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Control Theory and Analysis CONTROL THEORY K<0, a>0 AND ANALYSIS LECTURES PRESENTED AT AN INTERNATIONAL SEMINAR COURSE K<0, a<0 TRIESTE, 11 SEPTEMBER - 29 NOVEMBER 1974 co-*0-f ORGANIZED BY THE INTERNATIONAL CENTRE \ FOR THEORETICAL PHYSICS i TRIESTE VWÍ INTERNATIONAL ATOMIC ENERGY AGENCY, VIENNA, 1 976 CONTROL THEORY AND TOPICS IN FUNCTIONAL ANALYSIS Vol. I INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, TRIESTE CONTROL THEORY AND TOPICS IN FUNCTIONAL ANALYSIS LECTURES PRESENTED AT AN INTERNATIONAL SEMINAR COURSE AT TRIESTE FROM 11 SEPTEMBER TO 29 NOVEMBER 1974 ORGANIZED BY THE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS, TRIESTE In three volumes VOL. I INTERNATIONAL ATOMIC ENERGY AGENCY VIENNA, 1976 THE INTERNATIONAL CENTRE FOR THEORETICAL PHYSICS (ICTP) in Trieste was established by the International Atomic Energy Agency (IAEA) in 1964 under an agreement with the Italian Government, and with the assistance of the City and University of Trieste. The IAEA and the United Nations Educational, Scientific and Cultural Organization (UNESCO) subsequently agreed to operate the Centre jointly from 1 January 1970. Member States of both organizations participate in the work of the Centre, the main purpose of which is to foster, through training and research, the advancement of theoretical physics, with special regard to the needs of developing countries. CONTROL THEORY AND TOPICS IN FUNCTIONAL ANALYSIS IAEA, VIENNA, 1976 STI/PUB/415 ISBN 92-0 —130076-X Printed by the IAEA in Austria March 1976 FOREWORD The International Centre for Theoretical Physics has maintained an interdisciplinary character in its research and training programmes in different branches of theoretical physics and related applied mathematics. In pursuance of this objective, the Centre has - since 1964 - organized extended research courses in various disciplines; most of the Proceedings of these courses have been published by the International Atomic Energy Agency. The present three volumes record the Proceedings of the 1974 Autumn Course on Control Theory and Topics in Functional Analysis held from 11 September to 29 November 1974. The first volume consists of fundamental courses on differential systems, functional analysis and optimization in theory and applications; the second contains lectures on control theory and optimal control of ordinary differential systems; the third volume deals with infinite­ dimensional (hereditary, stochastic and partial differential) systems. The programme of lectures was organized by Professors R. Conti (Florence, Italy), L. Markus (Warwick, United Kingdom) and C. Olech (Warsaw, Poland). Abdus Salam CONTENTS OF VOL. I Basic concepts of control theory (IAEA-SMR-17/6).................................................................. 1 L. Markus Concepts of stability and control (IAEA-SMR-17/1)................................................................. 53 P.C. Parks Foundations of functional analysis theory (IAEA-SMR-17/2)................................................. 101 Ruth F. Curtain Control theory and applications (IAEA-SMR-17/3) .................................................................. 179 A.J. Pritchard Finite-dimensional optimization (IAEA-SMR-17 /4 ).................................................................. 223 D. Q. Mayne Reachability of sets and time- and norm-minimal controls (IAEA-SMR-17/37).................... 261 A. Marzollo Existence theory in optimal control (IAEA-SMR-17/61) ......................................................... 291 C. Olech Asymptotic control (IAEA-SMR-17/48) ..................................................................................... 329 R. Conti Controllability of non-linear control dynamical systems (IAEA-SMR-17/71) ...................... 361 C. Lobry Introduction to convex analysis (IAEA-SMR-17/63)................................................................. 385 J.P. Cecconi An introduction to probability theory (IAEA-SMR-17/29)...................................................... 419 J. Zabczyk Secretariat of Seminar..................................................................................................................... 