Differential Equations: Classical to Controlled (Mathematics in Science and Engineering, 162)

Differential Equations Classical to Controlled

This is Volume 162 in MATHEMATICS IN SCIENCE AND ENGINEERING A Series of Monographs and Textbooks Edited by RICHARD BELLMAN, University of Southern California The complete listing of books in this series is available from the Publisher upon request.

Differential Equations Classical to Controlled Dahlard L. Lukes Department of Applied Mathematics and Computer Science University of Virginia Charlottesville, Virginia

1982

ACADEMIC PRESS A subsidiary of Harcourt Brace Jovanovich, Publishers

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United Kingdom Edition published by ACADEMIC PRESS, INC. {LONDON) LTD. 24/28 Oval Road, London NWI

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Library of Congress Cataloging in Publication Data lukes, Dahlard l. Differential eq~ations. (Mathematics in science and engineering) Bibl iography: p. includes index. I. Differential equations, Linear. 2. Differential equations, Nonlinear. 3. Control theory. I. Title. I i. Series.

QA372.L84 515.3'5 ISBN O-12-459980-X

82-6797 AACR2

PRINTED IN THE UNITED STATES OF AMERICA

82 83 84 85

987654321

To my parents, Lawrence and Josephine

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Contents

xi

Preface

Chapter 1 Introduction 1.1. 1.2. 1.3. 1.4.

Chapter 2

Origin and Evolution Sources of First-Order Equations Classical Questions Control Questions

1

4

12 15

Matrix Algebra-The Natural Language of Linear Systems 2.1. 2.2. 2.3. 2.4. 2.5. 2.6. 2.7. 2.8.

Terminology Addition, Multiplication, and Scalar Multiplication The n-Vector Spaces ?}in Systems of Linear Algebraic Equations Matrix Inversion Large-Scale Matrix Computations Using Computers Determinant Functions Eigenvalues, Eigenspaces, Characteristic and Minimal Polynomials 2.9. The Jordan Form of a Matrix

19

20 23

25 29

31 35 39

43

Chapter 3 Linear Constant-Coeffident Equations: Computation 3.1. The Initial-Value Problem 3.2. The ABC Algorithm 3.3. A Vector Method for Computing the Matrix C

47 48

54

viii

Contents 3.4. Explicit Formulas for C(c) and eIl(t) 3.5. Additional Comments about Computing and Changing Variables

Chapter 4

59 60

Linear Constant-Coefficient Equations: Theory 4.1. Preliminary Remarks 4.2. Some Elementary Matrix Analysis 4.3. Existence, Uniqueness, and Solution Representations Using etA. 4.4. The Structure of etA 4.5. The Theoretical Foundation of the ABC Algorithm for Computing etA. 4.6. Explicit Formulas for C(c) and etA.

64 65 72 77 78

83

Chapter 5 Linear Constant-Coefficient Equations: Qualitative Behavior 5.1. Stability and Periodic Oscillation 5.2. The Lyapunov Approach to Asymptotic Stability

92 95

Chapter 6 General Linear Equations with Time-Varying Coefficients 6.1. 6.2. 6.3. 6.4. 6.5. 6.6. 6.7. 6.8.

Widening the Scope Preliminaries The Fundamental Matrix Series Defining ell Existence and Uniqueness of Solutions to Vector Differential Equations and Their Duals Variation-of-Parameters Formulas for Vector and Matrix Differential Equations Estimates of Solution Norms Fundamental Matrices of Equations with Time-Varying Coefficients Continuous Time, Discrete State Markov Processes

106

107 108 112 114 115 128 132

Chapter 7 Commutative Linear Differential Equations 7.1. Introductory Comments 7.2. The Definition and an Example of Commutative Equations

138 139

ix

Contents 7.3. Computation of Closed-Form Solutions to Linear Commutative Equations 7.4. Some Sufficient Conditions for Commutivity 7.5. Matrix Functions and Operations Preserving Commutivity 7.6. The Special Case n = 2 7.7. A Criterion for Exponential Decay of Solutions

140 145 148 157 160

Chapter 8 Periodic Linear Equations 8.1. Periodic Homogeneous Equations 8.2. Qualitative Behavior 8.3. Nonhomogeneous Equations 8.4. Periodic Commutative Equations

Chapter 9

162 167

172

175

Local Existence and Uniqueness Theory of Nonlinear Equations 9.1. Complications That Can Occur 9.2. Local Existence and Uniqueness

Chapter 10 Global Solutions

10.1. Introduction 10.2. Maximal Solutions and Extensions of Local Solutions 10.3. The Behavior of Solutions on Maximal Intervals

181 182

190 190

192

Chapter11 The General Solutio~Dependence of Solutions on Parameters 11.1. Mathematical Preliminaries 11.2. A Space of Curves 11.3. The General Solution (D, 'f) to the Cauchy Problem (D, f) and Continuous Dependence on Parameters 11.4. Differential Dependence of the General Solution (D, 'f) on Parameters and Related Variational Equations

200

202

205

212

Chapter12 Limit Properties of Solutions 12.1. 12.2. 12.3. 12.4. 12.5.

Limit Sets and Invariance Stability of Nonlinear Equations Partial Stability Local Behavior and Linearization about Rest Points Planar Systems

223

231

236

243

250

Contents

x

Chapter 13 Applications in Control Theory 13.1. 13.2. 13.3. 13.4. 13.5. 13.6. 13.7. 13.8. 13.9.

The Controlled and Uncontrolled Equations Transient and Steady-State Parts of the Solution The Frequency Response Function The ABC Approach to Computing Frequency Response Matrices Controllability Altering System Dynamics by Linear Feedback Decoupling and Recoupling-Noninteractive Design Controllability of Nonlinear Equations The Domain C of Null Controllability

261 262 263 266 269 275 284 307 313

References

317

Index

319

Preface

This book deals with the classical theory of differential equations, showing some of the impact that computers and control theory are having on the subject. It is based on notes that I developed in recent years while teaching an advanced-undergraduate beginning-graduate course aimed at applied mathematics and engineering students. These notes succeeded better than the numerous textbooks that I tried in dealing with the somewhat heterogeneous background of my students. All had studied calculus and beginning differential equations and had some familiarity with matrix manipulations. Additional courses in advanced calculus or analysis proved helpful but were not crucial. The book provides a basis for a onesemester course for such students. To complete the text, say in a secondsemester course, the students should first have acquired some knowledge of analysis, linear algebra, and topology. The historical development of differential equations and control theory is sketched in Chapter 1, in which motivation and an overview of the subjects are also provided. Chapter 2 contains a brief review of matrix theory and notions of linear algebra that are drawn on throughout much of the book. It can be skipped and used only as a reference if the instructor wishes to require a course in matrix theory as a prerequisite. On the basis of a heuristic matrix approach, which I have dubbed the ABC method, I introduce in Chapter 3 the computational aspects of linear constant-coefficient systems. The theoretical underpinning and the move to progressively more general equations are found in succeeding chapters, in contrast to the usual approach of treating nonlinear equations first. The loss in efficiency seemed to be outweighed by the pedagogical advantage gained, wherein the applied science students first see the theory building upon real analysis at a less abstract level and only after they are well into xi

xii

Preface

the course. At the same time the more mathematically inclined students get a better picture of where the loss of theoretical detail occurs during the transition to more general equations and more abstract concepts. In recent years, great progress has been made in "the use of computers to solve large systems of linear algebraic equations and to compute eigenvalues. The ABC approach developed in this book reflects my efforts to reduce the problem of generating the analytical (closed-form) solutions to large systems of linear .differential equations to the first problem and hence to exploit this progress . The applications of the ABC formula made here demonstrate its theoretical as well as computational value. The study of the variable coefficient systems that I call commutative equations, which appears in Chapters 7 and 8, grew out of this renewed interest in computing. The theorem concerning linearization of nonlinear equations about a hyperbolic rest point, first studied by Poincare in the context of analytic systems and later settled by Hartman, is given an elementary proof in Chapter 12in order to add further significance to linear equations. In Chapters 9-12, I cover many of the standard topics dealing with nonlinear equations. Dependence of solutions on parameters and associated variational equations are given particular attention. Although Chapter 13 is intended to provide only an introduction to control theory, the material covered is of fundamental importance to anyone with limited opportunities for studying modern control theory. In many respects the feedback theory presented in this chapter goes far beyond the old-fashioned root locus and transfer function methods still being taught in many courses: I believe it is necessary to devote special attention to the computational aspect of the decoupling problem since the abstract theory of the past ten years does not seem to have percolated down into the engineering texts, in spite of the recognized importance of the problem in applications. It should be noted that the sections of Chapter 13 on control theory (1-7) could be covered after finishing Chapters 1-5, and in general the book can be used in several ways. For example, a one-semester introductory level systems-oriented course might cover Chapters 1-6 and 8, and Sections 1-7 of Chapter 13. If students have had matrix theory and some complex function theory, then Chapter 2 could be skimmed over and Chapter 7 inserted into the course coverage. If more emphasis were to be placed on nonlinear equations, using Chapters 1-6,9, and 10 would leave some time to pick and choose from the topics in Chapters 11 and 12. If the text were to be used for a two-semester sequence, the chapters would be covered in the normal order, with the option of skipping Chapter 7. Numerous examples and exercise sections have been inserted at appropriate points throughout the book.

Preface

xiii

I wish both to thank Ruth Nissley for so graciously contributing her energy and professional skill in typing the final manuscript and to express my gratitude to Carolyn Duprey and Faye O'Neil for their help in preparing the earlier drafts.

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Chapter

1

Introduction

This chapter begins with a historical sketch of events that shaped the development of the subject. Its simple examples illustrate the well-established applicability of differential equations. They also provide motivation for the choices of the model equations studied in later chapters and indicate some of the new directions that control ideas have given to the classical theory. 1.1

Origin and Evolution

Since its invention by Isaac Newton circa 1666, the theory of ordinary differential equations has occupied a central position in the development of mathematics. One reason for this is its widespread applicability in the sciences. Another is its natural connectivity with other areas of mathematics. Newton conceived the idea of a gravitational field and concluded that the rate of change of the momentum of a free particle of mass in such a field must equal the gravitational force exerted upon it by that field. His equality is an example of what is now called a differential equation. The necessity for defining the meaning of the rate of change of a non uniformly changing variable prompted his discovery of differential and integral calculus. With these new concepts he was able to explain the tides and place the motion of the moon and planets upon a firm mathematical foundation. The phenomenal success of extensions of his ideas into engineering, physics, mathematics, and other branches of science now has a rich history. The profundity of Newton's original discovery is reflected in the fact that the continually expanding notions of derivative, integral and differential equation still underlie much of modern mathematical research. Following Newton, a number of great mathematicians, including Cauchy, Peano, and Riemann, contributed to what is now referred to as the classical 1

2

1.

Introduction

foundations of differential equations-a major topic of this text. The theory provides an elegant and powerful treatment of existence and uniqueness problems as well as other questions. In effect, it established the logical legitimacy of differential equations as models for scien.tifi.c phenomena.The scientist's concern is thus reduced to selecting the appropriate equations. Control. theory, a relatively modern development, attempts to go a step further. Control phenomena, in contradistinction to control theory, have their primordial origins in nature. The biologist Richard Dawkins regards the gene as a fundamental control element in his view of Darwin's theory of natural selection [6]. He envisions a gene as arising from a special configuration of organic molecules that somehow appeared in the primordial soup with the stability property of attracting sequences of elementary molecules from the environment and then splitting into two copies of the original after a threshold of complexity had been reached-a kind of primitive cell division. Another biologist, Lyle Watson, entertains the idea that these seed molecules might have originated from beyond the earth, possibly from the tails of comets through whose debris the earth passed [26]. At any rate, this replicator soon populated and in the process altered its environment by depleting the supply of elementary molecules. The variability in the replication produced a family of competing replica tor types. Some complexes of replicator molecules stumbled upon adaptive control processes that allowed them to survive the competition and the changing environment. Those discovering reproductive behavior and possessing other strong stability characteristics persisted through" successive generations. Thus, nature advanced toward creating a biological invariant called life. After some 4 billion years, this scenario culminates in the fundamental problem of biology; to determine how the information stored in the DNA of the cell's genes is used in controlling the cellular development of the surviving organisms. Dawkins regards living organisms as elaborate, highly successful survival structures built about their genes. Proceeding from this view, he goes on to offer new explanations for altruism and other unusual forms of behavior exhibited by some species. Watson suggests that man may have reached such a complex state of development that, in some profound manner, the genes' control is linked with and perhaps is superseded by the subconscious mind. Some scientists speculate that man's behavior, institutions, perception of himself, and possibly physical reality itself evolve according to some higher levels of Darwinian natural selection. Whatever the origin and evolution of living things, the present world abounds with organisms, each operating sophisticated control mechanisms. The human body's system for maintaining its temperature within the narrow

