Bilinear hereditary differential systems: State space models in Hilbert space

We analyze here state space models for bilinear hereditary differential systems of various levels of complexity. It is shown that for many reasons it is more convenient to develop state space models in Hilbert spaces, following the method of Delfour and Mitter. In certain cases explicit bilinear operational differential equations are obtained that illustrate the way the controls influence the spectrum of the infinitesimal generator of the system. Certain connections with the theory of Lie semigroups are also discussed.