Stability of the Turnpike Phenomenon in Discrete-Time Optimal Control Problems

114 Pages·2014·0.593 MB·English

by  Alexander J. Zaslavski

SpringerBriefs in Optimization
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Alexander J. Zaslavski
Stability of the Turnpike
Phenomenon in
Discrete-Time Optimal
Control Problems
2123
AlexanderJ.Zaslavski
DepartmentofMathematics
Technion-IsraelInstituteofTechn
Haifa
Israel
ISSN2190-8354 ISSN2191-575X(electronic)
ISBN978-3-319-08033-8 ISBN978-3-319-08034-5(eBook)
DOI10.1007/978-3-319-08034-5
SpringerChamHeidelbergNewYorkDordrechtLondon
LibraryofCongressControlNumber:2014943649
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Preface
The monograph is devoted to the study of the structure of approximate solutions
ofnonconvex(nonconcave)discrete-timeoptimalcontrolproblems.Itcontainsnew
resultsonpropertiesofapproximatesolutionswhichareindependentofthelength
of the interval, for all sufficiently large intervals. These results deal with the so-
calledturnpikepropertyofoptimalcontrolproblems.Thetermwasfirstcoinedby
P.Samuelsonin1948whenheshowedthatanefficientexpandingeconomywould
spendmostofthetimeinthevicinityofabalancedequilibriumpath(alsocalleda
von Neumann path). To have the turnpike property means, roughly speaking, that
the approximate solutions of the problems are determined mainly by the objective
functionandareessentiallyindependentofthechoiceofintervalandendpointcondi-
tions,exceptinregionsclosetotheendpoints.Nowitiswell-knownthattheturnpike
propertyisageneralphenomenonwhichholdsforlargeclassesofvariationaland
optimalcontrolproblems.UsingtheBairecategory(generic)approach,itwasshown
thattheturnpikepropertyholdsforageneric(typical)variationalproblem[45]and
for a generic optimal control problem [56].According to the generic approach we
saythatapropertyholdsforageneric(typical)elementofacompletemetricspace
(orthepropertyholdsgenerically)ifthesetofallelementsofthemetricspacepos-
sessing this property contains a G? everywhere dense subset of the metric space
which is a countable intersection of open everywhere dense sets. In [55] we were
interestedinindividual(non-generic)turnpikeresultsandinsufficientandnecessary
conditionsfortheturnpikephenomenoninthecalculusofvariations.Inourrecent
research [46-51, 54] we were are also interested in individual turnpike results but
fordiscrete-timeoptimalcontrolproblemswhich, inparticular, describeageneral
modelofeconomicdynamics.Fortheseproblemsweestablishedtheturnpikeprop-
ertyforapproximatesolutionswithasingleton-turnpikeandstudiedthestabilityof
theturnpikephenomenonundersmallperturbationsofobjectivefunctions.
In this book we continue to study the discrete-time optimal control problems
considered in [46-51, 54]. Some results of these works are discussed in Chap. 1.
In Chaps. 2 and 3 we show the stability of the turnpike phenomenon under small
perturbationsofobjectivefunctionsandundersmallperturbationsofcontrolmaps.
TheoptimalcontrolproblemswithoutdiscountingarestudiedinChap.2whilethe
discountcaseisconsideredinChap.3.InChap.4weestablishtheturnpikeproperty
v
vi Preface
anditsstabilityfordiscrete-timeproblemswithnonsingleton-turnpikes.Notethatthe
stabilityoftheturnpikepropertyiscrucialinpractice.Onereasonisthatinpractice
wedealwithaproblemwhichconsistsofaperturbationoftheproblemwewishto
consider.Anotherreasonisthatthecomputationsintroducenumericalerrors.
RishonLeZion AlexanderJ.Zaslavski
December30,2013
Contents
1 Introduction................................................... 1
1.1 TheTurnpikePhenomenon................................... 1
1.2 Discrete-TimeProblems..................................... 3
1.3 Examples ................................................. 6
2 OptimalControlProblemswithSingletonTurnpikes ............... 9
2.1 PreliminariesandStabilityResults ............................ 9
2.2 Extensions ................................................ 14
2.3 ThreeLemmata ............................................ 16
2.4 AuxiliaryResults........................................... 20
2.5 ProofsofTheorems2.2and2.3............................... 34
2.6 ProofsofTheorems2.4and2.5............................... 37
2.7 ProofofTheorem2.6 ....................................... 40
2.8 ProofofTheorems2.7and2.8................................ 43
3 OptimalControlProblemswithDiscounting ...................... 47
3.1 StabilityoftheTurnpikePhenomenon ......................... 47
3.2 AuxiliaryResults........................................... 50
3.3 ProofsofTheorems3.2and3.3............................... 57
3.4 ProofofTheorem3.4 ....................................... 61
4 OptimalControlProblemswithNonsingletonTurnpikes............ 65
4.1 Discrete-TimeOptimalControlSystems ....................... 65
4.2 TheTurnpikeProperty ...................................... 67
4.3 Preliminaries .............................................. 70
4.4 AuxiliaryResults........................................... 76
4.5 ProofofTheorem4.5 ....................................... 96
References........................................................ 105
Index ............................................................ 109
vii
List of Symbols
A,65
a(x),65
A,67
¯
A,67
B(M),65–69,79
B(x,r),9
Card,6,13,14,39,40,44
c(f),66
cl(E),66
C(M),65
c¯,3,10,17
d ,66,67
1
dist,66,68,70
¯
E,66
E(λ),12,13
H(f),68
(K,d),65
M,9,12,65
M ,47
0
M ,67,68
reg
r¯,5,11
Uf(q,y,z),70
U({f }T2−1,y,z),66
i i=T1
V ,67
f
(cid:2)w(cid:2),6
X,3
X ,4
M
ix
x ListofSymbols
x¯,4
Y({(cid:3) }T2−1,T ,T ),12
t t=T1 1 2
Y¯({(cid:3) }T2−1,T ,T ),12
t t=T1 1 2
zf,67
j
Z,65
Z ,65
p
Zq,65
p
γ(f),67,69
¯
λ,11
μ(f),67
ρ,1,3
ρ ,9
1
σ(w,T,x,y),10
σ({u }T2−1,{(cid:3) }T2−1,T ,T ),11
t t=T1 t t=T1 1 2
σ({u }T2−1,{(cid:3) }T2−1,T ,T ,x),11
t t=T1 t t=T1 1 2
σ({u }T2−1,{(cid:3) }T2−1,T ,T ,x,y),11
t t=T1 t t=T1 1 2
(cid:3)({x }∞ ),66
i i=0
ω({x }∞ ),66
i i=0