Delay-independent incremental stability in time-varying monotone systems satisfying a generalized condition of two-sided scalability

Abstract

Monotone systems generated by delay differential equations with explicit time-variation are of importance in the modeling of a number of significant practical problems, including the analysis of communications systems, population dynamics, and consensus protocols. In such problems, it is often of importance to be able to guarantee delay-independent incremental asymptotic stability, whereby all solutions converge toward each other asymptotically, thus allowing the asymptotic properties of all trajectories of the system to be determined by simply studying those of some particular convenient solution. It is known that the classical notion of quasimonotonicity renders time-delayed systems monotone. However, this is not sufficient alone to obtain such guarantees. In this work we show that by combining quasimonotonicity with a condition of scalability motivated by wireless networks, it is possible to guarantee incremental asymptotic stability for a general class of systems that includes a variety of interesting examples. Furthermore, we obtain as a corollary a result of guaranteed convergence of all solutions to a quantifiable invariant set, enabling time-invariant asymptotic bounds to be obtained for the trajectories even if the precise values of time-varying parameters are unknown.