Uncertainty And Optimal Control

This book introduces the theory and applications of uncertain optimal control, and establishes two types of models including expected value uncertain optimal control and optimistic value uncertain optimal control. These models, which have continuous-time forms and discrete-time forms, make use of dynamic programming. The uncertain optimal control theory relates to equations of optimality, uncertain bang-bang optimal control, optimal control with switched uncertain system, and optimal control for uncertain system with time-delay. Uncertain optimal control has applications in portfolio selection, engineering, and games. The book is a useful resource for researchers, engineers, and students in the fields of mathematics, cybernetics, operations research, industrial engineering, artificial intelligence, economics, and management science.

An examination of how the rational expectations revolution and game theory have enhanced the understanding of how an economy functions.

The success of so-called optimal control is investigated, by comparing the performance of an optimal control system to the performance of a control system considered as suboptimal, for non-trivial situations where uncertainties regarding the future behavior of the system or its environment exist. It is shown by analytical considerations that optimal control is successful only in limited and restricted circumstances and, more so, that the introduction of optimal control can deteriorate rather than improve the over-all system performance. Some criteria are suggested upon which a decision can be based regarding an optimal control mode. A number of numerical examples are presented in order to indicate the regions in uncertainty space where the optimal control is successful and the regions where the control considered as suboptimal is better than the so-called optimal control. (Author).

This book provides a direct and comprehensive introduction to theoretical and numerical concepts in the emerging field of optimal control of partial differential equations (PDEs) under uncertainty. The main objective of the book is to offer graduate students and researchers a smooth transition from optimal control of deterministic PDEs to optimal control of random PDEs. Coverage includes uncertainty modelling in control problems, variational formulation of PDEs with random inputs, robust and risk-averse formulations of optimal control problems, existence theory and numerical resolution methods. The exposition focusses on the entire path, starting from uncertainty modelling and ending in the practical implementation of numerical schemes for the numerical approximation of the considered problems. To this end, a selected number of illustrative examples are analysed in detail throughout the book. Computer codes, written in MatLab, are provided for all these examples. This book is adressed to graduate students and researches in Engineering, Physics and Mathematics who are interested in optimal control and optimal design for random partial differential equations.

This book highlights new developments in the wide and growing field of partial differential equations (PDE)-constrained optimization. Optimization problems where the dynamics evolve according to a system of PDEs arise in science, engineering, and economic applications and they can take the form of inverse problems, optimal control problems or optimal design problems. This book covers new theoretical, computational as well as implementation aspects for PDE-constrained optimization problems under uncertainty, in shape optimization, and in feedback control, and it illustrates the new developments on representative problems from a variety of applications.

The research being performed under AFOSR Grant 76-2923 is concerned with optimally controlling a system to a specified target when disturbances or uncertainties enter the system. Two general problems in this area are being investigated. One problem is that of determining if there exists a control that assures that the system reaches a specified target set for all possible disturbances. If such a control exists, then the next problem is to determine a control which guarantees that the target is reached and it also optimal in the sense of minimizing a specified measure of the system's performance. Our research effort has been directed toward obtaining methods for answering the questions raised by these problems and thereby aiding in the design of controllers for uncertain systems. (Author).

This coherent introduction to the theory and methods of robust control system design clarifies and unifies the presentation of significant derivations and proofs. The book contains a thorough treatment of important material of uncertainties and robust control otherwise scattered throughout the literature.

From the reviews: "The style of the book reflects the author’s wish to assist in the effective learning of optimal control by suitable choice of topics, the mathematical level used, and by including numerous illustrated examples. . . .In my view the book suits its function and purpose, in that it gives a student a comprehensive coverage of optimal control in an easy-to-read fashion." —Measurement and Control