Feedback stabilization of a 1D linear reaction-diffusion equation with\n delay boundary control

Abstract

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