Forward completeness does not imply bounded reachability sets and global asymptotic stability is not necessarily uniform for time-delay systems

TL;DR

This paper presents a counterexample showing that forward completeness does not imply bounded reachability sets and that global asymptotic stability is not necessarily uniform in time-delay systems, challenging existing conjectures.

Contribution

It provides a counterexample to two conjectures and introduces a new characterization of robust forward completeness for systems with finite delays.

Findings

Counterexample disproves equivalence of forward completeness and bounded reachability

Counterexample shows global asymptotic stability is not always uniform

New characterization links robust forward completeness to nondelayed systems

Abstract

An example of a time-invariant time-delay system that is uniformly globally attractive and exponentially stable, hence forward complete, but whose reachability sets from bounded initial conditions are not bounded over compact time intervals is provided. This gives a negative answer to two current conjectures by showing that (i) forward completeness is not equivalent to robust forward completeness (i.e. boundedness of reachability sets) and (ii) global asymptotic stability is not equivalent to uniform global asymptotic stability. In addition, a novel characterization of robust forward completeness for systems having a finite number of discrete delays is provided. This characterization relates robust forward completeness of the time-delay system with the forward completeness of an associated nondelayed finite-dimensional system.