We consider a deterministic two-player linear-state differential game, where Player 1 uses piecewise continuous controls, while Player 2 implements impulse controls. When the impulse instants are not the decision variables for Player 2, but provided exogenously, we recover the classical result that both open-loop and feedback Nash equilibria coincide for this class of games. When the number and timing of impulse instants are decision variables of Player 2, we show that the classical result no longer holds, that is, open-loop and feedback Nash equilibria are different.

We show that the impulse level is constant in both equilibria. More importantly, in the open-loop case, we show that the equilibrium number of impulses is at most three, while there can be at most two impulses in the feedback case.