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Published in: Dynamic Games and Applications 3/2019

23-01-2019

On Time-Optimal Problems with Lifeline

Authors: Nataly V. Munts, Sergey S. Kumkov

Published in: Dynamic Games and Applications | Issue 3/2019

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Abstract

This paper discusses time-optimal games with lifeline and corresponding boundary value problems for Hamilton–Jacobi equation as well. Existence of the value function for the time-optimal games with lifeline is proved. Existence of a minimax solution and its coincidence with the value function are shown.

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Literature
1.
go back to reference Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, BostonCrossRefMATH Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser, BostonCrossRefMATH
2.
go back to reference Bardi M, Falcone M, Soravia P (1999) Numerical methods for pursuit-evasion games via viscosity solutions. In: Bardi M, Parthasarathy T, Raghavan TES (eds) Annals of the international society of dynamic games. Stochastic and differential games, vol 6. Birkhäuser, Boston, pp 105–175CrossRef Bardi M, Falcone M, Soravia P (1999) Numerical methods for pursuit-evasion games via viscosity solutions. In: Bardi M, Parthasarathy T, Raghavan TES (eds) Annals of the international society of dynamic games. Stochastic and differential games, vol 6. Birkhäuser, Boston, pp 105–175CrossRef
3.
go back to reference Bardi M, Koike S, Soravia P (2000) Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations. Discrete Contin Dyn Syst 6(2):361–380MathSciNetCrossRefMATH Bardi M, Koike S, Soravia P (2000) Pursuit-evasion games with state constraints: dynamic programming and discrete-time approximations. Discrete Contin Dyn Syst 6(2):361–380MathSciNetCrossRefMATH
4.
go back to reference Breitner M, Pesch H, Grimm W (1993) Complex differential games of pursuit-evasion type with state constraints, part 1: necessary conditions for optimal open-loop strategies. J Optim Theory Appl 78(3):419–441MathSciNetCrossRefMATH Breitner M, Pesch H, Grimm W (1993) Complex differential games of pursuit-evasion type with state constraints, part 1: necessary conditions for optimal open-loop strategies. J Optim Theory Appl 78(3):419–441MathSciNetCrossRefMATH
5.
go back to reference Breitner M, Pesch H, Grimm W (1993) Complex differential games of pursuit-evasion type with state constraints, part 2: numerical computation of optimal open-loop strategies. J Optim Theory Appl 78(3):443–463MathSciNetCrossRefMATH Breitner M, Pesch H, Grimm W (1993) Complex differential games of pursuit-evasion type with state constraints, part 2: numerical computation of optimal open-loop strategies. J Optim Theory Appl 78(3):443–463MathSciNetCrossRefMATH
6.
go back to reference Cardaliaguet P, Quincampoix M, Saint-Pierre P (1994) Some algorithms for differential games with two players and one target. RAIRO—Modélisation mathématique et analyse numérique 28(4):441–461MathSciNetMATH Cardaliaguet P, Quincampoix M, Saint-Pierre P (1994) Some algorithms for differential games with two players and one target. RAIRO—Modélisation mathématique et analyse numérique 28(4):441–461MathSciNetMATH
7.
go back to reference Cardaliaguet P, Quincampoix M, Saint-Pierre P (1999) Set-valued numerical analysis for optimal control and differential games. In: Bardi M, Raghavan TES, Parthasarathy T (eds) Annals of the international society of dynamic games. Stochastic and differential games, vol 4. Birkhäuser, Boston, pp 177–247CrossRef Cardaliaguet P, Quincampoix M, Saint-Pierre P (1999) Set-valued numerical analysis for optimal control and differential games. In: Bardi M, Raghavan TES, Parthasarathy T (eds) Annals of the international society of dynamic games. Stochastic and differential games, vol 4. Birkhäuser, Boston, pp 177–247CrossRef
8.
go back to reference Cardaliaguet P, Quincampoix M, Saint-Pierre P (2001) Pursuit differential games with state constraints. SIAM J Control Optim 39(5):1615–1632MathSciNetCrossRefMATH Cardaliaguet P, Quincampoix M, Saint-Pierre P (2001) Pursuit differential games with state constraints. SIAM J Control Optim 39(5):1615–1632MathSciNetCrossRefMATH
9.
go back to reference Cardaliaguet P, Quincampoix M, Saint-Pierre P (2007) Differential games through viability theory: old and recent results. In: Jorgensen S, Quincampoix M, Vincent TL (eds) Annals of the international society of dynamic games. Advances in dynamic game theory, vol 9. Birkhäuser, Boston, pp 3–35CrossRef Cardaliaguet P, Quincampoix M, Saint-Pierre P (2007) Differential games through viability theory: old and recent results. In: Jorgensen S, Quincampoix M, Vincent TL (eds) Annals of the international society of dynamic games. Advances in dynamic game theory, vol 9. Birkhäuser, Boston, pp 3–35CrossRef
10.
go back to reference Fisac J, Sastry S (2015) The pursuit-evasion-defense differential game in dynamic constrained environments. In: 2015 IEEE 54th annual conference on decision and control (CDC), 15–18 Dec 2015. Osaka, Japan, pp 4549–4556 Fisac J, Sastry S (2015) The pursuit-evasion-defense differential game in dynamic constrained environments. In: 2015 IEEE 54th annual conference on decision and control (CDC), 15–18 Dec 2015. Osaka, Japan, pp 4549–4556
11.
12.
go back to reference Krasovskii NN, Subbotin AI (1974) Positional differential games. Nauka, Moscow (in Russian)MATH Krasovskii NN, Subbotin AI (1974) Positional differential games. Nauka, Moscow (in Russian)MATH
13.
go back to reference Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New YorkCrossRef Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New YorkCrossRef
14.
go back to reference Petrosjan LA (1965) A family of differential survival games in the space \(R^{n}\). Sov Math Dokl 6:377–380 Petrosjan LA (1965) A family of differential survival games in the space \(R^{n}\). Sov Math Dokl 6:377–380
15.
go back to reference Rakhmanov A, Ibragimov G, Ferrara M (2016) Linear pursuit differential game under phase constraint on the state of evader. Discrete Dyn Nat Soc 2016:1–6MathSciNetCrossRefMATH Rakhmanov A, Ibragimov G, Ferrara M (2016) Linear pursuit differential game under phase constraint on the state of evader. Discrete Dyn Nat Soc 2016:1–6MathSciNetCrossRefMATH
16.
go back to reference Subbotin AI (1995) Generalized solutions of first order PDEs: the dynamical optimization perspective. Birkhäuser, BostonCrossRef Subbotin AI (1995) Generalized solutions of first order PDEs: the dynamical optimization perspective. Birkhäuser, BostonCrossRef
Metadata
Title
On Time-Optimal Problems with Lifeline
Authors
Nataly V. Munts
Sergey S. Kumkov
Publication date
23-01-2019
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 3/2019
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-018-00295-8

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