Numerical methods for controlled stochastic delay systems:
Gespeichert in:
| 1. Verfasser: | |
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| Format: | Buch |
| Sprache: | English |
| Veröffentlicht: |
Boston [u.a.]
Birkhäuser
2008
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| Schriftenreihe: | Systems & control: foundations & applications
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| Online-Zugang: | Inhaltstext Inhaltsverzeichnis Inhaltsverzeichnis Klappentext |
| Beschreibung: | XIX, 281 S. graph. Darst. |
| ISBN: | 9780817646219 |
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MARC
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| 245 | 1 | 0 | |a Numerical methods for controlled stochastic delay systems |c Harold J. Kushner |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2008 | |
| 300 | |a XIX, 281 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 490 | 0 | |a Systems & control: foundations & applications | |
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Datensatz im Suchindex
| _version_ | 1805091332054581248 |
|---|---|
| adam_text |
Contents
Preface
. xi
Examples and Introduction
. 1
1.0
Outline of the Chapter
. 1
1.1
An Introductory Example: Controlling the Temperature of a
Fluid Flow
. 1
1.2
An Example from Internet Regulation
. 5
1.3
Models With Finite-Dimensional Equivalences
. 10
Weak Convergence and Martingales
. 13
2.0
Outline of the Chapter
. 13
2.1
Weak Convergence
. 14
2.1.1
Basic Theorems of Weak Convergence
. 15
2.1.2
The Function Spaces D(S; I)
. 17
2.2
Martingales and the Martingale Method
. 18
2.2.1
Martingales
. 18
2.2.2
Verifying That a Process Is a Martingale
. 20
Stochastic Delay Equations: Models
. 23
3.0
Outline of the Chapter
. 23
3.1
The System Model: Boundary Absorption
. 25
3.2
Reflecting Diffusions
. 30
3.2.1
The Reflected Diffusion
. 30
3.2.2
Delayed Control, Reflection Term, and/or Wiener
Process
. 35
3.2.3
Neutral Equations
. 37
3.2.4
Controlled Variance and Jumps
. 37
3.3
The Girsanov Transformation
. 38
3.4
Cost Functions
. 40
3.5
Existence of an Optimal Control
. 43
Contents
3.5.1
Reflecting or Absorbing Boundary: Discounted Cost
. 43
3.6
Singular and Impulsive Controls
. 50
3.6.1
Singular Controls
. 50
3.6.2
Definition of and Existence of Solutions
. 51
3.6.3
Existence of an Optimal Control
. 56
3.6.4
Impulsive Controls
. 58
Approximations to the Dynamical Models
. 61
4.0
Outline of the Chapter
. 61
4.1
Approximations of the Dynamical Systems
. 63
4.1.1
A Basic Approximation
. 63
4.2
Approximations by Time-Varying Delays
. 69
4.2.1
Discretized Delays
. 69
4.2.2
Periodic Delays
. 71
4.2.3
Randomly Varying Delays
. 73
4.2.4
Periodic-Erlang Delays
. 75
4.2.5
Convergence of Costs and Existence of Optimal Controls
77
4.2.6
Differential Operator for the Periodic-Erlang
Approximation
. 78
4.3
Simulations Illustrating the Model Approximations
. 78
4.3.1
Simulations Based on the Periodic Approximation
. 78
4.3.2
Simulations Based on the Periodic-Erlang Approximation
82
4.4
Approximations: Path and Control Delayed
. 84
4.5
Singular Controls
. 90
4.6
Rapidly Varying Delays
. 92
The Ergodic Cost Problem
. 97
5.0
Outline of the Chapter
. 97
5.1
The Basic Model
. 98
5.1.1
Relaxed Feedback Controls
. 98
5.1.2
Density Properties and Preliminary Results
.101
5.2
The Doeblin Condition
.105
5.3
Approximations of the Models
.107
5.4
Approximations with Periodic Delays
.110
5.4.1
Limit and Approximation Results for Periodic Delays
. . 110
5.4.2
Smoothed Nearly Optimal Controls
.115
5.4.3
Delays in the Variance Term
