Controllability and observability of linear retarded systems in Banach spaces

Controllability and observability are studied for linear retarded systems in a Banach space X. Both the X-exact and the X-approximate controllability are considered and a number of necessary and sufficient conditions in terms of the coefficient operators and/or the fundamental solution are presented. Various corresponding concepts for observability are introduced and the duality between controllability and observability is clarified. An applicable condition for the X-approximate controllability and the X*-observability is established under the assumptions that the control is a finite dimensional one and the system of generalized eigenspaces is projectively complete. The condition is stated in terms of eigenvectors and controllers.