On the Lack of Ideal Observability in Hilbert Space

Abstract

Some remarks are given on the lack of ideal observability for linear systems in Hilbert space with bounded operators, under the action of a disturbance generating a measured output on some finite interval of time. A criterion is established in this context by means of a range inclusion of operators according to a well known Douglas theorem. The ideal observability subspace is described as the intersection of a special collection of Kalman observable subspaces.