Boundary control for nonlinear parabolic PDEs by Volterra feedback linearization

Abstract

Boundary control of nonlinear parabolic PDEs is an open problem with applications that include fluids, thermal, chemically-reacting, and plasma systems. In this paper we present stabilizing control designs for a broad class of nonlinear parabolic PDEs in 1-D. Our approach is a direct infinite dimensional extension of the finite-dimensional feedback linearization/backstepping approaches and employs spatial Volterra series nonlinear operators both in the transformation to a stable linear PDE and in the feedback law. The control law design consists of solving a recursive sequence of linear hyperbolyc PDEs for the gain kernels of the spatial Volterra nonlinear control operator. These PDEs evolve on domains T <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> of increasing dimensions n+1 and with a domain shape in the form of a “hyper-pyramid,” 0 ≤ ξ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n</sub> ≤ ξ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">n-1</sub> ... ≤ ξ <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sub> ≤ x ≤ 1. We illustrate our design method and the class of tractable nonlinear plants with several examples. One of the examples is analytical, while in the remaining two examples the controller is numerically approximated. For all the examples we include simulations, showing kernel computations, plant blow up in open loop, and stabilization for large initial conditions in closed loop.