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 Mat. Teor. Igr Pril., 2013, Volume 5, Issue 2, Pages 105–136 (Mi mgta111)

Necessary conditions of overtaking equilibrium for infinite horizon differential games

Dmitry V. Khlopin

Krasovskii Institute of Mathematics and Mechanics, Ural Branch, Russian Academy of Sciences

Abstract: This paper is devoted to refining the necessary conditions of Nash equilibrium for infinite horizon games. The right-hand endpoint of the trajectories is free; the search for equilibrium is conducted in the class of open-loop strategies. For adjoint variables of each player, the necessary asymptotic condition of overtaking Nash equilibrium is obtained. For certain cases (in particular, when the 'strictly optimal solution' criterium is employed), it allows to explicitly specify the expression for the adjoint variable. S. M. Aseev, A. V. Kryazhimskii, and V. M. Veliov used such an expression as a necessary condition of optimality for certain infinite horizon control problems. In this paper we also specify an example of linear game with the opposite objective functionals. In this game, there exist multiple Nash equilibria that depend on the choice of the sequence along which the players optimize their actions. These equilibria produce absolutely different results. Also Sorger version of the 'Lanchester-type' differential game is considered.

Keywords: dynamic optimization, differential games, infinite horizon problem, transversality condition for infinity, necessary conditions of equilibrium, overtaking equilibrium, unbounded cost.

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Citation: Dmitry V. Khlopin, “Necessary conditions of overtaking equilibrium for infinite horizon differential games”, Mat. Teor. Igr Pril., 5:2 (2013), 105–136

Citation in format AMSBIB
\Bibitem{Khl13} \by Dmitry~V.~Khlopin \paper Necessary conditions of overtaking equilibrium for infinite horizon differential games \jour Mat. Teor. Igr Pril. \yr 2013 \vol 5 \issue 2 \pages 105--136 \mathnet{http://mi.mathnet.ru/mgta111} 

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Erratum
• On Sorger game
Dmitry V. Khlopin
Mat. Teor. Igr Pril., 2013, 5:3, 115–119

This publication is cited in the following articles:
1. Dmitrii V. Khlopin, “Ob igre Zorgera”, MTIP, 5:3 (2013), 115–119
2. Khlopin D., “Necessity of Vanishing Shadow Price in Infinite Horizon Control Problems”, J. Dyn. Control Syst., 19:4 (2013), 519–552
3. Khlopin D.V., “Necessity of Limiting Co-State Arcs in Bolza-Type Infinite Horizon Problem”, Optimization, 64:11 (2015), 2417–2440
4. A. J. Zaslavski, “Structure of approximate solutions of Bolza variational problems on large intervals”, Discret. Contin. Dyn. Syst.-Ser. S, 11:6, SI (2018), 1283–1316
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