Delay periodic Lyapunov equation
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arXiv
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| Format: | Preprint |
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2026
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| author | Aleksandrova, Irina V. Velázquez, Juan J. L. |
| author_facet | Aleksandrova, Irina V. Velázquez, Juan J. L. |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2605_15926 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Delay periodic Lyapunov equation Aleksandrova, Irina V. Velázquez, Juan J. L. Dynamical Systems 34K06, 34K13, 34G10, 37L05, 93D05 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. |
| title | Delay periodic Lyapunov equation |
| topic | Dynamical Systems 34K06, 34K13, 34G10, 37L05, 93D05 |
| url | https://arxiv.org/abs/2605.15926 |