On a Connection between Spectral Factorization and Geometric Control Theory

We investigate here how the geometric control theory of Basile, Marro and Wonham can be obtained in a Hilbert space context, as the byproduct of the factorization of a spectral density with no zeros on the imaginary axis. We show how controlled invariant subspaces can be obtained as images of orthogonal projections of coinvariant subspaces onto a semiinvariant (markovian) subspace of the Hardy space of square integrable functions analytic in the right half plane. Output nulling subspaces are then related to a particular spectral factorization problem. A similar construction is presented for controllability subspaces, and a new algorithm for the computation of these subspaces is presented.