Time-Varying Vector Fields and Their Flows

125 Pages·2014·1.821 MB·English

by  Saber Jafarpour, Andrew D. Lewis (auth.)

SPRINGER BRIEFS IN MATHEMATICS
Saber Jafarpour
Andrew D. Lewis
Time-Varying
Vector Fields and
Their Flows
123
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Saber Jafarpour Andrew D. Lewis
(cid:129)
Time-Varying Vector Fields
and Their Flows
123
SaberJafarpour AndrewD.Lewis
DepartmentofMathematicsandStatistics DepartmentofMathematicsandStatistics
Queen’sUniversity Queen’sUniversity
Kingston,ON,Canada Kingston,ON,Canada
ISSN2191-8198 ISSN2191-8201(electronic)
ISBN978-3-319-10138-5 ISBN978-3-319-10139-2(eBook)
DOI10.1007/978-3-319-10139-2
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014952196
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©TheAuthors2014
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Preface
In this monograph, time-varying vector fields on manifolds are considered
with measurable time dependence and with varying degrees of regularity in
state;e.g.,Lipschitz,finitelydifferentiable,smooth,holomorphic,andrealanalytic.
Classesofsuchvectorfieldsaredescribedforwhichtheregularityoftheflow’sde-
pendenceoninitialconditionmatchestheregularityofthevectorfield.Whileresults
of this sort are known for continuousdependence,some results exist for differen-
tiabledependence,andsmoothdependenceseemstobeapartofthefolklore,such
results have certainly not been established in the importantreal analytic case. We
treat all regularity classes, including the real analytic case. Moreover,it is shown
that the way in which one characterises these classes of vector fields corresponds
exactly to the vector fields, thought of as being vector field valued functions of
time,havingmeasurabilityandintegrabilitypropertiesassociatedwith appropriate
topologies.Forthisreason,asubstantialpartofthedevelopmentisconcernedwith
descriptionsoftheseappropriatetopologies.Tothisend,geometricdescriptionsare
providedoflocallyconvextopologiesforLipschitz,finitelydifferentiable,smooth,
holomorphic,and real analytic sections of vector bundles. In all but the real ana-
lyticcase,thesetopologiesareclassicallyknown.Thedescriptiongivenforthereal
analytictopologyisnew.Itisthisdescriptionthatallows,forthefirsttime,acharac-
terisationofthosetime-varyingrealanalyticvectorfieldswhoseflowsdependreal
analyticallyoninitialcondition.
ThisresearchwasfundedinpartbyagrantfromtheNaturalSciencesandEngi-
neeringResearchCouncilofCanada.ThesecondauthorwasaVisitingProfessorin
the Departmentof Mathematicsat University of Hawaii, Manoa,when the mono-
graph was written, and would like to acknowledge the hospitality of the depart-
ment,particularlythatofMoniqueChybaandGeorgeWilkens.Thesecondauthor
wouldalsoliketothankhisdepartmentalcolleagueMikeRothfornumeroususeful
conversationsoverthe years. While conversationswith Mike did notlead directly
v
vi Preface
to results in this monograph,Mike’s willingness to chat about complex geometry
and to answer ill-informed questions was always appreciated and ultimately very
helpful.
Kingston,ON,Canada SaberJafarpour
Kingston,ON,Canada AndrewD.Lewis
Contents
1 Introduction................................................... 1
1.1 Motivation ................................................ 1
1.2 AnOutlineoftheMonograph ................................ 3
1.3 Notation,ConventionsandBackground ........................ 4
References..................................................... 11
2 FibreMetricsforJetBundles.................................... 15
2.1 ADecompositionfortheJetBundlesofaVectorBundle.......... 15
2.2 FibreMetricsUsingJetBundleDecompositions................. 24
2.3 RealAnalyticConnections................................... 25
References..................................................... 43
3 The Compact-Open Topologies for the Spaces of Finitely
Differentiable,Lipschitz,andSmoothVectorFields ................ 45
3.1 GeneralSmoothVectorBundles .............................. 46
∞
3.2 PropertiesoftheCO -Topology.............................. 46
3.3 TheWeak-L TopologyforSmoothVectorFields ............... 48
3.4 TopologiesforFinitelyDifferentiableVectorFields .............. 53
3.5 TopologiesforLipschitzVectorFields ......................... 55
