Delay-independent global convergence in time-varying monotone systems of delay differential equations satisfying a scalability condition

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 communication systems and consensus protocols. In such problems, it is often of importance to be able to guarantee delay-independent incremental global convergence, whereby all solutions converge towards each other asymptotically. Such guarantees allow the asymptotic properties of all trajectories of the system to be determined by simply studying those of some particular convenient solution. A class of these systems was recently studied in the context of wireless networks through the imposition of a scalability condition. In this work, we seek to weaken the notions of monotonicity and system structure that were employed in that setting so as to extend this analysis to a more general context and make explicit exactly which system properties are required. 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.

I. Introduction

Monotone systems represent an important class of dynamical systems that are of interest both for their applicability to a number of practical problems and for their rich mathematical structure. The order-preserving structure of these systems allows strong results concerning their stability properties to be obtained. In the celebrated work [1], Hirsch established results of generic convergence, guaranteeing convergence of almost every bounded solution of any such system for which the monotonicity property holds strongly to the equilibrium set, provided this set is nonempty. In systems of differential equations that are not autonomous, however, the equilibrium set of even a monotone system will frequently be empty, so the generic convergence results are not directly applicable. Instead, a property that is often of interest is the concept of global convergence. A solution is said to be globally convergent if it is stable and all other solutions converge to it as time tends to infinity, and so this idea is of particular use in problems of system tracking or prediction. A system that admits a globally convergent solution, and hence for which all trajectories converge to one another, is said to be incrementally globally convergent.