Reach-Avoid Games with a Time Limit and Detection Range: A Geometric Approach

The reach-avoid game theory is an ideal tool to handle the conflicts among intelligent agents and has been previously studied assuming full state information and no time limits on the players in the past decades. In this article, we extend the problem by requiring the defender to detect the attacker and adding maximum operation time constraints to the attacker. The attacker aims to reach the target region without being captured or reaching its time limit. The defender can employ strategies to intercept the attacker only when the attacker is detected. A geometric method is proposed to solve this game qualitatively. By analyzing the geometric property of the Apollonian circle and the detection range, we give the barrier under the condition that the attacker is initially detected and the attacker's shortest route which guarantees its arrival at the target region when it is initially outside the detection range. Then, a barrier that separates the game space into two respective winning regions of the players is constructed based on the shortest route and the time limit of the attacker. The main contributions of this work are that this paper provides the first attempt to introduce the abovementioned two concepts simultaneously, which makes the game more practical, and we provide the complete solution of the game in all possible situations.