Calculus of variations and optimal control theory: a concise introduction
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Princeton [u.a.]
Princeton Univ. Press
2012
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| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XV, 235 S. graph. Darst. 26 cm |
| ISBN: | 9780691151878 0691151873 |
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MARC
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| 100 | 1 | |a Liberzon, Daniel |d 1973- |e Verfasser |0 (DE-588)14097637X |4 aut | |
| 245 | 1 | 0 | |a Calculus of variations and optimal control theory |b a concise introduction |c Daniel Liberzon |
| 264 | 1 | |a Princeton [u.a.] |b Princeton Univ. Press |c 2012 | |
| 300 | |a XV, 235 S. |b graph. Darst. |c 26 cm | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
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Record in the Search Index
| _version_ | 1804148755198377984 |
|---|---|
| adam_text | Contents
Preface
xiii
1
Introduction
1
1.1
Optimal
control problem
1
1.2
Some background on finite-dimensional optimization
3
1.2.1
Unconstrained optimization
.....,.,..,..., 4
1.2.2
Constrained optimization
................. 11
1.3
Pre àew
of infinite-dimensional optimization
17
1.3.1
Function spaces, norms, and local minima
....... 18
1.3.2
First variation and first-order necessary condition
... 19
1.3.3
Second variation and second-order conditions
..... 21
1.3.4
Global minima and convex problems
.......... 23
1.4
Notes and references for Chapter
1 24
2
Calculus of Variations
26
2.1
Examples of variational problems
26
2.1.1
Dido s isoperimetric problem
.............. 26
2.1.2
Light reflection and refraction
.............. 2
Y
2.1.3
Catenary
......................... 28
2.1.4
Brachistochrone
..................... 30
2.2
Basic calculus of variations problem
32
2.2.1
Weak and strong
extrema
................ 33
2.3
First-order necessary conditions for weak
extrema
34
2.3.1 Euler-Lagrange
equation
.................
Зо
2.3.2
Historical remarks
................... . 39
2.3.3
Technical remarks
.................... 40
VII
viii CONTENTS
2.3.4
Two special cases
..................... 41
2.3.5
Variable-endpoint problems
............... 42
2.4
Hamiltonian formalism and mechanics
44
2.4.1
Hamilton s canonical equations
............. 45
2.4.2
Legendre transformation
................. 46
2.4.3
Principle of least action and conservation laws
.... 48
2.5
^ariational problems with constraints
51
2.5.1
Integral constraints
.................... 52
2.5.2
Non-integral constraints
................. 55
2.6
Second-order conditions
58
2.6.1
Legendre s necessary condition for a weak minimum
. 59
2.6.2
Sufficient condition for a weak minimum
........ 62
2.7
Notes and references for Chapter
2 68
3
From Calculus of Variations to Optimal Control
71
3.1
Necessary conditions for strong
extrema
71
3.1.1
Weierstrass-
Erdmann
corner conditions
........ 71
3.1.2
Weierstrass
excess function
............... 76
3.2
Calculus of variations versus optimal control
81
3.3
Optimal control problem formulation and assumptions
83
3.3.1
Control system
...................... 83
3.3.2
Cost functional
...................... 86
3.3.3
Target set
......................... 88
3.4
Variational approach to the fixed-time, free-endpoint problem
89
3.4.1
Preliminaries
....................... 89
3.4.2
First variation
...................... 92
3.4.3
Second variation
.................... . 95
3.4.4
Some comments
..................... 96
3.4.5
Critique of the variational approach and preview of
the maximum principle
................. 98
3.5
Notes and references for Chapter
3 100
CONTENTS
їх
4 The Maximum
Principle
102
4.1 Statement
of the maximum principle
102
4.1.1
Basic fixed-endpoint control problem
.......... 102
4.1.2
Basic variable-endp
oint
control problem
........ 104
4.2
Proof of the maximum principle
105
4.2.1
From
Lagrange
to Mayer form
............. 107
4.2.2
Temporal control perturbation
............. 109
4.2.3
Spatial control perturbation
............... 110
4.2.4
Variational equation
................... 112
4,2.
