Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds
Pursuit-evasion games with simple motion on two-dimensional (2D) manifolds are considered. The analysis embraces the game spaces such as 2D surfaces of revolution (cones and hyperboloids of one and two sheets, ellipsoids), a Euclidean plane with convex bounded obstacle, two-sided Euclidean plane with hole(s), and two-sided plane bounded figures (disc, ellipse, polygon). In a two-sided game space players can change the side at the boundary (through the hole).
In all cases the game space is a 2D surface or a figure in 3D Euclidean space, while the arc length is induced by the Euclidean metric of the 3D space. Due to simple motion, optimal trajectories of the players generally consist of geodesic lines of the game space manifolds. For the game spaces under consideration there may exist two or more geodesic lines with equal lengths, connecting the players. In some cases this gives rise to a singular surface consisting of the trajectories, which are envelopes of a family of geodesics.
In this chapter we investigate the necessary and sufficient conditions for this and some other types of singularity, we specify the game spaces where the optimal pursuit-evasion strategies do not contain singularities and are similar to the case of a Euclidean plane, and we give a short review and analysis of the solutions for the games in several game spaces-manifolds.
In the analysis we use viscosity solutions to the Hamilton-Jacobi-Bellman-Isaacs equation, variation calculus and geometrical methods.We also construct the 3D manifolds in the game phase space representing the positions with two or more geodesic lines with equal lengths, connecting the players.
The investigation of 2D games on the manifolds has several direct applications, and it may also represent an approximate solution for more complicated games as an abstraction.
KeywordsOptimal Control Problem Optimal Path Optimal Trajectory Differential Game Euclidean Plane
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