Generalising the capture the flag scenario to active target defence

Abstract

This manuscript examines the TAD differential game. Here there are three drones, the Drone A, Drone T and Drone D, all obeying simple-motion. This game doesn’t terminate at some predefined time tf, rather tf is the first time in which Drone A collides with either of the other two drones. The objective of Drone A is to minimise the distance between itself and Drone T at termination time; Drone T and D on the other-hand work together as a team to maximise the aforementioned distance. The present manuscript expands upon the analysis previously given in the work of [1], here we study the game in the general case where VT < VA < VD (denoting the speeds of Drone T, Drone A and Drone D respectively), and the drones move in n-dimensional space. The previous work identified and rigorously proved the SFNE. Most of the proofs given in that work held for any VT < VA < VD, however the proof of the non-decreasing property of the value function made the restrictive assumption of VT = 0, as the machinery required to prove it for the general case VT ≥ 0 was not known at the time. VT = 0 corresponds with the capture the flag scenario since Drone T cannot move. The present manuscript brings to light new symmetries in the value function, which are used to complete the missing proof so that the results now hold generally for any VT < VA < VD.