In the present paper, we study a two-player, zero-sum, deterministic differential game with both players adopting impulse controls in infinite-time horizon, under rather weak assumptions on the cost functions. We prove by means of the dynamic programming principle that the lower and upper value functions are continuous and viscosity solutions to the corresponding Hamilton-Jacobi-Bellman-Isaacs (HJBI) quasi-variational inequality (QVI). We define a new HJBI-QVI for which, under a proportional property assumption on the maximizing player cost, the value functions are the unique viscosity solution. We then prove that the lower and upper value functions coincide.

在本文中，我们研究了一种两人零和确定性微分博弈，在成本函数的假设相当弱的情况下，双方都在无限时间范围内采用脉冲控制。我们利用动态规划原理证明了下值函数和上值函数是相应的 Hamilton-Jacobi-Bellman-Isaacs (HJBI) 拟变分不等式 (QVI) 的连续解和粘性解。我们定义了一个新的 HJBI-QVI，在最大化玩家成本的比例属性假设下，价值函数是唯一的粘度解。然后我们证明下值函数和上值函数是一致的。