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For time-invariant (nonimpulsive) systems, it is already well-known that the input-to-state stability (ISS) property is strictly stronger than integral input-to-state stability (iISS). Very recently, we have shown that under suitable uniform boundedness and continuity assumptions on the function defining system dynamics, ISS implies iISS also for time-varying systems. In this paper, we show that this implication remains true for impulsive systems, provided that asymptotic stability is understood in a sense stronger than usual for impulsive systems. extension to classes of systems for which Lyapunov characterizations do not exist, such as switched systems under restricted switching or impulsive systems. Very recently, Haimovich and Mancilla-Aguilar (2019) proved that ISS implies iISS for families of time-varying and switched nonlinear systems without resorting to any Lyapunov converse theorem, and, in this way, opening the door to proving the implication for other types of systems.

Several control design strategies for sampled-data systems are based on a discrete-time model. In general, the exact discrete-time model of a nonlinear system is difficult or impossible to obtain, and hence approximate discrete-time models may be employed. Most existing results provide conditions under which the stability of the approximate discrete-time model in closed-loop carries over to the stability of the (unknown) exact discrete-time model but only in a practical sense, meaning that trajectories of the closedloop system are ensured to converge to a bounded region whose size can be made as small as desired by limiting the maximum sampling period. In addition, some sufficient conditions exist that ensure global exponential stability of an exact model based on an approximate model. However, these conditions may be rather stringent due to the global nature of the result. In this context, our main contribution consists in providing rather mild conditions to ensure semiglobal exponential input-to-state stability of the exact model via an approximate model. The enabling condition, which we name the Robust Equilibrium-Preserving Consistency (REPC) property, is obtained by transforming a previously existing consistency condition into a semiglobal and perturbation-admitting condition. As a second contribution, we show that every explicit and consistent Runge-Kutta model satisfies the REPC condition and hence control design based on such a Runge-Kutta model can be used to ensure semiglobal exponential input-to-state stability of the exact discrete-time model in closed loop.

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