Dynamic polynomial stabilization of a 1D wave equation

Abstract

We study observer-based dynamic stabilization of a one-dimensional wave equation with boundary control and distributed observation. The control system we consider is exponentially stabilizable but not exponentially detectable. Consequently, exponential energy decay is not achievable with dynamic output feedback. As our main result we design an observer-based controller which achieves rational decay of energy for a class of initial conditions. The controller design relies on helpful results on polynomial stability of semigroups generated by block operator matrices.