NONLINEAR CONTROL OF PDES: ARE FEEDBACK LINEARIZATION AND GEOMETRIC METHODS APPLICABLE?

Feedback linearization is a natural first step in developing nonlinear control design methods for nonlinear PDEs. Feedback linearization procedures for nonlinear ODEs recursively absorb all the plant nonlinearities into a feedback transformation. The resulting transformation often involves nonlinearities of much higher growth than the plant nonlinearities. For example, systems with n states and only quadratic nonlinearities lead to feedback linearizing controllers of polynomial power up to n +1. Intuitively, one would worry that an infinite-step feedback linearization procedure may result in polynomial powers that go to infinity. We present a framework developed over the last eight years in which a boundary control design for nonlinear parabolic PDEs has a well defined limit. This approach opens the door for future possible developments of differential geometric tests of linearizability and controllability for nonlinear PDEs and for generalizations to various other classes of PDEs, including nonlinear Navier-Stokes models of fluid turbulence. As a starting point in this direction, we review the current status of the Lyapunov stabilization techniques for the linearized Navier-Stokes and magnetohydrodynamic PDEs.