limits is more practical. Also, this work provides the foundation for analyzing general multiple-attacker-multiple-defender games.