Feedback-Invariant Subspaces in Infinite-Dimensional Systems

We consider single-input single-output systems on a Hilbert space X, with infinitesimal generator A, bounded control element b, and bounded observation element c. Let c^⊥be the subspace of X perpendicular to c. We consider the problem of finding the largest feedback-invariant subspace of c^⊥. If b is in c^⊥, and c ∉ D(A^*), a largest feedback-invariant subspace does not exist in general.