Finite-Dimensional Boundary Control of a Wave Equation With Viscous Friction and Boundary Measurements

Recently, a constructive approach to the design of finite-dimensional observer-based controller has been proposed for parabolic partial differential equations (PDEs). This article extends it to hyperbolic PDEs. Namely, we design a finite-dimensional, output-feedback, boundary controller for a wave equation with in-domain viscous friction. The control-free system is unstable for any friction coefficient due to an external force. Our approach is based on modal decomposition: an observer-based controller is designed for a finite-dimensional projection of the wave equation on
$N$
eigenfunctions (modes) of the Sturm–Liouville operator. The danger of this approach is the “spillover” effect: such a controller may have a deteriorating effect on the stability of the unconsidered modes and cause instability of the full system. Our main contribution is an appropriate Lyapunov-based analysis leading to linear matrix inequalities (LMIs) that allow one to find a controller gain and number of modes,
$N$
, guaranteeing that the “spillover” effect does not occur. An important merit of the derived LMIs is that their complexity does not change when
$N$
grows. Moreover, we show that appropriate
$N$
always exists and, if the LMIs are feasible for some
$N$
, they remain so for
$N+1$
.