Characterizations of Complete Stabilizability

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 $C_0$-semigroup Hilbert space $X$ and $B$ is linear bounded operator from another $U$ to $X$. then extend aforementioned characterizations in two directions: first, unbounded; second, time periodic. also give sufficient conditions, perspective of spectral projections, ensure inequalities. As applications, we provide examples, which are not null controllable, but can be verified, completely stabilizable.