Approximate controllability of linear retarded systems in Hilbert spaces

The object of this paper is to construct some results on the control problems for retarded functional differential equations of parabolic type with unbounded principal operators in Hilbert spaces. In order to establish relations between controllability of the given equation and observability of the adjoint system, we investigate the equivalent relation for the completeness of generalised eigenspaces of the infinitesimal generators. Finally, when the control space is a finite-dimensional space, a necessary and sufficient condition for the approximate controllability of retarded equations is given by the so-called rank condition.