Linear Differential Games with Multi-Dimensional Terminal Target Set: Geometric Approach

The paper deals with linear differential games with a fixed terminal instant, convex geometric constraints of the players’ controls, and convex terminal target set. The first player tries to guide the system to the target set at the terminal instant, the second one hinders this. In the 1960’s, L. S. Pontryagin proposed a theoretic geometric procedure for approximate constructing time sections of the maximal stable bridge for games of this type. This procedure is known as the second Pontryagin’s method. At the beginning of the 1980’s in the Krasovskii Institute of Mathematics and Mechanics (Yekaterinburg, Rus- sia), a computational algorithm for the procedure has been suggested and implemented as a computer program. However, this algorithm is suitable only for games with two- dimensional equivalent phase vector. The authors suggest a procedure suitable for games with a multi-dimensional phase vector. For an implementation of this method, one needs implementations of convex hull construction, Minkowski sum and difference. The authors have taken known algorithms for convex hull construction and Minkowski sum. An al- gorithm for Minkowski difference as well as some procedures for conversion of different representations of multi-dimensional polytopes to each other have been suggested. All these algorithms have been implemented as a computer library in C# by the authors. A series of model differential games has been computed.