Some Variation Calculus Problems in Dynamic Games on 2D Surfaces

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.

This work was partially supported by the grant No.01-01-00376 of the Russian Foundation for Basic Research.