Robust stabilization of the wave equation against small delays

Abstract

In this paper we consider a system which can be modeled by (undamped) wave equation in a bounded domain. We assume that the system is fixed at one end and is controlled by a boundary controller at the other end. We also considered two damped versions of this system, both parameterized by a nonnegative damping constant. We study two problems for these models, namely the stabilization by means of a boundary controller, and the stability robustness of the closed-loop system against small time delays in the feedback loop. We propose a class of finite dimensional dynamic boundary controllers to solve these problems. One basic feature of these controllers is that the corresponding controller transfer functions are required to be strictly positive real functions. We show that these controllers stabilize both damped and undamped models and solve the stability robustness problem for the damped models. It is also shown that while strict positive realness of the controller transfer functions is important for closed-loop stability, the strict properness is important for the stability robustness against small time delays in the feedback loop.