seibert1995reduction

On the Reduction to a Subspace of Stability Properties of Systems in Metric Spaces

Authors: P. Seibert, J. S. Florio · Year: 1995 · Venue: Annali di Matematica Pura ed Applicata (Ser. IV), Vol. 169, pp. 291–320 · Raw: md · pdf

Summary

Establishes reduction theorems: sufficient conditions under which stability and asymptotic stability (local and global) of a compact invariant set for a full (semi-)dynamical system on a metric space follow from the corresponding properties of the reduced system induced on an invariant subset. It generalizes the classical “reduce to an invariant subspace” idea (Lyapunov, Malkin, the centre-manifold line) away from any dependence on a linear part. In this corpus it is the root reference for the R5 charter and the theorem that elhawwary2013reduction builds its reduction machinery on.

Key Claims

• Reduction principle (metric-space form). For $M\subset Y\subset X$ with $M$ compact invariant: stability of $M$ under the whole system follows from asymptotic stability of $M$ under the reduced system on $Y$ together with a stability/Lagrange-stability condition on $Y$ — with no assumption on any linearization.
• Both regimes. Local and global versions are given, for dynamical and semidynamical systems (the latter admitting one-sided time).
• Lineage. Framed against Vidyasagar’s triangular-form (cascade) result and Sontag’s CIBS condition — the reduction viewpoint subsumes the cascade-stability question.

Method

Abstract dynamical-systems theory on metric spaces; proofs via invariant sets, attraction, and Lyapunov (Lagrange) stability of the reduced flow. No control design.

Regime note. General stability theory — no manipulator/spacecraft model — so the free-flying vs free-floating axis does not apply; a foundational text.

Relevance to thesis

R5’s charter roots the project’s reduction arguments here: proving stability of a target set for the free-flying manipulator by reducing to an invariant subset (e.g. a submanifold on which part of the error has already converged). El-Hawwary–Maggiore’s backstepping/reduction results, already in the corpus, cite Seibert–Florio directly. This page supplies the resolving bibkey.

Connections

Topics: cascaded systems · Lyapunov stability · configuration-dependent stability domain Sources: elhawwary2013reduction · panteley2001growth

Key Equations / Quotes

Abstract, verbatim from the corpus text layer (prose extracts cleanly):

“The objects studied are dynamical and semidynamical systems defined on arbitrary metric spaces. Sufficient conditions for stability and asymptotic stability (both local and global) of a compact invariant set are established, using the corresponding properties of the (semi-) dynamical systems induced on a (positively) invariant subset.”

Open Questions

• The corpus PDF is a scanned offprint (PageGenie / 0100.tif) with an OCR text layer; prose extracts cleanly but the math symbols are mangled by the old fonts (∀/∃/γ/limsup → junk). A LightOnOCR smoke of pp. 3–5 exists at Docs/ocr_out/seibert1996_lightonocr_pp3-5_smoke.md. Per corpus policy this stays a search-only md — escalate to full VLM OCR only if the equations are needed verbatim.
• The ctrllib task supplied the starting bibkey seibert1996reduction; the journal masthead, the Springer record, and DOI 10.1007/BF01759358 all date the volume 1995, so the page / bibkey / year use 1995. Reversible if a downstream cite already assumed 1996.