This paper investigates the input-to-state contraction for impulsive systems. The notion of input-to-state contraction (ISC) is proposed via incremental input-to-state stability (δISS). By using the method of variation, the equivalence between global ISC and exponential δISS is established. Then by the technique of the average dwell-time (ADT), some criteria for global ISC and exponential δISS are derived for impulsive systems. Moreover, by using impulse frequency (I.F.) to replace ADT, the criteria are improved and less conservative criteria via I.F. are established for global ISC and exponential δISS. The criteria are used to derive global ISC and exponential δISS for continuous and discrete time systems. Discussions are given to relax the differentiable vector fields. And the criteria of exponential δISS via the Lyapunov-like function are derived for impulsive systems satisfying time-varying Lipschitz condition. Finally, three examples are worked out for illustrations, where impulsive control is designed for chaotic systems to achieve global ISC.

本文研究了脉冲系统的输入到状态收缩。输入到状态收缩 (ISC) 的概念是通过增量输入到状态稳定性 ( δ ISS) 提出的。利用变分法，建立了全局ISC与指数δISS的等价性。然后通过平均停留时间（ADT）技术，推导出脉冲系统的全局 ISC 和指数δ ISS 的一些标准。此外，通过使用脉冲频率 (IF) 代替 ADT，标准得到改进，并且通过 IF 为全局 ISC 和指数δ ISS 建立了不太保守的标准。该标准用于推导全局 ISC 和指数δISS 用于连续和离散时间系统。讨论放宽可微向量场。并通过类李雅普诺夫函数导出了满足时变 Lipschitz 条件的脉冲系统的指数δ ISS 判据。最后给出了三个例子进行说明，其中脉冲控制是针对混沌系统设计的，以实现全局ISC。