Zero-sum and nonzero-sum differential games without Isaacs condition

In this paper we study the zero-sum and nonzero-sum differential games with not assuming Isaacs condition. Along with the partition $\pi$ of the time interval $[0,T]$, we choose the suitable random non-anticipative strategy with delay to study our differential games with asymmetric information. Using Fenchel transformation, we prove that the limits of the upper value function $W^\pi$ and lower value function $V^\pi$ coincide when the mesh of partition $\pi$ tends to 0. Moreover, we give a characterization for the Nash equilibrium payoff (NEP, for short) of our nonzero-sum differential games without Isaacs condition, then we prove the existence of the NEP of our games. Finally, by considering all the strategies along with all partitions, we give a new characterization for the value of our zero-sum differential game with asymmetric information under some equivalent Isaacs condition.
Comment: Juan Li gave a talk on this paper in the International Conference on Mathematical Control Theory-- In Memory of Professor Xunjing Li for His 80th Birthday(16-19 July 2015, Sichuan University, Chengdu)