Harmonic Lyapunov equations in continuous-time periodic systems: solutions and properties

First, the harmonic Lyapunov equations claimed on the Hilbert space I₂ are restricted to the Banach space I₁ for asymptotic stability of finite-dimensional linear continuous-time periodic (FDLCP) systems. Second, solutions to the harmonic Lyapunov equations are scrutinised. Third, the harmonic Lyapunov equations are connected with periodic matrix differential (PMD) Lyapunov equations. The connection sheds new light on periodic solutions to the PMD Lyapunov equations. Fourth, the trace formula of the H₂ norm for FDLCP systems is shaped with periodic solutions to the PMD Lyapunov equations. Finally, an algorithm for computing periodic solutions to the PMD Lyapunov equations is proposed, which involves only solutions to algebraic Lyapunov equations. There are numerical examples to illustrate the results.