Asymptotic behavior of solutions in dynamical bimatrix games with discounted indices

The paper is devoted to the analysis of dynamical bimatrix games with integral indices discounted on an infinite time interval. The system dynamics is described by differential equations in which players' behavior changes according to incoming control signals. For this game, a problem of construction of equilibrium trajectories is considered in the framework of minimax approach proposed by N. N. Krasovskii and A. I. Subbotin in the differential games theory. The game solution is based on the structure of dynamical Nash equilibrium developed in papers by A. F. Kleimenov. The maximum principle of L. S. Pontryagin in combination with the method of characteristics for Hamilton–Jacobi equations are applied for the synthesis of optimal control strategies. These methods provide analytical formulas for switching curves of optimal control strategies. The sensitivity analysis for equilibrium solutions is implemented with respect to the discount parameter in the integral payoff functional. It is shown that equilibrium trajectories in the problem with the discounted payoff functional asymptotically converge to the solution of a dynamical bimatrix game with average integral payoff functionals examined in papers by V. I. Arnold. Obtained results are applied to a dynamical model of investments on financial markets.