References..................................................... 63
4 TheCOhol-TopologyfortheSpaceofHolomorphicVectorFields .... 65
4.1 GeneralHolomorphicVectorBundles ......................... 65
4.2 PropertiesoftheCOhol-Topology ............................. 68
4.3 TheWeak-L TopologyforHolomorphicVectorFields........... 68
References..................................................... 71
ω
5 TheC -TopologyfortheSpaceofRealAnalyticVectorFields....... 73
5.1 ANaturalDirectLimitTopology ............................. 74
5.1.1 Complexifications.................................... 74
vii
viii Contents
5.1.2 GermsofHolomorphicSectionsoverSubsetsofaReal
AnalyticManifold ................................... 75
5.1.3 TheDirectLimitTopology ............................ 77
5.2 TopologiesforGermsofHolomorphicFunctionsAboutCompact
Sets ...................................................... 77
5.2.1 The Direct Limit Topologyfor the Space of Germs
AboutaCompactSet................................. 78
5.2.2 A Weighted Direct Limit Topology for Sections
ofBundlesofInfiniteJets ............................. 79
5.2.3 SeminormsfortheTopologyofSpacesofHolomorphic
Germs ............................................. 82
5.2.4 AnInverseLimitTopologyfortheSpaceofRealAnalytic
Sections............................................ 85
ω
5.3 PropertiesoftheC -Topology................................ 85
5.4 TheWeak-L TopologyforRealAnalyticVectorFields .......... 87
References..................................................... 91
6 Time-VaryingVectorFields ..................................... 93
6.1 TheSmoothCase .......................................... 94
6.2 TheFinitelyDifferentiableorLipschitzCase ...................103
6.3 TheHolomorphicCase......................................105
6.4 TheRealAnalyticCase .....................................110
6.5 MixingRegularityHypotheses ...............................118
References.....................................................118
Chapter 1
Introduction
1.1 Motivation
Inthismonographweconsidervectorfieldswithmeasurabletimedependence.Such
vectorfieldsarenotwellstudiedinthedifferentialequationanddynamicalsystem
literature,butareimportantincontroltheory.Thereareatleasttworeasonsforthis:
1. In the theory of optimal control, it can often happen that optimal trajectories
correspondto controls that switch infinitely often in a finite duration of time.
ThisseemstohavefirstbeenobservedbyFuller[16],andhassincebeenstudied
bymanyauthors.Adetaileddiscussionofthesefacetsofoptimalcontroltheory
canbefoundin[41].
2. In the study of controllability, it is often necessary to use complex control
variations where there are an increasing number of switches in a finite time.
This was noticed by Kawski [28] and examined in detail by Agrachev and
Gamkrelidze[3].
In any event, vector fields with measurable time dependence do not have as ext-
ensivea basictheoryas,say,time-independentvectorfields. Oneplacewherethis
especially holds true is where regularity of flows with respect to initial condi-
tionsisconcerned.Fortime-independentvectorfields,thebasicregularitytheorems
are well established for all the standard degrees of regularity [9], including real
analyticity, e.g., [38, Proposition C.3.12]. In the time-varying case, the standard
Carathe´odory existence and uniqueness theorem for ordinary differential equa-
tions depending measurably on time is a part of classical text treatments, e.g.,
[9, Theorem 2.2.1]. Continuity with respect to initial condition in this setting is
notconsistentlymentioned,e.g.,itisnotprovedbyCoddingtonandLevinson[9].
Thiscontinuityisproved,forexample,bySontag[38,Theorem55].However,con-
ditionsfordifferentiabilityofflowsbecomeextremelyhardtocomebyinthecaseof
measurabletimedependence.Thesituationhere,however,isdealtwithcomprehen-
sivelyin[37].Thedifferentiablehypothesesareeasilyextendedtoanyfinitedegree
of differentiability. For smooth dependence on initial conditions with measurable
time dependence, we are not aware of the desired result being stated and proved
©TheAuthors2014 1
S.Jafarpour,A.D.Lewis,Time-VaryingVectorFieldsandTheirFlows,
SpringerBriefsinMathematics,DOI10.1007/978-3-319-10139-2 1