1.1 Origin and Evolution

3

range necessary for survival is a good example. Another is its immunological system for identifying and destroying invading bacteria and other foreign substances. Watson discusses the possibility that unstable cell growth is a common occurrence in physiology and that cancer develops when the identification process fails or when for some other reason the stabilizing immunological feedback system malfunctions. Examples of such automatic control systems can easily be multiplied. Civilization has long been concerned with control at various levels of consciousness. Childrearing, education, religion, and politics all involve a high degree of psychological control. The manifold institutions of government-including the maverick education-are highly evolved control devices intended to help manage (but unfortunately often dictating) man's affairs. Thus it appears that control phenomena have pervaded all aspects of life since its beginning and undoubtedly will continue to do so. Only in relatively recent times have engineers finally injected the element of control into their equations. The manner in which this came about is rather interesting. In 1769 James Watt, a Scottish engineer, invented the governor for steam engines. This early example of a feedback control system was a device for maintaining a specific speed of the flywheel of an engine operating under variable load. Before that- time, the applications of the steam engine were quite limited. The Watt governor dramatically transformed the steam engine into a feasible mechanism for delivering steady power and consequently was an important factor in the occurrence of the industrial revolution. Control phenomena attracted very little mathematical interest before 1900, with the exception of J. C. Maxwell's 1868 study of the steady-state error which occurred in applications of the Watt governor [17]. A great period of-industrialization took place from about 1900 up to World War II. It witnessed the development of large-scale power generation and transmission, aeronautics, the chemical industry, communications, and electronic engineering. This activity gave rise to the design of many pneumatic, hydraulic, electromechanical, and electrical regulators. The period, sometimes referred, to as the era of handbook engineering, was narrowly directed toward a few specific applications, relied heavily upon low-order linear models in Laplace transform form, and generally employed rather limited areas of mathematics. This status began to change rapidly during World War II, stimulated by such problems as fire control (which involves the rapid and accurate positioning of guns). The great advances in electronics, followed by the emergence of analog and digital computers, greatly expanded the horizons of control. The upgrading of the mathematical training of engineers was another

4

1.

Introduction

important factor involved in the transformation of control engineering into a science. These engineers were better prepared to define problems mathematically and to develop new concepts of control. This is tum allowed mathematicians to bring to bear upon the problems the power of abstract mathematics. Control theory is now regarded as one of the major areas of mathematics. Increasing numbers of scientists and laymen are recognizing the controltheoretic aspect of many of the serious problems of mankind that arise from the expansion and growing complexity of the world's evolving systems with their seemingly inextricable coupling. A major difficulty is that the actual dynamics and coupling of these systems are often poorly understood and sometimes ignored for economic, political, or other reasons. One important example of this is the problem of controlling insect damage to crops [1 J. It is estimated that in 1977 at least 250 million pounds of pesticides were dumped on California alone in an effort to combat insect damage to farm crops. (The complexity of the problem is indicated by the fact that in spite of these chemicals there remain 70 major insect pests that plague corn.) With Darwin's natural selection operating as predicted, the intended victims often develop resistant varieties, and these survivors are even more difficult to control. Moreover, the chemicals often create harmful imbalances in the ecosystem that trigger other infestations, contaminate drinking water, and generate many other problems. Again, the examples can be multiplied. The critical nature of these problems sorely awaits applications of control science outside classical areas of engineering. During the past 15 years, a great deal has been learned about the mathematical control of finite-dimensional systems, although the theory is far from complete. Currently there is considerable interest in the control theory of systems with time-delay and in systems modeled by partial differential equations (see [20J). Many of these problems can be treated by the theory of functional analysis as ordinary differential equations in infinite-dimensional (function) spaces. In this book, we treat primarily the classical theory of differential equations at the more modest finite-dimensional level where, ordinarily, more detailed results are available. This provides a firm foundation for more advanced study of the control aspects of such equations for those wishing to pursue the subject. 1.2 Sources of First-Order Equations

Although only limited space can be devoted to discussing practical applications in this book, some feeling for simple kinds of problem situations

5

1.2 Sources of First-Order Equations

-

rl rl

Tank

u(t)1 - -

I

rz

Tank 3

Tank 2

f---

I--

rl

Fig. 1.2.1 A heat-flow system with input.

which can be modeled by differential equations is desirable. The following examples serve that purpose.

Example 1.2.1 Flow and Mixing. Suppose that water is pumped between three thermally insulated tanks through connecting pipes at rates '1 = 1, '2 = 2 gallons per minute, configured as in Fig. 1.2.1. Initially each tank contains 100 gallons of water at specified temperatures. The temperatures of the water in the tanks change with time due to the pumping between tanks. Tank 1 has a heat exchanger which allows heat to be injected or extracted directly at a rate of u(t) British thermal units (Btu) per minute. Thus u(t) is regarded as an input to the system and the output of interest is the temperatures of the water in the tanks. Since the temperature of a gallon of water is proportional to the heat that it contains, it is sufficient to work with xlt), the amount of heat in the ith tank at time t (i = 1, 2, 3). By taking into account the heat flow both into and out of each tank and using the fact that the heat flow rate through a pipe equals the water flow rate times the heat per unit volume of water, it is easy to see that the appropriate system of differential equations satisfied by the x;(t) (assuming the heat loss and the contents of the pipes to be negligible) is

2[x100 + u(t), 2(t)] dX2(t) = l[x (t) ]_l[x100 100 ' dt dX3(t) = l[x (t)] 1[x2(t)] _ 2[x 3(t)] 100 + 100 100 . dt

dx 1(t) . dt

= 2[x 3(t)] _ 100 1

1(t)]

(1.2.1)

1

Letting x(t) denote a column-vector variable with scalar coordinate variables X2(t), X3(t), system (1.2.1) can be rewritten as a single vector-matrix

Xl (t),

1. Introduction

6

differential equation

x=

Ax

+

f(t), where

- lo A=

f(t) =

[

1

0

50

1

TOO

-TOO

1 100

1

TOO

o1 ] , --10

U(t)] [~ .

(1.2.2)

(1.2.3)

The same model would apply to other mixing problems such as the flow of pollutants in a water supply system, with the tanks being lakes or reservoirs. Example 1.2.2 Mechanics. Consider the forced spring-mass system with friction indicated in Fig. 1.2.2. The mass m moves in response to the spring force - k 2 x generated by displacement of the mass x units from equilibrium, friction force - k 1 x, and external driving force u(t). The k 1 and k 2 are nonnegative constants of proportionality associated with the assumed standard models of springs and friction. Application of Newton's law of motion leads directly to the second-order scalar equation

(1.2.4) By introducing variables Xl = x and X 2 = dx.ldt, Eq. (1.2.4) can be transformed into the equivalent first-order system (1.2.5) This system can likewise be written in the vector-matrix form x = Ax

u(t)

Fig. 1.2.2

A forced spring-mass system with friction.

+ f(t),

1.2

7

Sources of First-Order Equations

where (1.2.6) (1.2.7) Example 1.2.3 Electrical Circuits. Differential equations are often used for modeling electrical circuits. Recall that the voltage drop across an idealized resistor, inductor, or capacitor is vR = Ri, vL = L dildt, or uc = (1/C) i dt, respectively, where R, L, and C are physical constants and i(t)is the current passing through the circuit, Thus application of Kirchhoff s law, which says that the total voltage drop around a simple closed circuit must be zero, shows that the current i ( t ) in the circuit of Fig. 1.2.3, containing a timevarying voltage source e(t), must satisfy the-equation di Ldt

+ Ri +

AJ:

i(z)d z = e(t)

(1.2.8)

(assuming the charge on the capacitor to be zero at t = 0). Introducing the variables x 1 = i, x 2 = di/dt, and u = (1/L) deldt, the differentiated form of (1.2.8) transforms into the first-order system (1.2.9)

which once more has the vector-matrix form f

=

0

Ax

+ f (t),this time with (1.2.10) (1.2.11)

More complicated forms of Examples 1.2.1-1.2.3 can easily be obtained by R

C

I

Fig. 1.2.3 A-imple circuit.

8

1. Introduction

considering more tanks, springs and masses, or circuit elements connected in countless complex patterns. All the resultant systems of differential equations as well as the equations arising from many other areas of application are subsumed by the general equation of the form

x=

Ax

+ f(t),

(1.2.12)

in which x is a column-vector of n real variables, A is an n x n real matrix, and f(t) is a prescribed column-vector of n real-valued functions of t. In Chapters 3- 5, we develop the classical theory of linear differential equations with constant-coefficient matrix as given by (1.2.12). Obviously a substantial amount of matrix algebra is involved in such a treatment and this is anticipated by the review of matrix algebra that is presented in Chapter 2. The classical theory of (1.2.12) presented will be found to be in a rather satisfactory state of completion. It provides, amongst other results, an effective method for computing all solutions to (1.2.12) in a closed form. A moment of reflection makes it clear that only slight changes in the physical problems underlying Examples 1.2.1-1.2.3 produce systems of equations requiring a model more general than (1.2.12). For example, suppose that the value of the flow rate rl of the system presented in Example 1.2.1 were changed from r 1 = 1 to r 1 = 2. It is easy to check that this would require replacement of the real matrix (1.2.2) in the model equation by the matrix function of t, . [- 2/(50 - t) 1/(50 - r) 1/(50 - t)

0

-lo

1/(50 + t) ] 0 . 1/(50 + t)

to -

(1.2.13)

Clearly, something as simple as an error in a pumping rate or valve setting could turn the original system into one with time-dependent coefficients. One can as well envision the situation for which in Example 1.2.2 heat generated from the mass sliding along the surface or from other sources causes the' coefficient of friction k 1 to change with time. Likewise the resistor in Example 1.2.3 might heat up with the operation of the circuit and thereby result in a time-dependent value of resistance R. Again, these time-dependent effects would appear in the matrix coefficient of (1.2.12) and thus it would seem appropriate to expand the theory to cover the more general equation

x=

A(t)x

+ f(t),

(1.2.14)

in which the coefficient A(t) is an n x n real matrix-valued function of t. This indeed is on the program. It turns out that important questions concerning the behavior of many nonlinear differential equations as well can be studied

9

1.2 Sources of First-Order Equations

through associated (variational) linear time-dependent equations. This provides additional cause for interest in equations of type (1.2.14). Differential equations of the form (1.2.12) and, more generally, (1.2.14) are called linear first-order equations. These equations are sufficiently general to cover many important applications. However there are problems where they are not appropriate. For example, in the study of large-amplitude oscillations of the system in Example 1.2.2, some springs might be better modeled with the spring force - k2 x replaced by - k2 x 3 • The resultant differential equations replacing (1.2.5) would. then be (1.2.15) which is no longer of the linear type (1.2.14). Similarly, if the smoothness of the surface upon which the mass slides were not uniform, then the model of the friction force might require alteration; e.g., - k 1 Xmight be replaced by -k 1x2 X. This too would destroy the linearity. The same kinds of remarks apply to Example 1.2.3. In fact the famous Van der Pol equation .2 d2i dt 2 + (I

-

di.

1) dt

+1=

u(t)

(1.2.16)

arose from attempts to account for self-excited electrical oscillations in circuits by assuming the circuit resistance dependent upon the current. The first-order form of this equation, of course, is again not linear. Examples of such nonlinearities led researchers to the study of equations of the type

d2x dt2

dx

+ f(x) dt + g(x) = u(t)

(1.2.17)

called Lienard equations and, more generally, 2x

d dt2

+f

(dX) dx x, dt dt + g(x) = u(t),

(1.2.18)

the Levinson-Smith equation, in which the coefficient function f is allowed to depend on the derivative dxldt as well as on x. Discussion of these and other nonlinear equations will reappear in later chapters. For some applied problems a linear or slightly altered (perturbed) linear model is altogether inappropriate; e.g., long-term motion of bodies in gravitational fields (Newton's equations were nonlinear) or population models. In the final design stage of engineering it is frequently necessary to include numerous nonlinear terms in the simulations in order to fix the hardware specifications accurately.

10

1.

Introduction

Fortunately a substantial part of the classical theory applies to equations of the very general form dx (1.2.19) dt = f(t, x), with only rather mild assumptions imposed on f; for example, that f be a real n vector-valued function satisfying some continuity or differentiability condition. The precise details and consequences of the nonlinearities will emerge in later chapters. EXERCISES

1. Rewrite the following equations in first-order form, identifying the matrix coefficient and forcing term of the form (1.2.14) for those cases in which the system is linear. (a) x + f(x) X + g(x) = u(t), (b) x + f(x, x) x + g(x) = u(t), (c) x + (sin t) x + 2x = u(t), (d) {m 1 ~1 + k 3 ~1 + (k 1 + k 2 ) X l - k 2 X2 = ul(t),

m2 X 2 + k 3 X 2 - k 2X l + k2X2 = U2(t). 2. Find a scalar differential equation whose first-order form is x = Ax f(t) in which

+

y const.

