Algebraic and dynamic Lyapunov equations on time scales

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, called “time scales”, e.g. nonuniform discrete domains or domains consisting of a mixture of discrete and continuous components. In particular, we compare and contrast a generalization of the algebraic Lyapunov equation and the dynamic Lyapunov equation in this time scales setting.