The Game of Two Identical Cars: an Analytical Description of the Barrier

TL;DR

This paper provides a comprehensive analytical description of the barrier surface in a pursuit-evasion game involving two identical cars, revealing a critical capture radius where the barrier's nature changes and deriving optimal feedback controls.

Contribution

It extends previous work by analytically characterizing the barrier surface for all capture radii, including a critical value where the barrier's structure changes, and derives optimal controls.

Findings

Identified a critical capture radius where the barrier's structure changes.

Provided an analytical description of the barrier surface for all capture radii.

Derived optimal feedback control strategies on the barrier.

Abstract

In this paper, a pursuit-evasion game of two players known as the game of two identical cars is examined. It is assumed that the game proceeds on a two-dimensional plane. Both players have constant speeds and limited turn radii. The goal of the first player (pursuer) is to ensure that the second player (evader) is guaranteed to be within a capture circle as quickly as possible. The goal of the evader is to avoid the capture or delay it as long as possible. The kinematics of both players are described by the same equations. Thus, the game has only one free parameter, the capture radius. This work aims at a fully analytical description of the barrier surface for all values of the capture radius. Previously, A.W.~Merz analytically investigated the barrier of the game of two identical cars. In this work, it is found that there is a certain critical value of the capture radius, above which the barrier occurs qualitatively differently from Merz's example. Also, we obtained an analytical description of the optimal feedback controls on the barrier.