A Unified Floquet Theory for Discrete, Continuous, and Hybrid Periodic Linear Systems

TL;DR

This paper introduces a comprehensive Floquet theory applicable to discrete, continuous, and hybrid periodic linear systems on time scales, enabling unified stability analysis and spectral characterization.

Contribution

It develops a unified Floquet framework on time scales, including stability-preserving transformations, spectral mapping, and stability criteria applicable to hybrid systems.

Findings

Unified Floquet theorem established for various time scales.

Spectral mapping theorem connects Floquet multipliers and exponents.

Stability can be assessed via pole placement of associated systems.

Abstract

In this paper, we study periodic linear systems on periodic time scales which include not only discrete and continuous dynamical systems but also systems with a mixture of discrete and continuous parts (e.g. hybrid dynamical systems). We develop a comprehensive Floquet theory including Lyapunov transformations and their various stability preserving properties, a unified Floquet theorem which establishes a canonical Floquet decomposition on time scales in terms of the generalized exponential function, and use these results to study homogeneous as well as nonhomogeneous periodic problems. Furthermore, we explore the connection between Floquet multipliers and Floquet exponents via monodromy operators in this general setting and establish a spectral mapping theorem on time scales. Finally, we show this unified Floquet theory has the desirable property that stability characteristics of the original system can be determined via placement of an associated (but time varying) system's poles in the complex plane. We include several examples to show the utility of this theory.