On a game problem for point control near the surface of the Moon

A game control problem in which the first player controls the material point of variable composition is considered. The second player controls the point that can move with a limited speed. It is assumed that the material point of variable composition, along with the controlled reactive power, is exposed to a constant force, the value of which is proportional to the mass of the point. This situation occurs, for example, when we consider the motion of a material point near the surface of the Moon, where there is no atmospheric resistance. It is considered that the point of variable composition has constant relative velocity of separating fuel particles, and the value of thrust is limited from above with a given positive number. The first player tries to minimize the distance between the points in a set moment, consuming as little resources as possible. The formulated two-criterion problem, with the use of weight coefficients, gets reduced to a differential game, the payoff of which is the sum of both terminal and integral components. By changing variables, the problem is reduced to a single-type game in which vectograms of players are balls with time-dependent radii. The function of the game price is calculated, and optimal control of the players is determined.