Global analysis of one-dimensional variational problems

Philip John Schrader

    Research output: ThesisDoctoral Thesis

    141 Downloads (Pure)

    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 languageEnglish
    QualificationDoctor of Philosophy
    Awarding Institution
    • The University of Western Australia
    Award date15 Jul 2016
    Publication statusUnpublished - 2016

    Fingerprint

    Global Analysis
    Variational Problem
    XPath
    Topological Properties
    Differentiable
    Cost Function
    Critical point
    Gradient
    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",
    author = "Schrader, {Philip John}",
    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

    AU - Schrader, Philip John

    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 -