A small gain type result with respect to the induced $L_\infty$ norm

Abstract

The aim of this paper is to present a small gain type result with respect to the $L_1$ or induced $L_\infty$ norm of stable systems. This norm representing the peak--to--peak gain obtained for magnitude bounded inputs is determined using a method based on invariant ellipsoids. A version of Bounded Real Lemma providing necessary and sufficient conditions of boundedness of this norm is derived. Further, based on the necessary conditions of this result a Small Gain Theorem with respect to the induced $L_\infty$ norm is proved.