Boundary feedback stabilization of a controlled viscous Burgers´ equation

Boundary feedback stabilization of a nonlinear control system governed by Burgers equation on a finite interval is considered. A zero-dynamics approach is used. The authors first introduce boundary control via a colocated sensor and actuator at the left end of the interval and apply a simple proportional error feedback law. In this case, they show that, for all positive values of the gain parameter k, the nonlinear closed-loop system is locally exponentially stable on the zero dynamics subspace. Applying root locus principles it follows that the margin of stability is monotone increasing with increasing k and converges to that obtained for the zero dynamics (i.e., infinite gain). Pointwise convergence of trajectories of the closed-loop Burgers system to those of the zero dynamics for initial data in the zero dynamics subspace is also proved. Also considered is an example with two boundary controls, a flux control at each end of the rod, and two sensors given as the temperature at each end of the rod