Functional analysis, calculus of variations and optimal control:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
London
Springer
[2013]
|
| Series: | Graduate texts in mathematics
264 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis Klappentext |
| Physical Description: | xiv, 591 Seiten Illustrationen, Diagramme |
| ISBN: | 9781447148197 9781447162100 |
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MARC
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Record in the Search Index
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| adam_text | Contents
Part I Functional Analysis
1
Normed Spaces
................................................ 3
1.1
Basic definitions
........................................... 3
1.2
Linear mappings
........................................... 9
1.3
The dual space
............................................. 15
1.4
Derivatives, tangents, and normals
............................ 19
2
Convex sets and functions
....................................... 27
2.1
Properties of convex sets
.................................... 27
2.2
Extended-valued functions, semicontinuity
..................... 30
2.3
Convex functions
........................................... 32
2.4
Separation of convex sets
.................................... 41
3
Weak topologies
................................................ 47
3.1
Induced topologies
......................................... 47
3.2
The weak topology of a normed space
......................... 51
3.3
The weak* topology
........................................ 53
3.4
Separable spaces
........................................... 56
4
Convex analysis
................................................ 59
4.1
Subdifferential calculus
..................................... 59
4.2
Conjugate functions
........................................ 67
4.3
Polarity
................................................... 71
4.4
The minimax theorem
....................................... 73
5
Banach spaces
................................................. 75
5.1
Completeness of normed spaces
.............................. 75
5.2
Perturbed minimization
..................................... 82
5.3
Open mappings and surjectivity
............................... 87
5.4
Metric regularity
........................................... 90
5.5
Reflexive spaces and weak compactness
........................ 96
xii
Contents
6
Lebesgue spaces
................................................105
6.1
Uniform convexity and duality
...............................105
6.2
Measurable multifunctions
...................................114
6.3
Integral functional and semicontinuity
........................121
6.4
Weak sequential closures
....................................128
7
Hilbert spaces
.................................................133
7.1
Basic properties
............................................134
7.2
A smooth minimization principle
.............................140
7.3
The proximal subdifferential
.................................144
7.4
Consequences of proximal density
............................151
8
Additional exercises for Part I
...................................157
Part
Π
Optimization and Nonsmooth Analysis
9
Optimization and multipliers
....................................173
9.1
The multiplier rule
..........................................175
9.2
The convex case
............................................182
9.3
Convex duality
.............................................187
10
Generalized gradients
..........................................193
10.1
Definition and basic properties
...............................194
10.2
Calculus of generalized gradients
.............................199
10.3
Tangents and normals
.......................................210
10.4
A nonsmooth multiplier rule
.................................221
11
Proximal analysis
..............................................227
11.1
Proximal calculus
..........................................227
11.2
Proximal geometry
.........................................240
11.3
A proximal multiplier rule
...................................246
11.4
Dini
and viscosity subdifferentials
............................251
12
Invariance
and monotonicity
....................................255
12.1
Weak
invariance
............................................256
12.2
Weakly decreasing systems
..................................264
12.3
Strong
invariance
...........................................267
13
Additional exercises for Part II
..................................273
Part
Ш
Calculus of Variations
14
The classical theory
............................................287
14.1
Necessary conditions
.......................................289
14.2
Conjugate points
...........................................294
14.3
Two variants of the basic problem
.............................302
Contents xiü
15 Nonsmooth
extremals...........................................
307
15.1 The integral Euler
equation
..................................308
15.2
Regularity of Lipschitz solutions
..............................312
15.3
Sufficiency by convexity
.....................................314
15.4
The
Weierstrass
necessary condition
...........................317
16
Absolutely continuous solutions
..................................319
16.1
Tonelli s theorem and the direct method
........................321
16.2
Regularity via growth conditions
.............................326
16.3
Autonomous Lagrangians
....................................330
17
The multiplier rule
.............................................335
17.1
A classic multiplier rule
.....................................336
17.2
A modern multiplier rule
....................................338
17.3
The isoperimetric problem
...................................344
18
Nonsmooth Lagrangians
........................................347
18.1
The Lipschitz problem of Bolza
..............................347
18.2
Proof of Theorem
18.1......................................351
18.3
Sufficient conditions by convexity
.............................360
18.4
Generalized Tonelli-Morrey conditions
........................363
19
Hamilton- Jacobi methods
.......................................367
19.1
Verification functions
.......................................367
19.2
The logarithmic Sobolev inequality
............................376
19.3
The Hamilton-Jacobi equation
................................379
19.4
Proof of Theorem
19.11.....................................385
20
Multiple integrals
..............................................391
20.1
The classical context
........................................392
20.2
Lipschitz solutions
.........................................394
20.3
Hilbert-Haar theory
.........................................398
20.4
Solutions in Sobolev space
...................................407
21
Additional exercises for Part III
.................................415
Part IV Optimal
Contro/
22
Necessary conditions
...........................................435
22.1
The maximum principle
.....................................438
22.2
A problem
affine in
the control
...............................445
22.3
Problems with variable time
..................................449
22.4
Unbounded control sets
.....................................454
22.5
A hybrid maximum principle
.................................457
22.6
The extended maximum principle
.............................463
xiv Contents
23
Existence
and regularity
........................................473
23.1
Relaxed trajectories
.........................................473
23.2
Three existence theorems
....................................478
23.3
Regularity of optimal controls
................................486
24
Inductive methods
..............................................491
24.1
Sufficiency by the maximum principle
.........................491
24.2
Verification functions in control
...............................494
24.3
Use of the Hamilton-Jacobi equation
..........................500
25
Differential inclusions
..........................................503
25.1
A theorem for Lipschitz multifunctions
........................504
25.2
Proof of the extended maximum principle
......................514
25.3
Stratified necessary conditions
................................520
25.4
The multiplier rule and mixed constraints
......................535
26
Additional exercises for Part IV
..................................545
Notes, solutions, and hints
...........................................565
References
.........................................................583
Index
.............................................................585
Francis
Clarke
Functional Analysis, Calculus of Variations and Optimal Control
Functional analysis owes much of its early impetus to problems that arise in the calculus
of variations. In turn, the methods developed there have been applied to optimal control,
an area that also requires new tools, such as nonsmooth analysis. This self-contained
textbook gives a complete course on all these topics. It is written by a leading specialist
who is also a noted expositor.