.117
5.5
The Periodic-Erlang Approximation
.118
Markov Chain Approximations: Introduction
.125
6.0
Outline of the Chapter
.125
6.1
The System Model
.126
6.2
Approximating Chains and Local Consistency
.127
6.3
Continuous-Time Interpolations
.131
6.3.1
The Continuous-Time Interpolation
ξΗ(·)
.131
Contents ix
6.3.2
A Markov Continuous-Time Interpolation
.133
6.4
The "Explicit" Approximation Procedure
.137
6.5
The "Implicit" Approximating Processes
.140
6.5.1
The General Implicit Approximation Method
.142
6.5.2
Continuous-Time Interpolations
.144
6.5.3
Representations of the Cost Function
.146
6.5.4
Asymptotic Equivalence of the Timescales
.147
6.5.5
Convergence
.149
6.6
Singular and Impulsive Controls
.149
6.6.1
Singular Controls
.149
6.6.2
Impulsive Control
.153
6.7
The Ergodic Cost Function
.153
6.7.1
Introduction
.153
6.7.2
The Markov Chain Approximation Method
.154
Markov Chain Approximations: Path and Control Delayed
159
7.0
Outline of the Chapter
.159
7.1
The Model and Local Consistency
.161
7.1.1
The Models
.161
7.1.2
Delay in Path Only: Local Consistency and
Interpolations
.162
7.1.3
Delay in the Path and Control
.167
7.1.4
Absorbing Boundaries and Other Cost Functions
.171
7.1.5
Approximations to the Memory Segments
.172
7.2
Computational Procedures
.175
7.2.1
Delay in the Path Only: State Representations and
the Bellman Equation
.175
7.2.2
Delay in Both Path and Control
.178
7.2.3
A Comment on Higher-Dimensional Problems
.179
7.3
The Implicit Numerical Approximation: Path Delayed
.180
7.3.1
Local Consistency and the Memory Segment
.180
7.3.2
The Cost Function and Bellman Equation
.185
7.3.3
The Use of Averaging in Constructing the Path
Memory Approximation
.186
7.3.4
Timescales
.187
7.3.5
Convergence Theorems
.188
7.4
The Implicit Approximation Procedure and the Random
Delay Model
.189
Path and Control Delayed: Continued
.193
8.0
Outline of the Chapter
.193
8.1
Periodic Approximations to the Delay: Path Delayed
.194
8.2
A Periodic-Erlang Model
.196
8.3
The Number of Points in the State Space: Path Only Delayed
. 200
χ
Contents
8.3.1
The Implicit and Periodic-
Erlang
Approximation
Methods: Reduced Memory
.200
8.4
Control and Path Delayed
.203
8.4.1
A Periodic Approximating Memory Segment
.204
8.4.2
A Periodic-Erlang Approximation
.207
8.5
Proofs of Convergence
.213
8.5.1
Proofs of Theorems from Chapter
7.213
8.5.2
Proof of Theorem
4.1.218
8.6
Singular Controls
.219
8.7
Neutral Equations
.220
8.8
The Ergodic Cost Problem
.222
9
A Wave Equation Approach
.227
9.0
Outline of the Chapter
.227
9.1
The Model and Assumptions
.228
9.2
A Key Representation of x(-)
.230
9.2.1
A Representation of the Solution
.230
9.2.2
Comments on the Dimension and the System State
. 231
9.2.3
Proof of the Representation
.232
9.2.4
Extensions
.235
9.3
A Discrete-Time Approximation
.236
9.4
The Markov Chain Approximation
.240
9.4.1
Preliminaries and Boundaries
.242
9.4.2
Transition Probabilities and Local Consistency: An
Implicit Approximation Procedure
.242
9.4.3
Dynamical Representations, the Cost Function and
Bellman Equation
.248
9.5
Size of the State Space for the Approximating Chain
.250
9.6
Proof of Convergence: Preliminaries
.252
9.6.1
The Randomization Errors
.252
9.6.2
Continuous Time Interpolations
.255
9.7
Convergence of the Numerical Algorithm
.259
9.8
Alternatives: Periodic and Periodic-Erlang Approximations
. . . 261
9.8.1
A Periodic Approximation
.261
9.8.2
The Effective Delay and Numerical Procedures
.265
9.9
Singular and Impulsive Controls
.265
References
.267
Index
.273
Symbol Index
.277
Harold
J.
