ABSTRACT
Certainly the most interesting and challenging problems in regulation of distributed parameter systems arise in applications to nonlinear systems. For finite-dimensional nonlinear systems one of the most important theoretical results in geometric regulation is the 1990 paper by C.I. Byrnes and A. Isidori [56] in which an application of center manifold theory was used to prove necessary and sufficient conditions for solvability of the state (and error) feedback regulator problem in terms of a set of regulator equations analogous to those given in (1.1) for linear systems. While there has been progress in extending the nonlinear theory to distributed parameters systems (cf. Theorem (4.1) below), at the time of writing of this book there is no complete analog of the results of Byrnes and Isidori due in large part to the technical complications that arise due to unbounded control input operators. As there are currently several research groups working towards such a theory there is little doubt that a comprehensive theory will emerge in the not-too-distant future. In this work we hope to encourage this research and demonstrate the utility of the geometric methods for nonlinear distributed parameters systems.
