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arxiv: 2605.15926 · v1 · pith:GJUGJ6ZQnew · submitted 2026-05-15 · 🧮 math.DS

Delay periodic Lyapunov equation

Irina V. Aleksandrova , Juan J.L. Vel\'azquez This is my paper

Pith reviewed 2026-05-19 19:08 UTC · model grok-4.3

classification 🧮 math.DS
keywords delay Lyapunov matrixperiodic systemsLyapunov equationmonodromy operatorHilbert spacedelay systemsevolution familiesstability
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The pith

The delay Lyapunov matrix exists and is unique precisely when the monodromy operator has no reciprocal eigenvalues.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper generalizes classical results on periodic Lyapunov equations from finite-dimensional systems to periodic evolution families on Hilbert spaces, with applications to linear systems that include constant delays. It connects the existence of quadratic periodic Lyapunov functionals to the solvability of a discrete operator Lyapunov equation built around the monodromy operator. This connection supplies an alternative definition of the delay Lyapunov matrix and yields the uniqueness theorem that the matrix exists and is unique if and only if the monodromy operator has no reciprocal eigenvalues. The same framework also permits the construction of Lyapunov-Krasovskii functionals for periodic delay systems without first assuming exponential stability.

Core claim

For periodic evolution families on a Hilbert space the existence and uniqueness of a quadratic periodic Lyapunov functional is equivalent to the existence and uniqueness of a solution to the discrete operator Lyapunov equation involving the monodromy operator. This equivalence furnishes an alternative definition of the delay Lyapunov matrix for linear periodic delay systems and establishes that the matrix exists and is unique if and only if the monodromy operator possesses no reciprocal eigenvalues. As a direct consequence the construction of Lyapunov-Krasovskii functionals for such systems no longer requires a preliminary exponential-stability assumption.

What carries the argument

The discrete operator Lyapunov equation with the monodromy operator, which translates the existence of periodic quadratic Lyapunov functionals into a spectral condition on the monodromy operator.

If this is right

  • Existence of a unique periodic positive definite solution to the Lyapunov equation becomes equivalent to exponential stability for the infinite-dimensional periodic case.
  • Lyapunov-Krasovskii functionals for periodic delay systems can be assembled directly from the solution of the operator equation.
  • Stability criteria for the systems reduce to verifying that the monodromy operator has no reciprocal eigenvalues.
  • The finite-dimensional periodic Lyapunov theory is recovered exactly when the underlying space is finite-dimensional.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Numerical schemes that approximate the monodromy operator could be used to compute the delay Lyapunov matrix in practice.
  • Similar operator-theoretic arguments might apply to other functional equations that admit a periodic evolution family structure.
  • The uniqueness result could be tested on systems with slowly varying delays if an appropriate evolution family can be defined.

Load-bearing premise

A well-defined periodic evolution family on the Hilbert space must exist and generate a monodromy operator to which the discrete operator Lyapunov equation can be applied.

What would settle it

A concrete periodic delay system in which the monodromy operator has at least one reciprocal eigenvalue yet a unique delay Lyapunov matrix still exists.

read the original abstract

For linear periodic finite-dimensional systems, it is well-known that, first, exponential stability is equivalent to the existence of a unique periodic positive definite solution to the Lyapunov equation, and second, the Lyapunov equation admits a unique periodic solution, if and only if the monodromy matrix has no reciprocal eigenvalues. In the present paper, we generalize these results to the case of periodic evolution families on a Hilbert space, with application to the stability theory of linear periodic systems with constant delays. More precisely, we first link the existence and uniqueness of a quadratic periodic Lyapunov functional with the existence and uniqueness of the solution to a discrete operator Lyapunov equation with the monodromy operator involved. Second, we show that the presented theory on a Hilbert space gives rise to an alternative definition of the delay Lyapunov matrix, the concept previously appeared in the construction of quadratic Lyapunov-Krasovskii functionals for a class of linear periodic delay systems. An explicit connection between the infinite-dimensional Hilbert setting and the previously developed delay Lyapunov matrix framework is established. An important consequence is the uniqueness theorem: the delay Lyapunov matrix exists and is unique, if and only if the monodromy operator has no reciprocal eigenvalues. As a by-product, our framework enables the construction of Lyapunov-Krasovskii functionals for periodic delay systems without a preliminary exponential stability assumption as in earlier theory.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

1 major / 2 minor

Summary. The paper generalizes the equivalence of exponential stability with a unique periodic positive definite solution to the Lyapunov equation, and the uniqueness of periodic solutions iff the monodromy has no reciprocal eigenvalues, from finite-dimensional periodic linear systems to periodic evolution families on Hilbert spaces. It links quadratic periodic Lyapunov functionals to solutions of the discrete operator Lyapunov equation involving the monodromy operator, applies the framework to linear periodic constant-delay systems to give an alternative definition of the delay Lyapunov matrix, and derives the uniqueness theorem that this matrix exists and is unique precisely when the monodromy operator has no reciprocal eigenvalues. As a by-product, the approach permits construction of Lyapunov-Krasovskii functionals for such delay systems without a preliminary exponential stability assumption.

