Drop theorem, variational principle and their applications in locally convex spaces : a bornological approach
Abstract of thesis entitled
DROP THEOREM, VARIATIONAL PRINCIPLE
AND THEIR APPLICATIONS IN LOCALLY CONVEX
SPACES : A BORNOLOGICAL APPROACH
for the degree of Doctor of Philosophy at The University of Hong Kong
in August 2004
In the past decades, infinite-dimensional optimization problems have been of increasing interest; and necessary conditions for an optimum, being a very first step in locating cxtrema, are of primary significance.
Since Ekeland proposed his renowned variational principle (EVP) in 1974, it has been widely used in various branches of mathematics. One triumph was an elegant proof of Pontryagin's Maximum Principle (PMP). Though PMP is no longer valid in infinite-dimensional spaces, Fattorini and later, Li and Yong, successfully employed EVP and deduced variants of PMP in Banach spaces under extra conditions.
However, problems arise when one goes beyond Banach spaces. Due to the topological complexity, calculus is not 'well-defined' there: various notions of differentiability have been introduced but they were complicated enough to be applied. A breakthrough was made in 1980s by Frolicher, Kriegl and Michor who established a simple but useful calculus on a general class of infinite-dimensional spaces: convenient spaces. This calculus depends only on the homology of the underlying space where, however, only a very weak notion of completeness, namely c?-complete, is assumed and therefore the classical EVP is not applicable.
The first goal of this thesis is to establish a drop theorem in M-complete bornological vector spaces where topology plays no role. Thereafter, it is
possible to dedxice from it an Ekeland-type variational principle. Then the work is diverted into two directions.
On one hand, von Neumann homology induces a much weaker notion of completeness in locally convex space, namely weakly-M-complete, which is apparently a weaker notion than c?-completeness. As a result, generalized drop theorem and EVP are obtained under this notion of weakly-M-completeness. In addition, generalized Caristi-type fixed point theorem is proved and its equivalence with EVP is established. A sufficient condition for the normal solvability of an equation and those e-critical points of diffcrcntiable real-valued functions on convenient spaces are investigated.
On the other hand, Ekeland's method is extended to investigate control problems in convenient spaces. After establishing some preliminary results on Lipschitz functions and differential equations on a convenient space, an e-mmimum principle (e-MP) for a free-endpoint control problem on a convenient space is proved. It should be noted that this e-MP did not make any assumptions essentially concerning the boundedness of the control set.
With the aid of this e-MP, an attainability criterion of a control system to some target set could be derived provided that the 'normal' of the target set is 'non-degenerate' and is 'finite-dimensional', the precise meaning of which will be given in the context. This criterion is intimately related to PMP of a constraint endpoint control problem in Rn. This result serves two purposes: firstly, this type of results is rare in the literature and it is hoped that this could draw more attention to this approach. Secondly, it also sheds new light on deriving other necessary/sufficient conditions of optimum, say PMP for constraint endpoint control problem, of infinite-dimensional optimization problems.
School:The University of Hong Kong
School Location:China - Hong Kong SAR
Source Type:Master's Thesis
Keywords:variational principles convexity spaces bornological
Date of Publication:01/01/2005