EQUILIBRIUM TRAJECTORIES FOR CONTROL SYSTEMS WITH HETEROGENEOUS DYNAMICS

Nikolay A. Krasovskii     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108, Russian Federation)
Alexander M. Tarasyev     (Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, 16 S. Kovalevskaya Str., Ekaterinburg, 620108; Ural Federal University, 19 Mira Str., Ekaterinburg, 620002, Russian Federation)

Abstract


The paper considers the construction of equilibrium in bimatrix games with heterogeneous dynamics of players' interaction. Heterogeneity of dynamics is connected with difference in maximal rates of the participants. In such a formulation, the switching curves of players' controls are represented by fractional rational functions and are constructed on the basis of N.N. Krasovskii's guaranteed strategies using elements of L.S. Pontryagin's maximum principle. Equilibrium trajectories are generated within the framework of the concept of the dynamic Nash equilibrium introduced by A.F. Kleimenov and are obtained by pasting together the characteristics of the Hamilton-Jacobi equations expressed as exponential functions. The sensitivity analysis is carried out for the shapes of control switching curves with respect to the proportions of players' maximal rates. The comparative analysis is implemented for the values of players' payoffs calculated on equilibrium trajectories of the dynamic game.

Keywords


2-Terminal Reliable Problem, Dantzig–Wolfe decomposition, Branch-and-price

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References


  1. Arnold V.I. Optimization in mean and phase transitions in controlled dynamical systems. Funct. Anal. Appl., 2002. Vol. 36, No. 2. P. 83–92. DOI: 10.1023/A:1015655005114
  2. He Y., Petrosyan L.A. The τ-value in differential games with pairwise interactions. Vestnik St.-Petersburg Univ. Ser. 10. Prikl. Mat. Inform. Prots. Upr., 2025. Vol. 21, No. 2. P. 246–254.
  3. Hofbauer J., Sigmund K. The Theory of Evolution and Dynamical Systems. Cambridge etc.: Cambridge Univ. Press, 1988. 341 p. DOI: 10.1002/zamm.19900700210
  4. Kleimenov A.F. Neantagonisticheskiye pozitsionniye differentsial’niye igry [Non-antagonistic Positional Differential Games]. Yekaterinburg: Nauka, 1993. 185 p. (in Russian)
  5. Krasovskii A.N., Krasovskii N.N. Control Under Lack of Information. Boston: Burkhäauser, 1995. 322 p. DOI: 10.1007/978-1-4612-2568-3
  6. Krasovskii N.A., Tarasyev A.M. Equilibrium trajectories in dynamical bimatrix games with average integral payoff functionals. Autom. Remote Control, 2018. Vol. 79, No. 6. P. 1148–1167. DOI: 10.1134/S0005117918060139
  7. Krasovskii N.A., Tarasyev A.M. Trajectories of dynamic equilibrium and replicator dynamics in coordination games. Ural Math. J., 2024. Vol. 10, No. 2. P. 92–106. DOI: 10.15826/umj.2024.2.009
  8. Krasovskii N.N. Upravleniye dinamicheskoy sistemoy [Control of Dynamic System]. Moscow: Nauka, 1985. 520 p. (in Russian)
  9. Krasovskii N.N., Subbotin A.I. Game–Theoretical Control Problems. New York: Springer–Verlag, 1988. 517 p. URL: https://link.springer.com/book/9781461283188
  10. Mazalov V.V., Rettieva A.N. Application of arbitration schemes for determining equilibria in dynamic games. Mat. Teor. Igr Pril., 2023. Vol. 15., No. 2. P. 75–88. (in Russian)
  11.  Mertens J.-F., Sorin S., Zamir S. Repeated Games. Cambridge: Cambridge University Press, 2015. 567 p. DOI: 10.1017/CBO9781139343275
  12. Pontryagin L.S., Boltyanskii V.G., Gamkrelidze R.V., Mischenko E.F. The Mathematical Theory of Optimal Processes. New York: Wiley Int., 1962. 360 p.
  13.  Vinnikov E.V., Davydov A.A., Tunitskiy D.V. Existence of maximum of time-averaged harvesting in the KPP-model on sphere with permanent and impulse harvesting. Dokl. Math., 2023. Vol. 108, No. 3. P. 472–476. DOI: 10.1134/S1064562423701387
  14. Vorobyev N.N. Teoriya igr dlya ekonomistov–kibernetikov [The Theory of Games for Economists–Cyberneticians]. Moscow: Nauka, 1985. 272 p. (in Russian)




DOI: http://dx.doi.org/10.15826/umj.2025.2.010

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