Kamke functions for stability of interconnected systems

Abstract

It is known in control theory that the problem how an interconnection of ISS (abbreviation of input-to-state stability) subsystems remains ISS leads to a cycle condition, which is a set of inequalities involving the composition of unbounded Kamke functions. In this paper we convert the problem on those inequalities to functional equations and obtain \(\mathcal {K}_\infty \) solutions by finding fixed points in a complete metric space or a locally convex topological linear space. We also give \(\mathcal {K}_\infty \) solutions by piecewise construction. Our results can be applied to discussing ISS of interconnection.