Advances in mathematical economics. Vol.18

148 Pages·2014·1.338 MB·English

by  Shigeo Kusuoka, Toru Maruyama (ed.)

Shigeo Kusuoka
Toru Maruyama Editors
18
Volume
ManagingEditors
ShigeoKusuoka ToruMaruyama
TheUniversityofTokyo KeioUniversity
Tokyo,JAPAN Tokyo,JAPAN
Editors
RobertAnderson Jean-MichelGrandmont KunioKawamata
UniversityofCalifornia, CREST-CNRS KeioUniversity
Berkeley Malakoff,FRANCE Tokyo,JAPAN
Berkeley,U.S.A.
HiroshiMatano
NorimichiHirano
CharlesCastaing TheUniversityofTokyo
YokohamaNational
Universite´MontpellierII Tokyo,JAPAN
University
Montpellier,FRANCE
Yokohama,JAPAN
KazuoNishimura
FrancisH.Clarke
KyotoUniversity
Universite´deLyonI
TatsuroIchiishi Kyoto,JAPAN
Villeurbanne,FRANCE
TheOhioStateUniversity
EgbertDierker Ohio,U.S.A. MarcelK.Richter
UniversityofVienna UniversityofMinnesota
Vienna,AUSTRIA AlexanderD.Ioffe Minneapolis,U.S.A.
IsraelInstituteof
DarrellDuffie Technology YoichiroTakahashi
StanfordUniversity Haifa,ISRAEL TheUniversityofTokyo
Stanford,U.S.A. Tokyo,JAPAN
LawrenceC.Evans SeiichiIwamoto
AkiraYamazaki
UniversityofCalifornia, KyushuUniversity
HitotsubashiUniversity
Berkeley Fukuoka,JAPAN
Tokyo,JAPAN
Berkeley,U.S.A.
KazuyaKamiya MakotoYano
TakaoFujimoto
TheUniversityofTokyo KyotoUniversity
FukuokaUniversity
Tokyo,JAPAN Kyoto,JAPAN
Fukuoka,JAPAN
Aims and Scope. The project is to publish Advances in Mathematical
EconomicsonceayearundertheauspicesoftheResearchCenterforMath-
ematical Economics. It is designed to bring together those mathematicians
whoareseriouslyinterestedinobtainingnewchallengingstimulifromeco-
nomictheoriesandthoseeconomistswhoareseekingeffectivemathematical
toolsfortheirresearch.
The scope of Advances in Mathematical Economics includes, but is not
limitedto,thefollowingfields:
– Economictheoriesinvariousfieldsbasedonrigorousmathematicalrea-
soning.
– Mathematical methods (e.g., analysis, algebra, geometry, probability)
motivatedbyeconomictheories.
– Mathematicalresultsofpotentialrelevancetoeconomictheory.
– Historicalstudyofmathematicaleconomics.
Authorsareaskedtodeveloptheiroriginalresultsasfullyaspossibleand
alsotogiveaclear-cutexpositoryoverviewoftheproblemunderdiscussion.
Consequently,wewillalsoinvitearticleswhichmightbeconsideredtoolong
forpublicationinjournals.
Shigeo Kusuoka • Toru Maruyama
Editors
Advances in
Mathematical Economics
Volume 18
123
Editors
ShigeoKusuoka
Professor
GraduateSchoolofMathematicalSciences
TheUniversityofTokyo
3-8-1Komaba,Meguro-ku
Tokyo153-8914,Japan
ToruMaruyama
Professor
DepartmentofEconomics
KeioUniversity
2-15-45Mita,Minato-ku
Tokyo108-8345,Japan
ISSN1866-2226 1866-2234(electronic)
ISBN978-4-431-54833-1 978-4-431-54834-8(eBook)
DOI10.1007/978-4-431-54834-8
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Table of Contents
ResearchArticles
CharlesCastaing,ChristianeGodet-Thobie,LeXuanTruong,and
BiancaSatco
OptimalControlProblemsGovernedbyaSecondOrder
OrdinaryDifferentialEquationwithm-PointBoundary
Condition 1
ShigeoKusuokaandYusukeMorimoto
StochasticMeshMethodsforHo¨rmanderTypeDiffusion
Processes 61
SurveyArticle
AlexanderJ.Zaslavski
TurnpikePropertiesforNonconcaveProblems 101
Note
