Strong iISS: Combination of iISS and ISS with respect to small inputs

This paper studies the notion of Strong iISS, which imposes both integral input-to-state stability (iISS) and input-to-state stability (ISS) with respect to small inputs. This combination characterizes the robustness property, exhibited by many practical systems, that the state remains bounded as long as the magnitude of exogenous inputs is reasonably small but may diverge for stronger disturbances. We provide three Lyapunov-type sufficient conditions for Strong iISS. One is based on iISS Lyapunov functions admitting a radially non-vanishing (class K) dissipation rate. However we show that it is not a necessary condition for Strong iISS. Two less conservative conditions are then provided, which are used to demonstrate that asymptotically stable bilinear systems are Strongly iISS. Finally, we discuss cascade and feedback interconnections of Strong iISS systems.