Analytical Methods of Optimization

D. F. Lawden

• Publisher: Dover Publications
• Year: 2012
• Print ISBN: 9780486450346
• eBook ISBN: 9780486145815
• Topic: Mathematics
• Subtopic: Game Theory
• Index: Mathematics

Suitable for advanced undergraduates and graduate students, this text surveys the classical theory of the calculus of variations. It takes the approach most appropriate for applications to problems of optimizing the behavior of engineering systems. Two of these problem areas have strongly influenced this presentation: the design of the control systems and the choice of rocket trajectories to be followed by terrestrial and extraterrestrial vehicles.
Topics include static systems, control systems, additional constraints, the Hamilton-Jacobi equation, and the accessory optimization problem. Prerequisites include a course in the analysis of functions of many real variables and a familiarity with the elementary theory of ordinary differential equations, especially linear equations. Emphasis throughout the text is placed upon methods and principles, which are illustrated by worked problems and sets of exercises. Solutions to the exercises are available from the publisher upon request.

1 STATIC SYSTEMS

1.1 Statement of the problem

The fundamental problem which we shall study can be expressed in the following terms: Suppose the state of a given physical system can be changed by varying the values of N physical quantities denoted by u₁, u₂, . . . , u_N. This process whereby the system’s state is influenced by an external agent or operator will be referred to as controlling the system and the quantities u_r (r = 1,2,..., N) will be called the control variables. The behaviour of the system or the configuration it adopts as a consequence of any particular choice of control variables, we shall designate the system’s response. It will be desired to control the system in such a manner that its response is as close as possible to a standard the operator has his own reasons for striving to attain. It will always be assumed that the operator’s success in achieving this object can be measured by giving the value of a quantity C called the performance criterion (or index). Thus, a response resulting in a large value of C may be judged superior to any other response which gives a smaller value. In these circumstances, our problem is to choose values for the control variables which maximize C. The system response which results from such a choice will be said to be optimal and our problem may accordingly be described as the optimization of systems.

In some cases, it is more natural to assess the performance of a system by reference to the cost to the operator of eliciting a particular response. The quantity C will then be termed the cost or penalty index and the optimization problem will be to minimize its value.

In this chapter, we shall investigate the optimization of systems whose responses to control are of a particularly simple character. Thus, having set the control variables to chosen values, it will be assumed that the system under consideration responds by adopting a corresponding fixed configuration and that the excellence or otherwise of this configuration for the operator’s purpose can be assessed by calculation of an index C; i.e. C will be a function of the control variables. For example, if the system is a yacht, the angles made by the sails and boat axis with the wind could be taken as control variables; having set these, the sailing posture becomes determinate and the boat’s velocity is calculable; ...

Table of contents

• Title Page
• Copyright Page
• PREFACE
• Table of Contents
• 1 STATIC SYSTEMS
• 2 CONTROL SYSTEMS
• 3 ADDITIONAL CONSTRAINTS
• 4 HAMILTON-JACOBI EQUATION
• 5 THE ACCESSARY OPTIMIZATION PROBLEM
• BIBLIOGRAPHY
• INDEX