Feedback invariance of SISO infinite-dimensional systems

We consider a linear single-input single-output system on a Hilbert space X, with infinitesimal generator A, bounded control element b, and bounded observation element c. We address the problem of finding the largest feedback invariant subspace of X that is in the space c ^⊥ perpendicular to c. If b is not in c ^⊥, we show this subspace is c ^⊥. If b is in c ^⊥, a number of situations may occur, depending on the relationship between b and c.