YuhkiHosoya
A Characterization of Quasi-concave Function in View
oftheIntegrabilityTheory 135
SubjectIndex 141
InstructionsforAuthors 143
v
Adv.Math.Econ.18,1–59(2014)
Optimal Control Problems Governed
by a Second Order Ordinary Differential
Equation with m-Point Boundary Condition
CharlesCastaing1,ChristianeGodet-Thobie2,LeXuanTruong3,
andBiancaSatco4
1 De´partementdeMathe´matiquesdeBrest,Case051,Universite´MontpellierII,
PlaceE.Bataillon,34095Montpelliercedex,France
(e-mail:[email protected])
2 LaboratoiredeMathe´matiquesdeBrest,CNRS-UMR6205,Universite´de
BretagneOccidentale,6,avenueLeGorgeu,CS93837,29238
BrestCedex3,France
(e-mail:[email protected])
3 DepartmentofMathematicsandStatistics,
UniversityofEconomicsofHoChiMinhCity,
59CNguyenDinhChieuStr.Dist.3,HoChiMinhCity,Vietnam
(e-mail:[email protected])
4 StefancelMareUniversityofSuceava,Suceava,Romania
(e-mail:[email protected])
Received:August22,2013
Revised:November20,2013
JELclassification:C61,C73
MathematicsSubjectClassification(2010):34A60,34B15,47H10,45N05
Abstract. Using a new Green type function we present a study of optimal control
problemwherethedynamicisgovernedbyasecondorderordinarydifferentialequa-
tion(SODE)withm-pointboundarycondition.
Keywords: Differentialgame,Greenfunction,m-Pointboundary,Optimalcontrol,
Pettis,Strategy,Sweepingprocess,Viscosity
S.KusuokaandT.Maruyama(eds.),AdvancesinMathematicalEconomics 1
Volume18,DOI:10.1007/978-4-431-54834-8 1,
(cid:2)c SpringerJapan2014
2 C.Castaingetal.
1. Introduction
Thepioneeringworksconcerningcontrolsystemsgovernedbysecondorder
ordinarydifferentialequations(SODE)withthreepointboundarycondition
aredeveloped in[2,16].Inthispaper wepresentsomenewapplications of
the Green function introduced in [11] to the study of viscosity problem in
OptimalControlTheorywherethedynamicisgovernedby(SODE)withm-
point boundary condition. The paper is organized as follows. In Sect.2 we
recall and summarize the properties of a new Green function (Lemma 2.1)
withapplicationtoasecondorderdifferentialequationwithm-pointbound-
aryconditioninaseparableBanachspaceEoftheform
⎧
⎪⎪⎨u¨τ,x,f(t)+γu˙τ,x,f(t)=f(t), t ∈[τ,1]
m(cid:6)−2
(SODE)
⎪⎪⎩uτ,x,f(τ)=x,uτ,x,f(1)= αiuτ,x,f(ηi).
i=1
Here γ is positive, f ∈ L1([0,1]), m is an integer number > 3, 0 ≤ τ <
E
η1 < η2 < ··· < ηm−2 < 1, αi ∈ R (i =1,2,...,m−2) satisfying the
condition
m(cid:6)−2 m(cid:6)−2
α −1+exp(−γ(1−τ))− α exp(−γ(η −τ))(cid:5)=0 (1.1.1)
i i i
i=1 i=1
andu isthetrajectoryW2,1([τ,1])-solutionto(SODE)associatedwith
τ,x,f E
f ∈L1([0,1])startingatthepointx ∈Eattimeτ ∈[0,1[.ByLemma2.1,
E
u andu˙ arerepresented,respectively,by
τ,x,f τ,x,f
⎧ (cid:7)
⎪⎪⎨u (t)=e (t)+ 1G (t,s)f(s)ds, ∀t ∈[τ,1]
τ,x,f τ,x τ
(cid:7)0
⎪⎪⎩u˙ (t)=e˙ (t)+ 1 ∂Gτ(t,s)f(s)ds, ∀t ∈[τ,1]
τ,x,f τ,x
∂t
0
whereG istheGreenfunctiondefinedinLemma2.1with
τ
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨eτ,x(t)=x+A(cid:8)τ(1−m(cid:6)−m(cid:6)i=2−12α(cid:9)i)(1−exp(−γ(t −τ)))x, ∀t ∈[τ,1]
e˙ (t)=γA 1− α exp(−γ(t −τ))x, ∀t ∈[τ,1]
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩Aττ,x=(cid:8)m(cid:6)−2ατi −1+ie=x1p(−i γ(1−τ))−m(cid:6)−2αiexp(−γ(ηi −τ))(cid:9)−1.
