Differential Game: ‘‘Life Line’’ for Non-Stationary Geometric Constraints On Controls

We consider the differential game with ‘‘Life line’’ of R. Isaacs that occupies a special place as an example of differential game with phase constraint. In the present paper, the problem of one pursuer and one evader is studied, in which case controls of players are subjected to non-stationary geometric constraints of different types. The notion of strategy of parallel pursuit (briefly -strategy) was introduced and used to solve the quality problem for ‘‘The game with a life line’’ by L. A. Petrosjan. Dynamics of changing of the attainability domains of the players is studied by the properties of theory of multi-valued mapping and a simple proof of the main lemma is given. This work develops and extends the works of Isaacs, Petrosjan, Pshenichnyi, Azamov and other researchers, including the authors.