ISS inequalities for vector versions of Halanay’s inequality and of the trajectory-based approach

Abstract

Halanay’s inequality is a powerful and widely used tool to prove asymptotic convergence properties of functions that arise in the study of systems with delays or continuous-discrete features. In its standard form, it applies to scalar valued functions that satisfy decay conditions, with overshoots depending on suprema of the functions over suitable intervals. Then it provides an exponential decay estimate on the scalar functions. Here, we provide vector versions of Halanay’s inequality, and of the so-called trajectory based approach, both yielding input-to-state stability (or ISS) inequalities. Our proofs of the inequalities use the theory of positive systems. We apply our results to prove ISS for interval observers and other cases.