This paper investigates the input-to-state stability (ISS) problem for a type of time-varying nonlinear systems with random delayed impulses. On the basis of the Lyapunov method, it provides sufficient conditions to obtain the ISS properties by indefinite Lyapunov functions with time-varying conditions. Besides, the mixed effect of random and delayed impulse is also fully considered. On this basis, we examine the ISS issue for impulsive systems in two different scenarios. One scenario is that the impulsive intensity is random, and the impulsive intensity is limited by the average impulsive interval (AII) and the mode-dependent average impulsive interval (MDAII). The other scenario is that the impulsive intensity is random, and the impulsive density satisfies certain stochastic conditions. Finally, two examples are provided to test the validity of the stability criterion proposed.
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