i=1 i=1
OptimalControlProblemsGovernedbyaSecondOrderOrdinary... 3
Westressthatbothexistenceanduniquenessandtheintegralrepresentation
formulas of solution and itsderivative for (SODE) viathe new Green func-
tion are of importance of this work. Indeed this allows to treat several new
applications to optimal control problems and also some viscosity solutions
for the value function governed by (SODE) with m-point boundary condi-
tion.InSect.3,wetreatanoptimalcontrolproblemgovernedby(SODE)in
aseparableBanachspace
⎧
⎪⎪⎨u¨f(t)+γu˙f(t)=f(t), f ∈S(cid:7)1
(SODE) m(cid:6)−2
(cid:7)⎪⎪⎩uf(0)=x, uf(1)= αiuf(ηi)
i=1
where (cid:7) is a measurable and integrably bounded convex compact valued
mapping and S1 is the set of all integrable selections of Γ. We show the
(cid:7)
compactness of the solution set and the existence of optimal control for the
problem ⎧
⎪⎪⎨u¨f(t)+γu˙f(t)=f(t), f ∈S(cid:7)1
m(cid:6)−2
⎪⎪⎩uf(0)=x, uf(1)= αiuf(ηi),
i=1
(cid:7)
1
inf J(t,u (t),u˙ (t),u¨ (t))dt.
f f f
f∈S1 0
(cid:7)
These results lead naturally to the problem of viscosity for the value func-
tion associated with this class of (SODE) which is presented in Sect.4. In
Sect.5 we deal with a class of (SODE) with Pettis integrable second mem-
ber. Existence and compactness of the solution set are also provided. Open
problemsconcerningdifferentialgamegovernedby(SODE)and(ODE)with
strategies are given in Sect.6. We finish the paper by providing an applica-
tiontothedynamicprogrammingprinciple(DPP)andviscositypropertyfor
thevaluefunctionassociatedwithasweepingprocessrelatedtoamodelin
MathematicalEconomics[25].
2. ExistenceandUniqueness
∗
Let E be a separable Banach space. We denote by E the topological dual
ofE;B istheclosedunitballofE;L([0,1])istheσ algebraofLebesgue
E
measurablesetson[0,1];λ=dt istheLebesguemeasureon[0,1];B(E)is
theσ algebra of Borel subsets ofE. By L1([0,1]),we denote the space of
E
all Lebesgue–Bochner integrable E-valued functions defined on [0,1]. Let
4 C.Castaingetal.
C ([0,1])betheBanach spaceofallcontinuous functions u : [0,1] → E
E
endowed with the sup-norm and let C1([0,1]) be the Banach space of all
E
functionsu∈C ([0,1])withcontinuousderivative,endowedwiththenorm
E
(cid:10) (cid:11)
max max (cid:9)u(t)(cid:9), max (cid:9)u˙(t)(cid:9) .
t∈[0,1] t∈[0,1]
We also denote W2,1([0,1]) the space of all continuous functions in
E
C ([0,1]) such that their first derivatives are continuous and their second
E
weakderivativesbelongtoL1([0,1]).
E
WerecallandsummarizeanewGreentypefunctiongivenin[11]thatis
akeyingredientinthestatementoftheproblemsunderconsideration.
Lemma2.1.Let0≤τ <η1 <η2 <···<ηm−2 <1,γ >0,m>3bean
integernumber,andα ∈R(i =1,...,m−2)satisfyingthecondition
i
m(cid:6)−2 m(cid:6)−2
α −1+exp(−γ(1−τ))− α exp(−γ(η −τ))(cid:5)=0. (1.1.1)
i i i
i=1 i=1
LetE beaseparableBanachspaceandletG : [τ,1]×[τ,1] → Rbethe
τ
functiondefinedby
⎧
⎨ 1
(1−exp(−γ(t −s))),τ ≤s ≤t ≤1
Gτ(t,s)=⎩γ
0, τ ≤t <s ≤1
A
+ τ (1−exp(−γ(t −τ)))φ (s), (2.1)
τ
γ
where
⎧
⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨1−exp(−γ(1−s))−mm(cid:6)(cid:6)i=−−122αi(1−exp(−γ(ηi−s))),τ ≤s<η1
φτ(s)=⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩1..−..e.x.p(−γ(1−s))− i=2αi(1−exp(−γ(ηi−s))),η1≤s≤η2
1−exp(−γ(1−s)), ηm−2≤s≤1,
(2.2)