We present respectively necessary and sufficient conditions on the exponential stabilizability and the complete stabilizability for abstract linear control systems in Hilbert spaces. These conditions are characterized by the weak observability inequalities for the dual systems. Several examples, which are not null controllable but are complete stabilizable, are given. The results in this talk are provided by joint works with E. Trelat and Y. Xu (2020) and with H. Liu, Y. Xu and H. Yu. (2022).