On properties of one functional used in software constructions for solving differential games

Nonlinear differential game (DG) is investigated; relaxations of the game problem of guidance are investigated also. The variant of the program iterations method realized in the space of position functions and delivering in limit the value function of the minimax-maximin DG for special functionals of a trajectory is considered. For every game position, this limit function realizes the least size of the target set neighborhood for which, under proportional weakening of phase constraints, the player interested in a guidance yet guarantees its realization. Properties of above-mentioned functionals and limit function are investigated. In particular, sufficient conditions for realization of values of given function under fulfilment of finite iteration number are obtained.

Nonlinear differential game “pursuit-evasion”: information aspect.

The article deals with the information aspects of the nonlinear differential game “pursuit-evasion”. The mini- max quality criterion (the price of the game), mathematical models of the movement of the same type of players are set. The main problem that arises in the practical consideration of a nonlinear differential game is the solution of a boundary value problem. It is proved that: 1) a differential game can be represented as an envelope of a family of singular curves (“instantaneous solutions”) constructed from each of its points, 2) the control of the players can be found on this family. The method of constructing instant solutions is justified. The principle of information dualism in the differential game “pursuit-evasion” is proved. Its use facilitates the solution of information and computational problems that arise during the implementation of the game.