Forward completeness does not imply bounded reachability sets and global asymptotic stability is not necessarily uniform for time-delay systems
Jose L. Mancilla-Aguilar, Hernan Haimovich
TL;DR
The paper investigates the relationship between forward completeness ($FC$), robust forward completeness ($RFC$), and uniform/global stability notions for time-delay systems. By constructing a time-delay system with a single discrete delay that is forward complete, exponentially stable, and uniformly globally attractive, yet not $RFC$, it demonstrates that $FC$ does not collapse to $RFC$ and that global asymptotic stability ($GAS$) does not imply uniform global asymptotic stability ($UGAS$) for time-delay models. A key contribution is a novel characterization: for systems with a finite number of discrete delays, $RFC$ is equivalent to the forward completeness of an associated nondelayed finite-dimensional system under Lebesgue-measurable and locally essentially bounded inputs. The results imply that nondelayed reductions to finite dimensions do not rescue forward completeness from non-robust behavior, and they establish a clear boundary between $GAS$ and $UGAS$ in the time-delay setting, with significant implications for analysis and control design of delayed 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.