  We present several characterizations, via some weak observability
inequalities, on the complete stabilizability for a control system $[A,B]$,
i.e., $y'(t)=Ay(t)+Bu(t)$, $t\geq 0$, where $A$ generates a $C_0$-semigroup on
a Hilbert space $X$ and $B$ is a linear and bounded operator from another
Hilbert space $U$ to $X$. We then extend the aforementioned characterizations
in two directions: first, the control operator $B$ is unbounded; second, the
control system is time-periodic. We also give some sufficient conditions, from
the perspective of the spectral projections, to ensure the weak observability
inequalities. As applications, we provide several examples, which are not null
controllable, but can be verified, via the weak observability inequalities, to
be completely stabilizable.
