Equilibrium trajectories in dynamical bimatrix games with average integral payoff functionals

Models of evolutionary nonzero-sum games are considered on the infinite time interval. Methods of differential games theory are used for the analysis of game interactions between two groups of participants. It is assumed, that participants in groups are subject to control through signals for the behavior change. Payoffs of coalitions are determined as average integral functionals on the infinite horizon. The problem of constructing a dynamical Nash equilibrium is posed for the considered evolutionary game. Ideas and approaches of non-antagonistic differential games are applied for the determination of the Nash equilibrium solutions. The results are based on dynamic constructions and methods of evolutionary games. The great attention is paid to the formation of the dynamical Nash equilibrium with players strategies, that maximize the corresponding payoff functions and have the guaranteed properties according to the minimax approach. The application of the minimax approach for constructing optimal control strategies synthesizes trajectories of the dynamical Nash equilibrium that provide better results in comparison to static solutions and evolutionary models with the replicator dynamics. The dynamical Nash equilibrium trajectories for evolutionary games with the average integral quality functionals are compared with trajectories for evolutionary games based on the global terminal quality functionals on the infinite horizon.