Description : This book presents the first part of a planned two-volume series devoted to a systematic exposition of some recent developments in the theory of discrete-time Markov control processes (MCPs). Interest is mainly confined to MCPs with Borel state and control (or action) spaces, and possibly unbounded costs and noncompact control constraint sets. MCPs are a class of stochastic control problems, also known as Markov decision processes, controlled Markov processes, or stochastic dynamic pro grams; sometimes, particularly when the state space is a countable set, they are also called Markov decision (or controlled Markov) chains. Regardless of the name used, MCPs appear in many fields, for example, engineering, economics, operations research, statistics, renewable and nonrenewable re source management, (control of) epidemics, etc. However, most of the lit erature (say, at least 90%) is concentrated on MCPs for which (a) the state space is a countable set, and/or (b) the costs-per-stage are bounded, and/or (c) the control constraint sets are compact. But curiously enough, the most widely used control model in engineering and economics--namely the LQ (Linear system/Quadratic cost) model-satisfies none of these conditions. Moreover, when dealing with "partially observable" systems) a standard approach is to transform them into equivalent "completely observable" sys tems in a larger state space (in fact, a space of probability measures), which is uncountable even if the original state process is finite-valued.
Description : Devoted to a systematic exposition of some recent developments in the theory of discrete-time Markov control processes, the text is mainly confined to MCPs with Borel state and control spaces. Although the book follows on from the author's earlier work, an important feature of this volume is that it is self-contained and can thus be read independently of the first. The control model studied is sufficiently general to include virtually all the usual discrete-time stochastic control models that appear in applications to engineering, economics, mathematical population processes, operations research, and management science.
Description : In a unified form, this monograph presents fundamental results on the approximation of centralized and decentralized stochastic control problems, with uncountable state, measurement, and action spaces. It demonstrates how quantization provides a system-independent and constructive method for the reduction of a system with Borel spaces to one with finite state, measurement, and action spaces. In addition to this constructive view, the book considers both the information transmission approach for discretization of actions, and the computational approach for discretization of states and actions. Part I of the text discusses Markov decision processes and their finite-state or finite-action approximations, while Part II builds from there to finite approximations in decentralized stochastic control problems. This volume is perfect for researchers and graduate students interested in stochastic controls. With the tools presented, readers will be able to establish the convergence of approximation models to original models and the methods are general enough that researchers can build corresponding approximation results, typically with no additional assumptions.
Description : This book focuses on two-time-scale Markov chains in discrete time. Our motivation stems from existing and emerging applications in optimization and control of complex systems in manufacturing, wireless communication, and ?nancial engineering. Much of our e?ort in this book is devoted to designing system models arising from various applications, analyzing them via analytic and probabilistic techniques, and developing feasible compu- tionalschemes. Ourmainconcernistoreducetheinherentsystemcompl- ity. Although each of the applications has its own distinct characteristics, all of them are closely related through the modeling of uncertainty due to jump or switching random processes. Oneofthesalientfeaturesofthisbookistheuseofmulti-timescalesin Markovprocessesandtheirapplications. Intuitively,notallpartsorcom- nents of a large-scale system evolve at the same rate. Some of them change rapidly and others vary slowly. The di?erent rates of variations allow us to reduce complexity via decomposition and aggregation. It would be ideal if we could divide a large system into its smallest irreducible subsystems completely separable from one another and treat each subsystem indep- dently. However, this is often infeasible in reality due to various physical constraints and other considerations. Thus, we have to deal with situations in which the systems are only nearly decomposable in the sense that there are weak links among the irreducible subsystems, which dictate the oc- sional regime changes of the system. An e?ective way to treat such near decomposability is time-scale separation. That is, we set up the systems as if there were two time scales, fast vs. slow. xii Preface Followingthetime-scaleseparation,weusesingularperturbationmeth- ology to treat the underlying systems.
Description : The intent of this book is to present recent results in the control theory for the long run average continuous control problem of piecewise deterministic Markov processes (PDMPs). The book focuses mainly on the long run average cost criteria and extends to the PDMPs some well-known techniques related to discrete-time and continuous-time Markov decision processes, including the so-called ``average inequality approach'', ``vanishing discount technique'' and ``policy iteration algorithm''. We believe that what is unique about our approach is that, by using the special features of the PDMPs, we trace a parallel with the general theory for discrete-time Markov Decision Processes rather than the continuous-time case. The two main reasons for doing that is to use the powerful tools developed in the discrete-time framework and to avoid working with the infinitesimal generator associated to a PDMP, which in most cases has its domain of definition difficult to be characterized. Although the book is mainly intended to be a theoretically oriented text, it also contains some motivational examples. The book is targeted primarily for advanced students and practitioners of control theory. The book will be a valuable source for experts in the field of Markov decision processes. Moreover, the book should be suitable for certain advanced courses or seminars. As background, one needs an acquaintance with the theory of Markov decision processes and some knowledge of stochastic processes and modern analysis.
Description : This volume provides a general overview of discrete- and continuous-time Markov control processes and stochastic games, along with a look at the range of applications of stochastic control and some of its recent theoretical developments. These topics include various aspects of dynamic programming, approximation algorithms, and infinite-dimensional linear programming. In all, the work comprises 18 carefully selected papers written by experts in their respective fields. Optimization, Control, and Applications of Stochastic Systems will be a valuable resource for all practitioners, researchers, and professionals in applied mathematics and operations research who work in the areas of stochastic control, mathematical finance, queueing theory, and inventory systems. It may also serve as a supplemental text for graduate courses in optimal control and dynamic games.
Description : This book is concerned with a class of discrete-time stochastic control processes known as controlled Markov processes (CMP's), also known as Markov decision processes or Markov dynamic programs. Starting in the mid-1950swith Richard Bellman, many contributions to CMP's have been made, and applications to engineering, statistics and operations research, among other areas, have also been developed. The purpose of this book is to present some recent developments on the theory of adaptive CMP's, i. e. , CMP's that depend on unknown parameters. Thus at each decision time, the controller or decision-maker must estimate the true parameter values, and then adapt the control actions to the estimated values. We do not intend to describe all aspects of stochastic adaptive control; rather, the selection of material reflects our own research interests. The prerequisite for this book is a knowledgeof real analysis and prob ability theory at the level of, say, Ash (1972) or Royden (1968), but no previous knowledge of control or decision processes is required. The pre sentation, on the other hand, is meant to beself-contained,in the sensethat whenever a result from analysisor probability is used, it is usually stated in full and references are supplied for further discussion, if necessary. Several appendices are provided for this purpose. The material is divided into six chapters. Chapter 1 contains the basic definitions about the stochastic control problems we are interested in; a brief description of some applications is also provided.
Description : Continuous-time Markov decision processes (MDPs), also known as controlled Markov chains, are used for modeling decision-making problems that arise in operations research (for instance, inventory, manufacturing, and queueing systems), computer science, communications engineering, control of populations (such as fisheries and epidemics), and management science, among many other fields. This volume provides a unified, systematic, self-contained presentation of recent developments on the theory and applications of continuous-time MDPs. The MDPs in this volume include most of the cases that arise in applications, because they allow unbounded transition and reward/cost rates. Much of the material appears for the first time in book form.
Description : One of the first books in the timely and important area of heavy traffic analysis of controlled and uncontrolled stochastics networks, by one of the leading authors in the field. The general theory is developed, with possibly state dependent parameters, and specialized to many different cases of practical interest.