Equilibrium trajectories for control systems with heterogeneous dynamics

Nikolay A. Krasovskii^a, Alexander M. Tarasyev<sup>ab

a</sup> N.N. Krasovskii Institute of Mathematics and Mechanics, Ural Branch of the Russian Academy of Sciences, Ekaterinburg
^b Ural Federal University named after the First President of Russia B. N. Yeltsin, Ekaterinburg

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: Dynamic bimatrix games, Heterogeneous dynamics, Average integral payoffs, Characteristics of Hamilton–Jacobi equations, Equilibrium trajectories

Language: English

DOI: 10.15826/umj.2025.2.010

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