Necessary and Sufficient Small Gain Conditions for Integral Input-to-State Stable Systems: A Lyapunov Perspective

This paper is concerned with conditions for the stability of interconnected nonlinear systems consisting of integral input-to-state stable (iISS) systems with external inputs. The treatment of iISS and input-to-state stable (ISS) systems is unified. Both necessary conditions and sufficient conditions are investigated using a Lyapunov formulation. In the presence of model uncertainty, this paper proves that, for the stability of the interconnected system, at least one subsystem is necessarily ISS which is a stronger stability property in the set of iISS. The necessity of a small-gain-type property is also demonstrated. This paper proposes a common form of smooth Lyapunov functions which can establish the iISS and the ISS of the interconnection comprising iISS and ISS subsystems whenever the small-gain-type condition is satisfied. The result covers situations more general than the earlier study and removes technical conditions assumed in the previous literature. Global asymptotic stability is discussed as a special case.