The paper deals with the controllability and exponential stabilizability for the control system
-semigroup on a Hilbert space
, U being a Hilbert space. When
, the equivalence between controllability, exponential stabilizability, an observability inequality and a frequency domain condition is presented. A related result for boundary control systems has been given by V. Komornik
[SIAM J. Control Optim. 35, No. 5, 1591-1613 (1997)]. When
generates a group and the system is exactly controllable, it is shown that every
can be transferred to the origin by smooth controls. A frequency domain characterization for exactly controllable second order elastic systems is then given. The above results are applied to the study of the control and damping of the wave, Schrödinger and Petrovsky equations. Precisely, one obtains a sufficient condition for the subregion on which the application of the control (resp. damping) gives exact controllability (resp. the exponential decay property) on a not necessarily smooth region. Finally, a relationship between the controllability of the three control systems mentioned above is established.