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.