We show that a Lyapunov-Krasovskii functional whose dissipation rate involves solely the current instantaneous value of the state norm is enough to guarantee integral input-to-state stability (iISS). This result generalizes existing sufficient conditions for iISS, where the dissipation rate involves the whole Lyapunov-Krasovskii functional itself, and simplifies their applicability. Moreover, it provides a more natural bridge with the classical condition for global asymptotic stability of input-free systems. The proof strategy we employ relies on a novel characterization of global asymptotic stability, which may be of interest on its own.
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