Delay periodic Lyapunov equation

TL;DR

This paper extends the theory of Lyapunov equations to infinite-dimensional Hilbert spaces, providing new insights into the stability analysis of linear periodic delay systems and establishing conditions for the existence and uniqueness of delay Lyapunov matrices.

Contribution

It generalizes classical results to Hilbert spaces, linking quadratic Lyapunov functionals with operator equations, and introduces an alternative delay Lyapunov matrix framework.

Findings

Existence and uniqueness of delay Lyapunov matrices depend on monodromy operator eigenvalues.

Established a connection between Hilbert space theory and delay Lyapunov matrices.

Constructed Lyapunov-Krasovskii functionals without requiring exponential stability.

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.