Feedback Nash Equilibria in Differential Games with Impulse Control

In this article, we study a class of deterministic finite-horizon two-player nonzero-sum differential games where players are endowed with different kinds of controls. We assume that Player 1 uses piecewise-continuous controls, whereas Player 2 uses impulse controls. For this class of games, we seek to derive conditions for the existence of feedback Nash equilibrium strategies for the players. More specifically, we provide a verification theorem for identifying such equilibrium strategies, using the Hamilton-Jacobi-Bellman equations for Player 1 and the quasi-variational inequalities for Player 2. Furthermore, we show that the equilibrium number of interventions by Player 2 is upper bounded. Furthermore, we specialize the obtained results in a scalar two-player linear-quadratic differential game. In this game, Player 1's objective is to drive the state variable toward a specific target value, and Player 2 has a similar objective with a different target value. We provide, for the first time, an analytical characterization of the feedback Nash equilibrium in a linear-quadratic differential game with impulse control. We illustrate our results using numerical experiments.