Kushner
Numerical Methods for Controlled Stochastic Delay Systems
The Markov chain approximation methods are widely used for the numerical solution
of nonlinear stochastic control problems in continuous time. This book extends the
methods to stochastic systems with delays. Because such problems are infinite-
dimensional, many new issues arise in getting good numerical approximations
and in the convergence proofs. Useful forms of numerical algorithms and system
approximations are developed in this work, and the convergence proofs are
given. All of the usual cost functions are treated as well as singular and impulsive
controls. A major concern is on representations and approximations that use
minimal memory.
Features and topics include:
•
Surveys properties of the most important stochastic dynamical models,
including singular control, and those for diffusion and reflected diffusion
models.
•
Gives approximations to the dynamical models that simplify the numerical
problem, but have only small effects on the behavior.
•
Develops an ergodic theory for reflected diffusions with delays, as well as
model simplifications useful for numerical approximations for average cost per
unit time problems.
•
Provides numerical algorithms for models with delays in the path, or path and
control, with reduced memory requirements.
•
Develops transformations of the problem that yield more efficient
approximations when the control, driving Wiener process, and/or reflection
processes might be delayed, as well as the path.
•
Presents examples with applications to control and modern communications
systems.
The book is the first on the subject and will be of interest to all those who work
with stochastic delay equations and whose main interest is in either the use of
the algorithms or the underlying mathematics. An excellent resource for graduate
students, researchers, and practitioners, the work may be used as a graduate-
level textbook for a special topics course or seminar on numerical methods in
stochastic control. |
| adam_txt |
Contents
Preface
. xi
Examples and Introduction
. 1
1.0
Outline of the Chapter
. 1
1.1
An Introductory Example: Controlling the Temperature of a
Fluid Flow
. 1
1.2
An Example from Internet Regulation
. 5
1.3
Models With Finite-Dimensional Equivalences
. 10
Weak Convergence and Martingales
. 13
2.0
Outline of the Chapter
. 13
2.1
Weak Convergence
. 14
2.1.1
Basic Theorems of Weak Convergence
. 15
2.1.2
The Function Spaces D(S; I)
. 17
2.2
Martingales and the Martingale Method
. 18
2.2.1
Martingales
. 18
2.2.2
Verifying That a Process Is a Martingale
. 20
Stochastic Delay Equations: Models
. 23
3.0
Outline of the Chapter
. 23
3.1
The System Model: Boundary Absorption
. 25
3.2
Reflecting Diffusions
. 30
3.2.1
The Reflected Diffusion
. 30
3.2.2
Delayed Control, Reflection Term, and/or Wiener
Process
. 35
3.2.3
Neutral Equations
. 37
3.2.4
Controlled Variance and Jumps
. 37
3.3
The Girsanov Transformation
. 38
3.4
Cost Functions
. 40
3.5
Existence of an Optimal Control
. 43
Contents
3.5.1
Reflecting or Absorbing Boundary: Discounted Cost
. 43
3.6
Singular and Impulsive Controls
. 50
3.6.1
Singular Controls
. 50
3.6.2
Definition of and Existence of Solutions
. 51
