  This paper is concerned with the regulation control of a one-dimensional
reaction-diffusion equation in the presence of a state-delay in the reaction
term. The objective is to achieve the PI regulation of the right Dirichlet
trace with a command selected as the left Dirichlet trace. The control design
strategy consists of the design of a PI controller on a finite dimensional
truncated model obtained by spectral reduction. By an adequate selection of the
number of modes of the original infinite-dimensional system, we show that the
proposed control design procedure achieves both the exponential stabilization
of the original infinite-dimensional system as well as the setpoint regulation
of the right Dirichlet trace.