Verify that the matrix function A(t) given by (1.2.13) is correct. Show that if x(t) is a solution to the homogeneous vector-matrix equation x = Ax, where A is an n x n constant matrix, then J& x(t) dt = 0 implies that x(T) = x(O). (Although its proof requires a result not yet derived, the stronger conclusion that x(t + T) = x(t) for all t is valid, i.e., x(t) is periodic . of period T.) 5. A small drop of oil is placed at the rest point of the mass in Fig. 1.2.2. Which of the following alterations of (1.2.4) best reflects that phenomenon? (a) m x + 2k l X + k2 x = u(t), X2 (b) m x + k l (1 - e- ) x + k 2 x = u(t), (c) m x - k l X + k 2 x = u(t), (d) m x + k 1 X + k 2 x 3 = u(t). 6. Derive the differential equations model for the forced spring-mass system indicated in Fig. 1.2.4, letting Xl and X2 denote the displacement of 3. 4.

11

1.2 Sources of First-Order Equations

k3

Fig. 1.2.4 A double forced spring-mass system with friction.

the respective masses from equilibrium and using linear modeling as in Example 1.2.2. 7. Three tanks are connected by pipes through which salt brines flow with xi(t)denoting the number of pounds of salt in tank i ( i = 1,2,3) at time t . The variables u1 and u2 are pure salt flow rates. Determine the configuration of pipes, flow rates, and volume of water initially in each tank that would be modeled by the equations

What is the significance of the negative coefficient of u2 in the equation for i2? 8. Derive the differential equation model for the currents i l ( t ) and i z ( t ) passing through the respective resistors in the electrical circuit indicated in Fig. 1.2.5. 9. The populations x1and x2 of two predator-prey species of fish occupying a lake are modeled by the nonlinear differential equations i 1

=

~ 1 x 1

klXlxz,

i 2

=~

2 x 2 k2XIX2,

in which p l , p z , k l , k2 are positive constants. Discuss the possible interpretation of the terms in the equations in context of reproduction and losses due to predation. Think of situations in which alternative models of birth and predation might be appropriate.

Fig. 1.2.5 A two-stage electrical network.

12

1. Introduction

1.3 Classical Questions

A major part of the classical theory is concerned with the pair of equations

dx

dt = f(t, x),

X(t) = ~,

(1.3.1)

called the Cauchy or initial-value problem. Roughly speaking, the function f and parameters r, f are presumed given, with a solution x( .) being any continuous function defined on an open interval that contains t and satisfying (1.3.1) on that interval. In the text, we deal with the existence and uniqueness of solutions to (1.3.1) and study their dependence on the parameters r and ~. The general results obtained are particularly significant since, with the exception of the special equations of type (1.2.12), it is usually impossible to represent solutions to such problems in terms of elementary functions. In Chapter 3 we present a method (the ABC method) for constructing an explicit representation of the solution to the initial-value problem associated with (1.2.12) using only the operations of finding roots of a polynomial (i.e., the eigenvalues of A), solving systems of linear algebraic equations, and performing elementary integration. Even for this simplest class of equations, an independent existence proof is needed to substantiate critical steps in the algebraic construction. In applications of differential equations, any information about the qualitative behavior of the solutions is usually extremely valuable. For equations of type (1.2.12), a rather complete picture is presented in Chapter 3. The solution is shown to be the sum of a forced solution and the unforced solution. In particular, it turns out that for a fixed matrix A, a coordinate function Xi(t) of the unforced solution x(t) to (1.2.12) can exhibit one of the forms indicated in Fig. 1.3.1 (neglecting transients), and computations on A will predict which can be expected. In applications in which xi(t) denotes the difference (error) between the actual value and the ideal value of some variable in an operating system, forms (a) or (c), having decreasing amplitude, are usually the most desirable, with the periodic form (e)sometimes tolerable but unstable forms (b) and (d) assuredly disastrous. The theory also provides estimates of the rate of attenuation of (a) and (c) as well as information concerning the frequencies of oscillation that might occur. Another property of the solutions that is of importance in applications is the extent of the coupling between the coordinate variables of the solution; i.e., the extent to which a change in initial values of some of the xi(t) willaffect the others. (For example, a well-designed automatic control system for an airplane would not allow one

13

1.3 Classical Questions

I

(el

Fig. 1.3.1 Qualitative response forms of unforced solutions.

orienting variable to respond to a disturbance in another.) Here again the theory is helpful. The computational and' qualitative theory of the time-varying equation (1.2.14) is less complete, as might be expected due to the increased generality of the equation. Although it turns out that a certain finite dimensionality of the solutions persists, the variety of possible qualitative behavior is so rich that it is difficult to select any one set of properties upon which to develop the theory and it appears impossible to derive a comprehensive notion of closedform solution that approaches the relative simplicity of that of constantcoefficient equations. Specifications arising from applications provide some insight into which properties might be most important, but presently that source likewise appears overwhelmed by the possibilities. It is clear that any results which will predict periodicity in the solutions or detect the stability property that x ( t ) 0 as t 00 are of utmost interest. Some rather detailed results concerning the structure and qualitative behavior of the solutions are provided in this book for the special cases in which A@)is periodic or satisfies a certain commutivity property. In applications it is often necessary to resort --+

--f

14

1. Introduction

to numerical integration or computer simulation of the equations to obtain an adequate understanding of the behavior of the solutions. The broader generalization from linear time-varying equations to nonlinear equations (1.3.1) creates much greater difficulties. Even for autonomous equations for which f in (1.3.1) is a function of the variable x alone, the finite dimensionality disappears and new complications arise. One of these is what we call the immortality problem; that is, the phenomenon in which even though f in (1.3.1) is defined for all values of its argument, the solutions to the initial-value problem can fail to exist for t outside a finite interval containing r. An example would be the nonlinear equation (1.3.1) associated with the second-order nonlinear equation ii

+ uu

- u 3 = 0,

(1.3.2)

which has a solution u(t) = 1/(t - 1) that cannot be extended beyond the interval - 00 < t < 1. In spite of the incomplete state of the theory, later chapters of this book do contain some often useful results dealing with the immortality problem and other aspects of qualitative behavior. These results are largely based upon perturbation techniques, differential inequalitites, and linear variational equations. In passing, it should be mentioned that modern mathematicians have made serious attempts to deal with the question of what form the qualitative theory should take when the equations are generalized beyond the linear constantcoefficient type modeled by (1.2.12). One idea considered by Smale [24] and others is to define some choice of equivalence relation on, roughly speaking, all differential equations on some region of (t, x)-space (more generally, on a manifold) which in some sense preserves the qualitative behavior of the solutions. Next some choice of a subset of differential equations is made, satisfying two conditions: The subset must be sufficiently rich to provide approximations to all differential equations, on one hand, and, on the other hand, if a differential equation is approximated sufficiently accurately by another from the prescribed set, then both will be in the same equivalence class. Ideally, all the choices would be made so that the equivalence classes could be distinguished by numerical and algebraic invariants. This would be a reasonable compromise with the unrealistic goal of trying to in some sense solve and completely understand the behavior of the solutions to all differential equations. Thus far, such a program has not proved to be of much practical value in studying specific types of nonlinear equations. To a considerable degree the technical difficulties encountered in the study of nonlinear equations have turned the attention of many engineers and mathematicians toward new questions about linear equations, namely, toward the control theory of linear systems. Needless to say, nonlinear equations reappear as an obstacle in control theory.

15

1.4 Control Questions

I

--J----..

s

u - -........

... x

Fig. 1.4.1 An input-output system.

1.4 Control Questions

A natural setting for the application of control theory is any situation involving one or more input-output systems, as depicted in Fig. 1.4.1, involving input or control variable u and output or state variable x. The intuitive notion of an input-output system is quite general and ordinarily applies to any phenomenon, real or imagined, which conforms to a causeeffect relationship. Examples would include problems modeled by difference, algebraic, ordinary differential, partial differential, or functional-differential equations. There is no need for a formal definition here, since the example to be discussed shortly will serve the purpose adequately. Control theory deals with the problem of connecting compatible inputoutput systems to create new input-output systems satisfying some criterion of performance or otherwise directed toward some goal. As discussed in earlier sections, nature seems to be highly successful in this process. To illustrate the idea, consider two input-output systems Sl and S2 of compatible types, connected as indicated in Fig. 1.4.2. The resultant system S3 with input variable U3 and output X 3 is described as the consequence of wrapping a feedback system S2 about Sl' An example of why and how this might happen will now be discussed. The control-theoretic nature of Examples 1.2.1-1.2.3 becomes apparent under closer examination of the term f(t) in their differential equation models. In Example 1.2.1,

J

U(t)l

[ ~

f(t) =

= u(t)

[1]

(1.4.1)

~ .

The differential equations (1.2.1) can be written as

d [Xl]

dt

X2 X3

=

[-lo °:~o lo0-L][Xl] + 1~0 -

100100

X2

-50

X3

[1]

u(t) 0 . 0

(1.4.2)

Here the scalar control or input function u(t) provides an opportunity to influence the vector output or state vector variable X as it evolves with time. Observe that the value of the control function u(t) at time t directly affects the

16

1-------,

"'1 C

1"'·1

5,

5,

L

1. Introduction

hr" J

Fig. 1.4.2 Connecting input-output systems using feedback.

Xl

rate of change of (t), whereas the numbers in the coefficient matrix A direct the manner in which this control influence on Xl (t) indirectly feeds into the other state variables. Control theory attempts to illuminate the consequences of the manner in which the control variables enter the equations and their capabilities for altering the uncontrolled dynamics of the system. It tends to focus upon a limited number of mathematical problems abstracted from the study of applied problems. Undoubtedly other important concepts and fundamental problems will emerge as the field of applications expands. It will be sufficient, in this introductory chapter, to mention a few which occur in the continued discussion of Example 1.2.1. One fruitful concept is that of system controllability. Roughly speaking, the idea is to assess those output states which can be achieved by applying all allowed inputs. For Example 1.2.1, a typical controllability question would ask whether, over a prescribed finite interval of operation to ::s;; t ::s;; t l and for arbitrarily prescribed initial and final temperature values of water in each tank, there exists some appropriate control input function u(t) which will drive the initial temperatures at time to to their final prescribed values at time t i - It turns out that if the only constraint upon u(t) is that it be continuous, then it is in fact always possible, whatever the values of to < t 1 and prescribed temperatures, although this is not at all obvious. Similar questions could be asked concerning Examples 1.2.2 and 1.2.3, whose respective control differential equations are

[Xl] [0 1 ][Xl] + u()t[OJi ' -k /m -kdm d [Xl] [0 1 J[XlJ +ut()[OJLl' dt = -1/LC -R/L

d dt

X2

X2

=

2

X2

X2

(1.4.3) (1.4.4)

Again the answer is in the affirmative, and thus the mass in Fig. 1.2.2 can be driven from any state of position and velocity to any other such state over any

17

1.4 Control Questions

finite time interval by appropriate choice of input function u(t). From the similarity between (1.4.3) and (1.4.4) it is clear that the result translates into a corresponding valid statement about the driven circuit of Fig. 1.2.3. Other examples complicated by more state variables and multiple control inputs are easily imagined. Obviously the controllability can fail for those systems involving subsystems that are totally disconnected from inputs and controllable state variables, and in more subtle ways whose discussion is deferred to later chapters. The concept of controllability has strong mathematical appeal and obvious utility in mapping out an upper limit of the extent to which a system might be controlled. Interest in the phenomenon of an even more compelling sort has been stimulated through discovery of its remarkable implications concerning other important aspects of control. Stabilizability is one such notion, and automatic regulation is another. The general meaning of these terms is most easily conveyed by further discussion of Example 1.2.1. Intuition suggests that if the input u(t) into the system of Fig. 1.2.1 eventually became and remained zero, then the heat in the system would tend to a uniform distribution due to the circulating flow. Hence the water in each tank would approach the same limiting temperature

Too = _1_ [x? 300e

+ x~ + x~ + roo u(a) daJ Jo

(1.4.5)

as t - 00, in which e is the specific heat of water in the compatible units of calories per gallon per degree. Suppose that the intended purpose of the system were to drive the temperature of the water in the tanks to some prescribed value Too' If the initial temperatures were measured, the values of x? (i = 1, 2, 3) would be known, and then upon assuming some particular form of u(t) (such as constant for an appropriate initial interval and then forever zero), Eq. (1.4.5) could be employed to solve for the appropriate u(t). Each choice of Too and each "run" of the system would require recalculation of u(t) and an operator to turn the heat-exchanger on and then off. This raises the question of whether it might be possible to design a (regulator) device that would sense the tank temperatures and operate upon these along with a prescribed temperature command to drive the temperatures toward the prescribed value automatically. A simple form of regulator that might be considered would be of the linear type (1.4.6) in which the k, (i

