Do 0-GAS-Guaranteeing Impulse Sequences Preserve ISS or iISS Properties? Not Always

Based on the recent work on determination of classes of impulse/switching signals, uniformly over which a switched or impulsive system is input-to-state stable (ISS) or integral input-to-state stable (iISS), it is conjectured that for a switched system with all ISS or iISS subsystems, as long as the impulse/switching signal makes the system 0-input globally asymptotically stable (0-GAS), then it also ensures ISS and iISS, respectively. This work disproves this conjecture by showing examples where the impulse sequence amplifies the input, causing 0-GAS impulsive system to lose the ISS or iISS properties. As a compensation, we provide sufficient conditions on the impulse sequences which guarantee ISS or iISS. It turns out that these conditions are strictly stronger than those ensuring 0-GAS; the differences between them are in fact necessary conditions for the impulsive system to have the asymptotic gain (AG) property or uniformly bounded energy bounded state (UBEBS) property.