Geodesic
Numerical construction of Nash and Stackelberg solutions in a~two-player linear nonantagonistic positional differential game
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 4, pp. 120-133

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Numerical methods are proposed for constructing Nash and Stackelberg solutions in a two-player linear nonantagonistic positional differ rential game with terminal quality indices 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 antagonistic differential games developed by N. N. Krasovskii and his school. It is assumed that the game is reduced to a plane game and the constraints on the players' controls are given in the form of convex polygons. The problem of finding solutions of the game is reduced to solving nonstandard optimal control problems. For the construction of approximate trajectories in these problems, several computational geometry algorithms are used, in particular, the algorithms for constructing the convex hull, the union and intersection of polygons, and the algebraic sum of polygons.
Keywords: nonantagonistic positional differential game, Nash solution, Stackelberg solution, numerical algorithm. 
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     title = {Numerical construction of {Nash} and {Stackelberg} solutions in a~two-player linear nonantagonistic positional differential game},
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A. F. Kleimenov; D. R. Kuvshinov; S. I. Osipov. Numerical construction of Nash and Stackelberg solutions in a~two-player linear nonantagonistic positional differential game. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 15 (2009) no. 4, pp. 120-133. http://geodesic.mathdoc.fr/item/TIMM_2009_15_4_a10/