A Note on Open Loop Nash Equilibrium in Linear-State Differential Games

In this paper we focus on non-cooperative two-player linear-state differential games. In the standard definition this family is introduced assuming that there is no multiplicative interaction among state and control variables. In this paper we show that a multiplicative interaction between the state and the control of one player does not destroy the analytical features of the linear-state differential games if it appears in the objective functional of the other player. We prove that this slightly new definition preserves not only the solvability of the differential game, but also the subgame perfectness of an Open Loop Nash Equilibrium.