# Issue 185, article 5

KVT, 2016, Issue 185, pp.48-59

UDC 517.977

GROUP PURSUIT IN DIFFERENTIAL-DIFFERENCE GAMES WITH VARIABLE DELAY

Liubarshchuk I.A.

Department of Mathematics and Informatics, Yuriy Fedkovych Chernivtsi National University, 58012 Chernivtsi, Ukraine

finvara@gmail.com

Introduction. A variety of interesting examples stimulated the development of the Mathematical Control Theory, in particular, the Dynamic Games Theory. First fundamental results in Differential Games Theory were obtained by R. Isaacs. Some others directions of research are Pontryagin’s procedures and Krasovskii’s extremal aiming principle. The further development of Pontryagin’s ideas by his disciples and followers resulted in the Method of Resolving Functions, one of the most powerful methods of dynamic game theory. The essence of the Method of Resolving Functions is in the construction of some numeric resolving function on the known parameters of the process. The resolving function outlines the course of the process. At the moment at which its integral turns into unit the trajectory of the process hits the terminal set. This method was used by Baranovskaya for local convergence problems with fixed time, which are described by a system of differential-difference equations of delay-type.

The purpose of the article is to investigate group problem, which is described by a system of differential-difference equations with variable delay. The necessary and sufficient conditions for solvability of such problems are established.

Results. We considered a pursuit problem in 2-person differential game, one player is a pursuer and another one is an evader. The problem was given by the system of the differential-difference equations of delay-type and for such a conflict-controlled process we presented conditions on its parameters and initial state, which were sufficient for capturing the evader. For differential-difference games with time lag we generalized Pontryagin’s First Direct Method. That gave us a possibility to compare results obtained by the Method of Resolving Functions for such conflict-controlled processes to Pontryagin’s First Direct Method. The necessary and sufficient conditions for group problem solvability were established.

Conclusions. A general scheme of the Method of Resolving Functions for the local convergence problem with fixed time is presented . The conflict-controlled process is described by a system of differential-difference equations of delay-type with variable delay. For differential-difference games with variable delay we generalized Pontryagin’s First Direct Method. We also considered the group pursuit problem in differential game. For such conflict-controlled process we obtained and investigated general scheme of the Method of Resolving Functions.

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