= 1,2,3) are parameters to be determined and

Uc

is a

18

1. Introduction

constant input command. Thus the control problem is transformed into the study of the system with parameters kl> k 2 , k 3 , ue

~ [Xl] _ [- (k~ dt

X2 X3

-

100 1

TOO

lo)

3] [Xl] ~k12 fa -k 0 100 1

TOO

X2

-

1 50

X3

+

Ue

[1]0

(1.4.7)

0

resulting from substitution of (1.4.6) into (1.4.2). This equation is a concrete example of an input-output system with input Ue and output X = (Xl' X2' X3) obtained by wrapping a feedback system around the system of Example 1.2.1 in order to solve a control problem. It fits into the abstract diagram of Fig. 1.4.2 with U3 = Ue , 8 1 the input-output system of Fig. 1.2.1,and 8 2 the affine operator defined by Eq. (1.4.6). The stabilization problem is that of determining which values of the parameters k, (i = 1,2,3) in (l.4.7) will ensure that each x;(t) approaches a limit as t --+ 00 for every initial value X O of x. One elegant result of control theory states that the possibility for stabilization is ensured by the system's controllability. Hence the search for the appropriate values of the k;s is not futile. This is but one of many interesting relationships between various concepts of control theory. The question of how the limiting values ofthe X; depend on the kis and Ue is also of vital interest. For example, if the regulator were intended to drive all tank temperatures to the same prescribed value automatically, then it would be desirable to arrange for the limiting temperature to be some scalar multiple of U e if possible. The qualitative behavior of the responses of the feedback system to the initial X O and Ue , as well as certain quantitative properties such as the rate of convergence, also would likely bear upon the final choice of the values of the k;s in completion of the design. It is hoped that this discussion of the examples has given the reader at least a glimpse of the manner in which control problems can lead to interesting questions about differential equations. It should also be evident that a firm foundation in the classical theory of differential equations is a logical prerequisite for the pursuit of such questions. Hence, in Chapter 2 we shall begin the study of the classical theory by reviewing some basic matrix algebra.

Chapter

2

Matrix Algebra-The Natural Language of Linear Systems

The invention of vectors and matrices arose from the need to manage large numbers of variables and equations efficiently. Although eventually they must be pursued to a somewhat higher level of abstraction, the basic ideas are quite visible in the problem of solving systems of linear algebraic equations. That problem is taken as the starting point for a rapid review of the standard theory discussed in the context of finite-dimensional real and complex number spaces. This chapter paves the way for the elementary treatment oflinear systems of differential equations with constant coefficients that appears in Chapters 3 and 4. It also provides a convenient intermediate point from which to complete the transition to the abstract axiomatic view of vector spaces as its need begins to materialize at the end of Chapter 4. 2.1 Terminology

An m x n matrix A over afield!F is defined as any function with domain {(i,j)ll ~ i ~ m, 1 ~j ~ n} and range in a field fli. In this book, the fields

employed in the discussions of differential equations are all subfields of the complex number field re, i.e., subsets of re closed under the operations a - b and ab - 1, b :1= 0 on its elements a, b. The most frequently occurring examples are fli = re or the field of real numbers !F = fJl. Elements of fli are called scalars.

The standard notational convention denotes the value of A at (i,j) by A;j' called the (i,j)th element of A. When the elements of A are arranged in the 19

20

2. Matrix Algebra-The Natural Language of Linear Systems

rectangular array

(2.1.1) A m2· · ·

the index (variable) i is interpreted as a row position and j as the column position. Conversely, every such array offield elements determines a matrix A over IF. If m = n, the A is called square. The positions in A occupied by Au (i = 1, 2, ... , n) are referred to as the main diagonal. A square matrix is called a diagonal matrix if all its elements off the main diagonal are zero, and a diagonal matrix A with all A ii = c is called a scalar matrix, which is identified with the scalar c in IF. An upper-triangular matrix is one whose elements Aij satisfy Aij = 0 for i > j, and a lower-triangular matrix is one satisfying that equality for i < j. 2.2

Addition, Multiplication, and Scalar Multiplication

Problems of working with large numbers of scalar variables and numerical data motivated development of matrix algebra as an extension of the one-dimensional scalar case. The following are the standard operations which arose:

Definition Let A and B be m x n matrices over a field IF. The matrix sum + B is the m x n matrix defined by

A

(A

+

B)ij = Aij

+

Bij'

(2.2.1)

It is a simple matter to verify the following consequences of definition (2.2.1): A A

+ (B

+B = + C) =

+ A; (A + B) + C B

(2.2.2) (2.2.3)

for all m x n matrices A, B, C over IF; A+Z=A

(2.2.4)

has a unique solution for each m x n A, namely, Zij = 0 for all i.j; Z is called the m x n zero matrix and is denoted by 0; and A+N=O

(2.2.5)

has a unique solution N for each m x n matrix A, namely, N i j = - A ij for all i,j; N is called the additive inverse of A. The standard notation for N is - A.

21

2.1 Addition, Multiplication, and Scalar Multiplication

Definition A second operation on an m x n matrix A over f/i called scalar multiplication by c in f/i is defined by

for all

(CA)ij = cA ij

(2.2.6)

i,j.

As a simple consequence of the definitions it follows that (cIC2)A

=

(2.2.7)

CI(c2A),

+ c2)A = cl(A + B) =

cIA

(CI

cIA

+ C2 A, + c .B,

(2.2.8) (2.2.9) (2.2.10)

lA = A

for all m x n matrices A, B over f/i and scalar

Cl' C2

in f/i.

Definition Let A and B be m x rand r x n matrices, respectively, over f/i. The matrix product AB is the m x n matrix defined by (AB)ij

=

r

I

k=1

(2.2.11)

AikB kj

= 1,2, .. 0' m;j = 1,2,. 0" n). Note the compatibility requirement, namely, that A must have the same number of columns as B has rows.

(i

The appropriateness of the definitions of matrix addition and multiplication for producing extensions of familiar algebraic operations on scalar variables to matrices is supported by the following consequent properties: (2.2.12)

(AB)C = A(BC),

+ B)C = C(A + B) =

(A

AC CA

+ BC, + CB

(2.2.13) (2.2.14)

for all matrices over f/i of appropriately compatible sizes. The manner in which algebraic properties of f/i extend to matrices via definitions (2.2.1) and (2.2.11) is illuminated by the following calculation: [(A

+ B)C] ij =

I

(A

k=l

+ B)ikCkj

r

=

I

k=l

(A ik

+ Bik)Ckj

r

=

I

k=l

= (AC

AikCkj

+

+ BC)ij'

I

k=1

=

I

k=1

BikC kj

(AikCkj

=

(AC)ij

+ BikCkj) + (BC)ij (2.2.15)

22

2. Matrix Algebra- The Natural language of linear Systems

Since (2.2.15) holds for all i, j, obviously the corresponding elements of (A + B)C and AC + BC are the same elements of:F and (2.2.13) is thereby proved. There is a matrix generalization of the multiplicative identity 1 in :F. That is, (2.2.16)

IA = A = AI

has a unique n x n matrix solution I over:F holding for all n x n A over :F; namely, (I)ij =. bijwhere bij takes on the value 1 in :F when i = j and the value 0 in :F when i '" j. I is called the n x n (multiplicative) identity matrix over:F and is denoted by In when its size needs to be emphasized. For n x n matrices A over:F and c in:F, cl; is a scalar matrix and (2.2.17) Thus it is natural to identify each c in:F with the matrix cl.; Scalar matrices commute, under multiplication, with all square matrices of corresponding size and, more generally, for all scalars c in :F, (2.2.18)

(cA)B = A(cB)

for all matrices A, B over :F of compatible size. Thus the definitions of matrix addition, multiplication, and scalar multiplication provide extensions of many of the formal rules of algebra in :F to matrices over :F. A word of warning: Even with square matrices, not all the formal manipulations and inferences of elementary algebra carry over to matrices. For example,

(2.2.19)

AB", BA

for some square matrices A and B; and AB = 0

(2.2.20)

does not imply that at least one of A, B is the zero matrix. Hence while a factorization such as (2.2.21) is valid for all n x n matrices A over gj, the factorization A2

-

B 2 = (A - B)(A

+ B)

(2.2.22)

fails for matrices satisfying (2.2.19), and it would be erroneous to conclude

23

2.3 The n-Vector Spaces 9'"

from the equation (2.2.23) that A = ±2In • EXERCISES

1.. Prove the following matrix identities directly from the definitions of matrix sum, product, and scalar product: (a) A + B == B + A, (c) A(BC) = (AB)C, (e) (- c)A = - (cA),

(g) cO

(b) (d) (f)

= O.

A + (B + C) = (A + B) C(A + B) = CA + CB,

+ C,

OA = 0,

for matrices A, B, C and scalars c in fF. 2.. A nonstandard product A * B of an r x n matrix A with an m x r matrix B, both over a field fF, is defined by (A

* B)ij =

r

L AkjB ik· k=l

(a) Do the identities corresponding to (c) and (d) of Exercise 1 hold for the nonstandard product? (b) Solve the equations X * A = A and A * X = A for the unknown n x n matrix X with A likewise n x n. (c) Find matrices A, B for which A * B i= AB. (d) Find an example 'for which A * B i= B * A. 3. Find an example of a square matrix A for which A 4 i= 0 but AS = O. 4. Evaluate (A - 2B)2 - A 2 + 4AB - 4B 2 in which A =

101]

[o -1

1 2, 1 3

B=

[~

-1

~ ~].

1 0

2.3 The n-Vector Spaces 17-

The n x 1 matrices over a field fF are called n-uectors over fF, with the collection of all such n-vectors being denoted by fFn. As matrices, the elements of fFn can be added and multiplied by elements of fF to again produce n-vectors in fFn. These two operations thus conform to Eqs. (2.2.1)(2.2.10). fFn together with the two operations is called the n-oector space

24

2. Matrix Algebra-The Natural Language of linear Systems

over :IF. Often n-vectors are called column-vectors or simply vectors. The superfluous column index 1 is dropped, and thus the equations defining addition and scalar multiplication of n-vectors are written as

Xl] [Yl] [Xl +-Yl] X2 Y2 X + Y2 ·· + .. = 2 .. , . . [· X" Yn X" + Y" C

Xl] X2

.. •

[

Xn

=

[CXl] CX2 .

..

,

(2.3.1)

(2.3.2)

CX"

respectively, in which Xi' Yi' and c are all elements of:IF (i = 1,2, ... , n). If v1 , V 2' ... , Vm are n-vectors over :IF and c l' C2, ... , Cm are scalars in :IF, then the vector ClV l + C2V2 + .. , + CmV mis unambiguously defined in:IF" (since matrix addition was found to be associative) and is called a (finite) linear combination ofvl' V 2, ••• , V m with respective coefficients Cl' c 2 , ••• , Cm • A subspace of :IF" is any nonempty subset of :IF" containing all linear combinations of vectors taken from the subset. (The subset is said to be closed under linear combinations.) A subset!/' of distinct vectors in .'F" is called linearly independent if the only linear combinations of distinct elements taken from!/' that yield the zero vector are those with all coefficientszero. If!/' is not linearly independent, then it is called linearly. dependent. If !/' is a subset of :IF", then the collection of all linear combinations of vectors taken from!/' is called the span of !/' and is denoted by span(!/'). Note that if !/' is not empty, then span(!/') is a subspace of :IF". Let!/' be a subspace of :lFn • A subset of!/' is called a basis for!/' if it is linearly independent and spans !/'. A basis for!/' is called an ordered basis if its elements have been placed in a one-to-one correspondence with a collection of consecutive positive integers, including the integer 1. Important facts concerning the above concepts in :IF" are summarized by the following exercises. EXERCISES

Prove the following statements: 1. A collection {Vl' V2" .., vm} of distinct vectors in:IF" is linearly dependent ifand only ifat least one of the vectors in the collection is a linear combination of the others.

25

2.4 Systems of linear Algebraic Equations

2. If g is a subset of distinct vectors in a subspace of IF n , then it is a basis for the subspace if and only if each vector in the subspace is a linear combination of distinct vectors in g with the choice of nonzero coefficients being unique. 3. The single zero n-vector over IF constitutes a subspace of IF n called the trivial subspace of IF n. It has no basis and is said to have dimension zero. Each nontrivial subspace of IF n has a basis; the basis can be ordered; and any two ordered bases for the subspace have the same (finite) number of elements, called the dimension of the subspace. 4. IFn is a subspace of itself having the consecutive columns of the identity matrix In = (en e2' ... , en) as an ordered basis called the standard ordered basis for IF n.

