Time-delay systems that defy intuition: nonrobust forward completeness and related (non)properties

An example of a time-invariant time-delay system that is uniformly globally attractive and Lyapunov 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 and (ii) global asymptotic stability is not equivalent to uniform global asymptotic stability. In addition, a novel characterization of robust forward completeness for usually encountered classes of time-delay systems is provided. This characterization relates robust forward completeness of the time-delay system with the forward completeness of an associated nondelayed finite-dimensional system.Comment: Submitted to Automatica. Print ISSN: 0005-1098, Online ISSN: 1873-283