Differential game with “lifeline” for Pontryagin's control example

The main purpose of this work is to solve one of the main problems of Isaacs, i.e., a game with a “lifeline” for Pontryagin’s control example when both players have the same movement dynamics. To solve this problem, the pursuer is offered a strategy of parallel pursuit (briefly, $\Pi$-strategy), which ensures the fastest convergence of the players and the capture of the evader within a certain closed ball. In addition, for the differential game under consideration, an explicit analytical formula for the players' attainability domain is given and the main lemma is generalized (L.A. Petrosjan's lemma on monotonicity of the players' attainability domain with respect to embedding for a game of simple pursuit). Using this main lemma, we find conditions for the solvability of the game with a “lifeline” for Pontryagin's control example as well. For clarity, at the end of the work, examples are given for some special cases.