An Approach for Solving Discrete Game Problems with Total Constraints on Controls

Abstract

We consider a linear pursuit game of one pursuer and one evader whose motions are described by different-type linear discrete systems. Controls of the players satisfy total constraints. Terminal setMis a subset ofℝnand it is assumed to have nonempty interior. Game is said to be completed ifyk-xk∈Mat some stepk. To construct the control of the pursuer, at each stepi, we use positions of the players from step 1 to stepiand the value of the control parameter of the evader at the stepi. We give sufficient conditions of completion of pursuit and construct the control for the pursuer in explicit form. This control forces the evader to expend some amount of his resources on a period consisting of finite steps. As a result, after several such periods the evader exhausted his energy and then pursuit will be completed.

Linear Pursuit Differential Game under Phase Constraint on the State of Evader

“…In [2-5, 7, 10, 13, 15, 17-19, 22, 24], various pursuit and evasion differential game problems were investigated under phase constraints. Sufficient conditions of completion of pursuit were obtained in [4,5,19], when controls of both players are subjected to geometric constraints [19] or integral constraints [4], and phase constraint is being imposed on the state of the evader. The objectives of the researches [8,9,12] are to find value of the game and construct optimal strategies of players.…”

“…The objectives of the researches [8,9,12] are to find value of the game and construct optimal strategies of players. The paper [5] is devoted to discrete pursuit games with phase constraints.…”

“…The present paper is continuation of [4,5,19] and deals with the linear pursuit differential games where pursuit control is subjected to integral constraints and evasion control is subjected to geometric constraint, with phase constraint being imposed on the state of the evader. For such constraints on controls of players, to obtain solution of pursuit problem for all initial states of phase space is complicated because of boundedness of pursuit resources.…”