Algebraic and Dynamic Lyapunov Equations on Time Scales

TL;DR

This paper generalizes Lyapunov matrix equations to systems on diverse time domains, including nonuniform discrete and mixed continuous-discrete systems, extending classical stability analysis methods.

Contribution

It introduces a unified framework for algebraic and differential Lyapunov equations on arbitrary time scales, broadening their applicability beyond classical continuous and discrete cases.

Findings

Extended Lyapunov equations to nonuniform discrete domains

Compared classical and generalized Lyapunov theories

Provided insights into stability analysis on mixed domains

Abstract

We revisit the canonical continuous-time and discrete-time matrix algebraic and matrix differential equations that play a central role in Lyapunov based stability arguments. The goal is to generalize and extend these types of equations and subsequent analysis to dynamical systems on domains other than $\R$ or $\Z$, e.g. nonuniform discrete domains or domains consisting of a mixture of discrete and continuous components. We compare and contrast the standard theory with the theory in this general case.