BOUNDARY CONTROL LAWS FOR PARABOLIC PDES WITH VOLTERRA NONLINEARITIES—PART III: ANALYSIS

This is the final part of a three-part paper; in the first two parts we introduced a class of stabilizing boundary controllers for nonlinear 1-D parabolic PDEs and presented several examples. In this part, we derive bounds for the gain kernels of our nonlinear Volterra controllers, show how the series in the feedback laws converge, and establish the stability properties of the closed-loop system. We show that the state transformation is at least locally invertible. Using the inverse, we show L2 and H1 exponential stability and explicitly construct the exponentially decaying closed-loop solutions. We then illustrate the theoretical results on an analytically tractable example from the second part.