2.4

Systems of Linear AlgebraicEquations

A system of m linear algebraic equations all xl + al2x2 + a21 xl + a22x2 +

+ alnXn = bl , + a2nXn = b2,

(2.4.1)

with the aij, b, (i = 1,2, ... , m; j = 1,2, ... , n) prescribed elements of a field IF, can be written as the single matrix equation Ax

= b.

(2.4.2)

The m x n matrix A with elements Au = aij is called the coefficient matrix;

(2.4.3)

is regarded as an n-vector variable in IFn; and the m-vector b is defined as

(2.4.4)

Equations (2.4.1) and (2.4.2) are called homogeneous if b = O. It is possible to write down examples of (2.4.2) that have no solution, precisely one solution, or an infinite number of solutions. Generally, counting

26

2.

Matrix Algebra-The

Natural Language of Linear Systems

equations and unknowns provides no information about which is the case. Fortunately, there exists a systematic method of computing that generates any and all solutions and detects the case for which there is no solution. The technique involves performance of elementary row operations on the augmented matrix (Alb),obtained by adjoining b as an (n -t 1)st column to A . The three allowed operations are (I) addition of any scalar multiple of one row to any other row, (11) multiplication of any row by any nonzero scalar, and (111) interchanging any two rows. (It is an easy exercise to show that a type I11 operation can be accomplished by an appropriate sequence of elementary row operations of types I and 11.) The method is based on the fact that the solutions to the system (2.4.2) are preserved (invariant) under the three elementary row operations. Hence an appropriate finite sequence of elementary row operations is applied to (Alb), transforming it into its so-called row-reduced echelon form.A matrix is in its unique row-reduced echelon form if the following conditions are met: 1. All the zero rows (if there are any) occupy the bottom rows. 2. The leftmost nonzero term in each nonzero row is a one, called a leading one. 3. Each column containing a leading one has all its other elements zero. 4. The column occupied by a leading one is to the right of the column occupied by the leading one in the row (if there is one) directly above it.

Example 2.4.1 The solution to the system with row-reduced echelon form of its augmented matrix 0

1 0 - 1 (2.4.5)

0 0 0 is x2 = x4

+ 3,

0

x3 = -2x4 - 1,

x1 and x4 arbitrary.

(2.4.6)

That is, the solutions are all vectors of the form

[[

:[2:

ll

a,b arbitrary].

1]

(2.4.7)

2.4

27

Systems of Linear Algebraic Equations

Example 2.4.2 The unique solution to the system having row-reduced echelon form of its augmented matrix (2.4.8)

[:x ;],

(2.4.9)

0 0 0 0

It is easy to show that the row space of A is invariant under elementary row operations on A . Hence the nonzero rows of the row-reduced echelon form of A are a basis for the row space of A and the number of such rows is the rank of A . In particular, row reduction provides a method for testing for the It . is also true linear independence of a finite collection of m-vectors in 9'" but not obvious that rank ( A T )= rank ( A ) .

CRITERIA FOR THE SOLVABILITY O F SYSTEMS

The following results are quite transparent when (2.4.2) is looked at in the form for which its augmented matrix is in row-reduced echelon form. With A an m x n matrix, the homogeneous equation AX = 0

(2.4.11)

has a nontrivial (i.e., nonzero) solution if and only if rank(A) < n. A sufficient

28

2. Matrix Algebra-The Natural Language of Linear Systems

condition for (2.4.1 I) to have a nontrivial solution is that m < n, i.e.,that there be fewer equations than unknowns. In general, (2.4.2) will have at least one solution if and only if rank(Alb) = rank(A).

EXERCISES

1. Find all solutions to the following systems by row reducing the associated augmented matrices: (a)

(b)

(c)

3Xl + 2X2 + 3X3 + 12x4 = 9, -Xl + X2 - 3X3 - 3X4 = -5, Xl + X2 + 3X4 = 2, Xl + X2 + X3 + 5X4 = 3. Xl + X2 + 4X3 = 3, 3Xl + 2x 2 + IOx3 = 4, 2Xl + X2 + 6X3 = 2. Xl - X2 - X3 = - 2, Xl + 2x 2 + X3 = -3, Xl + 2x 2 + 2X3 = 1, Xl + 4X2 + 2x 3 = -5, Xl + 3x 2 + X3 = -6.

2. Compute the ranks of the following matrices. (a)

(c)

[1 2-1]

1 1 2, 231

[ 12]u: -1 0 1 1 1

(b)

[l

2 0 1 2 1 3

(d)

[~

3 2 1 2

~l -~l

234 3. Determine all the subsets of rows which are bases for the row space for the matrix (c) of Exercise 2. 4. Prove that the transpose operation has the following properties: (a) (b) (c)

(AT)T = A, (AB)T = B TAT, (A + B)T = AT

+ BT.

29

2.5 Matrix Inversion

2.5 Matrix Inversion

In the scalar case for which in (2.4.2) m = n = 1, an elementary method of solving (2.4.2) is to multiply that equation by the solution L to LA

= In

(2.5.1)

(at least in the case for which the coefficient of x in (2.4.2) is not zero.) Observe that if (2.5.1) has a solution L, the same technique solves (2.4.2) for general n. By writing out (2.5.1) as n column equations, it is apparent that (2.5.1) is equivalent to a system of type (2.4.2) but of a special form and larger size. Equation (2.5.1) plays such a prominent role in matrix theory that its solution is given the status of a definition.

Definition If A is an n x n matrix over a field:#' for which (2.5.1) has a matrix solution Lover :#', then A is called invertible or nonsingular and L, ordinarily denoted by A-I, is called the (multiplicative) inverse of A. Remark 2.5.1 It is possible to prove that if L is a solution to (2.5.1), then it is also a solution to AL

= In

(2.5.2)

and conversely, if a matrix L satifies (2.5.2), then it satisfies (2.5.1) as well. Remark 2.5.2 As a corollary of Remark 2.5.1, it is an easy exercise to show that if A has an inverse, then the inverse is unique and is itself invertible with (A- 1 ) - 1 = A. COMPUTING A-'

The problem of inverting a matrix is one of solving a system of linear algebraic equations. To see that this is indeed true, let A be an n x n invertible matrix and denote the columns of A-I by Cl' C 2, ... , Cn and those of In by e 1 , e2' ... , en' Equation (2.5.2) can then be written as (2.5.3) which is the same as (2.5.4) Thus the problem of computing A-I = (c., C2, problem of solving the system of vector equations Ac; = e;

(i

... ,

= 1,2, ... , n).

cn) is precisely the

(2.5.5)

30

Matrix Algebra-The

2.

Natural Language of Linear Systems

But all these equations have the same coefficient matrix A. Hence by row reducing (Ale,, e2, . . ., en),all equations (2.5.5) are solved simultaneously, and clearly the row-reduced form of the matrix (Ale,, e2, . . ., en)will be U,IA

-

(2.5.6)

).

Hence for any n x n matrix A, either row reduction of (AII,) will produce a matrix with the first n columns constituting I,, in which case A is invertible and A-' can be read off the reduced form (2.5.6), or else the row-reduced form will have a string of n zeros at the left end of the bottom row, indicating that A is not invertible. This method for computing A-' is called the Gauss-Jordan method.

Example 2.5.1 Consider the matrix A=

[ ' :]. -1 -1 -1 -2

(2.5.7)

-2

To study its invertibility, row reduce the associated augmented matrix

[ [

1 2 3 -1-1 2 -1 -2 -2

1 2 0 0 1 0 0 0 1

1 0 0 0 1 0 0 0 1-

I

(2.5.8)

- 2 0 - 3 - 4 1 - 5 1 0 1

The conclusion drawn is that A is invertible and the inverse is read off to be

(2.5.9)

Example 2.5.2 Consider the matrix A=

['

-1 -1 2 5

:],

11

(2.5.10)

31

2.6 Large-Scale Matrix Computations Using Computers

whose augmented matrix row reduces as follows:

H

2 3 -1 2 5 11

1 0 0 1 0 0

~]-[~ -[~

2 3 1 5 1 5 2 3 1 5 0 0

1 0 1 1 -2 0 1 1

0 1 -3 -1

~]

(2.5.11 )

~]

and the calculation can be terminated, since the three zeros in the bottom row indicate that A is not invertible.

2.6 Large-Scale MatrixComputations Using Computers

In recent years the capability for doing numerical matrix computations on digital computers has progressed to the point where large-scale problems can now be solved quickly and accurately. This success is attributable to the numerical analyst's careful attention to the accuracy and efficiency of the algorithms developed for doing the computing and the programming effort expended in implementing those algorithms. Any scientist contemplating extensive numerical work is advised to consult the references to learn about the capabilities and limitations of available packaged routines such as ElSPACK and UNPACK (see [5, 18]). Much of Chapter 3 concerns derivation of appropriate forms of the equations for treating large-scale systems of differential equations. For a discussion of the large-scale systems of linear algebraic equations to which they are reduced, the reader should consult the cited references. Only one aspect of the latter problem is mentioned here. Programs for solving large systems of linear algebraic equations ordinarily do not reduce the augmented matrix to the echelon form described earlier in this chapter. The more widely used method of Gauss elimination row reduces the matrix to an upper-triangular form and then completes the computation by back substitution, starting with the bottom nonzero row, rather than proceeding with back elimination to reach the echelon form. To compare the efficiency of algorithms, suppose that each division and each multiplication-addition is counted as one operation. For an n x n nonsingular system, it is easy to show that Gauss elimination takes the order

32

2,

Matrix Algebra-The Natural Language of Linear Systems

of !-n3 operations to reach the upper-triangular form and the order of another in2 operations to complete the back substitutions-a total still of the order tn3 • (The back substitution takes fewer operations, since it is done to a triangular matrix.) Clearly the multiplication of two n x n matrices done in the standard way takes n 3 multiplications-the same order of operations required to invert an n x n matrix using the Gauss-Jordan method described in Section 2.5. Thus Gauss elimination compares favorably to the method of solution based on inverting the coefficient matrix. Suppose that the task were to solve the system of equations Ax;

= b,

(2.6.1)

= 1,2, ... , m) in which A is an n x n invertible matrix. A novice might make the mistake of applying Gauss elimination to reduce each matrix (Alb;), with a total operation count of order !-mn 3 • If all the data b, were available at the start, a more efficient strategy would be to apply the Gauss elimination to the augmented matrix (Alb 1 , b2 , ••• , bm ) , which could be done with the order of j-n3 + mn 2 operations-a worthwhile improvement if n were large and m large but relatively small compared with n. If m were much larger than n, then the latter method again would be more efficient. Of course, in an application in which some of the initial XiS had to be computed before the later bjs arrived, the reduction of (Alb 1 , b2 , ••• , bm ) just one time would not be possible. However, there is another efficient solution to the problem that the uninitiated most likely would overlook! Suppose that (2.6.1) were to be solved for m relatively large compared with n. Obviously the only reason for row reducing A in (2.6.1) is to generate the row operations that need to be applied to each b.. This information is available after solving (2.6.1) for i = 1. The problem reduces to the question of how best to record those operations so that they can be repeated on the other biS in whatever order and whenever they arise. A solution to this problem is well-known in numerical analysis. It is called triangular (i

factorization.

Triangular factorization is particularly well adapted to problems of type (2.6.1) in which the coefficient matrix A is common to all the equations. The idea is to compute a lower-triangular matrix L and an upper-triangular matrix U such that A = LU. Then each equation (2.6.1) splits into the pair LYi = b.,

(2.6.2)

Ux, =Yi

(2.6.3)

(i = 1,2, ..., m). The solution Xi to (2.6.3)-and hence to (2.6.I)-is achieved by first solving (2.6.2) to get Yi and then substituting Yi into (2.6.3) to get Xi'

33

2.6 Large-Scale Matrix Computations Using Computers

The advantage gained by the pair is that each takes but i n 2 operations, since L and U are triangular. The computation of L and U is accomplished once and for all in i n 3 operations. Thus the total count for LU factorization is about i n 3 + mn'. Notice the improvement in efficiency over the first approach, which took the order of 3mn3 operations. An advantage of the LU approach over the second method is that one equation in (2.6.1) can be solved at a time with no penalty in computation. The matrix U is the one obtained by doing Gauss elimination in reducing A to upper-triangular form. L is simply a bookkeeping matrix that records the multipliers involved in the triangularization of A and requires no operations. For a more detailed discussion of LU factorization and related topics see [ 5 ] .

