Variational Methods In Optimum Control Theory

Variational Methods in Optimum Control Theory

"The purpose of this book is to elucidate the variational methods underlying optimum control theory, especially for the electrical engineer, and to acquaint the reader with the application of these methods to the solution of specific technical problems. The book assumes that the reader has the proper mathematical background such as the customary mathematical curriculum of the majority of engineering schools."--Preface.

This book focuses on how to implement optimal control problems via the variational method. It studies how to implement the extrema of functional by applying the variational method and covers the extrema of functional with different boundary conditions, involving multiple functions and with certain constraints etc. It gives the necessary and sufficient condition for the (continuous-time) optimal control solution via the variational method, solves the optimal control problems with different boundary conditions, analyzes the linear quadratic regulator & tracking problems respectively in detail, and provides the solution of optimal control problems with state constraints by applying the Pontryagin’s minimum principle which is developed based upon the calculus of variations. And the developed results are applied to implement several classes of popular optimal control problems and say minimum-time, minimum-fuel and minimum-energy problems and so on. As another key branch of optimal control methods, it also presents how to solve the optimal control problems via dynamic programming and discusses the relationship between the variational method and dynamic programming for comparison. Concerning the system involving individual agents, it is also worth to study how to implement the decentralized solution for the underlying optimal control problems in the framework of differential games. The equilibrium is implemented by applying both Pontryagin’s minimum principle and dynamic programming. The book also analyzes the discrete-time version for all the above materials as well since the discrete-time optimal control problems are very popular in many fields.

This work describes all basic equaitons and inequalities that form the necessary and sufficient optimality conditions of variational calculus and the theory of optimal control. Subjects addressed include developments in the investigation of optimality conditions, new classes of solutions, analytical and computation methods, and applications.

This is a textbook written for use in a graduate-level course for students of mechanics and engineering science. It is designed to cover the essential features of modern variational methods and to demonstrate how a number of basic mathematical concepts can be used to produce a unified theory of variational mechanics. As prerequisite to using this text, we assume that the student is equipped with an introductory course in functional analysis at a level roughly equal to that covered, for example, in Kolmogorov and Fomin (Functional Analysis, Vol. I, Graylock, Rochester, 1957) and possibly a graduate-level course in continuum mechanics. Numerous references to supplementary material are listed throughout the book. We are indebted to Professor Jim Douglas of the University of Chicago, who read an earlier version of the manuscript and whose detailed suggestions were extremely helpful in preparing the final draft. He also gratefully acknowledge that much of our own research work on variational theory was supported by the U.S. Air Force Office of Scientific Research. He are indebted to Mr. Ming-Goei Sheu for help in proofreading. Finally, we wish to express thanks to Mrs. Marilyn Gude for her excellent and pains taking job of typing the manuscript. J. T. ODEN J. N. REDDY Table of Contents PREFACE 1. INTRODUCTION 1.1 The Role of Variational Theory in Mechanics. 1 1.2 Some Historical Comments ......... . 2 1.3 Plan of Study ............... . 5 7 2. MATHEMATICAL FOUNDATIONS OF CLASSICAL VARIATIONAL THEORY 7 2.1 Introduction . . . . . . . .

This classic book covers the solution of differential equations in science and engineering in such as way as to provide an introduction for novices before progressing toward increasingly more difficult problems. The Method of Weighted Residuals and Variational Principles describes variational principles, including how to find them and how to use them to construct error bounds and create stationary principles. The book also illustrates how to use simple methods to find approximate solutions, shows how to use the finite element method for more complex problems, and provides detailed information on error bounds. Problem sets make this book ideal for self-study or as a course text.

When the Tyrian princess Dido landed on the North African shore of the Mediterranean sea she was welcomed by a local chieftain. He offered her all the land that she could enclose between the shoreline and a rope of knotted cowhide. While the legend does not tell us, we may assume that Princess Dido arrived at the correct solution by stretching the rope into the shape of a circular arc and thereby maximized the area of the land upon which she was to found Carthage. This story of the founding of Carthage is apocryphal. Nonetheless it is probably the first account of a problem of the kind that inspired an entire mathematical discipline, the calculus of variations and its extensions such as the theory of optimal control. This book is intended to present an introductory treatment of the calculus of variations in Part I and of optimal control theory in Part II. The discussion in Part I is restricted to the simplest problem of the calculus of variations. The topic is entirely classical; all of the basic theory had been developed before the turn of the century. Consequently the material comes from many sources; however, those most useful to me have been the books of Oskar Bolza and of George M. Ewing. Part II is devoted to the elementary aspects of the modern extension of the calculus of variations, the theory of optimal control of dynamical systems.