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Some Variation Calculus Problems in Dynamic Games on 2D Surfaces

In: ICM Millennium Lectures on Games

Author

Listed:
  • Arik Melikyan

    (Russian Academy of Sciences, Institute for Problems in Mechanics)

  • Naira Hovakimyan

    (School of Aerospace Engineering, Georgia Institute of Technology)

Abstract

Summary When the game space is a manifold rather than a Euclidean space, and there exist two or more minimal geodesic lines, connecting the players for certain positions, with the same lengths, the construction of optimal phase portraits cannot be done based on Euclidean solution. This paper demonstrates how variational calculus, including the focal point technique, can be used in differential games on surfaces. In particular, it was shown that in games on the two-dimensional surfaces, the existence of two equal geodesics in for some positions of the players gives rise to so-called secondary domain where the optimal motion of the players is a motion along a geodesic. Each player exploits his own geodesic line, which is different from the one connecting them. Optimal phase portraits in differential games on manifolds are constructed and the structure of the game values are derived.

Suggested Citation

  • Arik Melikyan & Naira Hovakimyan, 2003. "Some Variation Calculus Problems in Dynamic Games on 2D Surfaces," Springer Books, in: Leon A. Petrosyan & David W. K. Yeung (ed.), ICM Millennium Lectures on Games, pages 287-296, Springer.
    • Handle: RePEc:spr:sprchp:978-3-662-05219-8_17
    DOI: 10.1007/978-3-662-05219-8_17
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