On exact controllability and complete stabilizability for linear systems

This work is concerned with the relations between exact controllability and complete stabilizability for linear systems in Hilbert spaces. We give an affirmative answer to the open problem posed by Rabah and Karrakchou [R. Rabah, J. Karrakchou, Exact controllability and complete stabilizability for linear systems in Hilbert spaces, Appl. Math. Lett. 10 (1997) 35–40]. More precisely, if the C 0 -semigroup S ( t ) generated by A is surjective and the pair ( A , B ) with a bounded operator B is completely stabilizable, then ( A , B ) is exactly controllable without any additional condition.