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This paper studies the periodic feedback stabilization for a class of linear$T$-periodic evolution equations.Several equivalent conditions on the linearperiodic feedback stabilization are obtained. These conditions are related withthe following subjects: the attainable subspace of the controlled evolutionequation under consideration; the unstable subspace (of the evolution equationwith the null control) provided by the Kato projection; the Poincar$\acute{e}$map associated with the evolution equation with the null control; and twounique continuation properties for the dual equations on different timehorizons $[0,T]$ and $[0,n_0T]$ (where $n_0$ is the sum of algebraicmultiplicities of distinct unstable eigenvalues of the Poincar$\acute{e}$ map).It is also proved that a $T$-periodic controlled evolution equation is linear$T$-periodic feedback sabilizable if and only if it is linear $T$-periodicfeedback sabilizable with respect to a finite dimensional subspace. Someapplications to heat equations with time-periodic potentials are presented.

Equivalent Conditions on Periodic Feedback Stabilization for Linear Periodic Evolution Equations