Nonlinear evolution equations and applications in optimal control theory
Doctoral thesis
English
OPEN
Plant, Andrew Thomas
This thesis is an attempt to tackle two related problems in in nonlinear functional analysis.\ud \ud The study of abstract evolution equations started in the early 1950's with the development of the theory of linear contraction semigroups and holomorphic semigroups. The power of the Dunford integral made the holomorphic theory the more attractive, and only in the middle 1960's was it realized that the contraction theory could easily be generalized to semigroups with dissipative nonlinear infinitesimal generators.\ud \ud Since then the corresponding theory for evolution operators has been greatly studied, Kato probably being the first to do so in 1967. A Holder type continuity assumption on the time dependence of the generators is common to all this work. It is the purpose of Chapters I and IV to weaken this condition to allow a certain amount of discontinuity in the time dependence. A bounded variation condition replaces Lipschitz continuity in Chapter I. A Riemann integrability condition replaces a continuity condition in Chapter IV. The original motivation to do this came from Control Theory where discontinuous controls play a major role.\ud \ud The second purpose of this thesis is to give a rigorous derivation of Pontryagin's Maximum Principle with fixed endpoint for nonlinear evolution operators in Banach space. Because the unit ball is not compact we replace Pontryagin's elegant use of the Browder Fixed Point Theorem by an abstract controllability condition which seems appropriate for the particular dissipative systems discussed earlier. We have to derive a first order variational theory for these systems 'from scratch'.\ud Finally we have had to show the 'perturbation cone' is convex, a trivial result in finite dimensions.\ud

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