Nash equilibria in nonzero-sum differential games with impulse control

In this paper, we introduce a class of deterministic finite-horizon two-player nonzero-sum differential games where one player uses ordinary controls while the other player uses impulse controls. We use the word ‘ordinary’ to mean that Player 1 uses control strategies that are piecewise continuous functions of time. We formulate the necessary and sufficient conditions for the existence of an open-loop Nash equilibrium for this class of differential games. We specialize these results to linear-quadratic games, and show that the open-loop Nash equilibrium strategies can be computed by solving a constrained non-linear optimization problem. In particular, for the impulse player, the equilibrium timing and level of impulses can be obtained. Furthermore, for the special case of linear-state differential games, we obtain analytical characterization of equilibrium number, timing, and the level of impulse in terms of the problem data. We illustrate our results using numerical experiments.