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Bertsekas D.P., Shreve S.E. — Stochastic Optimal Control: The Discrete-Time Case
Bertsekas D.P., Shreve S.E. — Stochastic Optimal Control: The Discrete-Time Case



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Название: Stochastic Optimal Control: The Discrete-Time Case

Авторы: Bertsekas D.P., Shreve S.E.

Аннотация:

Preface:

This monograph is the outgrowth of research carried out at the University of Illinois over a three-year period beginning in the latter half of 1974. The objective of the monograph is to provide a unifying and mathematically rigorous theory for a broad class of dynamic programming and discrete-time stochastic optimal control problems. It is divided into two parts, which can be read independently.

Part I provides an analysis of dynamic programming models in a unified framework applicable to deterministic optimal control, stochastic optimal control, minimax control, sequential games, and other areas. It resolves the structural questions associated with such problems, i.e., it provides results that draw their validity exclusively from the sequential nature of the problem. Such results hold for models where measurability of various objects is of no essential concern, for example, in deterministic problems and stochastic problems defined over a countable probability space. The starting point for the analysis is the mapping defining the dynamic programming algorithm. A single abstract problem is formulated in terms of this mapping and counterparts of nearly all results known for deterministic optimal control problems are derived. A new stochastic optimal control model based on outer integration is also introduced in this part. It is a broadly applicable model and requires no topological assumptions. We show that all the results of Part I hold for this model.

Part II resolves the measurability questions associated with stochastic optimal control problems with perfect and imperfect state information. These questions have been studied over the past fifteen years by several researchers in statistics and control theory. As we explain in Chapter 1, the approaches that have been used are either limited by restrictive assumptions such as compactness and continuity or else they are not sufficiently powerful to yield results that are as strong as their structural counterparts. These deficiencies can be traced to the fact that the class of policies considered is not sufficiently rich to ensure the existence of everywhere optimal or epsilon-optimal policies except under restrictive assumptions. In our work we have appropriately enlarged the space of admissible policies to include universally measurable policies. This guarantees the existence of epsilon-optimal policies and allows, for the first time, the development of a general and comprehensive theory which is as powerful as its deterministic counterpart.

We mention, however, that the class of universally measurable policies is not the smallest class of policies for which these results are valid. The smallest such class is the class of limit measurable policies discussed in Section 11.1. The sigma-algebra of limit measurable sets (or C-sets) is defined in a constructive manner involving transfinite induction that, from a set of theoretic point of view, is more satisfying than the definition of the universal sigma-algebra. We believe, however, that the majority of readers will find the universal sigma-algebra and the methods of proof associated with it more understandable, and so we devote the main body of Part II to models with universally measurable policies.

Parts I and II are related and complement each other. Part II makes extensive use of the results of Part I. However, the special forms in which these results are needed are also available in other sources (e.g., the textbook by Bertsekas [B4]). Each time we make use of such a result, we refer to both Part I and the Bertsekas textbook, so that Part II can be read independently of Part I. The developments in Part II show also that stochastic optimal control problems with measurability restrictions on the admissible policies can be embedded within the framework of Part I, thus demonstrating the broad scope of the formulation given there.