Example 2.6.1 This example carries out the solution of (2.6.1)by the method utilizing L U factorization for A=

[ J 2 1 3 - 2 - 10 1

bl =

[-;I9 [;I. b,

=

" ' 3 " ["3

(2.6.4)

By Gauss elimination A is reduced to upper-triangular form to get 1 3 -2 0 1 - 0 1 4 + 0 1 4 = u , 4 -1 2 0 - 3 -4 0 0 8

(2.6.5)

and the multipliers used are recorded in the corresponding positions below the diagonal of the identity matrix to get 1

L=

.]

0 0 1 2 -3 1

[-'

(2.6.6)

Note that LU = A as intended. The solution to (2.6.1) for i = 1 is accomplished by doing the forward and backward substitutions that solve (2.6.2) and (2.6.3):

[-:

0 0 1 0 2 -3 1

[

2 1 3 0 1 4 0 0 8

(2.6.7)

(2.6.8)

34

2. Matrix Algebra-The Natural Language of LinearSystems

The reader is invited to repeat the substitutions following replacement of the augmented column in (2.6.7) by b 2 to get X2 = (-i, -1, !)T. The problem of efficiency is just one of those that must be dealt with in computerized numerical linear algebra. A. more extensive discussion of it and other problems such as roundoff errors can be found in the cited references. EXERCISES

1. Determine which of the following matrices are invertible and compute the inverse in those cases where it exists.

(a)

H

(b)

~],

-1

(c)

2.

[~

(d)

[~

1

3

5

o 1 o -1 o 0

fl

Prove the following statements: (a) If A and B are invertible n x n matrices, then AB is invertible and

(AB)-l = B-IA- I. (b) If A is invertible, then so is A-I and (A-l)-l = A. (c) If A is invertible, then so is AT and (AT)-l = (A-I)T. (d) rank(AA T ) = rank(A) for every m x n matrix A. (e) An n x n matrix A is invertible if and only if rank(A) = n. 3. Compute all the matrix solutions X to the matrix equation AX = B,

where

1 - 1 1

2]

A= 3 02 -1, [2 1 1 3 (Hint:

4 0 4]

B= 2 6 7 .

[ 409

minimize the calculations by row reducing the appropriate matrix.)

35

2.7 Determinant Functions

2.7

Determinant Functions

In the following discussion A is an n x n matrix variable over a field !F written out in terms of its columns A = (al> a2'" ., an)'

Definition A determinant function det(A) is defined to be an fF-valued function (of the columns) of A, det(A) = dia«, az, ... , an), with the properties (1) d(el' ez,· .. , en) = 1, where In = (el' ez,"" en);

d(a 1, az,"" an) = -d(al' az,.··, an), where the matrix (ai, a z, , an) arises from the interchange of any two adjacent columns of (ai' az, , an); (3) dia«, az,···, an)la,=ktbt +k2 b2 = k 1d(al> az,···, an)!a,=bt + k zd(al' az,···, an)la,=b2 for every kj in IF, bj in IFn (j = 1,2; i = 1,2, ... , n). (2)

A function satisfying (2) is called alternating and one satisfying (3) is said to be n-linear. Hence a determinant function is an n-linear, alternating IFvalued function of an n x n matrix variable A over fF with value 1 at the identity matrix. Case n = 1. For n = 1 the function det(a) = a is clearly a determinant function and there is no other, since if d(a) is a determinant function, by (1)-(3), d(a) = d(a'l Case n

+ 0,1) = ad(l) + Od(l) =

a.

= 2. For n = 2 and A

11 12] det[a a a21 a22

==

[aau11 a22 a12 ] ,

the standard function,

= a11 aZ2 - a12a21

can easily be checked to be a determinant function, and there is but one, since if dia-, a2) is a determinant function, properties (1)-(3) can be applied to compute d[:::

:::]

= d[ a11 =

a11d[~

[~]

+

: ~]

a2{~J[: :] +

a21d[~

: ~] (continues)

36

2. Matrix Algebra-The Natural Language of Linear Systems

=

+

a2Id[ ~1

a12

[~J

all

{a12d[~

~J

+ a22d[~

+ a21

{a12d[~

~J

= alla22d[~

~J

= alla22-

+ a22

[~J] .~J}

+ a22d[~

~J}

+ a2IaI2d[~

a2Ia12d[~

~J

~J =

alla22 - a21 a12'

(2.7.1)

Case n General. Expansion by Minors Formulas. Let A be an n x n matrix and let AOJj) denote the(n - 1) x (n -l)matrixobtainedbystriking out the ith row and jth column of A. Let D be a determinant function on (n - 1) x (n - 1) matrices. Then the formulas n

det(A) =

I

( - l r j AijDA(iU)

(2.7.2)

i= I

can easily be shown to define a determinant function on n x n matrices for each fixedj (j = 1,2, ... , n). In this way, by induction, a determinant function can be shown to exist for each n = 1,2, .... With some work it can be shown that for each n there is but one determinant function, which is then simply called the determinant of order n. Hence in (2.7.2) D can be replaced by det and the resultant formula is one of the expansion by minors formulas. Thus the right-hand side of (2.7.2) is actually independent of j. This and other consequent results concerning determinants can now be summarized. DERIVED PROPERTIES OF DETERMINANTS

det(a l, a2""'~)

=

0 if a, = aj

for some

det(al' a2' ... , an)la, =kb = k det(al' a2' ... , an)1 a, =b for each k in IF, b in IFn (i

i "# j.

(2.7.3) (2.7.4)

= 1,2, ... , n).

det(AR) = det(A) deteR). det(A- I } = [det(A)]-1

(2.7.5) (2.7.6)

37

2.7 Determinant Functions

for nonsingular A. det(AT ) = det(A).

(2.7.7)

n

L (-l)i+iAijdetAUIj)

det(A) =

(j = 1,2, .. . ,n).

(2.7.8)

(i = 1,2, ... , n).

(2.7.9)

i= 1 n

det(A) =

L (-l)i+iAijdetA(ilj)

i> 1

(2.7.10) in which the sum is over all permutations a of {I, 2, ... , n} and sgn(a) takes on the value 1 or -1 as a is even or odd, respectively. Formula (2.7.10) is sometimes taken as the definition of the determinant function and is logically equivalent to that employed in Section 2.7. Frequent use will be made of the following standard result concerning determinants. Theorem 2.7.1 For A an n x n matrix over afield iF, the followinq statements are equivalent:

det(A) = O. Ax

=0

has a nontrivial solution A

is not invertible.

(2.7.11)

x in

(2.7.12) (2.7.13)

COMPUTING DETERMINANTS

Formula (2.7.10) shows that the determinant of a matrix variable is a very special polynomial in the elements of the matrix. Note that if the matrix is triangular, then the value of its determinant is just the product of the diagonal elements. Determinant functions are used primarily in theoretical work. In numerical work their use should be avoided when possible, since the number of arithmetical operations involved in their evaluation tends to grow rapidly with the size of the matrix. When the determinant of a sizable matrix must be computed, it is generally advisable to base the calculation on triangularization of the matrix using elementary operations on rows and columns. It is apparent that every square matrix can be triangularized by row operations. The calculations often utilize (2.7.4) and (2.7.7) as well as the invariance of the value of a determinant under type I row operations (which follows readily from the defining properties of the determinant function). Another useful operation is the interchange of two rows or columns, which simply alters the

38

2. Matrix Algebra-The

Natural Language of Linear Systems

sign of the value of the determinant. The latter is a consequence of the alternating property. For notational convenience det(A) is occasionally denoted by [ A [ . -

Example 2.7.1 0 1 3-1 2 1 3-1

1-2 2 2 3 1 2 - - _1 0 1 2 1 3 3 2 2 2 1 4 3 - 1 1

4 3 2 4

1-2 2 4 - - _1 0 1 2 3 2 0 5 -1 - 6 O 5 - 5 -8

1 0

0 0 -15

1 0 0 0 1 1 0 1 0 1 0 0 = - 2 0 2 0 0-11 1 0 0 -15 7 0 1 - _1 0 2 0 0

0 0 1 0 0 6 2 0 0

0

0 1 0 0 6

0 0 0 2

0

-23 0 0 1 7

0 0 = 31. 7 1

(2.7.14)

EXERCISES

1. Derive the formula det(A) = a l l % ~ a 3 3+ a13a2lU32+ a12a23a31 - a11a23a32

- a12a21a33 - 413u22u31

for a 3 x 3 matrix: (a) directly from the defining properties (1)-(3), (b) directly from formula (2.7.10), (c) directly from (2.7.9) and the standard formula for a 2 x 2 matrix. 2. Prove that an interchange of any two rows of a matrix results in the change in sign of the value of the determinant. 3. Evaluate the determinant of each of the square matrices appearing in Exercise 1, Section 2.6.

39

2.8 Eigenvalues, Eigenspaces, Characteristic and Minimal Polynomials

4. Show that (2.7.5) implies (2.7.6). 5. Apply the defining properties (1)-(3) of a determinant to prove (2.7.3) and (2.7.4). 6. Deduce (2.7.7) from (2.7.10). 7. Prove Theorem 2.7.1. 8. Verify that (2.7.2) defines a determinant function of an n x n matrix.

2.8

Eigenvalues, Eigenspaces, Characteristic and Minimal Polynomials

Definition Let A be any n x n matrix over a field iF and v be an n-vector in iF n • The scalar A in iF is called an eigenvalue (or characteristic value) of A if (2.8.1)

Av = AV n

for some v :I: O. If Ais an eigenvalue of A, then any v in iF satisfying (2.8.1) is called an eigenvector (or characteristic vector) associated with A. The collection "11/;. of eigenvectors associated with an eigenvalue Ais clearly a subspace of iF n• Note that if 1 is an eigenvalue of A, then the problem of computing "11/;. is the problem of finding all solutions v in iF n to the system of linear algebraic equations

(2.8.2) (Un - A)v = O. A polynomial p(x) = Co + CtX + ... + cnxn of degree n is said to be over the field iF if its coefficients are elements of iF. Such a polynomial is called monic if c; = 1.

Definition The characteristic polynomial p..i x) of an n x n matrix A over a field iF is the monic polynomial defined as PA(X) = det(xl n

-

A).

(2.8.3)

In view of (2.8.2) and Theorem 2.7.1, it is apparent that the eigenvalues of A are precisely the-roots of PA in iF. By the fundamental theorem of algebra, for iF the complex field, A will have n eigenvalues, possibly with repetitions. A real matrix may be regarded as over the real field or over the complex field. If A is an n x n matrix with real elements, it might not have any eigenvalues when regarded as a matrix over the real field, e.g., A = [_? aJ. The main point to be made is that the problem of computing the eigenvalues and eigenvectors of a matrix over a field is a problem of computing the roots of a polynomial and solving the associated systems of linear algebraic equations.

40

2. Matrix Algebra-The Natural Language of Linear Systems

This is not to say that eigenvalues should be computed by numerically computing with PA(X). Numerical analysts have a variety of algorithms which work more directly with A, each with its virtues and disadvantages. For f(x) = Co + c ix + .. , + cmxm a polynomial over a field ff define f(A) = col + ciA + '" + cmAm called a matrix polynomial. If f(A) = 0 then f is said to annihilate A. These notions appear in the following theorem. Theorem 2.8.1 Cayley-Hamilton Let A be any n x n matrix over afield ff and PA be its characteristic polynomial. Then PA(A) = 0, i.e.; PA annihilates A.

Definition The minimal polynomial qA of an n x n matrix A over a field ff is defined to be the monic polynomial over ff annihilating A and ofleast degree. It can be shown that qA divides PA and that both polynomials have the same distinct roots in ff. Thus qA can be computed by finding PA which is then factored over ff into the product of powers of distinct prime polynomials. The minimal polynomial is then obtained by the finite process of deleting one prime factor at a time and testing to see whether the resultant polynomial annihilates A. The proof of these facts concerning PA, qA, and Theorem 2.8.1 can be found in [11].

A demonstration of the calculations involved in obtaining the characteristic polynomial, eigenvalues, and associated eigenvectors is given for Example 2.8.1

A =

2 2 -2] -4 3 1, [ 413

(2.8.4)

where A is regarded as a matrix over the real field ~. First it is necessary to compute the characteristic polynomial PA(X) = Ix13

=

-

AI =

(x - 4)

x-2 2 -2 -1 4 x-3 -1 x-3 -4

x-2

-2

4

x-3

o

1

= (x - 4)[(x - 2)2 + 16].

=

2 x-2 -2 -1 4 x-3 0 x-4 x -4

2 -1 = (x - 4)

1

x - 2

-4

4

x-2

0

o

2 -1 1

(2.8.5)

From PA it is apparent that A has but the single eigenvalue A = 4 in ~. The associated eigenvectors are obtained by row reducing the matrix of (2.8.2)

41

2.8 Eigenvalues, Eigenspaces, Characteristic and Minimal Polynomials

[

2 -2

-44 - 11

3-t

2

1 - 1

-;

0

1

5 - 50 0.

3 (2.8.7)

which is noted to be one-dimensional.

