A two-person non-antagonistic positional differential game with integral and vector payoffs of the players

A two-person non-antagonistic positional differential game is considered on a given time interval. The payoff functional of the first player is the sum of integral and terminal components, whereas the second player has a vector terminal payoff functional. Both players know the value of the state vector of the system at the current time. In addition, the first player knows the value of the control of the second player. The second player acts in the class of pure positional strategies, and the first player acts in the class of positional counter-strategies. The notion of Nash-type solution of the game is introduced. A problem whose solutions are the basis for the construction of Nash-type equilibria is formulated.