  Given a linear time-periodic control system in a Hilbert space with a bounded
control operator, we present a characterization of periodic stabilization in
terms of a detectability inequality. Similar characterizationwas built up in
[E. Trelat, G. Wang, Y. Xu, Characterization by observability inequalities of
controllability and stabilization properties, Pure and Appl. Anal., 2 (2020),
93-122] for time-invariant systems.

Optimization and Control