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Mat. Sb., 2001, Volume 192, Number 10, Pages 95–122 (Mi msb604)  

This article is cited in 23 scientific papers (total in 24 papers)

An algorithm for the numerical solution of linear differential games

E. S. Polovinkin, G. E. Ivanov, M. V. Balashov, R. V. Konstantinov, A. V. Khorev

Moscow Institute of Physics and Technology

Abstract: A numerical algorithm for the construction of stable Krasovskii bridges, Pontryagin alternating sets, and also of piecewise program strategies solving two-person linear differential (pursuit or evasion) games on a fixed time interval is developed on the basis of a general theory. The aim of the first player (the pursuer) is to hit a prescribed target (terminal) set by the phase vector of the control system at the prescribed time. The aim of the second player (the evader) is the opposite. A description of numerical algorithms used in the solution of differential games of the type under consideration is presented and estimates of the errors resulting from the approximation of the game sets by polyhedra are presented.

DOI: https://doi.org/10.4213/sm604

Full text: PDF file (400 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2001, 192:10, 1515–1542

Bibliographic databases:

UDC: 517.977.8
MSC: Primary 49N75, 90C05, 93C15; Secondary 93C55
Received: 19.01.2001

Citation: E. S. Polovinkin, G. E. Ivanov, M. V. Balashov, R. V. Konstantinov, A. V. Khorev, “An algorithm for the numerical solution of linear differential games”, Mat. Sb., 192:10 (2001), 95–122; Sb. Math., 192:10 (2001), 1515–1542

