Linear Discrete Game Under Quadratic Constraints on Controls

Introduction. In studies concerning the problems of approaching moving objects, the authors, as a rule, use continuous dynamic models under integral constrains on controls. However, only discrete models under quadratic or resource constraints are suitable for practical applications. The purpose of the paper is to develop a discrete analog of the method of time dilation for solving the problem of guaranteed approaching a terminal set by a discrete conflict-controlled system trajectory. Results. We introduce the concept of integer function of time dilation. Its using, in the frames of the discrete analog of the Pontryagin First Direct method, makes it possible to deduce sufficient conditions for bringing the trajectory of the discrete conflict-controlled process to the terminal set. We outline the way of constructing current pursuer’s control, which brings the object trajectory to the terminal set under arbitrary admissible counteraction of the evader. It differs from the pursuer control choice in the continuous case, when the pursuer chooses his current control in view of the evader’s control at a certain moment of time in the past. In the discrete case, the pursuer constructs his control on the basis of information about the evader’s controls on a whole discrete interval of time in the past. We prove that such control satisfies original quadratic constraints. Conclusions. We derive conditions for approaching the trajectory of conflict-controlled discrete process a terminal set. In so doing, quadratic constraints on controls are fulfilled. The terminal set is supposed to be a subset that corresponds to the catching the evader by the pursuer. Keywords: linear discrete game of approach, quadratic constraints, pursuer, evader, integer function of time dilation, admissible control.