Strategies for Alternative Pursuit Games

Abstract

A pursuit game is called alternative if it may be terminated by the pursuer on one of several given terminal manifolds and corresponding payoffs of Boltza type associated with those variants differ only in their terminal parts. For the case of only two alternatives, we describe an approach for construction of the strategies when solutions for all games with fixed targeted terminal alternatives are known. Some alternative dominates the second if it has a less cost for the pursuer.We assume that the pursuer targets a particular alternative only if the dominating condition is stable along the respective optimal trajectory. Thus, it remains to study alternative pursuit games only in the subdomains where the dominating conditions are unstable. When the playing space has a specific structure, the state has to leave these subdomains through a manifold with matched alternatives. To get a solution, three interconnected auxiliary games are to be set up. We present the Hamilton-Jacobi-Isaacs equations for these games. Also, we describe a procedure to construct locally gradient strategies that optimize the local growth of approximations for the minimum of the alternative value functions.