Example 2.8.2 Consider the same problem as in Example 2.8.1 but with A now regarded as a matrix over the complex field V. The characteristic polynomial computes exactly as in (2.8.5) but now factors into a product of linear polynomials over V, PA(x)

= (X -

4 ) [ ( ~- 2)’

= (x - 4)[x - (2

+ 161

+ 4i)] [x - (2 - 4i)l.

(2.8.8)

Hence the eigenvalues of A are I = 4, 2 f 4i. From the calculations of Example 2.8.1 it is possible to immediately write (2.8.9)

4i 4 [-4

-2 -1+4i -1

’

2 -1 -1+4i

0

1

0

(2.8.10)

42

2. Matrix Algebra-The Natural Language of Linear Systems

From (2.8.10) it follows directly that (2.8.11) which also has dimension 1. Similar calculations show (2.8.12) Remark 2.8.1 If A is a real n x n matrix with a complex eigenvalue A. = a + ico, the associated eigenvectors can be computed by real arithmetic. Let v = x + iy. Equation (2.8.1) splits into the pair of equations over fJl, (rxI - A)x

=

(2.8.13)

wy,

(2.8.14)

(rxI - A)y = - wx,

which can be written as [(IXI - A)Z

+ wZI]x = 0,

(2.8.15) (2.8.16)

Y = (Ilw)(rxI - A)x.

The former is a homogeneous linear system over fJl that can be solved for x, and then y can be computed by substitution into (2.8.16), doing only real arithmetic.

EXERCISES

1. Find the characteristic and minimal polynomials of the following matrices:

. (a)

(c)

[~ [~

21]

-1

o

1 2 0

1, 3

(b)

[~,

1 0 kz k3

~l

n H ~J (d)

2 1 0 0

0 0 2 1 . 0 2

43

2.9 The Jordan Form of a Matrix

2. Compute the eigenvalues and associated eigenspaces of the matrices in Exercise 1, first regarding the matrices as over 9 and then over W. 3. Let A be a real n x n matrix having eigenvalue u + iw with associated eigenvector x + iy where x and y are in P.Show that u - i o is likewise an eigenvalue of A with associated eigenvector x - iy.

2.9

The Jordan Form of a Matrix

Let A be an n x n matrix over a field 9 in which the characteristic polynomial pA “splits.” That is, assume that PA(X)

= (x - n,)ml(x- J-2)m2..

‘

(x - n

~)~‘

(2.9.1)

with the Ai in 9distinct. The positive integer mi is called the multiplicity of Li. As remarked earlier, the minimal polynomial qA will have the same linear factors but with possibly lower multiplicities qA(X) = (x - n’)l’(x - n2)12 ’. ( x - &)r., 1 5 ri 5 mi

(i = 1 , 2 , .. .,s).

(2.9.2)

Theorem 2.9.1 Jordan Form. There exists a nonsingular n x n matrix P over 9 such that PAP-’ has the block diagonal form J

=

diag[J,, J z , . . ., Js]

(2.9.3)

in which block Jiassociated with ,Ii is mi x mi. Each Ji is itselfa block diagonal matrix whose subblocks are of the form 1 J-i

0

1

. .. .

(2.9.4)

with all diagonal positions occupied by li,ones along the superdiagonal, and all other elements zero. ‘The uppermost subblock of Ji of type (2.9.4) has size ri x ri where ri is the multiplicity of Ai in the minimal polynomial qA of A. IThe sizes o f t h e subblocks (2.9.4) of Ji form a nonincreasing sequence down the diagonal of Ji. IThe number of subblocks of Ji equals the dimension of the eigenspace W n ,associated with ,Ii.

44

2. Matrix Algebra-The Natural language of Linear Systems

Remark 2.9.1 If the multiplicity of A.i in qA is one, then dim "I(/' Ai = m., and if this is true for each i = I, 2, ... , S, then PAP - 1 is diagonal. This then gives a necessary and sufficient condition for the diagonalizability of A by a similarity transformation over :F. A sufficient condition for diagonalizability is that PA have no repeated roots. . Remark 2.9.2 The characteristic polynomial PA always splits over :F = 'IJ. (This is a consequence of the fundamental theorem of algebra.) Sometimes P can be taken over fJl or some other subfield of ce. For example, if the elements of A are all real and PA has only real roots, then P can be taken to have real elements. Other examples of subfields of'IJ are [J,ta, the rational numbers, the subfield obtained by adjoining to [J,ta, etc.

±.J2

Remark 2.9.3 The Jordan form of a matrix A as defined by Theorem 2.9.1 is unique up to the order of the numbering of the distinct eigenvalues AI' A2' ... , As of A. Any such ordering can be selected. Generally, the characteristic and minimal polynomials of a matrix do not carry enough information about the matrix to determine its Jordan form. In those cases where they do or where J is somehow known, P can be determined by solving the system of linear algebraic equations JP - PA = 0

(2.9.5)

for a nonsingular P. Generally, such a P is not unique. Example 2.9.1 Suppose that A is a real matrix with pix) = (x - 2)2(X - 1) and qA(X) = (x - 2)2(X - 1). In this case the Jordan form of A can be written as

J =

Moreover, dim

"1(/'2 =

dim

"1(/'1 =

2 1I 0] . [~--=-l~ o 0 I 1

(2.9.6)

1.

Example 2.9.2 Consider a matrix A whose characteristic and minimal polynomials are PA(X) = (x - i)3(X + i?(x + 2)2 and qA(X) = (x - i)2(X + i)2(X + 2), respectively. Here again, Theorem 2.9.1 determines

4.5

2.9 The lordan Form of a Matrix

J , which would be

(2.9.7)

EXERCISES

1. Consider the matrix

A=

I

1 -2 0 0 -1

0 0 3 -1 0 2 0 1 1-1

1 1 0 2 0

0 1 1 0 0

over the rational field. Compute (a) the characteristic polynomial pa, (b) the minimal polynomial qa , (c) the Jordan form J , (d) dim WA,. 2. Suppose that all that is known of a matrix A over the field W is its characteristic polynomial and minimal polynomial

respectively. What can be said concerning the Jordan form of A? What can be said about the dimensions of the eigenspaces? If A itself were given, how could it be used in solving the problem? 3. Consider the matrix

'1

A = [ ' k2 k l

46

2. Matrix Algebra-The Natural language of linear Systems

in which k 2 = - (a 2 + ( 2 ), k1 = -20(, with a and ca real and t» #- O. (a) Determine J, the Jordan form of A. (b) Determine all nonsingular 2 x 2 matrices P for which PAP- 1 = J. 4. Suppose that A is a matrix with pix) = (x - Ar, qA(X) = (x - A)3, and dim 11/;. = 3. Is this enough information to determine the Jordan form of A ? 5. If A and Bare n x n matrices over a field /#', then they are called similar if A = PBP- 1 for some n x n nonsingular matrix P over /#'. Prove that if A and B are similar, then (a) det(A) = det(B), (b)

PA = PB'

(c) tr(A) = tr(B), where the trace of a matrix A is defined by tr(A) = I:?= 1 Au. 6. Prove that if A is an n x n matrix over ~, then (a) tr(A) = I:i = 1 m;Ai' (b) det(A) = ni= 1 ).i', where Al' ... , As are the distinct eigenvalues of A and m, is the multiplicity of Ai in the characteristic polynomial PA • 7. Prove that each n x n matrix A over ~ is similar to its transpose AT.

Chapter

3

Linear Constant-Coefficient Equations: Computation

In this chapter, we present several versions of a step-by-step procedure (the ABC algorithm) for computing the closed-form solution to the general,

linear, first-order differential equation with constant-coefficient matrix. In solving the homogeneous equation, all forms of the computation require determination of the eigenvalues of the coefficient matrix (more precisely, the roots of the characteristic polynomial with their multiplicities). One version involves the solution of a finite system of recurrent linear algebraic vector equations. Another is based on solution of a single matrix equation with nonsingular coefficient. Both require a certain amount of matrix-vector multiplication and addition. All operations are adaptable to standard computer routines using either real or complex arithmetic. The solution to the nonhomogeneous problem is accomplished by an additional integration. Although heuristic derivations of the methods are given, their rigorous justification is deferred to Chapter 4.

3.1

The Initial-Value Problem

The procedure to be discussed deals with the initial-value problem in $'",

x = Ax + f(t), x(O) = x",

(3.1.1) (3.1.2)

in which A is an n x n matrix over $' and f: fYt -- $'" is assumed continuous with Xo in $'" prescribed. In a step-by-step manner it constructs a closed-form 47

48

3.

Linear Constant-Coefficient Equations: Computation

solution to (3.1.1) and (3.1.2) in terms of elementary functions and an integral involving f. The procedure includes the option of avoiding complex arithmetic when F = fJl. The following discussion treats the real case first and requires only real arithmetic. The problem in which A is complex or A is real but the facility for doing complex arithmetic is available is covered by Remark 3.2.2.

3.2

The

ABC Algorithm

Step 1. Compute the eigenvalues {/l'l' A2' ... , An} of A (some possibly repeated and/or complex) regarded as a matrix over re. This is equivalent to finding all the real and complex solutions to the equation (3.2.1)

in which PA is the characteristic polynomial of A. This is done analytically or numerically. More often than not, the eigenvalues would be found by using a computer. A number of programmed algorithms are available, several of which work with A directly rather than with (3.2.1). A numerical analysis or text should be consulted about which is most appropriate and about difficulties that can occur (see [5]). When A is real, complex eigenvalues will occur in conjugate pairs. Step 2.

Apply the real correspondence rule

(3.2.2)

which assigns to the collection of eigenvalues a collection of n distinct realvalued functions arranged as an n-vector function according to the scheme 2

m-l

' ).t t Al t Al t Al rea 1I\. - - e , 1! e '2! e , ... , (m _ I)! e , complex A =

Ct.

± uo -- e" cos tot, e" sin cot; I t e~ Ism ' cot . -t e~ cos cot I! ' 1! '

tm - l tm - l ~I t ~/' t (m _ I)! e cos w '(m _ I)! e sm w .

(3.2.3)

3.2

49

The ABC Algorithm

In (3.2.3) m is the multiplicity of the eigenvalue A. The algorithm adopts the convention that in writing down y(t), (3.2.3) is applied to each distinct real eigenvalue and then to each distinct complex conjugate pair. Otherwise the order is immaterial.

Step 3.

Determine the real n x n matrix B such that

By(t)

= y(t).

(3.2.4)

This can be done by inspection, due to the fact that the derivative of each Yi(t) assigned by the correspondence rule is a linear combination of the coordinate functions composing y(t).

Step 4. With A given and B computed in step 3, solve the linear homogeneous matrix equation (3.2.5) AC - CB = 0 to obtain its general n x n matrix solution C. Equation (3.2.5) can be regarded as a system of n2linear algebraic equations in n2 unknowns, and the standard technique of solution discussed in Chapter 2 applies. However, B is a sparse matrix (i.e., it has many zero elements) with special structure. More practical techniques for solving (3.2.5) which utilize the special form of B are presented later in this chapter. Whatever approach is followed, the general solution C to (3.2.5) will have elements that are linear combinations of n arbitrary constants C 1, C2' ••• , Crt which can be arranged as the n-vector (3.2.6)

Hence, by inspection it is possible to read off a unique n x n matrix function Y(t) such that Y(t)C = Cy(t). (3.2.7) (The matrix function Y(t) is an example of what will later be called a fundamental matrix function for (3.1.1).)

Step 5. Using the matrix function Y(t) obtained in step 4, evaluate Y(O) and invert to obtain the matrix function (t) = Y (t)Y (0)- 1. The solution to the homogeneous form of (3.1.1), (3.1.2), oX = Ax,

(3.2.8) (3.2.9)

is now x(t) = (t)xo.

50

3.

Linear Constant-Coefficient Equations: Computation

Step 6. The matrix function (t) likewise provides a solution (later shown to be unique) to the nonhomogeneous problem (3.1.1), (3.1.2),

x(t) = (t)XO

+ f~

(t -:- a)f(a) da,

(3.2.10)

called the oariation-of-parameters formula for the solution. Example 3.2.1 The six steps of the ABC algorithm are demonstrated by applying the procedure to solve the simple example problem

:t[;:] = [-~ [;:](0) [;~J

~J[;J

+

[e~J

(3.2.11) (3.2.12)

=

In this example, (3.2.13) Step 1.

The characteristic polynomial is computed as

PA(X)=IXI2-AI=I~

-~I=X2+1,

whose roots are the eigenvalues {A'l' ),2} Step 2.

(3.2.14)

= {± i}.

Application of the real correspondence rule provides

yet) = [c~ssm rtJ.

(3.2.15)

Step 3. The equation for B is

B[cosm. stJt = [-sintJ, cos t

(3.2.16)

whose solution is read off to be B

=

[°1 -olJ.

