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CAPTURE PURSUIT GAMES ON UNBOUNDED DOMAINS S. ALEXANDER, R. BISHOP, AND R. GHRIST
 

Summary: CAPTURE PURSUIT GAMES ON UNBOUNDED DOMAINS
S. ALEXANDER, R. BISHOP, AND R. GHRIST
ABSTRACT. We introduce simple tools from geometric convexity to analyze capture-
type (or "Lion and Man") pursuit problems in unbounded domains. The main result is
a necessary and sufficient condition for eventual capture in equal-speed discrete-time
multi-pursuer capture games on convex Euclidean domains of arbitrary dimension and
shape. This condition is presented in terms of recession sets in unit tangent spheres.
The chief difficulties lie in utilizing the boundary of the domain as a constraint on the
evader's escape route. We also show that these convex-geometric techniques provide
sufficient criteria for pursuit problems in non-convex domains with a convex decom-
position.
1. INTRODUCTION
Games of pursuit and evasion are among the oldest and most elegant problems in
game theory, osculating differential equations, control theory, differential geometry,
and graph theory. This paper focuses on global geometric features of capture-type
pursuit problems. The primary contribution is an introduction of tools from geometric
convexity which allow for results so general as to be independent of the number of
pursuers, and the dimension and (to a lesser extent) the geometry of the playing field.
1.1. Of lions and men. The history of pursuit-evasion games is rich, with the earliest
formal problems being inspired by naval exploits [3]. Isaac's text [12] is the classical

  

Source: Alexander, Stephanie - Department of Mathematics, University of Illinois at Urbana-Champaign
Ghrist, Robert W. - Department of Electrical Engineering, University of Pennsylvania

 

Collections: Mathematics