Backstepping boundary control for first-order hyperbolic PDEs and application to systems with actuator and sensor delays

We consider a problem of boundary feedback stabilization of first-order hyperbolic partial differential equations (PDEs). These equations serve as a model for physical phenomena such as traffic flows, chemical reactors, and heat exchangers. We design controllers using a backstepping method, which has been recently developed for parabolic PDEs. With the integral transformation and boundary feedback the unstable PDE is converted into a “delay line” system which converges to zero in finite time. We then apply this procedure to finite-dimensional systems with actuator and sensor delays to recover a well-known infinite-dimensional controller (analog of the Smith predictor for unstable plants). We also show that the proposed method can be used for the boundary control of a Korteweg–de Vries-like third-order PDE. The designs are illustrated with simulations.