Pursuit–evasion differential games with Gr-constraints on controls

In the paper, a pursuit–evasion differential game is considered when controls of the players are subject to differential constraints in the form of Grönwall's integral inequality. The strategy of parallel pursuit (briefly, $\Pi$-strategy) was introduced and used by L. A. Petrosyan to solve simple pursuit problems under phase constraints on the states of the players in the case when control functions of both players are chosen from the class $L_\infty$. In the present work, the $\Pi$-strategy is constructed for a simple pursuit problem in the cases when control functions of both players are chosen from different classes of the Grönwall type constraints, and sufficient conditions of capture and optimal capture time are found in these cases. To solve the evasion problem, we suggest a control function for the Evader and find sufficient conditions of evasion. In addition, an attainability domain of the players is constructed and its conditions of embedding in respect to time are given. Results of this work continue and extend the works of R. Isaacs, L. A. Petrosyan, B. N. Pshenichnyi, A. A. Chirii, A. A. Azamov and other researchers, including the authors.