Citation in format AMSBIB
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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. Ivanov G.Ye., “Differential games with ellipsoidal penalties”, J. Appl. Math. Mech., 68:5 (2004), 647–664  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    2. S. I. Kumkov, V. S. Patsko, S. G. Pyatko, A. A. Fedotov, “Construction of the solvability set in a problem of guiding an aircraft under wind disturbance”, Proc. Steklov Inst. Math. (Suppl.), 2005no. , suppl. 1, S163–S174  mathnet  mathscinet  zmath  elib
    3. SERGEY S. KUMKOV, VALERY S. PATSKO, JOSEF SHINAR, “ON LEVEL SETS WITH “NARROW THROATS” IN LINEAR DIFFERENTIAL GAMES”, Int. Game Theory Rev, 07:03 (2005), 285  crossref  mathscinet  scopus  scopus  scopus
    4. Kumkov S.S., Patsko V.S., “Level sweeping of the value function in linear differential games”, Advances in Dynamic Games - APPLICATIONS TO ECONOMICS, MANAGEMENT SCIENCE, ENGINEERING, AND ENVIRONMENTAL MANAGEMENT, Annals of the International Society of Dynamic Games, 8, 2006, 23–37  crossref  mathscinet  zmath  isi
    5. Ganebny S.A., Kumkov S.S., Patsko V.S., Pyatko S.G., “Constructing robust control in differential games: Application to aircraft control during landing”, Advances in Dynamic Game Theory - NUMERICAL METHODS, ALGORITHMS, AND APPLICATIONS TO ECOLOGY AND ECONOMICS, Annals of the International Society of Dynamic Games, 9, 2007, 69–92  crossref  mathscinet  zmath  isi
    6. S. A. Ganebnyi, V. S. Patsko, S. G. Pyatko, “Aircraft landing control under wind disturbances”, Proc. Steklov Inst. Math. (Suppl.), 268, suppl. 1 (2010), S112–S130  mathnet  crossref  isi  elib
    7. Ganebnyi S.A., Kumkov S.S., Patsko V.S., “Extremal aiming in problems with an unknown level of dynamic disturbance”, J. Appl. Math. Mech., 73:4 (2009), 411–420  crossref  mathscinet  zmath  isi  elib  elib  scopus
    8. Dias S., Smirnov G., “High Precision Attitude Control via Differential Game Techniques”, Numerical Analysis and Applied Mathematics, AIP Conference Proceedings, 1281, 2010, 469–472  crossref  adsnasa  isi  scopus  scopus  scopus
    9. Balashov M.V., Repovs D., “Polyhedral approximations of strictly convex compacta”, Journal of Mathematical Analysis and Applications, 374:2 (2011), 529–537  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    10. G. E. Ivanov, V. A. Kazeev, “Minimax algorithm for constructing an optimal control strategy in differential games with a lipschitz payoff”, Comput. Math. Math. Phys., 51:4 (2011), 550–574  mathnet  crossref  mathscinet  isi  elib
    11. Ganebny S.A., Kumkov S.S., Patsko V.S., Pyatko S.G., “Extremal Aiming in Problems with Unknown Level of Dynamic Disturbance”, Advances in Dynamic Games: Theory, Applications, and Numerical Methods for Differential and Stochastic Games: Dedicated to the Memory of Arik a. Melikyan, Annals of the International Society of Dynamic Games, 11, eds. Breton M., Szajowski K., Birkhauser Boston, 2011, 167–187  crossref  mathscinet  zmath  isi
    12. D. V. Kornev, “On numerical solution of positional differential games with nonterminal payoff”, Autom. Remote Control, 73:11 (2012), 1808–1821  mathnet  crossref  zmath  isi
    13. Ivanov G.E., “Algorithm for Constructing an Optimal Control Strategy in a Nonlinear Differential Game with Lipschitz Compactly Supported Payoff”, Differ. Equ., 48:4 (2012), 560–573  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    14. Polovinkin E.S., “Teorema suschestvovaniya reshenii differentsialnogo vklyucheniya s psevdo-lipshitsevoi pravoi chastyu”, Nelineinyi mir, 10:9 (2012), 571–578  elib
    15. Ivanov G.E., “Algoritm postroeniya optimalnoi strategii upravleniya v nelineinoi differentsialnoi igre s lipshitsevoi finitnoi platoi”, Differentsialnye uravneniya, 48:4 (2012), 551–551  mathscinet  zmath  elib
    16. Ipatov D.E., “Razreshimost differentsialnogo vklyucheniya dlya mnogosloinoi modeli obschei tsirkulyatsii okeana s mnogoznachnoi pravoi chastyu”, Almanakh sovremennoi nauki i obrazovaniya, 2012, 52–55  elib
    17. D. R. Kuvshinov, “Chislennoe postroenie reshenii po Neshu v lineinoi pozitsionnoi differentsialnoi igre dvukh lits s fazovym prostranstvom razmernosti bolshe dvukh”, Tr. IMM UrO RAN, 19, no. 1, 2013, 170–181  mathnet  mathscinet  elib
    18. M. I. Gomoyunov, D. V. Kornev, N. Yu. Lukoyanov, “On the numerical solution of a minmax control problem with a positional functional”, Proc. Steklov Inst. Math. (Suppl.), 291, suppl. 1 (2015), 77–95  mathnet  crossref  mathscinet  isi  elib
    19. M. I. Gomoyunov, “Lineino-vypuklye zadachi optimizatsii garantii pri zapazdyvanii v upravlenii”, Izv. IMI UdGU, 2015, no. 1(45), 37–105  mathnet  elib
    20. M. V. Balashov, O. V. Besov, B. I. Golubov, V. V. Goryainov, V. N. Diesperov, S. I. Dudov, G. E. Ivanov, S. P. Konovalov, R. V. Konstantinov, A. B. Kurzhanskii, S. R. Nasyrov, A. G. Sergeev, V. V. Starkov, V. M. Tikhomirov, M. I. Shabunin, “Evgenii Sergeevich Polovinkin (on his 70th birthday)”, Russian Math. Surveys, 71:5 (2016), 983–987  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    21. M. V. Balashov, “On polyhedral approximations in an $n$-dimensional space”, Comput. Math. Math. Phys., 56:10 (2016), 1679–1685  mathnet  crossref  crossref  isi  elib
    22. D. V. Kornev, “Chislennye metody resheniya differentsialnykh igr s neterminalnoi platoi”, Izv. IMI UdGU, 2016, no. 2(48), 82–151  mathnet  elib
    23. Kumkov S.S. Le Menec S. Patsko V.S., “Zero-Sum Pursuit-Evasion Differential Games With Many Objects: Survey of Publications”, Dyn. Games Appl., 7:4, SI (2017), 609–633  crossref  mathscinet  zmath  isi  scopus
    24. M. V. Balashov, “Inscribed balls and their centers”, Comput. Math. Math. Phys., 57:12 (2017), 1899–1907  mathnet  crossref  crossref  isi  elib
  • Математический сборник - 1992–2005 Sbornik: Mathematics (from 1967)
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