  In the present paper, the exponential stability of the mono-tubular heat
exchanger equation with boundary observation possessing a time delay and inner
control is investigated through a simply proportional feedback. Firstly, the
close-loop system is translated into an abstract Cauchy problem in the suitable
state space. A uniformly bounded $C_{0}$-semigroup generated by the close-loop
system, which means that the unique solution of the system exist, is shown.
Secondly, the spectrum configuration of the closed-loop system is analyzed and
the eventual differentiability and the eventual compactness of the semigroup
are shown by the resolvent estimates on some resolvent set. This implies that
the spectrum determined growth assumption hold. Finally, a sufficient
condition, which is on the physical parameters in the system and independent of
the time delay, of the exponential stability of the closed-loop system is
given.
