The differential game with inertial players under integral constraints on controls

We consider pursuit-evasion differential games between inertial players (a pursuer and an evader) whose controls are subject to integral constraints. The pursuer is believed to have captured the evader when their positions coincide. The key method used to provide a win for the pursuer in the pursuit differential game is the parallel pursuit strategy (in brief, the $\Pi$-strategy). We obtain sufficient conditions for the solvability of pursuit-evasion problems. Furthermore, we investigate Isaacs' “life-line” game in favor of the pursuer in the case of the identical initial velocities of the players. Here, the main lemma characterizing its monotonicity property provides an analytical formula for the players' meeting domain. This paper extends and continues the works of R. Isaacs, L. A. Petrosyan, B. N. Pshenichnyi, N. Yu. Satimov, the authors of this article, and other researchers.