Significance. If the operator-theoretic steps hold, the results would meaningfully extend Lyapunov theory to infinite-dimensional periodic systems and provide a useful bridge to the existing delay Lyapunov matrix literature for periodic delay equations. The ability to construct functionals without assuming stability a priori could simplify certain stability analyses, and the spectral characterization via the monodromy operator offers a concrete test that is independent of the finite-dimensional case.

major comments (1)
  1. [Abstract and the section deriving the link to the discrete operator Lyapunov equation] The uniqueness theorem (abstract and the main result linking the delay Lyapunov matrix to the monodromy operator) rests on applying the discrete operator Lyapunov equation to the monodromy operator on the chosen Hilbert space. The manuscript does not explicitly establish that the periodic evolution family is strongly continuous (or at least generates a bounded monodromy operator) for the constant-delay systems with periodic coefficients under consideration; without this, the spectral condition on reciprocal eigenvalues does not automatically guarantee uniqueness of the solution to the operator equation.
minor comments (2)
  1. [Introduction] The transition from the finite-dimensional Lyapunov equation to the operator version would benefit from an explicit statement of the state space (e.g., the precise Hilbert space of history functions) at the first appearance of the evolution family.
  2. [The section on quadratic periodic Lyapunov functionals] A short remark clarifying whether the quadratic Lyapunov functional is required to be positive definite or merely positive semi-definite would help readers compare the result with the classical finite-dimensional statement.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for the careful reading of the manuscript and for highlighting this important point about the foundations of the operator-theoretic framework. We address the major comment below and will incorporate the necessary clarifications in the revised version.

read point-by-point responses

  1. Referee: [Abstract and the section deriving the link to the discrete operator Lyapunov equation] The uniqueness theorem (abstract and the main result linking the delay Lyapunov matrix to the monodromy operator) rests on applying the discrete operator Lyapunov equation to the monodromy operator on the chosen Hilbert space. The manuscript does not explicitly establish that the periodic evolution family is strongly continuous (or at least generates a bounded monodromy operator) for the constant-delay systems with periodic coefficients under consideration; without this, the spectral condition on reciprocal eigenvalues does not automatically guarantee uniqueness of the solution to the operator equation.

    Authors: We agree that explicit verification strengthens the presentation. The general theory in the manuscript is developed for strongly continuous periodic evolution families on Hilbert spaces, which is the standard setting ensuring that the monodromy operator is a bounded linear operator (see, e.g., the well-posedness results for periodic evolution families in the literature). For the application to linear periodic systems with constant delays, the state-space formulation (typically in a product space incorporating the delay interval) yields a strongly continuous evolution family by the standard theory of retarded functional differential equations with periodic coefficients; consequently the monodromy operator is bounded. To make this fully explicit and address the referee's concern, we will add a short paragraph or remark in the relevant section (near the application to delay systems) recalling these well-posedness facts and confirming boundedness of the monodromy operator. This clarification does not change the main theorems but ensures the spectral condition applies rigorously. revision: yes

Circularity Check

0 steps flagged

No significant circularity

full rationale

The paper generalizes established finite-dimensional results on periodic Lyapunov equations and monodromy operators to periodic evolution families on Hilbert spaces. It links quadratic periodic Lyapunov functionals to discrete operator Lyapunov equations and derives an alternative definition of the delay Lyapunov matrix as a consequence, with the uniqueness theorem following directly from the spectral condition that the monodromy operator has no reciprocal eigenvalues. No steps reduce by construction to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations; the derivation relies on standard operator theory and provides independent mathematical content for the delay-system application without circular reduction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The paper relies on standard domain assumptions from functional analysis and delay differential equations without introducing new free parameters or postulated entities.

axioms (1)
  • domain assumption A periodic evolution family on a Hilbert space is strongly continuous and generates a well-defined monodromy operator.
    This premise is required to link the quadratic periodic Lyapunov functional to the discrete operator Lyapunov equation.

pith-pipeline@v0.9.0 · 5760 in / 1274 out tokens · 63182 ms · 2026-05-19T19:08:10.583026+00:00 · methodology

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