Compensating a String PDE in the Actuation or Sensing Path of an Unstable ODE

How to control an unstable linear system with a long pure delay in the actuator path? This question was resolved using ´predictor´ or ´finite spectrum assignment´ designs in the 1970s. Here we address a more challenging question: How to control an unstable linear system with a wave partial differential equation (PDE) in the actuation path? Physically one can think of this problem as having to stabilize a system to whose input one has access through a string. The challenges of overcoming string/wave dynamics in the actuation path include their infinite dimension, finite propagation speed of the control signal, and the fact that all of their (infinitely many) eigenvalues are on the imaginary axis. In this technical note we provide an explicit feedback law that compensates the wave PDE dynamics at the input of an linear time-invariant ordinary differential equation and stabilizes the overall system. In addition, we prove robustness of the feedback to the error in a priori knowledge of the propagation speed in the wave PDE. Finally, we consider a dual problem where the wave PDE is in the sensing path and design an exponentially convergent observer.