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, while 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 (HJB) equations for Player 1 and the quasi-variational inequalities (QVIs) for Player 2. Further, we show that the equilibrium number of interventions by Player 2 is upper bounded. Furthermore, we specialize the obtained results to a scalar two-player linear-quadratic differential game. In this game, Player 1's objective is to drive the state variable towards 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.

中文翻译：

---

带脉冲控制的微分博弈中的反馈纳什均衡

我们研究了一类确定性有限范围的两人非零和微分博弈，其中玩家被赋予了不同类型的控制。我们假设玩家 1 使用分段连续控制，而玩家 2 使用脉冲控制。对于这类博弈，我们寻求推导出玩家反馈纳什均衡策略存在的条件。更具体地说，我们提供了一个验证定理来识别这种均衡策略，使用玩家 1 的 Hamilton-Jacobi-Bellman (HJB) 方程和玩家 2 的准变分不等式 (QVI)。此外，我们证明了均衡数玩家 2 的干预是有上限的。此外，我们将获得的结果专门用于标量两人线性二次微分游戏。在这场比赛中，球员 1' 的目标是将状态变量驱动到特定的目标值，而玩家 2 有一个相似的目标，但目标值不同。我们首次提供了具有脉冲控制的线性二次微分游戏中反馈纳什均衡的分析表征。我们使用数值实验来说明我们的结果。