Approximate Capture in Gromov–Hausdorff Close Spaces

Olga Yufereva
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Olga Yufereva: Krasovskii Institute of Mathematics and Mechanics

Dynamic Games and Applications, 2022, vol. 12, issue 4, No 15, 1376-1387

Abstract: Abstract This paper addresses the robustness of the capture radii with respect to perturbation in the phase space on the example of the so-called Lion and Man game. This is a two-person pursuit-evasion game with equal players’ top speeds. The existence of $$\alpha $$ α -capture by a time T in one compact geodesic space is proved to yield the existence of $$\bigl (\alpha + (20T +8)\sqrt{\delta }\bigr )$$ ( α + ( 20 T + 8 ) δ ) -capture by the time T in any compact geodesic space that is $$\delta $$ δ -close to the given one. In this way, a pursuer’s strategy in one space is transferred to another space that is close to the given one in the sense of the Gromov–Hausdorff distance. It means that the capture radii (in similar spaces) tend to the given one as the distance between spaces tends to zero. In particular, this result justifies the consideration of Lion and Man game on the finite metric graphs instead of complicated original spaces.

Keywords: Pursuit–evasion game; Lion and Man problem; Robustness; Gromov–Hausdorff distance; Capture radius; Geodesic space; Finite metric graph; 49N75; 91A24; 53C23 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s13235-021-00419-7

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