Boundary control, stabilization and zero-pole dynamics for a non-linear distributed parameter system

In this work we show that the now standard lumped non-linear enhancement of root-locus design still persists for a non-linear distributed parameter bo undary control system governed by a scalar viscous Burgers' equation. Namel y, we construct a proportional error boundary feedback control law and show that closed-loop trajectories tend to trajectories of the open-loop zero d ynamics as the gain parameters are increased to infinity. We also prove a r obust version of this result, valid for perturbations by an unknown disturb ance with arbitrary L-2 norm. For the controlled Burgers equation forced by a disturbance we prove that, for all initial data in L-2(0, 1), the closed -loop trajectories converge in L-2(0, 1), uniformly in t is an element of [ 0, T] and in H-1(0, 1), uniformly in t is an element of [t(0), T] for any t (0) > 0, to the trajectories of the corresponding perturbed zero dynamics. We have also extended these results to include the case when additional bou ndary controls are included in the closed-loop system. This provides a proo f of convergence of trajectories in case the zero dynamics is replaced by a non-homogeneous; Dirichlet boundary controlled Burgers' equation. As an ap plication of our convergence of trajectories results, we demonstrate that o ur boundary feedback control scheme provides a semiglobal exponential stabi lizing feedback law in L-2, H-1 and L-infinity for the open-loop system con sisting of Burgers' equation with Neumann boundary conditions and zero forc ing term. We also show that this result is robust in the sense that if the open-loop system is perturbed by a sufficiently small non-zero disturbance then the resulting closed-loop system is 'practically semiglobally stabiliz able' in L-2-norm. Copyright (C) 1999 John Wiley & Sons, Ltd.