Applications of the  \(\Pi \) -Strategy When Players Move with Acceleration

In the present paper, we have carried out a fairly complete study of the pursuit problem when the players move by inertia, i.e. when the players carry out their movements with the help of acceleration vectors. The main tool for solving this pursuit problem is the parallel approach strategy ( \(\Pi \) -strategy), which, under an arbitrary action of the evader, allows the best approach of the players. It is known that for the simple pursuit problem, the set of meeting points of the players is the ball of Apollonius. In the present paper, it is shown that in the case of inertial motions of players, the set of meeting points of the players is a linear combination of two such Apollonius balls, the first of which is built from the initial states, and the second from the initial states of the velocities.