Pursuit and evasion problems with decreasing players' energy

The paper is devoted to study the pursuit and evasion problems in a differential game with two players, pursuer and evader, under exponential decreasing integral constraints on controls. First of all, we construct an attainability domain of each player for the proposed constraints, and show that the pursuer (also, the evader) can reach any point of the domain. The pursuit problem is solved by virtue of the method of parallel approach, and thus the parallel pursuit strategy ($\Pi$-strategy) is given to the pursuer, and a sufficient condition of capture is found. In order for the evasion problem to be solved, we give a constant control to the evader, and obtain a sufficient condition of evasion. Moreover, the lower bound of a distance between the players is determined.