463 IAEA-SMR-17/6 BASIC CONCEPTS OF CONTROL THEORY L. MARKUS School of Mathematics, University of Minnesota, Minneapolis, Minnesota, United States of America Abstract BASIC CONCEPTS OF CONTROL THEORY. After a philosophical introduction on control theory and its position among various branches of science, mathematical control theory and its connection with functional analysis are discussed. A chapter on system theory concepts follows. After a summary of results and notations in the general theory of ordinary differential equations, a qualitative theory of control dynamical systems and chapters on the topological dynamics, and the controllability of linear systems are presented. As examples of autonomous linear systems, the switching locus for the synthesis of optimal controllers and linear dynamics with quadratic cost optimization are considered. I. WHAT IS CONTROL THEORY - AND WHY? Just what is control theory? Who or what is to be controlled and by whom or by what, and why is it to be controlled? In a nutshell, control theory, sometimes called automation, cybernetics or systems theory, is a branch of applied mathematics that deals with the design of machinery and other engineering systems so that these systems work, arid work better than before. As an example, consider the problem of controlling the temperature in a cold lecture hall. This is a standard engineering problem fam iliar to us all. The thermal system consists of the furnace as the heating source, and the room thermometer as the record of the temperature of the hall. The external environment we assume fixed and not belonging to the thermo­ dynamic system under analysis. The basic heating source is the furnace, but the control of the furnace is through a thermostat. The thermostat device usually contains a thermometer to measure the current room temperature and a dial on which we set the desired room temperature. The control aspect of the thermostat is that it compares the actual and the desired temperatures at each moment and then it sends an electric signal or control command to the furnace to turn the fire intensity up or down. In this case, the job of the control engineer is to invent or design an effective thermostat. Let us next look at a control problem from biology. Parts of the world are being overrun by an increasing population of rats. Here the system consists of the living population of rats and the environmental parameters that affect that population. The natural growth of the rat population is to be controlled towards some desired number, say, zero. Here the job of the control engineer is to build a better mouse-trap. From this viewpoint control theory does not appear too sinister. On the other hand, it does not seem too profound. So let me elaborate on the 1 2 MARK US structure of control theory to indicate the reasons why UNESCO and the ICTP believe this subject is important. To organize these ideas, I shall discuss control theory from four viewpoints: (i) as an intellectual discipline within science and the philosophy of science; (ii) as a part of engineering, with industrial applications; (iii) as a part of the educational curriculum at university; (iv) as a force in the world related to technological, economic and social problems of the present and the future. First consider the philosophical position of the discipline of control theory. Within the framework of metaphysics, control theory is a teleological science. That is, the concepts of control theory involve ideas such as purpose, goal-seeking and ideal or desirable norms. These are terms of nineteenth century biology and psychology, terms of volition, will and motiva­ tion such as were introduced by Aristotle to explain the foundations of physics, but then carefully exorcized by Newton when he constructed an unhuman geometric mechanics. So control theory represents a synthesis of the philosophies of Aristotle and Newton showing that inanimate deterministic mechanisms can function as purposeful self-regulating organisms. Recall how the inanimate thermostat regulates the room temperature towards the agreed ideal. Another philosophical aspect of control theory is that it avoids the concepts of energy but, instead, deals with the phenomenon of information in physical systems. If we compare the furnace with the thermostat we note a great disparity of size and weight. The powerful furnace supplies quantities of energy: a concept of classical physics. However, the tiny but ingenious thermostat deals with information - an aspect of modern statistical physics and mathematics. Thus control theory rests on a new category of physical reality, namely information, which is distinct from energy or matter. Possibly, this affords a new approach to the conundrum of mind versus matter, concerning which the philosophical journal Punch once r e m a r k e d , "What is matter? — Never mind. What is mind? - No matter. " But what are the problems, methods and results of control theory as they are interpreted in modern mathematical physics or engineering? In this sense control theory deals with the inverse problem of dynamical systems. That is, suppose we have a dynamical system, for example many vibrating masses interconnected by elastic springs. Such a dynamical system is described mathematically by an array of ordinary differential equations that predict the evolution of the vibrations according to Newton's laws of motion. The customary
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