A Spectral Criterion for Exact Detectability of a Class of Continuous-Time and Discrete-Time Systems With Periodic Coefficients

This paper investigates exact detectability of systems with periodic coefficients in finite-dimensional real ordered Hilbert spaces, extending the classical framework of positive systems, which have applications in many fields. A spectral PBH-type criterion for detectability is established for both discrete and continuous-time systems in a unified treatment. We further propose a Barbasin–Krasovskii-type criterion for exponential stability by showing that the existence of a solution to a dual system, combined with detectability, ensures stability. The obtained results provide a Lyapunov-like framework and open directions for studying stability in optimal control.