SOME PROPERTIES OF THE BELLMAN–ISAACS EQUATION FOR THE GAMES ON SURFACES OF REVOLUTION

Arik Melikyan ()
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Arik Melikyan: Institute for Problems in Mechanics, Russian Academy of Sciences, Vernadsky Ave., 101-1, Moscow 117526, Russia

International Game Theory Review (IGTR), 2004, vol. 06, issue 01, 171-179

Abstract: Certain geometrical properties of a dynamic game may generate a situation when in some subspaceDof the game space the game value depends only on the coordinates of one of the players. Consequently, that player has a definite optimal behavior, while the other one may exploit an arbitrary control till reaching the boundary ofD. In other words, inDone has an optimal control problem rather than a dynamic game. We show that such phenomenon takes place in pursuit-evasion games on a family of 2D surfaces of revolution, including the cones and the hyperboloids of two sheets, due to special convexity properties of the reachability sets of the players. The complete solution of the games on cones one can find in Melikyan [1998], Melikyan,et al.[1998], the games on hyperboloids are investigated in Hovakimyan and Melikyan [2000]. This paper presents new results for the latter games. In the considerations an important role happen to play the focal points, conjugate to the pursuer's positions in the variational problem on geodesic lines.

Keywords: Pursuit–evasion game; hyperboloids; surfaces of revolution (search for similar items in EconPapers)
JEL-codes: B4 C0 C6 C7 D5 D7 M2 (search for similar items in EconPapers)
Date: 2004
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DOI: 10.1142/S0219198904000137

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