On Feedback Stabilizability of Linear Systems With State and Input Delays in Banach Spaces

The feedback stabilizability of a general class of well-posed linear systems with state and input delays in Banach spaces is studied in this paper. Using the properties of infinite dimensional linear systems, a necessary condition for the feedback stabilizability of delay systems is presented, which extends the well-known results for finite dimensional systems to infinite dimensional ones. This condition becomes sufficient as well if the semigroup of the delay-free system is immediately compact and the control space is finite dimensional. Moreover, under the condition that the Banach space is reflexive, a rank condition in terms of eigenvectors and control operators is proposed. When the delay-free state space and control space are all finite dimensional, a very compact rank condition is obtained. Finally, the abstract results are illustrated with examples.