Язык: en

Рубрика: Математика/

Статус предметного указателя: Готов указатель с номерами страниц

ed2k: ed2k stats

Год издания: 1996

Количество страниц: 333

Добавлена в каталог: 07.05.2010

Операции: Положить на полку | Скопировать ссылку для форума | Скопировать ID
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Предметный указатель
$F_{\sigma}$-set      102
$G_{\delta}$-set      102
A posteriori distribution      260ff
A priori distribution      260ff
Alexandroff's theorem      107
Analytic $\sigma$-algebra      171
Analytic measurability of a function      171
Analytic set      160
Assumption A.1      93
Assumption A.2      93
Assumption A.3      93
Assumption A.4      93
Assumption A.5      93
Assumption C      96
Assumption C (Contraction Assumption)      52
Assumption D (Uniform Decrease Assumption)      70
Assumption D.1      71
Assumption D.2      71
Assumption F.1      39
Assumption F.2      40 94
Assumption F.3      40 95
Assumption I (Uniform Increase Assumption)      70
Assumption I.1      71
Assumption I.2      71
Axiom of Choice      301
Baire null space      103 109
Borel $\sigma$-algebra      117
Borel isomorphism      121
Borel measurability of a function      120
Borel programmable      21
Borel space      118
C-sets      20
Cantor's Continuum Hypothesis      301
Completion of a $\sigma$-algebra      167
Completion of a metric space      114
Composition of measurable functions      298
Contraction assumption      52
Control, constraint      2 26 188 216 243 245 248 251 271
Control, space      2 26 188 216 243 245 248 251 271
Cost, corresponding to a policy      2 28 191 217 244 249 254
Cost, one-stage      2 189 216 243 245 248 251 271
Cost, optimal      2 29 191 217 244 246 250 254
Definition 10.1      243
Definition 10.2      245
Definition 10.3      248
Definition 10.4      249
Definition 10.5      249
Definition 10.6      250
Definition 10.7      251
Definition 10.8      256
Definition 7.1      104
Definition 7.10      122
Definition 7.11      133
Definition 7.12      134
Definition 7.13      146
Definition 7.14      157
Definition 7.15      157
Definition 7.16      160
Definition 7.17      161
Definition 7.18      167
Definition 7.19      171
Definition 7.2      105
Definition 7.20      171
Definition 7.21      177
Definition 7.3      107
Definition 7.4      112
Definition 7.5      114
Definition 7.6      117
Definition 7.7      118
Definition 7.8      120
Definition 7.9      121
Definition 8.1      188
Definition 8.2      190
Definition 8.3      191
Definition 8.4      194
Definition 8.5      195
Definition 8.6      206
Definition 8.7      208
Definition 8.8      210
Definition 9.1      213
Definition 9.10      229
Definition 9.2      214
Definition 9.3      214
Definition 9.4      216
Definition 9.5      217
Definition 9.6      217
Definition 9.7      217
Definition 9.8      217
Definition 9.9      218
Definition A.1      273
Definition B.1      290
Definition B.2      293
Definition B.3      298
Definition C.1      303
Definition C.2      310
Disturbance kernel      189 243 245 271
Disturbance space      189 243 245 271
Dynamic programming (DP) algorithm      3 6 39 57 80 198 229 259
Dynkin system      133
Dynkin system theorem      133
Epigraph      82
Exact Selection Assumption      95
Exponential topology      304
Filtering      261
Fixed point theorem (Banach)      55
Fixed Point, Theorem      55
Hausdorff metric      303
Hilbert cube      103
Homeomorphism      104
Horizon, finite      28 189 243 245 248 251 271
Horizon, infinite      70 213 216 243 245 248 251
Imperfect state information model      248
Indicator function      103
Information vector      248
Isometry      144
Jankov — von Neumann theorem      182
Kuratowski's theorem      121
Lemma 10.1      253
Lemma 10.2      255
Lemma 10.3      260
Lemma 10.4      261
Lemma 3.1      45
Lemma 5.1      75
Lemma 5.2      82
Lemma 7.1 (Urysohn's Lemma)      105
Lemma 7.10      131
Lemma 7.11      139
Lemma 7.12      144
Lemma 7.13      146
Lemma 7.14      147
Lemma 7.15      149
Lemma 7.16      150
Lemma 7.17      151
Lemma 7.18      151
Lemma 7.19      152
Lemma 7.2      105
Lemma 7.20      152
Lemma 7.21      154
Lemma 7.22      161
Lemma 7.23      162
Lemma 7.24      163
Lemma 7.25      164
Lemma 7.26      172
