  The goal of this work is to compute a boundary control of reaction-diffusion
partial differential equation. The boundary control is subject to a constant
delay, whereas the equation may be unstable without any control. For this
system equivalent to a parabolic equation coupled with a transport equation, a
prediction-based control is explicitly computed. To do that we decompose the
infinite-dimensional system into two parts: one finite-dimensional unstable
part, and one stable infinite-dimensional part. An finite-dimensional delay
controller is computed for the unstable part, and it is shown that this
controller succeeds in stabilizing the whole partial differential equation. The
proof is based on a an explicit form of the classical Artstein transformation,
and an appropriate Lyapunov function. A numerical simulation illustrate the
constructive design method.
