Some equivalent conditions for exponential stabilization of linear systems with unbounded control

It is proved that the equivalences among the exponential stabilizability of the control system, the existences of admissible controls for every initial state condition and the solvability of some Riccati equation or some LQ problem also hold for the case in Hilbert space with unbounded control. The results do not need the compactness assumption for the resolvents of the infinitesimal generator, and improve the results available. They can be applied to the parabolic systems with boundary control through Dirichlet, Neumann condition or pointwise control on not only bounded domain but also unbounded domain.