3.6.3
Existence of an Optimal Control
. 56
3.6.4
Impulsive Controls
. 58
Approximations to the Dynamical Models
. 61
4.0
Outline of the Chapter
. 61
4.1
Approximations of the Dynamical Systems
. 63
4.1.1
A Basic Approximation
. 63
4.2
Approximations by Time-Varying Delays
. 69
4.2.1
Discretized Delays
. 69
4.2.2
Periodic Delays
. 71
4.2.3
Randomly Varying Delays
. 73
4.2.4
Periodic-Erlang Delays
. 75
4.2.5
Convergence of Costs and Existence of Optimal Controls
77
4.2.6
Differential Operator for the Periodic-Erlang
Approximation
. 78
4.3
Simulations Illustrating the Model Approximations
. 78
4.3.1
Simulations Based on the Periodic Approximation
. 78
4.3.2
Simulations Based on the Periodic-Erlang Approximation
82
4.4
Approximations: Path and Control Delayed
. 84
4.5
Singular Controls
. 90
4.6
Rapidly Varying Delays
. 92
The Ergodic Cost Problem
. 97
5.0
Outline of the Chapter
. 97
5.1
The Basic Model
. 98
5.1.1
Relaxed Feedback Controls
. 98
5.1.2
Density Properties and Preliminary Results
.101
5.2
The Doeblin Condition
.105
5.3
Approximations of the Models
.107
5.4
Approximations with Periodic Delays
.110
5.4.1
Limit and Approximation Results for Periodic Delays
. . 110
5.4.2
Smoothed Nearly Optimal Controls
.115
5.4.3
Delays in the Variance Term
.117
5.5
The Periodic-Erlang Approximation
.118
Markov Chain Approximations: Introduction
.125
6.0
Outline of the Chapter
.125
6.1
The System Model
.126
6.2
Approximating Chains and Local Consistency
.127
6.3
Continuous-Time Interpolations
.131
6.3.1
The Continuous-Time Interpolation
ξΗ(·)
.131
Contents ix
6.3.2
A Markov Continuous-Time Interpolation
.133
6.4
The "Explicit" Approximation Procedure
.137
6.5
The "Implicit" Approximating Processes
.140
6.5.1
The General Implicit Approximation Method
.142
6.5.2
Continuous-Time Interpolations
.144
6.5.3
Representations of the Cost Function
.146
6.5.4
Asymptotic Equivalence of the Timescales
.147
6.5.5
Convergence
.149
6.6
Singular and Impulsive Controls
.149
6.6.1
Singular Controls
.149
6.6.2
Impulsive Control
.153
6.7
The Ergodic Cost Function
.153
6.7.1
Introduction
.153
6.7.2
The Markov Chain Approximation Method
.154
Markov Chain Approximations: Path and Control Delayed
159
7.0
Outline of the Chapter
.159
7.1
The Model and Local Consistency
.161
7.1.1
The Models
.161
7.1.2
Delay in Path Only: Local Consistency and
Interpolations
.162
7.1.3
Delay in the Path and Control
.167
7.1.4
Absorbing Boundaries and Other Cost Functions
.171
7.1.5
Approximations to the Memory Segments
.172
7.2
Computational Procedures
.175
7.2.1
Delay in the Path Only: State Representations and
the Bellman Equation
.175
7.2.2
Delay in Both Path and Control
.178
7.2.3
A Comment on Higher-Dimensional Problems
.179
7.3
The Implicit Numerical Approximation: Path Delayed
.180
7.3.1
Local Consistency and the Memory Segment
.180
7.3.2
The Cost Function and Bellman Equation
.185
7.3.3
The Use of Averaging in Constructing the Path
Memory Approximation
.186
7.3.4
Timescales
.187
7.3.5
Convergence Theorems
.188
7.4
The Implicit Approximation Procedure and the Random
Delay Model
.189
Path and Control Delayed: Continued
.193
8.0
Outline of the Chapter
.193
8.1
Periodic Approximations to the Delay: Path Delayed
