Optimal Control Theory (PDF)
Applications to Management Science and Economics
(Sprache: Englisch)
Sethi and Thompson have provided management science and economics communities with a thoroughly revised edition of their classic text on Optimal Control Theory. Central to the book is its extraordinarily wide range of optimal control theory applications....
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Sethi and Thompson have provided management science and economics communities with a thoroughly revised edition of their classic text on Optimal Control Theory. Central to the book is its extraordinarily wide range of optimal control theory applications. Chapter 5 covers finance; Chapter 6 considers production and inventory problems; Chapter 7 covers marketing problems; Chapter 9 treats machine maintenance and replacement; Chapter 10 deals with problems of optimal consumption of natural resources (renewable or exhaustible); and Chapter 11 discusses a number of applications of control theory to economics. The book has been successfully used as a professional reference tool and as a graduate course book. Its usefulness lies in its emphasis on building applied models of realistic problems faced in a variety of business management situations. The new edition has been completely refined with careful attention to the text and graphic material presentation. In Chapter 3, models have been added that use mixed (control and state) constraints, current value formulations, and terminal conditions. Chapter 4 now covers more advanced material on pure state constraints as they relate to mixed constraints. Each of these chapters contains new results that were not available when the first edition was published. Another important change is the expansion of the material on stochastic optimal control theory, which has become the new Chapter 13. This new chapter provides a brief introduction to stochastic optimal control problems, and it contains formulations of simple stochastic models in production, marketing and finance, and their solutions.Optimal control methods are used to determine optimal ways to control a dynamic system. The theoretical work in this field serves as a foundation for the book, which the authors have applied to business management problems developed from their research and classroom instruction.
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Chapter 13 Stochastic Optimal Control (p.341)In previous chapters we assumed that the state variables of the system were known with certainty. If this were not the case, the state of the system over time would be a stochastic process. In addition, it might not be possible to measure the value of the state variables at time t. In this case, one would have to measure functions of the state variables. Moreover, the measurements are usually noisy, i.e., they are subject to errors. Thus, a decision maker is faced with the problem of making good estimates of these state variables from noisy measurements on functions of them.
The process of estimating the values of the state variables is called optimal filtering, In Section 13.1, we will discuss one particular filter, called the Kalman filter and its continuous-time analogue caUed the Kalman- Bucy filter. It should be noted that while optimal filtering provides optimal estimates of the value of the state variables from noisy measurements of related quantities, no control is involved.
When a control is involved, we are faced with a stochastic optimal control problem. Here, the state of the system is represented by a controlled stochastic process. In Section 13.2, we shall formulate a stochastic optimal control problem which is governed by stochastic differential equations. We shall only consider stochastic differential equations of a type known as Ito equations. These equations arise when the state equar tions, such as those we have seen in the previous chapters, are perturbed by Markov diffusion processes. Our goal in Section 13.2 will be to synthesize optimal feedback controls for systems subject to Ito equations in a way that maximizes the expected value of a given objective function. In Section 13.3, we shall extend the production planning model Chapter 6 to allow for some uncertain disturbances. We shall obtain an optimal production policy for the stochastic production planning problem thus
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formulated.
In Section 13.4, we solve an optimal stochastic advertising problem explicitly. The problem is a modification as well as a stochastic extension of the optimal control problem of the Vidale-Wolfe advertising model treated in Section 7.2.4.
In Section 13.5, we wiU introduce investment decisions in the consumption model of Example 1.3. We will consider both risk-free and risky investments. Our goal will be to find optimal consumption and investment policies in order to maximize the discounted value of the utility of consumption over time. In Section 13.6, we shall conclude the chapter by mentioning other types of stochastic optimal control problems that arise in practice. In particular, production planning problems where production is done by machines that are unreliable or failure-prone, can be formulated as stochastic optimal control problems involving jimip Markov processes. Such problems are treated in Sethi and Zhang (1994a, 1994c). Karatzas and Shreve (1998) address stochastic optimal control problems in finance involving more general stochastic processes including jimip processes.
In Section 13.4, we solve an optimal stochastic advertising problem explicitly. The problem is a modification as well as a stochastic extension of the optimal control problem of the Vidale-Wolfe advertising model treated in Section 7.2.4.
In Section 13.5, we wiU introduce investment decisions in the consumption model of Example 1.3. We will consider both risk-free and risky investments. Our goal will be to find optimal consumption and investment policies in order to maximize the discounted value of the utility of consumption over time. In Section 13.6, we shall conclude the chapter by mentioning other types of stochastic optimal control problems that arise in practice. In particular, production planning problems where production is done by machines that are unreliable or failure-prone, can be formulated as stochastic optimal control problems involving jimip Markov processes. Such problems are treated in Sethi and Zhang (1994a, 1994c). Karatzas and Shreve (1998) address stochastic optimal control problems in finance involving more general stochastic processes including jimip processes.
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Bibliographische Angaben
- Autoren: Suresh P. Sethi , Gerald L. Thompson
- 2006, Englisch
- ISBN-10: 0387299033
- ISBN-13: 9780387299037
- Erscheinungsdatum: 02.06.2006
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