Stochastic Optimal Control: The Discrete-Time Case

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Athena Scientific, Dec 1, 1996 - Mathematics - 330 pages
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This research monograph, first published in 1978 by Academic Press, remains the authoritative and comprehensive treatment of the mathematical foundations of stochastic optimal control of discrete-time systems, including the treatment of the intricate measure-theoretic issues. It is an excellent supplement to the first author's Dynamic Programming and Optimal Control (Athena Scientific, 2018).

Review of the 1978 printing:"Bertsekas and Shreve have written a fine book. The exposition is extremely clear and a helpful introductory chapter provides orientation and a guide to the rather intimidating mass of literature on the subject. Apart from anything else, the book serves as an excellent introduction to the arcane world of analytic sets and other lesser known byways of measure theory."

Mark H. A. Davis, Imperial College, in IEEE Trans. on Automatic Control

Among its special features, the book:

1) Resolves definitively the mathematical issues of discrete-time stochastic optimal control problems, including Borel models, and semi-continuous models

2) Establishes the most general possible theory of finite and infinite horizon stochastic dynamic programming models, through the use of analytic sets and universally measurable policies

3) Develops general frameworks for dynamic programming based on abstract contraction and monotone mappings

4) Provides extensive background on analytic sets, Borel spaces and their probability measures

5) Contains much in depth research not found in any other textbook


 

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Contents

Introduction
1
Monotone Mappings Underlying Dynamic Programming Models
25
Finite Horizon Models
39
Infinite Horizon Models Under a Contraction Assumption
52
Infinite Horizon Models Under Monotonicity Assumptions
70
A Generalized Abstract Dynamic Programming Model
91
Borel Spaces and Their Probability Measures
101
The Finite Horizon Borel Model
188
The Imperfect State Information Model
242
Miscellaneous
266
The Outer Integral
273
Additional Measurability Properties of Borel Spaces
282
The Hausdorff Metric and the Exponential Topology
303
References
312
Table of Propositions Lemmas Definitions and Assumptions
317
Index
321

The Infinte Horizon Borel Models
213

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About the author (1996)

Dimitri Bertsekas is McAffee Professor of Engineering at the Massachusetts Institute of Technology (MIT), and Fulton Professor of Computational Decision Making, at Arizona State University (ASU).

Steven Shreve is Orion Hoch Professor of Mathematical Sciences at the Carnegie Mellon University.