# Global analysis of one-dimensional variational problems

Research output: ThesisDoctoral Thesis

### Abstract

[Truncated]

From the global analytical point of view a one-dimensionalvariational problem consists

in extremizing a differentiable action/cost function f: X →R, where X is an

infinite dimensional manifold of paths in a manifold M, overa subset Ω C X of

admissible paths, for example those satisfying someregularity conditions, boundary

conditions or other constraints. Thus a solution to thevariational problem is a

critical point of the restriction f|Ω.

A standard criterion for existence of critical points is thePalais-Smale condition.

If this condition is satisfied then the gradient flowassociated with f is well behaved,

and we are guaranteed not only existence of critical pointsbut also existence of a

minimum. Moreover it is then possible to relate the totalnumber of critical points

to topological properties of Ω

Original language English Doctor of Philosophy The University of Western Australia 15 Jul 2016 Unpublished - 2016

### Fingerprint

Global Analysis
Variational Problem
XPath
Topological Properties
Differentiable
Cost Function
Critical point
Restriction
Path
Subset

### Cite this

@phdthesis{00d99236fb92430babef500fda3fbdd2,
title = "Global analysis of one-dimensional variational problems",
abstract = "[Truncated] From the global analytical point of view a one-dimensionalvariational problem consistsin extremizing a differentiable action/cost function f: X →R, where X is aninfinite dimensional manifold of paths in a manifold M, overa subset Ω C X ofadmissible paths, for example those satisfying someregularity conditions, boundaryconditions or other constraints. Thus a solution to thevariational problem is acritical point of the restriction f|Ω.A standard criterion for existence of critical points is thePalais-Smale condition.If this condition is satisfied then the gradient flowassociated with f is well behaved,and we are guaranteed not only existence of critical pointsbut also existence of aminimum. Moreover it is then possible to relate the totalnumber of critical pointsto topological properties of Ω",
keywords = "Calculus of variations, Global analysis, Elastica, Morse theory, Geometric control, Riemannian cubics, Sub-Riemannian, Palais-Smale condition",
year = "2016",
language = "English",
school = "The University of Western Australia",

}

Schrader, PJ 2016, 'Global analysis of one-dimensional variational problems', Doctor of Philosophy, The University of Western Australia.

Global analysis of one-dimensional variational problems. / Schrader, Philip John.

2016.

Research output: ThesisDoctoral Thesis

TY - THES

T1 - Global analysis of one-dimensional variational problems

PY - 2016

Y1 - 2016

N2 - [Truncated] From the global analytical point of view a one-dimensionalvariational problem consistsin extremizing a differentiable action/cost function f: X →R, where X is aninfinite dimensional manifold of paths in a manifold M, overa subset Ω C X ofadmissible paths, for example those satisfying someregularity conditions, boundaryconditions or other constraints. Thus a solution to thevariational problem is acritical point of the restriction f|Ω.A standard criterion for existence of critical points is thePalais-Smale condition.If this condition is satisfied then the gradient flowassociated with f is well behaved,and we are guaranteed not only existence of critical pointsbut also existence of aminimum. Moreover it is then possible to relate the totalnumber of critical pointsto topological properties of Ω

AB - [Truncated] From the global analytical point of view a one-dimensionalvariational problem consistsin extremizing a differentiable action/cost function f: X →R, where X is aninfinite dimensional manifold of paths in a manifold M, overa subset Ω C X ofadmissible paths, for example those satisfying someregularity conditions, boundaryconditions or other constraints. Thus a solution to thevariational problem is acritical point of the restriction f|Ω.A standard criterion for existence of critical points is thePalais-Smale condition.If this condition is satisfied then the gradient flowassociated with f is well behaved,and we are guaranteed not only existence of critical pointsbut also existence of aminimum. Moreover it is then possible to relate the totalnumber of critical pointsto topological properties of Ω

KW - Calculus of variations

KW - Global analysis

KW - Elastica

KW - Morse theory

KW - Geometric control

KW - Riemannian cubics

KW - Sub-Riemannian

KW - Palais-Smale condition

M3 - Doctoral Thesis

ER -