  This paper studies the periodic feedback stabilization for a class of linear
$T$-periodic evolution equations.Several equivalent conditions on the linear
periodic feedback stabilization are obtained. These conditions are related with
the following subjects: the attainable subspace of the controlled evolution
equation under consideration; the unstable subspace (of the evolution equation
with the null control) provided by the Kato projection; the Poincar$\acute{e}$
map associated with the evolution equation with the null control; and two
unique continuation properties for the dual equations on different time
horizons $[0,T]$ and $[0,n_0T]$ (where $n_0$ is the sum of algebraic
multiplicities 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 $T$-periodic
feedback sabilizable with respect to a finite dimensional subspace. Some
applications to heat equations with time-periodic potentials are presented.
