Perturbation of controllable systems

Let a semi-dynamical system, (A,B): (A,B) x=Ax + Bu x(0) = 0 be given on a Hilbert space X. When A is self-adjoint and semi-bounded with spectral measure E(?) it is shown that controllability of the system (A,B) is equivalent to that of (f(A), B) where f(?) is any semi-bounded, Borel measurable E(?)-surjective function on-?C0 - semigroup on X and are applied to a system whose dynamics are governed by the singular integral operator generating the Poisson integral semigroup.