.194
8.2
A Periodic-Erlang Model
.196
8.3
The Number of Points in the State Space: Path Only Delayed
. 200
χ
Contents
8.3.1
The Implicit and Periodic-
Erlang
Approximation
Methods: Reduced Memory
.200
8.4
Control and Path Delayed
.203
8.4.1
A Periodic Approximating Memory Segment
.204
8.4.2
A Periodic-Erlang Approximation
.207
8.5
Proofs of Convergence
.213
8.5.1
Proofs of Theorems from Chapter
7.213
8.5.2
Proof of Theorem
4.1.218
8.6
Singular Controls
.219
8.7
Neutral Equations
.220
8.8
The Ergodic Cost Problem
.222
9
A Wave Equation Approach
.227
9.0
Outline of the Chapter
.227
9.1
The Model and Assumptions
.228
9.2
A Key Representation of x(-)
.230
9.2.1
A Representation of the Solution
.230
9.2.2
Comments on the Dimension and the System State
. 231
9.2.3
Proof of the Representation
.232
9.2.4
Extensions
.235
9.3
A Discrete-Time Approximation
.236
9.4
The Markov Chain Approximation
.240
9.4.1
Preliminaries and Boundaries
.242
9.4.2
Transition Probabilities and Local Consistency: An
Implicit Approximation Procedure
.242
9.4.3
Dynamical Representations, the Cost Function and
Bellman Equation
.248
9.5
Size of the State Space for the Approximating Chain
.250
9.6
Proof of Convergence: Preliminaries
.252
9.6.1
The Randomization Errors
.252
9.6.2
Continuous Time Interpolations
.255
9.7
Convergence of the Numerical Algorithm
.259
9.8
Alternatives: Periodic and Periodic-Erlang Approximations
. . . 261
9.8.1
A Periodic Approximation
.261
9.8.2
The Effective Delay and Numerical Procedures
.265
9.9
Singular and Impulsive Controls
.265
References
.267
Index
.273
Symbol Index
.277
Harold
J.
Kushner
Numerical Methods for Controlled Stochastic Delay Systems
The Markov chain approximation methods are widely used for the numerical solution
of nonlinear stochastic control problems in continuous time. This book extends the
methods to stochastic systems with delays. Because such problems are infinite-
dimensional, many new issues arise in getting good numerical approximations
and in the convergence proofs. Useful forms of numerical algorithms and system
approximations are developed in this work, and the convergence proofs are
given. All of the usual cost functions are treated as well as singular and impulsive
controls. A major concern is on representations and approximations that use
minimal memory.
Features and topics include:
•
Surveys properties of the most important stochastic dynamical models,
including singular control, and those for diffusion and reflected diffusion
models.
•
Gives approximations to the dynamical models that simplify the numerical
problem, but have only small effects on the behavior.
•
Develops an ergodic theory for reflected diffusions with delays, as well as
model simplifications useful for numerical approximations for average cost per
unit time problems.
•
Provides numerical algorithms for models with delays in the path, or path and
control, with reduced memory requirements.
•
Develops transformations of the problem that yield more efficient
approximations when the control, driving Wiener process, and/or reflection
processes might be delayed, as well as the path.
•
Presents examples with applications to control and modern communications
systems.
