Controllability of a semilinear control system by feedback control

We study controllability problem for an abstract control system: x' + Ax = f(x,v), v ∈ Vad. Assuming the existence of an invariance function vη for a state η (generalizing the invariance condition on reachable sets under affine perturbations for the case f(x,v) = f(x) + Bv), we can show the controllability of a subset which contains η.