Equilibria in Motion: Stability, Tracking, and Convergence

[2606.28422] Equilibria in Motion: Stability, Tracking, and Convergence Skip to main content Search Submit Donate Log in Mathematics > Dynamical Systems arXiv:2606.28422 (math) [Submitted on 25 Jun 2026 ( v1 ), last revised 29 Oct 2026 (this version, v3)] Title: Equilibria in Motion: Stability, Tracking, and Convergence Authors: Hassan Saoud View PDF HTML (experimental) Abstract: {We study nonautonomous systems whose equilibrium sets are nonisolated and move with time. Their motion is measured on each ball by a localized Attouch--Wets distance, which yields an equilibrium speed that may depend on the radius. The distance from a trajectory to the current equilibrium set is estimated in two ways: through a composite Lyapunov function compared with a power of this distance, and, without any Lyapunov function, through a decay condition on the frozen vector fields. In the first setting, we prove practical stability, and Lyapunov stability when the dissipation has no residual, a case that forces the equilibrium sets to be nested. In the second, we obtain, in the region where the decay condition holds, an explicit tracking estimate driven by the equilibrium speed and the inputs, and we prove asymptotic tracking whenever their averages over unit time intervals vanish, a condition that cannot be weakened among conditions on the speed alone. We also derive input-to-state stability bounds and explicit convergence rates. When the equilibrium speeds are integrable, the equilibrium sets converge in the Attouch--Wets sense to a limiting set at an explicit rate. If, moreover, the velocity is bounded by a multiple of the distance to the current equilibrium set, every trajectory remaining in the region considered has finite length and converges to a point of the limiting set; for a fixed equilibrium set, this yields semistability. The results are illustrated on a scalar example and on resource-allocation models with time-varying demand and with rotating weights. Comments: Revised and corrected version. Some statements and arguments have been corrected and strengthened; the main scope and conclusions of the paper remain unchanged Subjects: Dynamical Systems (math.DS) ; Classical Analysis and ODEs (math.CA); Optimization and Control (math.OC) MSC classes: 34D20, 34D05, 93D05, 93D09, 34C11 Cite as: arXiv:2606.28422 [math.DS] (or arXiv:2606.28422v3 [math.DS] for this version) https://doi.org/10.48550/arXiv.2606.28422 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Hassan Saoud [ view email ] [v1] Thu, 25 Jun 2026 17:44:27 UTC (56 KB) [v2] Sat, 22 Aug 2026 14:53:43 UTC (55 KB) [v3] Thu, 29 Oct 2026 14:10:56 UTC (229 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.DS < prev | next > new | recent | 2026-06 Change to browse by: math math.CA math.OC References & Citations SciX Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab)

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