Combining iISS and ISS With Respect to Small Inputs: The Strong iISS Property

This technical note studies the notion of Strong iISS, which is defined as the combination of input-to-state stability (ISS) with respect to small inputs, and integral input-to-state stability (iISS). This notion characterizes the robustness property that the state remains bounded as long as the magnitude of exogenous inputs is reasonably small, but may diverge for stronger disturbances. We provide several Lyapunov-based sufficient conditions for Strong iISS. One of them relies on iISS Lyapunov functions admitting a radially non-vanishing K dissipation rate. Although such dissipation inequality appears natural in view of the existing Lyapunov characterization of iISS and ISS, we show through a counter-example that it is not a necessary condition for Strong iISS. Less conservative conditions are then provided, as well as tools to estimate the tolerated input magnitude that preserves solutions´ boundedness.