Mathematical Theory of Optimal Processes

L.S. Pontryagin

• Publisher: Routledge
• Year: 2018
• ISBN:
  9781351433068
• Edition: 1
• Topic: Mathematics
• Subtopic: Mathematics General
• Index: Mathematics

The fourth and final volume in this comprehensive set presents the maximum principle as a wide ranging solution to nonclassical, variational problems. This one mathematical method can be applied in a variety of situations, including linear equations with variable coefficients, optimal processes with delay, and the jump condition. As with the three preceding volumes, all the material contained with the 42 sections of this volume is made easily accessible by way of numerous examples, both concrete and abstract in nature.

CHAPTER I

THE MAXIMUM PRINCIPLE

§1. ADMISSIBLE CONTROLS

We shall consider the behavior of an object whose state at any instant of time is characterized by n real numbers x¹, x², …, xⁿ (for example, these may be coordinates and velocities). The vector space X of the vector variable x = (x¹, …, xⁿ) is the phase space of the object under consideration. The behavior (motion) of the object (from a mathematical viewpoint) consists of the fact that the variables x¹, …, xⁿ change with time. It is assumed that the object’s motion can be controlled; i.e., that the object is equipped with certain “controllers” on whose position the motion of the object depends. The positions of the “controllers” are characterized by points u, of a certain control region U, which may be any set in some r-dimensional Euclidean space E_r. Giving a point

$u=\left(u^1,u^2,\dots u^r\right)\in U$

is equivalent to giving a numerical system of parameters u¹, u², …, u^r. In applications, the case where U is a closed region in E_r is important. In particular, the control region U may be a cube in the r-dimensional space of the variables u¹, u², …, u^r:

$\begin{array}{cc}\left|u^j\right|\le 1,& j=1,2,\dots ,r,\end{array}$

(1)

or any other closed and bounded set in this r-dimensional space. The physical meaning of considering a closed and bounded control region U (in the space of the variables u¹, u², …, u^r) is clear. The quantity of fuel being supplied to a motor, temperature, current, voltage, etc., which cannot take on arbitrarily large values, may serve as the control parameters u¹, u², …, u^r. In addition, because of the physical construction of the object’s control portion, relations given by one, or several equations of the form ϕ(u¹, u², …, u^r) = 0, may exist among the control parameters. In this case, U may have a more or less complicated geometric character. If, for example, there are two control parameters u¹ and u², which, because of the object’s construction, have the form u¹ = cos ϕ and u² = sin ϕ, where ϕ i...

Table of contents

• Cover
• Half Title
• Title Page
• Copyright Page
• Table of Contents
• Editor’s Preface
• Preface to the English Translation
• Introduction
• Chapter I. The Maximum Principle
• Chapter II. The Proof of the Maximum Principle
• Chapter III. Linear Time-Optimal Processes
• Chapter IV. Miscellaneous Problems
• Chapter V. The Maximum Principle and the Calculus of Variations
• Chapter VI. Optimal Processes with Restricted Phase Coordinates
• Chapter VII. A Statistical Optimal Control Problem
• References
• Index