TY - JOUR

T1 - Numerical construction of Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game

AU - Kleimenov, A. F.

AU - Kuvshinov, D. R.

AU - Osipov, S. I.

PY - 2010/7/1

Y1 - 2010/7/1

N2 - Numerical methods are proposed for constructing Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game with terminal cost functionals and geometric constraints on the players’ controls. The formalization of the players’ strategies and of the motions generated by them is based on the formalization and results from the theory of positional zero-sum differential games developed by N.N. Krasovskii and his school. It is assumed that the game is reduced to a planar game and the constraints on the players’ controls are given in the form of convex polygons. The problem of finding solutions of the game may be reduced to solving nonstandard optimal control problems. Several computational geometry algorithms are used to construct approximate trajectories in these problems, in particular, algorithms for constructing the convex hull as well as the union, intersection, and algebraic sum of polygons.

AB - Numerical methods are proposed for constructing Nash and Stackelberg solutions in a two-player linear non-zero-sum positional differential game with terminal cost functionals and geometric constraints on the players’ controls. The formalization of the players’ strategies and of the motions generated by them is based on the formalization and results from the theory of positional zero-sum differential games developed by N.N. Krasovskii and his school. It is assumed that the game is reduced to a planar game and the constraints on the players’ controls are given in the form of convex polygons. The problem of finding solutions of the game may be reduced to solving nonstandard optimal control problems. Several computational geometry algorithms are used to construct approximate trajectories in these problems, in particular, algorithms for constructing the convex hull as well as the union, intersection, and algebraic sum of polygons.

UR - https://gateway.webofknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcAuth=tsmetrics&SrcApp=tsm_test&DestApp=WOS_CPL&DestLinkType=FullRecord&KeyUT=000209701100013

UR - http://www.scopus.com/inward/record.url?partnerID=8YFLogxK&scp=84962427836

U2 - 10.1134/S0081543810060131

DO - 10.1134/S0081543810060131

M3 - Article

VL - 269

SP - 147

EP - 161

JO - Proceedings of the Steklov Institute of Mathematics

JF - Proceedings of the Steklov Institute of Mathematics

SN - 0081-5438

IS - S1

ER -