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

Asqar Raxmonov,Gafurjan I Ibragimov

doi:10.1155/2014/674651

Asqar Raxmonov, Gafurjan I Ibragimov

Open Access

https://doi.org/10.1155/2014/674651

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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.

Highlights

A large number of works are devoted to differential games where the position of the players changes continuously in time
The control of the pursuer is subjected to geometric constraint and that of the evader is subjected to total constraint
Necessary and sufficient conditions were obtained under which solvability of 0-controllability is equivalent to completion of pursuit

Summary

Introduction

A large number of works are devoted to differential games where the position of the players changes continuously in time (see, e.g., [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18]). We study a linear pursuit game with total constraints on controls. Such constraints are discrete analogues of integral constraints for differential games. In the paper of Satimov et al [15], motions of players are simple It was a starting point for multiperson differential games with integral constraints. Different from the above game, both controls of the players are subjected to total constraints They proved that if eigenvalues of the matrix A in absolute value are less than 1 and dim BS = n, where S is unit ball in Rn centered at the origin, pursuit can be completed from any initial position. We study a linear pursuit discrete game of one pursuer and one evader.

Statement of the Problem

Main Result

Conclusion