  This is the third part of four series papers, aiming at the delay
compensation for the abstract linear system (A,B,C). Both the input delay and
output delay are investigated. We first propose a full state feedback control
to stabilize the system (A,B) with input delay and then design a
Luenberger-like observer for the system (A,C) in terms of the delayed output.
We formulate the delay compensation in the framework of regular linear systems.
The developed approach builds upon an upper-block-triangle transform that is
associated with a Sylvester operator equation. It is found that the
controllability/observability map of system (-A,B)/(-A,-C) happens to be the
solution of the corresponding Sylvester equation. As an immediate consequence,
both the feedback law and the state observer can be expressed explicitly in the
operator form. The exponential stability of the resulting closed-loop system
and the exponential convergence of the observation error are established
without using the Lyapunov functional approach. The theoretical results are
validated through the delay compensation for a benchmark one-dimensional wave
equation.
