Geodesic
Application of Lyapunov–Poincaré method of small parameter for Nash and Berge equilibrium designing in one differential two-player game
Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 33 (2023) no. 4, pp. 601-624 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

The Poincaré small parameter method is actively used in celestial mechanics, as well as in the theory of differential equations and in its important section called optimal control. In this paper, the mentioned method is used to construct an explicit form of Nash and Berge equilibrium in a differential positional game with a small influence of one of the players on the rate of change of the state vector.
Keywords: small parameter method, differential linear-quadratic noncooperative game, Nash equilibrium, Berge equilibrium 
@article{VUU_2023_33_4_a4,
     author = {V. I. Zhukovskii and L. V. Zhukovskaya and S. N. Sachkov and E. N. Sachkova},
     title = {Application of {Lyapunov{\textendash}Poincar\'e} method of small parameter for {Nash} and {Berge} equilibrium designing in one differential two-player game},
     journal = {Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹ\^uternye nauki},
     pages = {601--624},
     year = {2023},
     volume = {33},
     number = {4},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/VUU_2023_33_4_a4/}
}
TY  - JOUR
AU  - V. I. Zhukovskii
AU  - L. V. Zhukovskaya
AU  - S. N. Sachkov
AU  - E. N. Sachkova
TI  - Application of Lyapunov–Poincaré method of small parameter for Nash and Berge equilibrium designing in one differential two-player game
JO  - Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
PY  - 2023
SP  - 601
EP  - 624
VL  - 33
IS  - 4
UR  - http://geodesic.mathdoc.fr/item/VUU_2023_33_4_a4/
LA  - en
ID  - VUU_2023_33_4_a4
ER  - 
%0 Journal Article
%A V. I. Zhukovskii
%A L. V. Zhukovskaya
%A S. N. Sachkov
%A E. N. Sachkova
%T Application of Lyapunov–Poincaré method of small parameter for Nash and Berge equilibrium designing in one differential two-player game
%J Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki
%D 2023
%P 601-624
%V 33
%N 4
%U http://geodesic.mathdoc.fr/item/VUU_2023_33_4_a4/
%G en
%F VUU_2023_33_4_a4
V. I. Zhukovskii; L. V. Zhukovskaya; S. N. Sachkov; E. N. Sachkova. Application of Lyapunov–Poincaré method of small parameter for Nash and Berge equilibrium designing in one differential two-player game. Vestnik Udmurtskogo universiteta. Matematika, mehanika, kompʹûternye nauki, Tome 33 (2023) no. 4, pp. 601-624. http://geodesic.mathdoc.fr/item/VUU_2023_33_4_a4/

[1] Gorelov M.A., “Logic in the study of hierarchical games under uncertainty”, Automation and Remote Control, 78:11 (2017), 2051–2061 | DOI | MR | Zbl

[2] Gorelov M.A., “Risk management in hierarchical games with random factors”, Automation and Remote Control, 80:7 (2019), 1265–1278 | DOI | DOI | MR | Zbl

[3] Malkin I.G., Some problems of theory of non-linear oscillations, URSS, Moscow, 2020

[4] Mishchenko E.F., Rozov N.Kh., Differential equations with small parameter and relaxation oscillations, Nauka, Moscow, 1975

[5] Ougolnitsky G.A., Usov A.B., “Dynamic hierarchical two-player games in open-loop strategies and their applications”, Automation and Remote Control, 76:11 (2015), 2056–2069 | DOI | MR | MR | Zbl

[6] Petrosjan L.A., Zenkevich N.A., “Conditions for sustainable cooperation”, Automation and Remote Control, 76:10 (2015), 1894–1904 | DOI | MR

[7] Petrosian O., Barabanov A., “Looking forward approach in cooperative differential games with uncertain stochastic dynamics”, Journal of Optimization Theory and Applications, 172:1 (2017), 328–347 | DOI | MR | Zbl

[8] Petrosian O., Tur A., Wang Zeyang, Gao Hongwei, “Cooperative differential games with continuous updating using Hamilton–Jacobi–Bellman equation”, Optimization Methods and Software, 36:6 (2021), 1099–1127 | DOI | MR | Zbl

[9] Voevodin V.V., Kuznetsov Yu.A., Matrices and calculations, Nauka, Moscow, 1984 | MR

[10] Zhukovskiy V.I., Lyapunov functions in differential games, CRC Press, London, 2003 | DOI | MR

[11] Zhukovskii V.I., Chikrii A.A., Linear-quadratic differential games, Naukova Dumka, Kyiv, 1994

[12] Zhukovskiy V.I., Zhukovskaya L.V., Kudryavtsev K.N., Larbani M., “Strong coalitional equilibria in games under uncertainty”, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp’yuternye Nauki, 30:2 (2020), 189–207 | DOI | MR

[13] Zhukovskiy V.I., Salukvadze M.E., The vector-valued maximin, Academic Press, Boston, 1994 | MR | Zbl

[14] Zhukovskiy V.I., Topchishvili A.T., “The explicit form of Berge strong guaranteed equilibrium in linear-quadratic non-cooperative game”, International Journal of Operations and Quantitative Management, 21:4 (2015), 265–273

[15] Zhukovskiy V., Topchishvili A., Sachkov S., “Application of probability measures to the existence problem of Berge–Vaisman guaranteed equilibrium”, Model Assisted Statistics and Applications, 9:3 (2014), 223–239 | DOI