General Spectral Theory of Nonlinear Systems

The spectral theory of linear (time-invariant) systems is, of course, the most important aspect of control theory in classical feedback design, and historically this was the approach taken by control engineers until the introduction of state-space theory. The techniques developed in the past include Nyquist and Bode diagrams, pole assignment and root locus methods. Frequency domain methods are therefore extremely important, especially for the suppression of resonant vibrations in mechanical systems.

In this chapter we shall outline a general spectral theory for nonlinear systems. First, a generalised transform theory is developed which can generate Volterra series kernels directly using Schwartz’ kernel theorem, without the need of a (somewhat arbitrary) definition of multi-dimensional Laplace transforms. This theory can then be directly applied, by using the iteration scheme developed in this book to derive a general spectral theory for nonlinear systems. We shall then briefly show how to apply the same ideas to generalise exponential dichotomies and the Sacker-Sell spectrum to nonlinear systems. We shall assume a basic knowledge of functional analysis and distribution theory.