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03-01-2022

Approximate Capture in Gromov–Hausdorff Close Spaces

Author: Olga Yufereva

Published in: Dynamic Games and Applications

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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 \(\alpha \)-capture by a time T in one compact geodesic space is proved to yield the existence of \(\bigl (\alpha + (20T +8)\sqrt{\delta }\bigr )\)-capture by the time T in any compact geodesic space that is \(\delta \)-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.

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Literature
1.
go back to reference Alexander S, Bishop R, Ghrist R (2006) Pursuit and evasion in non-convex domains of arbitrary dimensions. In: Robotics: science and systems Alexander S, Bishop R, Ghrist R (2006) Pursuit and evasion in non-convex domains of arbitrary dimensions. In: Robotics: science and systems
2.
go back to reference Alonso L, Goldstein AS, Reingold EM (1992) “Lion and Man’’: upper and lower bounds. ORSA J Comput 4(4):447–452 MathSciNetCrossRef Alonso L, Goldstein AS, Reingold EM (1992) “Lion and Man’’: upper and lower bounds. ORSA J Comput 4(4):447–452 MathSciNetCrossRef
3.
go back to reference Averboukh Y (2019) Krasovskii-subbotin approach to mean field type differential games. Dyn Games Appl 9(3):573–593 MathSciNetCrossRef Averboukh Y (2019) Krasovskii-subbotin approach to mean field type differential games. Dyn Games Appl 9(3):573–593 MathSciNetCrossRef
5.
go back to reference Bollobás B (2011) The lion and the christian, and other pursuit and evasion games. In: An invitation to mathematics. Springer, pp 181–193 Bollobás B (2011) The lion and the christian, and other pursuit and evasion games. In: An invitation to mathematics. Springer, pp 181–193
7.
go back to reference Bramson M, Burdzy K, Kendall W (2013) Shy couplings, CAT(0) spaces, and the lion and man. Ann Probab 41(2):744–784 MathSciNetCrossRef Bramson M, Burdzy K, Kendall W (2013) Shy couplings, CAT(0) spaces, and the lion and man. Ann Probab 41(2):744–784 MathSciNetCrossRef
8.
go back to reference Bramson M, Burdzy K, Kendall WS (2014) Rubber bands, pursuit games and shy couplings. Proc Lond Math Soc 109(1):121–160 MathSciNetCrossRef Bramson M, Burdzy K, Kendall WS (2014) Rubber bands, pursuit games and shy couplings. Proc Lond Math Soc 109(1):121–160 MathSciNetCrossRef
9.
go back to reference Bridson MR, Haefliger A (2013) Metric spaces of non-positive curvature, vol 319. Springer, Berlin MATH Bridson MR, Haefliger A (2013) Metric spaces of non-positive curvature, vol 319. Springer, Berlin MATH
10.
go back to reference Buckdahn R, Cardaliaguet P, Quincampoix M (2011) Some recent aspects of differential game theory. Dyn Games Appl 1(1):74–114 MathSciNetCrossRef Buckdahn R, Cardaliaguet P, Quincampoix M (2011) Some recent aspects of differential game theory. Dyn Games Appl 1(1):74–114 MathSciNetCrossRef
11.
go back to reference Burago D, Burago Y, Ivanov S (2001) A course in metric geometry, vol 33. American Mathematical Soc, London MATH Burago D, Burago Y, Ivanov S (2001) A course in metric geometry, vol 33. American Mathematical Soc, London MATH
12.
go back to reference Casini M, Criscuoli M, Garulli A (2019) A discrete-time pursuit-evasion game in convex polygonal environments. Syst Control Lett 125:22–28 MathSciNetCrossRef Casini M, Criscuoli M, Garulli A (2019) A discrete-time pursuit-evasion game in convex polygonal environments. Syst Control Lett 125:22–28 MathSciNetCrossRef
13.
go back to reference Dumitrescu A, Suzuki I, Żyliński P (2008) Offline variants of the “lion and man problem”: - Some problems and techniques for measuring crowdedness and for safe path planning. Theor Comput Sci 399(3):220–235 Dumitrescu A, Suzuki I, Żyliński P (2008) Offline variants of the “lion and man problem”: - Some problems and techniques for measuring crowdedness and for safe path planning. Theor Comput Sci 399(3):220–235
16.
go back to reference Kohlenbach U, López-Acedo G, Nicolae A (2020) A uniform betweenness property in metric spaces and its role in the quantitative analysis of the ‘lion-man game’. Pacific J Math Kohlenbach U, López-Acedo G, Nicolae A (2020) A uniform betweenness property in metric spaces and its role in the quantitative analysis of the ‘lion-man game’. Pacific J Math
17.
go back to reference Krasovskii NN, Kotel’nikova AN (2010) An approach-evasion differential game: Stochastic guide. Proc Steklov Inst Math 269(1):191–213 CrossRef Krasovskii NN, Kotel’nikova AN (2010) An approach-evasion differential game: Stochastic guide. Proc Steklov Inst Math 269(1):191–213 CrossRef
18.
go back to reference Krasovskii NN, Subbotin AI (1974) Pozitsionnye differentsial’nye igry. M.: Nauka Krasovskii NN, Subbotin AI (1974) Pozitsionnye differentsial’nye igry. M.: Nauka
19.
go back to reference Krasovskii NN, Subbotin AI (1987) Game-theoretical control problems. Springer, New York Krasovskii NN, Subbotin AI (1987) Game-theoretical control problems. Springer, New York
20.
go back to reference Littlewood JE (1953) A mathematicians miscellany. Methuen, London MATH Littlewood JE (1953) A mathematicians miscellany. Methuen, London MATH
21.
go back to reference López-Acedo G, Nicolae A, Pia̧tek B (2019) “Lion-Man” and the fixed point property. Geometriae Dedicata 202(1):69–80 López-Acedo G, Nicolae A, Pia̧tek B (2019) “Lion-Man” and the fixed point property. Geometriae Dedicata 202(1):69–80
22.
go back to reference López-Acedo G, Nicolae A, Pia̧tek B (2020) Geodesic rays, the “Lion-Man” game, and the fixed point property. J Math Anal Appl 487(1): 123948 López-Acedo G, Nicolae A, Pia̧tek B (2020) Geodesic rays, the “Lion-Man” game, and the fixed point property. J Math Anal Appl 487(1): 123948
23.
go back to reference Noori N, Isler V (2014) Lion and man with visibility in monotone polygons. Int J Robot Res 33(1):155–181 CrossRef Noori N, Isler V (2014) Lion and man with visibility in monotone polygons. Int J Robot Res 33(1):155–181 CrossRef
24.
25.
28.
go back to reference Yufereva O (2018) On limit-and infimum-captures in pursuit-evasion problems. IFAC-PapersOnLine 51(32):277–280 CrossRef Yufereva O (2018) On limit-and infimum-captures in pursuit-evasion problems. IFAC-PapersOnLine 51(32):277–280 CrossRef
Metadata
Title
Approximate Capture in Gromov–Hausdorff Close Spaces
Author
Olga Yufereva
Publication date
03-01-2022
Publisher
Springer US
Published in
Dynamic Games and Applications
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-021-00419-7

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