DIFFERENTIAL GAMES OF FRACTIONAL ORDER WITH DISTRIBUTED PARAMETERS

The article deals with the problem of pursuit in differential games of fractional order with distributed parameters. Partial fractional derivatives with respect to time and space variables are understood in the sense of Riemann - Liouville, and the Grunwald-Letnikov formula is used in the approximation. The problem of getting into some positive neighborhood of the terminal set is considered. To solve this problem, the finite difference method is used. The fractional Riemann-Liouville derivatives with respect to spatial variables on a segment are approximated using the Grunwald-Letnikov formula. Using a sufficient criterion for the existence of a fractional derivative, a difference approximation of the fractional-order derivative with respect to time is obtained. By approximating a differential game to an explicit difference game, a discrete game is obtained. The corresponding pursuit problem for a discrete game is formulated, which is obtained using the approximation of a continuous game. The concept of the possibility of completing the pursuit, a discrete game in the sense of an exact capture, is defined. Sufficient conditions are obtained for the possibility of completing the pursuit. It is shown that the order of approximation in time is equal to one, and in spatial variables is equal to two. It is proved that if in a discrete game from a given initial position it is possible to complete the pursuit in the sense of exact capture, then in a continuous game from the corresponding initial position it is possible to complete the pursuit in the sense of hitting a certain neighborhood. A structure for constructing pursuit controls is proposed, which will ensure the completion of the game in a finite time. The methods used for this problem can be used to study differential games described by more general equations of fractional order.