A Riesz Basis Methodology for Proportional and Integral Output Regulation of a One-Dimensional Diffusive-Wave Equation

In this article, we consider a dam-river system modeled by a diffusive-wave equation. This model is commonly used in hydraulic engineering to describe dynamic behavior of the unsteady flow in a river for shallow water when the flow variations are not important. In order to stabilize and regulate the system, we propose a proportional and integral boundary controller. Contrary to many physical systems, we end up with a nondissipative closed-loop system with noncollocated actuators and sensors. We show that the closed-loop system is a Riesz spectral system and generates an analytic semigroup. Then, we shall be able to assign the spectrum of the closed-loop system in the open left half-plane to ensure its exponential stability as well as the output regulation independently of any known or unknown constant perturbation. These results are illustrated by several numerical examples.