Stability and observer designs using new variants of Halanay’s inequality

We provide a generalization of Halanay’s inequality, where the decay rate is constant but gain multiplying delayed term time varying. While usual conditions require to be strictly larger than an upper bound on gain, our less restrictive results allow times when can exceed rate. This allows us prove asymptotic stability in significant cases that were not amenable previous Lyapunov function constructions, and violate contraction requirement was needed trajectory based results. apply work problems for linear continuous systems with switched delays, observers nonlinear discrete measurements.