The $\Pi$-strategy when players move under repulsive forces

In this paper we study differential pursuit game with a “Life line” for the case when the inertial movements of the players are carried out using controls subject to the action of repulsive forces. For solving the pursuit game with a “Life line”, the main tool remains the strategy of parallel pursuit (for brevity, the $\bf{\Pi}$-strategy). With the help of this $\bf{\Pi}$-strategy, necessary and sufficient conditions for completing the pursuit game are obtained, and for this case a set of capture points or a set of attainability of the evader in the pursuit game is constructed. For solving the problem with a “Life line” in favor of the pursuer we prove the monotonically decreasing (by inclusion) relative to time of this set of attainability.