Dead-time compensation for wave/string PDEs

Smith Predictor-like designs for compensation of arbitrarily long input delays are commonly available only for finite-dimensional systems. Only very few examples exist where such compensation has been achieved for PDE systems, including our recent result for a parabolic (reaction-diffusion) PDE. In this paper we address a more challenging wave PDE problem, where the difficulty is amplified by allowing all of this PDE´s eigenvalues to be a distance to the right of the imaginary axis. Anti-damping (positive feedback) on the uncontrolled boundary induces this dramatic form of instability. We develop a design which compensates an arbitrarily long delay at the input of the boundary control system and achieve exponential stability in closed loop. We derive explicit formulae for our controller´s gain kernel functions. They are related to the open-loop solutions of the anti-stable wave equation system over the time period of input delay (this simple relationship is the result of the design approach).