b
Terminal cone
....................... 115
4.2.6
Key topological lemma
.................. 117
4.2.7
Separating
hyperplane .................. 120
4.2.8
Adjoint equation
..................... 121
4.2.9
Properties of the
Hamüt
ordan
.............. 122
4.2.10
Transversality condition
................. 126
4.3
Discussion of the maximum principle
128
4.3.1
Changes of variables
................... 130
4.4
Time-optimal control problems
134
4.4.1
Example: double integrator
............... 135
4.4.2
Bang-bang principle for linear systems
......... 138
4.4.3
Nonlinear systems, singular controls, and Lie brackets
141
4.4.4
Fuller s problem
..................... 146
4.5
Existence of optimal controls
148
4.6
Notes and references for Chapter
4 153
Б
jL he Hamilton-
J
acobi-Belimaïi
Jülquation 156
ö.l Dynamic
programming and the HJB equation
156
5.1.1
Motivation: the discrete problem
............ 156
5.1.2
Principle of optimality
.................. 158
5.1.3
HJB equation
....................... 161
5.1.4
Sufficient condition for optimality
........... 165
5.1.5
Historical remarks
.................... 167
5.2
HJB equation versus the maximum principle
168
χ
CONTENTS
5.2.1
Example: nondifferentiable value function
....... 170
5.3
Viscosity solutions of the HJB equation
172
5.3.1
One-sided differentials
.................. 172
5.3.2
Viscosity solutions of PDEs
............... 174
5.3.3
HJB equation and the value function
.......... 176
5.4
Notes and references for Chapter
5 178
6
The Linear Quadratic Regulator
180
6.1
Finite-horizon LQR problem
180
6.1.1
Candidate optimal feedback law
............ 181
6.1.2
Riccati differential equation
............... 183
6.1.3
Value function and optimality
.............. 185
6.1.4
Global existence of solution for the RDE
. ....... 187
6.2
Infinite-horizon LQR problem
189
6.2.1
Existence and properties of the limit
.......... 190
6.2.2
Infinite-horizon problem and its solution
........ 193
6.2.3
Closed-loop stability
................... 194
6.2.4
Complete result and discussion
............. 196
6.3
Notes and references for Chapter
6 199
7
Advanced Topics
200
7.1
Maximum principle on manifolds
200
7.1.1
Differentiable manifolds
................. 201
7.1.2
Re-interpreting the maximum principle
........ 203
7.1.3
Symplectic geometry and Hamiltonian flows
...... 206
7.2
HJB equation, canonical equations, and characteristics
207
7.2.1
Method of characteristics
................ 208
7.2.2
Canonical equations as characteristics of the HJB
equation
..........................211
7.3
Riccati equations and inequalities in robust control
212
7.3.1
Ľ2
gain
.......................... 213
7.3.2
Tťoo
control problem
................... 216
7.3.3
Riccati inequalities and LMIs
.............. 219
7.4
Maximum principle for hybrid control systems
219
CONTENTS
xi
7.4.1
Hybrid
optimal
control problem
.............219
7.4.2
Hybrid
maximum
principle
...............221
7.4.3
Example: light reflection
.................222
7.5
Notes and references for Chapter
7 223
Bibliography
225
Index
231
|
| any_adam_object | 1 |
| author | Liberzon, Daniel 1973- |
| author_GND | (DE-588)14097637X |
| author_facet | Liberzon, Daniel 1973- |
| author_role | aut |
| author_sort | Liberzon, Daniel 1973- |
| author_variant | d l dl |
| building | Verbundindex |
| bvnumber | BV039820342 |
| classification_rvk | SK 880 SK 660 |
| classification_tum | MAT 490f |
| ctrlnum | (OCoLC)772627669 (DE-599)OBVAC08887030 |
| 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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| isbn | 9780691151878 0691151873 |
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| physical | XV, 235 S. graph. Darst. 26 cm |
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| spelling | Liberzon, Daniel 1973- Verfasser (DE-588)14097637X aut Calculus of variations and optimal control theory a concise introduction Daniel Liberzon Princeton [u.a.] Princeton Univ. Press 2012 XV, 235 S. graph. Darst. 26 cm txt rdacontent n rdamedia nc rdacarrier Optimale Kontrolle (DE-588)4121428-6 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Calculus of variations Control theory (DE-588)4123623-3 Lehrbuch gnd-content Variationsrechnung (DE-588)4062355-5 s Optimale Kontrolle (DE-588)4121428-6 s DE-604 Digitalisierung UB Passau - ADAM Catalogue Enrichment application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024680485&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Liberzon, Daniel 1973- Calculus of variations and optimal control theory a concise introduction Optimale Kontrolle (DE-588)4121428-6 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| subject_GND | (DE-588)4121428-6 (DE-588)4062355-5 (DE-588)4123623-3 |
| title | Calculus of variations and optimal control theory a concise introduction |
| title_auth | Calculus of variations and optimal control theory a concise introduction |
| title_exact_search | Calculus of variations and optimal control theory a concise introduction |
| title_full | Calculus of variations and optimal control theory a concise introduction Daniel Liberzon |
| title_fullStr | Calculus of variations and optimal control theory a concise introduction Daniel Liberzon |
| title_full_unstemmed | Calculus of variations and optimal control theory a concise introduction Daniel Liberzon |
| title_short | Calculus of variations and optimal control theory |
| title_sort | calculus of variations and optimal control theory a concise introduction |
| title_sub | a concise introduction |
| topic | Optimale Kontrolle (DE-588)4121428-6 gnd Variationsrechnung (DE-588)4062355-5 gnd |
| topic_facet | Optimale Kontrolle Variationsrechnung Lehrbuch |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=024680485&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
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