Lemma 7.27      173
Lemma 7.28      174
Lemma 7.29      174
Lemma 7.3      116
Lemma 7.30      177
Lemma 7.4      119
Lemma 7.5      119
Lemma 7.6      125
Lemma 7.7      125
Lemma 7.8      125
Lemma 7.9      127
Lemma 8.1      194
Lemma 8.2      196
Lemma 8.3      196
Lemma 8.4      197
Lemma 8.5      202
Lemma 8.6      205
Lemma 8.7      206
Lemma 9.1      220
Lemma 9.2      221
Lemma 9.3      230
Lemma A.1      273
Lemma A.2      274
Lemma A.3      275
Lemma B.1      285
Lemma B.2      285
Lemma B.3      286
Lemma B.4      287
Lemma B.5      288
Lemma B.6      294
Lemma B.7      295
Lemma B.8      298
Lemma C.1      306
Lemma C.2      310
Limit $\sigma$-algebra      293
Limit measurability      298
Lindeloef space      106
Lower semianalytic function      177
Lower semicontinuous function      146
Lower semicontinuous model      208
Lusin's theorem      167
Metrizable space      104
Monotonicity Assumption      27
Nonstationary model      243
Observation kernel      248
Observation space      248
Optimality equation      4 57 71 73 78ff 225
Optimality equation, nonstationary      246
Outer integral      273
Outer integral, monotone convergence theorem for      278
p-outer measure      166 274
Paved space      157
Policy      2 6 26 91 190 214 217 243 249
Policy, $p-\epsilon$-optimal      12
Policy, $\epsilon$-optimal      29 191 215 244
Policy, $\{\epsilon_{n}\}$-dominated convergence to optimality      29 191 245
Policy, analytically measurable      190 269ff
Policy, Borel-measurable      190
Policy, k-originating      243
Policy, limit-measurable      190 266ff
Policy, Markov      6 190
Policy, nonrandomized      190 249
Policy, optimal      29 191 215 244
Policy, q-optimal      256
Policy, semi-Markov      190
Policy, stationary      214
Policy, uniformly N-stage optimal      29 206
Policy, universally measurable      190
Policy, weakly $q-\epsilon$-optimal      256
Projection mapping      103
Proposition 10.1      246
Proposition 10.2      254
Proposition 10.3      256
Proposition 10.4      257
Proposition 10.5      262
Proposition 10.6      264
Proposition 11.1      266
Proposition 11.2      267
Proposition 11.3      267
Proposition 11.4      268
Proposition 11.5      269
Proposition 11.6      270
Proposition 11.7      272
Proposition 3.1      40
Proposition 3.2      43
Proposition 3.3      44
Proposition 3.4      46
Proposition 3.5      47
Proposition 3.6      50
Proposition 3.7      51
Proposition 4.1      53
Proposition 4.10      68
Proposition 4.11      69
Proposition 4.2      55
Proposition 4.3      56
Proposition 4.4      57
Proposition 4.5      59
Proposition 4.6      60
Proposition 4.7      62
Proposition 4.8      64
Proposition 4.9      64
Proposition 5.1      71
Proposition 5.10      86
Proposition 5.11      87
Proposition 5.12      88
Proposition 5.13      89
Proposition 5.14      90
Proposition 5.15      90
Proposition 5.2      73
Proposition 5.3      75
Proposition 5.4      78
Proposition 5.5      78
Proposition 5.6      79
Proposition 5.7      80
Proposition 5.8      81
Proposition 5.9      84
Proposition 6.1      95
Proposition 6.2      95
Proposition 6.3      96
Proposition 6.4      97
Proposition 6.5      97
Proposition 7.1      106
Proposition 7.10      117
Proposition 7.11      118
Proposition 7.12      119
Proposition 7.13      119
Proposition 7.14      120
Proposition 7.15 (Kuratowski's Theorem)      121
Proposition 7.16      121
Proposition 7.17      122
Proposition 7.18      124
Proposition 7.19      127
Proposition 7.2 (Urysohn's Theorem)      106
Proposition 7.20      127
Proposition 7.21      128
Proposition 7.22      130
Proposition 7.23      131
Proposition 7.24 (Dynkin System Theorem)      133
Proposition 7.25      133
Proposition 7.26      134
Proposition 7.27      135
Proposition 7.28      140
Proposition 7.29      144
Proposition 7.3 (Alexandroff's Theorem)      107
Proposition 7.30      145
Proposition 7.31      148
Proposition 7.32      148
Proposition 7.33      153
Proposition 7.34      154
Proposition 7.35      158
Proposition 7.36      161
Proposition 7.37      164
Proposition 7.38      165
Proposition 7.39      165
Proposition 7.4      108
Proposition 7.40      165
Proposition 7.41      166
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