Advances in Dynamic Game Theory pp 173-194 | Cite as

# Geometry of Pursuit-Evasion Games on Two-Dimensional Manifolds

## Abstract

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.

## Keywords

Optimal Control Problem Optimal Path Optimal Trajectory Differential Game Euclidean Plane## Preview

Unable to display preview. Download preview PDF.

## References

- [1]Isaacs, R.
*Differential Games*. JohnWiley, New York, 1965.MATHGoogle Scholar - [2]Krasovskii, N.N. and Subbotin, A.I.
*Game-Theoretical Control Problems*. Springer-Verlag, New York, 1988.MATHGoogle Scholar - [3]Subbotin, A.I.
*Generalized Solutions of First Order PDEs: the Dynamical Optimization Perspective*. Birkhäuser, Boston, 1995.Google Scholar - [4]Petrosjan, L.A. and Zenkevich, N.A.
*Game Theory*. World Scientific Publ., Singapore-London, 1996.Google Scholar - [5]Bernhard, P. Singular Surfaces in Differential Games: An Introduction. In:
*Differential Games and Applications*. Springer-Verlag, Berlin, 1977, pp. 1–33.CrossRefGoogle Scholar - [6]Patsko, V.S. and Turova, V.L. Level sets of the value function in differential games with the homicidal chauffeur dynamics. Intern. Game Theory Review. 2001, Vol. 3. N 1, pp. 67–112.MATHCrossRefMathSciNetGoogle Scholar
- [7]Melikyan, A.A.
*Generalized Characteristics of First Order PDEs: Applications in Optimal Control and Differential Games*. Birkhäuser, Boston, 1998.MATHGoogle Scholar - [8]Melikyan, A.A., Hovakimyan, N.V. and Harutunian, L.L. Games of simple pursuit and approach on two dimensional cone. JOTA, 1998, Vol. 98, N 3, pp. 515–543.MATHCrossRefGoogle Scholar
- [9]Hovakimyan, N.V. and Melikyan, A.A. Geometry of pursuit-evasion on second order rotation surfaces. Dynamics and Control, 2000, Vol. 10, pp. 297–312.MATHCrossRefMathSciNetGoogle Scholar
- [10]Hovakimyan, N.V. and Harutunian, L.L. Gameproblems on rotation surfaces. In: Petrosjan, L.A. and Matalov, V.V. (editors),
*Game Theory and Applications*, 1998, Vol. IV, pp. 62–74.Google Scholar - [11]Melikyan, A.A. Games of simple pursuit and approach on the manifolds. Dynamics and Control. 1994, Vol. 4. N 4. pp. 395–405.MATHCrossRefMathSciNetGoogle Scholar
- [12]Demyanov, V.F.
*Minimax: Directional Differentiability*. Leningrad State University Press, Leningrad, 1974 (in Russian).Google Scholar - [13]Melikyan, A.A. and Hovakimyan, N.V. Game Problem of Simple Pursuit on Two-Dimensional Cone. J. Appl. Math. Mech. (PMM), 1991, Vol. 55, No 5, pp. 741–751.CrossRefGoogle Scholar
- [14]Melikyan, A.A. Games of pursuit on two-sided plane with a hole. 8th Intern. Symp. on Dynamic Games and Applications. Maastricht, the Netherlands, 1998. pp. 394–396.Google Scholar
- [15]Vishnevetsky, L.S. and Melikyan, A.A. Optimal pursuit on a plane in the presence of an obstacle. J. Appl. Math. Mech. (PMM), 1982, Vol. 46, N 4, pp. 485–490.CrossRefGoogle Scholar
- [16]Pozharitsky, G.K. Isaacs’problem of island enveloping. J. Appl. Math. Mech. (PMM), 1982, Vol. 46, N 5, pp. 707–713.CrossRefGoogle Scholar
- [17]Gelfand, I.M. and Fomin, S.V.
*Calculus of Variations*. Prentice-Hall, Englewood Cliff, NJ, 1963.Google Scholar - [18]Melikyan, A.A. Some properties of the Bellman-Isaacs equation for the games on the surfaces of revolution. International Game Theory Review, 2001, Vol. 6, N 1, pp. 171–179.CrossRefMathSciNetGoogle Scholar
- [19]Pappas, G.J. and Simic, S. Consistent Abstractions ofAffine Control Systems. IEEE Trans. on Automatic Control, 2002, Vol. 47, N 5, pp. 745–756.CrossRefMathSciNetGoogle Scholar