"Why Overlooking Uniform Attractivity Can Break Stability Results in Time-Varying Systems"

What is it about?

This publication revisits a well-known 2017 study on the stability of complex systems that change over time. The original study made a key assumption error by not considering a condition called “uniform attractivity,” which ensures system behavior stabilizes consistently. We show through a counterexample why this condition is essential and how omitting it can lead to incorrect conclusions about a system’s stability. We also highlight mistakes in the original proofs and offer improved assumptions to fix the problem. This work contributes to a clearer understanding of when and how time-varying nonlinear systems can be trusted to behave stably.

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Why is it important?

This work highlights the essential role of uniform attractivity in establishing uniform asymptotic stability for time-varying nonlinear systems. While the condition might seem subtle, its absence can lead to misleading conclusions, especially in practical applications such as adaptive control, where uniform convergence guarantees are critical. By clarifying this point and illustrating it with a counterexample, the study helps refine the theoretical tools used in stability analysis and encourages more careful use.