Global Asymptotic Stability in Time-Delay Systems

Core Concepts

Global asymptotic stability in time-delay systems does not guarantee uniform global attractivity, challenging traditional assumptions.

Abstract

The content explores the relationship between global asymptotic stability (GAS) and uniform global attractivity (UGA) in time-delay systems. It presents a counter-example showing that GAS does not imply UGA, even when combined with local exponential stability. The example highlights the potential for arbitrarily slow convergence rates despite initial states being confined to bounded sets. The study delves into the intricacies of forward completeness, bounded reachability sets, and the implications of stability properties on system behavior. By providing detailed proofs and explanations, the authors shed light on the complexities of analyzing stability in infinite-dimensional systems.

Stats

For all t ≥ 0: |zi(t)| = |zi0(0)|e^-t
Lemma 1: There exists a constant c > 0 such that given any M > 0, there exists an initial condition z10 ∈ X satisfying ∥z10∥ ≤ 1 and some x0 ∈ X2 with ∥x0∥ ≤ 1 such that given any z20 ∈ X satisfying z20(t) = 1 for all t ∈ [−2, −1], the corresponding solution of (3) satisfies |x(1)| ≥ 2M.
Proposition 1: Given any c > 0, the time-delay system (3) is FC, LES, and GAS but is not BRS or UGA.
Lemma 1: There exists a constant c > 0 such that given any M > 0, there exists an initial condition z10 ∈ X satisfying ∥z10∥ ≤ 1 and some x0 ∈ X2 with ∥x0∥ ≤ 1 such that given any z20 ∈ X satisfying z20(t) = 1 for all t ∈ [−2, −1], the corresponding solution of (3) satisfies |x(1)| ≥ 2M.

Quotes

"Global asymptotic stability does not necessarily guarantee uniform global attractivity."
"The absence of uniform attractivity is due to additional dynamics introduced in the system."
"The study challenges traditional assumptions about stability in time-delay systems."

Key Insights Distilled From

For time-invariant delay systems, global asymptotic stability does not imply uniform global attractivity

by Antoine Chai... at arxiv.org 03-04-2024

https://arxiv.org/pdf/2403.00387.pdf

Deeper Inquiries

How can these findings impact real-world applications relying on stable systems

The findings of this research can have significant implications for real-world applications that rely on stable systems, especially those involving time-delay systems. Understanding that global asymptotic stability does not necessarily imply uniform global attractivity in these systems is crucial for various engineering fields like control systems, robotics, and telecommunications. For instance, in autonomous vehicles where stability is paramount for safe operation, knowing the limitations of certain stability properties can guide the design of more robust control algorithms. Similarly, in communication networks with delays, ensuring reliable and efficient data transmission requires a deep understanding of system stability beyond just global asymptotic stability.

What are potential limitations or criticisms of the approach taken by the authors

One potential limitation or criticism of the approach taken by the authors could be related to the generalizability of their results across different types of time-delay systems. While they provide a compelling counter-example demonstrating the lack of uniform global attractivity despite global asymptotic stability, it would be valuable to explore how these findings extend to a broader class of delay systems with varying characteristics. Additionally, further investigation into the practical feasibility and computational complexity of implementing strategies based on their results could offer more insights into real-world applicability.

How might insights from this research be applied to other fields beyond mathematics

Insights from this research in mathematics can be applied to various other fields beyond pure mathematical analysis. In particular:

Engineering: Concepts like forward completeness and bounded reachability sets are essential in designing stable control systems for complex engineering processes such as aircraft autopilots or industrial automation.
Biology: Understanding system stability is critical in biological models with feedback loops or regulatory mechanisms where delays play a role.
Economics: Time-delay dynamics are prevalent in economic models; applying principles from this study can enhance predictive accuracy and policy formulation.
By leveraging these mathematical findings across interdisciplinary domains, researchers and practitioners can improve system performance and reliability while accounting for inherent delays and uncertainties present in dynamic systems.