In this paper, we introduce a class of deterministic finite-horizon two-player non-zero-sum differential games where one player uses continuous control while the other player uses impulse control. We extend the Pontryagin maximum principle to continuous-time optimal-control problems that involve additional state-dependent costs as well as jumps in the state variable. Thereafter, we formulate the necessary and sufficient conditions for the existence of an open-loop Nash equilibrium in the dynamic games with impulse control. We show that the equilibrium timing of impulses can be obtained as a solution of a non-linear optimization problem for a linear-quadratic dynamic game. For the case of a linear-state dynamic game, we obtain analytical solutions for the equilibrium number, timing, and level of impulse. We illustrate our results using numerical experiments.
Published June 2019 , 25 pages