Existence of Value in Generalized Pursuit-Evasion Games

Abstract

The results presented here relate both to a pursuit-evasion game and to a game of fixed time duration. The dynamics of both games is governed by a system of differential inclusions, and the state variables x, y have to satisfy some general type constraints. Player I can apply any lower 17-strategy [15, p. 400] and player II can apply any strategy in the Varaiya–Lin sense. The main result is the existence of a value (in both the games) and an optimal strategy for player II. The key assumption is the condition that from any sequence of player II’s trajectories emanating from a convergent sequence of points one can choose a subsequence convergent on a fixed segment $[t_0 ,T]$ to some player II’s trajectory. The condition holds when the set $F_2 (t,y)$ is convex for any $(t,y) \in R^{1 + k} $. This paper is an outgrowth of work conducted by the author [14]–[17]; it extends the results obtained in [14], [16].