Differential game with a “life line” under the Grönwall constraint on controls

We study the pursuit-evasion and “life line” differential games of one pursuer and one evader, whose controls are subjected to constraints given by Grönwall type inequalities. It is said that an evader has been captured by a pursuer if the state of the pursuer coincides with the state of the evader. One of the main aims of this work is to formulate optimal strategies of players and define guaranteed capture time. Here a strategy of parallel convergence (briefly, $\Pi$-strategy) for the pursuer is suggested and proved that it is optimal for pursuit. To solve the “life line” problem we will investigate dynamics of the attainability domain of players by Petrosyan method, that is for the attainability domain, conditions of embedding in respect to time are given. This work grows and maintains the works of Isaacs, Petrosyan, Pshenichnyi, Azamov and other researchers.