The book is the first on the subject and will be of interest to all those who work
with stochastic delay equations and whose main interest is in either the use of
the algorithms or the underlying mathematics. An excellent resource for graduate
students, researchers, and practitioners, the work may be used as a graduate-
level textbook for a special topics course or seminar on numerical methods in
stochastic control. |
| any_adam_object | 1 |
| any_adam_object_boolean | 1 |
| author | Kushner, Harold J. 1933- |
| author_GND | (DE-588)11559163X |
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| discipline_str_mv | Informatik Mathematik Mess-/Steuerungs-/Regelungs-/Automatisierungstechnik / Mechatronik |
| format | Book |
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| id | DE-604.BV035130940 |
| illustrated | Illustrated |
| index_date | 2024-07-02T22:24:45Z |
| indexdate | 2024-07-20T09:54:00Z |
| institution | BVB |
| isbn | 9780817646219 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016798441 |
| oclc_num | 209333153 |
| open_access_boolean | |
| owner | DE-20 DE-473 DE-BY-UBG DE-19 DE-BY-UBM DE-703 DE-11 DE-824 |
| owner_facet | DE-20 DE-473 DE-BY-UBG DE-19 DE-BY-UBM DE-703 DE-11 DE-824 |
| physical | XIX, 281 S. graph. Darst. |
| publishDate | 2008 |
| publishDateSearch | 2008 |
| publishDateSort | 2008 |
| publisher | Birkhäuser |
| record_format | marc |
| series2 | Systems & control: foundations & applications |
| spelling | Kushner, Harold J. 1933- Verfasser (DE-588)11559163X aut Numerical methods for controlled stochastic delay systems Harold J. Kushner Boston [u.a.] Birkhäuser 2008 XIX, 281 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Systems & control: foundations & applications Stochastic systems Markov-Kette (DE-588)4037612-6 gnd rswk-swf Numerisches Verfahren (DE-588)4128130-5 gnd rswk-swf Stochastische Kontrolltheorie (DE-588)4263657-7 gnd rswk-swf Stochastische Differentialgleichung mit Gedächtnis (DE-588)4691382-8 gnd rswk-swf Stochastische Differentialgleichung mit Gedächtnis (DE-588)4691382-8 s Stochastische Kontrolltheorie (DE-588)4263657-7 s Numerisches Verfahren (DE-588)4128130-5 s Markov-Kette (DE-588)4037612-6 s DE-604 Erscheint auch als Online-Ausgabe 978-0-8176-4534-2 text/html http://deposit.dnb.de/cgi-bin/dokserv?id=2857142&prov=M&dok_var=1&dok_ext=htm Inhaltstext http://d-nb.info/981328547/04 Inhaltsverzeichnis Digitalisierung UB Bamberg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016798441&sequence=000002&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=016798441&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA Klappentext |
| spellingShingle | Kushner, Harold J. 1933- Numerical methods for controlled stochastic delay systems Stochastic systems Markov-Kette (DE-588)4037612-6 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Stochastische Kontrolltheorie (DE-588)4263657-7 gnd Stochastische Differentialgleichung mit Gedächtnis (DE-588)4691382-8 gnd |
| subject_GND | (DE-588)4037612-6 (DE-588)4128130-5 (DE-588)4263657-7 (DE-588)4691382-8 |
| title | Numerical methods for controlled stochastic delay systems |
| title_auth | Numerical methods for controlled stochastic delay systems |
| title_exact_search | Numerical methods for controlled stochastic delay systems |
| title_exact_search_txtP | Numerical methods for controlled stochastic delay systems |
| title_full | Numerical methods for controlled stochastic delay systems Harold J. Kushner |
| title_fullStr | Numerical methods for controlled stochastic delay systems Harold J. Kushner |
| title_full_unstemmed | Numerical methods for controlled stochastic delay systems Harold J. Kushner |
| title_short | Numerical methods for controlled stochastic delay systems |
| title_sort | numerical methods for controlled stochastic delay systems |
| topic | Stochastic systems Markov-Kette (DE-588)4037612-6 gnd Numerisches Verfahren (DE-588)4128130-5 gnd Stochastische Kontrolltheorie (DE-588)4263657-7 gnd Stochastische Differentialgleichung mit Gedächtnis (DE-588)4691382-8 gnd |
| topic_facet | Stochastic systems Markov-Kette Numerisches Verfahren Stochastische Kontrolltheorie Stochastische Differentialgleichung mit Gedächtnis |
| url | http://deposit.dnb.de/cgi-bin/dokserv?id=2857142&prov=M&dok_var=1&dok_ext=htm http://d-nb.info/981328547/04 http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016798441&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016798441&sequence=000004&line_number=0002&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT kushnerharoldj numericalmethodsforcontrolledstochasticdelaysystems |