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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 ( δ ISS). By using the method of variation, the equivalence between global ISC and exponential  δ ISS is established. Then by the technique of the average dwell-time (ADT), some criteria for global ISC and exponential  δ ISS 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  δ ISS. The criteria are used to derive global ISC and exponential  δ ISS for continuous and discrete time systems. Discussions are given to relax the differentiable vector fields. And the criteria of exponential  δ ISS 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.

Input-to-state contraction for impulsive systems