Functional analysis and aspects of non-linear control theory
This thesis studies the state reconstruction problem for a class of non-linear systems. This class is that of perturbed linear systems. The properties of the linear part are used to arrive at results for the complete system. Whilst this is a common technique in mathematics and physics its use in non-linear infinite dimensional systems theory has not been extensively investigated. The present work makes such an investigation with a view to indicating the successes, and limitations, of such a treatment. As to contribution, as far as the author is aware, many of the results are new both in precise statement and general approach. Chapter 1 introduces, and motivates, the formulation adopted. Chapter 2 provides some useful information on linear infinite dimensional control theory. Chapter 3 gives, subject to certain, perhaps restrictive, conditions, a rigorous statement, and proof, of the basic theorems. Here, as elsewhere, the standard fixed point results are used. Parts of this chapter are extracts from, as yet unpublished, joint work with A.J. Pritchard and M.D. Quinn. Chapter 4 relaxes some of the conditions in 3 and applies the same techniques to other areas. Chapter 5 surveys, in a formal fashion the more constructive, numerical aspects of the preceding results with a view to indicating directions for this important area of further research. It is concluded that the "perturbed linear" approach used here can give results that are both theoretically and computationally useful. The strength of the requirements placed on the linear part, however, indicates a challenging area for future investigations: a constructive approach to intrins1cally non-linear problems.