  The general context of this work is the feedback control of an
infinite-dimensional system so that the closed-loop system satisfies a
fading-memory property and achieves the setpoint tracking of a given reference
signal. More specifically, this paper is concerned with the Proportional
Integral (PI) regulation control of the left Neumann trace of a one-dimensional
reaction-diffusion equation with a delayed right Dirichlet boundary control. In
this setting, the studied reaction-diffusion equation might be either open-loop
stable or unstable. The proposed control strategy goes as follows. First, a
finite-dimensional truncated model that captures the unstable dynamics of the
original infinite-dimensional system is obtained via spectral decomposition.
The truncated model is then augmented by an integral component on the tracking
error of the left Neumann trace. After resorting to the Artstein transformation
to handle the control input delay, the PI controller is designed by pole
shifting. Stability of the resulting closed-loop infinite-dimensional system,
consisting of the original reaction-diffusion equation with the PI controller,
is then established thanks to an adequate Lyapunov function. In the case of a
time-varying reference input and a time-varying distributed disturbance, our
stability result takes the form of an exponential Input-to-State Stability
(ISS) estimate with fading memory. Finally, another exponential ISS estimate
with fading memory is established for the tracking performance of the reference
signal by the system output. In particular, these results assess the setpoint
regulation of the left Neumann trace in the presence of distributed
perturbations that converge to a steady-state value and with a time-derivative
that converges to zero. Numerical simulations are carried out to illustrate the
efficiency of our control strategy.
