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

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Abstract

In this study, a pursuit-evasion game of two players, known as a game of two identical cars, is examined. It is assumed that the game proceeds in a two-dimensional plane. Both players have a constant speed and a limited turn radius. The goal of the first player (pursuer) is to ensure that the second player (evader) enters the capture circle as quickly as possible. The goal of the evader is to avoid or delay capturing for as long as possible. The kinematics of both players are described using the same equations. Thus, the game has only one free parameter: capture radius. This study aims to provide an exhaustive analytical description of the barrier surface for all values of capture radius. Previously, Merz analytically investigated the barrier in a game of two identical cars. In this work, it was found that there is a certain critical value of the capture radius, above which the barrier is qualitatively different from Merz’s example. In addition, we obtained an explicit analytical description of the optimal feedback controls for the barrier.

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Data Availability

All data generated or analyzed during this study are included in this published article.

Notes

  1. Throughout the paper, \({\mathbb {B}}\) denotes the binary set \(\{-1, +1\}\). Additionally, we use the letter \(\upsilon \) (upsilon) to denote a binary variable. This letter looks like an evader’s control v, which will actually play this role.

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Acknowledgements

The research was supported by RSF (project No. 23-19-00134).

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  1. V. A. Trapeznikov Institute of Control Sciences of Russian Academy of Sciences, Moscow, Russia

    Maksim Buzikov & Andrey Galyaev

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  1. Maksim Buzikov
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Buzikov, M., Galyaev, A. The Game of Two Identical Cars: An Analytical Description of the Barrier. J Optim Theory Appl 198, 988–1018 (2023). https://doi.org/10.1007/s10957-023-02278-1

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