Volterra boundary control laws for a class of nonlinear parabolic partial differential equations

Abstract

The efforts on boundary control of general classes of nonlinear parabolic PDEs with nonlinearities of superlinear growth have so far only resulted in counterexamples-results that show that finite time blow up occurs for large initial conditions even for simple cases like quadratic nonlinearities, with or without control. In this paper we present results identifying a class of systems that is stabilizable. Our approach is a direct infinite dimensional extension of the feedback linearization/backstepping approach and employs Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. While the full detail of our general approach is left for a future publication without a page limit, in this paper we give an example with explicit solutions for the plant/controller pair, including an explicit construction of the inverse of the feedback linearizing Volterra transformation. This, in turn, allows us to explicitly construct the exponentially decaying closed loop solutions. We include also a numerical illustration, showing blow up in open loop, and stabilization for large initial conditions in closed loop.