This book provides a thorough introduction to functional analysis and includes many
novel elements as well as the standard topics. A short course on nonsmooth analysis
and geometry completes the first half of the book whilst the second half concerns the
calculus of variations and optimal control. The author provides a comprehensive course
on these subjects, from their inception through to the present. A notable feature is the
inclusion of recent, unifying developments on regularity, multiplier rules, and the Pon-
tryagin maximum principle, which appear here for the first time in a textbook. Other
major themes include existence and Hamilton-Jacobi methods.
The many substantial examples, and the more than three hundred exercises, treat such
topics as viscosity solutions, nonsmooth Lagrangians, the logarithmic Sobolev inequality,
periodic trajectories, and systems theory. They also touch lightly upon several fields of
application: mechanics, economics, resources, finance, control engineering.
Functional AnalysL·, Calculus of Variations and Optimal Control is intended to support
several different courses at the first-year or second-year graduate level, on functional
analysis, on the calculus of variations and optimal control, or on some combination.
For this reason, it has been organized with customization in mind. The text also has
considerable value as a reference. Besides its advanced results in the calculus of varia¬
tions and optimal control, its polished presentation of certain other topics (for example
convex analysis, measurable selections, metric regularity, and nonsmooth analysis) will
be appreciated by researchers in these and related fields.
Mathematics
ISBN
978-1-4471-4819-7
9
81447 148197
►
spnngercom
|
| any_adam_object | 1 |
| author | Clarke, Francis H. 1948- |
| author_GND | (DE-588)173744710 |
| author_facet | Clarke, Francis H. 1948- |
| author_role | aut |
| author_sort | Clarke, Francis H. 1948- |
| author_variant | f h c fh fhc |
| building | Verbundindex |
| bvnumber | BV040799816 |
| classification_rvk | SK 600 |
| classification_tum | MAT 496 |
| ctrlnum | (OCoLC)830898760 (DE-599)GBV737269413 |
| dewey-full | 515.64 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.64 |
| dewey-search | 515.64 |
| dewey-sort | 3515.64 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| institution | BVB |
| isbn | 9781447148197 9781447162100 |
| language | English |
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| spelling | Clarke, Francis H. 1948- Verfasser (DE-588)173744710 aut Functional analysis, calculus of variations and optimal control Francis Clarke London Springer [2013] © 2013 xiv, 591 Seiten Illustrationen, Diagramme txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 264 Funktionalanalysis (DE-588)4018916-8 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Optimale Kontrolle (DE-588)4121428-6 gnd rswk-swf Funktionalanalysis (DE-588)4018916-8 s Variationsrechnung (DE-588)4062355-5 s Optimale Kontrolle (DE-588)4121428-6 s DE-604 Erscheint auch als Online-Ausgabe 978-1-4471-4820-3 Graduate texts in mathematics 264 (DE-604)BV000000067 264 Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025779941&sequence=000003&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis Digitalisierung UB Bayreuth application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=025779941&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Clarke, Francis H. 1948- Functional analysis, calculus of variations and optimal control Graduate texts in mathematics Funktionalanalysis (DE-588)4018916-8 gnd Variationsrechnung (DE-588)4062355-5 gnd Optimale Kontrolle (DE-588)4121428-6 gnd |
| subject_GND | (DE-588)4018916-8 (DE-588)4062355-5 (DE-588)4121428-6 |
| title | Functional analysis, calculus of variations and optimal control |
| title_auth | Functional analysis, calculus of variations and optimal control |
| title_exact_search | Functional analysis, calculus of variations and optimal control |
| title_full | Functional analysis, calculus of variations and optimal control Francis Clarke |
| title_fullStr | Functional analysis, calculus of variations and optimal control Francis Clarke |
| title_full_unstemmed | Functional analysis, calculus of variations and optimal control Francis Clarke |
| title_short | Functional analysis, calculus of variations and optimal control |
| title_sort | functional analysis calculus of variations and optimal control |
| topic | Funktionalanalysis (DE-588)4018916-8 gnd Variationsrechnung (DE-588)4062355-5 gnd Optimale Kontrolle (DE-588)4121428-6 gnd |
| topic_facet | Funktionalanalysis Variationsrechnung Optimale Kontrolle |
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