Lyapunov Equations, Inequalities and Stability Theorems for Periodic Systems

Abstract

Lyapunov equations and inequalities are studied in connection with the stability of linear periodic systems. First, as preliminaries, existing results on Lyapunov equations in time-invariant systems are summarized in the form of equivalence theorems. Then, based on those results, existence conditions are derived for positive-definite or nonnegative-definite periodic solutions of periodic Lyapunov differential equations and inequalities with reference to the detectability and observability. Such conditions are stated all as equivalent conditions for the system to be either asymptotically stable or stable. Brief discussions are included also for periodic Lyapunov difference equations in discrete-time periodic systems.