(3.2.17)

Step 4. With A given in (3.2.13) and B as found in (3.2.17), Eq. (3.2.5), written out in terms of the elements of C, appears in the form

51

3.2 The ABC Algorithm

The resultant scalar equations reduce to the pair (3.2.19) It follows directly that the general solution is

(3.2.20) involving two arbitrary constants now computed, Eq. (3.2.7) reads Y(t)[cIJ Cl

=

Cl

and

[C I Cl

Cl

as anticipated. With y(t) and C

ClJ[C~S

-CI

(3.2.21)

tJ'

sm t

from which is read off Y(t) = [ Step 5. Setting t

c~s t sin tJ. -sm t cos t

(3.2.22)

= 0 and inverting give Y(O)-l =

[~ ~rl

=

[~ ~J

(3.2.23)

The matrices of (3.2.22) and (3.2.23) produce (t). This is apparent from (3.4.3), which would require on the order of n4 operations.

60

3.

Linear Constant-Coefficient Equations: Computation

EXERCISES

1.

Apply (3.4.4) and (3.4.5) to solve the initial-value problem Xl = X2 + X3' X2 X3

= Xl + X3' = Xl + X2

with Xl(O) = 1, X2(0) = -1, xiO) = 2. 2. Apply (3.4.4) and (3.4.5) to solve the initial-value problem

3.5 Additional Comments about Computing and Changing Variables

Earlier in this chapter it was shown that once the homogeneous problem (f = 0) has been solved, that is, the fundamental matrix (t) is computed, then the solution to the nonhomogeneous problem is obtained by an integration. Thus attention is returned to the homogeneous problem X

=

Ax,

(3.5.1) (3.5.2)

The change of variable z = r :». involving any nonsingular matrix P, transforms the equations into i =

Az,

z(O) = z",

(3.5.3) (3.5.4)

in which A = r:' AP and Zo = Px". Thus the change of variables induces a similarity transformation on the coefficient matrix. The rigorous analysis of (3.5.1) and (3.5.2) presented in the next chapter will utilize such a transformation. The approach will be to select P in a manner rendering the behavior of (3.5.3) transparent. This will be adequate since most dynamical behavior of interest is invariant under such a trans-

3.5 Additional Comments about Computing and Changing Variables

61

formation. Frequently, there is considerable freedom anyway in the choice of variables selected to model a physical system. One instance of this occurs when an engineer or scientist decides which system of physical units to use. To give some indication of the role of coordinate transformations in the subject, consider the special class of matrices A in (3.5.1) that have a full linearly independent system of eigenvectors WI' W2,"" Wn, corresponding to respective eigenvalues AI, A2, ... , An (not necessarily distinct or real). Recall that these are precisely those matrices for which each distinct eigenvalue has multiplicity one in the minimal polynomial qA' By the independence, it follows that the matrix (WI, W2, ..., Wn) is nonsingular. Thus suppose that the change of variable made in (3.5.1) and (3.5.2) uses P = (WI' W 2, .•• , wn ) :

A = P-1AP = P-I(AwI,Aw2,oo.,Awn) = P-1(AIWI,A2W2, ... ,AnWn)

= P- 1pD(AI,A2,oo.,An) (3.5.5)

in which D(AI' A2,"" An) is the diagonal matrix with AI' A2"'" An along the main diagonal. From the calculation (3.5.5), it is now clear that a solution to (3.5.3) and (3.5.4) is (3.5.6) and hence a solution to (3.5.1) and (3.5.2) is computed to be x(t) = (WI> W2,"" wn)D(e"'I, e"2 1, ... , e"nl)zO -- (e"t1w 1e"2l ' . w2'···, e"n1wn )ZO

(3.5.7) Moreover, it is apparent that the fundamental matrix (t) for (3.5.1), as discussed in Section 3.2, could be taken to be (3.5.8) Observe that if A is real, then although some of the AS and ws may be complex, (t) will remain real. Computation of (t), using (3.5.8) for those matrices to which it applies, is sometimes called the method of eigenvectors.

There is a notable case in which (t) can be computed with minimal effort. Consider the nth-order scalar constant coefficient equation

Remark 3.5.1

(3.5.9)

62

Linear Constant-Coefficient Equations: Computation

3.

The coefficient matrix of the first-order form of (3.5.9) is . 0

0

0 . ... 1 0 ...

1 0

0

0 0

(3.5.10)

A=

0 -a,

. ... -anFl . . . . . 0

1

0 -a2

0 1 -a,

For such a matrix, it will be shown in Chapter 4 that @ ( t ) = Y(t)Y-'(O),where Y ( t )is the matrix function whose first-row elements are those assigned to A by the appropriate correspondence rule (written in any order) and whose successive rows are generated by differentiation of the row immediately above it.

Remark 3.5.2 The solution to the initial-value problem in which the value xo of x is prescribed at t o , i= A x X(t0) =

+ f(t),

(3.5.11) (3.5.12)

xo,

is given by the modification of (3.2.10)

+

x ( t ) = @(t- t O ) x o

@(t - o)f(o)do.

(3.5.13)

EXERCISES

1. Verify that (3.5.13) satisfies (3.5.11) and (3.5.12). 2. Show that Eqs. (3.5.11) and (3.5.12) in complex n-space V" with A a matrix over 3 ' can be converted into a problem in g2"with a real 2n x 2n coefficient matrix. This points out that equations in complex n-space could be treated by the ABC approach, using real arithmetic. 3. Solve the initial-value problem (3.1.1), (3.1.2) in which

using the method of Section 3.3. 4. Solve the problem of Exercise 3, using the method of Section 3.4.

3.5 Additional Comments about Computing and Changing Variables

5.

63

Solve the initial-value problem (3.5.1), (3.5.2) in which 1

~J,

o

-8 -5

(a) by the ABC algorithm, employing Eq. (3.3.2), (b) by the method employing Eqs. (3.4.4) and (3.4.5), (c) by the method of Remark 3.5.1. Does the method of eigenvectors, employing (3.5.7), also provide the solution? 6. Compute the closed-form solution to (3.5.1), (3.5.2) in which

A

~

U_~ !]

(a) by the ABC algorithm employing Eqs. (3.3.2), (3.3.3), and (3.3.4), (b) by the method of Section 3.4, (c) by the method of eigenvectors (3.5.7). 7. Find the solution to x = Ax, x(1) = XO with A as in Exercise 6. 8. Solve the initial-value problem eb = AlP, lP(O) = 14 in which

A

=

0 1 1-1] [-~ -~ -~ ~. -4 -4 -8

9.

Solve the matrix differential equation problem eb = AlP + lPA with

A=[_~ and

0

~J

Chapter

4

Linear Constant-Coefficient Equations: Theory

The heuristic reasoning which led to the equations purportedly computing the closed-form solution as discussed in Chapter 3 will now be given a rigorous foundation. As is the case with many existence theorems, a solution to the initial-value problem will be shown to exist by means of a limit process, namely, through the exponential function of a matrix. This solution is proved to be unique. The requisite matrix solution C to the equation AC - CB = 0 is constructed from the solution to the initial-value problem, and a representation of the solution of the type computed by the ABC algorithm is thereby shown to exist. . The Jordan form of a matrix applied to the exponential representation of the solution illuminates the choice of elementary functions assigned by the correspondence rules and at the same time reveals precisely those characteristics of the coefficient matrix which govern the qualitative aspects of solution behavior of primary interest in applications. 4.1 Preliminary Remarks

A rigorous and thorough treatment of Eqs. (3.1.1) and (3.1.2) of Chapter 3 will now be developed. It is the homogeneous equation that requires the most attention since the nonhomogeneous problem involves only an additional integration. 64

65

4.2 Some Elementary Matrix Analysis

Assuming that the construction of the closed-form solution presented there is correct, one is struck by the fact that such a limited class of elementary functions is involved. A clue as to why this is the case is already present in the one-dimensional problem in which in beginning calculus the solution to the scalar equation x = ax is found to be the exponential function ce", usually defined as a power series. Hence, the notions of convergence of series, their differentiability, etc., arise. It will soon become apparent that one ofthe main reasons for working with differential equations in first-order form (at least for linear equations) is that the higher-dimensional problem can be treated along somewhat similar lines. (The analogy actually extends well beyond finite-dimensional problems.) The first step will be to extend some familiar notions of real analysis to matrices.

4.2 Some Elementary Matrix Analysis In this chapter, :F denotes any subfield of the complex numbers and, as before, :F" is the collection of n-vectors over :F with the standard algebraic operations. When there is a need to be more specific, :F is replaced by ~ when considering the complex numbers and by 9t when dealing with real numbers. Recall the basic properties of the absolute-value function on :F:

o s Ix I

with equality if and only if

lexl = lei I~I, [x - yl ~ Ixl + Iyl

x = 0,

(4.2.1) (4.2.2) (4.2.3)

for all e, x, and y in :F. A real-valued function with these properties can be extended to higher dimensions. Definition

For x in :F" with coordinates

Ixl =

max 1 ~i~n

Xi'

lXii,

define (4.2.4)

called the norm of x. Remark 4.2.1 Note that when n = 1, the norm of x and its absolute value agree. Moreover, the norm function defined by (4.2.4) continues to satisfy (4.2.1)--(4.2.3), where in (4.2.2) e denotes any element in :F while x is in :F". (The student should verify these claims.) To a considerable extent, the notion can be extended one step further to m x n matrices.

66

4. linear Constant-Coefficient Equations: Theory

Definition

For A an m x n matrix over

IAI

= max

1"1,, 1

§',

define

IAxl,

(4.2.5)

in which x denotes a variable in §'n. IAI as defined by (4.2.5) is called the operator norm of A relative to the norm defined on §'n.

Proposition 4.1

The operator norm of an m x n matrix A over §' is a realvalued function and satisfies the conditions

o ~ IAI if and

with equality holding

(4.2.6)

if A = O.

only

leAl

=

IcllAI

(4.2.7)

for every c in §'.

IA - BI s IAI + IBI

(4.2.8)

for all m x n matrices A and B over §'.

IAxl

~

IA/lxl

(4.2.9)

for all x in §'n.

IABI ~ IAIIBI

(4.2.10)

for all matrices A, B over §' of compatible size.

JAI = IIn l = Proof.

For all x in

§'n

n

L IAul·

max l~i~mj=l

l

(4.2.11)

(n=1,2, ...).

satisfying

Ixl

~ 1,

n

n

n

j;l

j;l

j;l

I L Auxjl ~ L IAul ~ L IAi.jl (i

=

(4.2.12)

(4.2.13)

1,2, ... ,m), where i* is chosen so that

max l~i~m

Defining y* in

§'n

n

n

j=l

j=l

L IAul = L IAiJ·

(4.2.14)

by

I

yj =

sgn(A i• j

(4.2.15)

)

(j = 1,2, ... , n), note that y*/ ~ 1, and furthermore n

n

n

L1 IA jI = L1 A,.jyj = Ii>L1 Ai.jYjl·

j=

i•

j;

(4.2.16)

67

4.2 Some Elementary Matrix Analysis

Thus it follows from (4.2.13) and (4.2.16) that n

max l~i~m

IL

n

j=l

Ai';xjl::; max l~i~m

IL

(4.2.17)

Aijyjl

j=l

for all xin fFn with Ixl ::; 1. From (4.2.17) it is apparent that the left-hand side achieves a maximum over x and moreover from (4.2.13) on one hand and (4.2.16) on the other it follows from (4.2.17) that ,

n

n

L

max max I Aijxj! Ixl,;;l l,;;i,;;m j=l

=

L IAi.jl, j=l

(4.2.18)

which, according to (4.2.2), (4.2.5), and (4.2.14), is the required inequality n

L IAijl·

IAI = max

(4.2.19)

j=l

l~i~m

Properties (4.2.6) and (4.2.7) are obvious consequences of (4.2.19). Since (4.2.20) for numbers n

L IAij j=l

n

Bijl ::;

n

L IAijl + j=l L IBijl, j=l

(4.2.21)

and thus by formula (4.2.19), etc., IA -

BI =

n

max l~i~m

s

L

j=l

IAij - Bij!::; max l~i~m

n

max l~i~mj=l

L IAijl +

n

n

j=l

j=l

[L IAijl + L IBijl]

n

max

L

IBijl = !AI

l~i~mj=l

+ IB!,

(4.2.22)

proving (4.2.8). To prove (4.2.9), apply (4.2.2) and (4.2.5) to get

IAx'l =

IxllA

-,xII::; Ixl lyl,;;l max IAyl x

=

IxllAj,

(4.2.23)

which is valid for x#-O in fFn. (Note that the maximum in (4.2.23) was shown to exist in the previous part of the proof.) Thus (4.2.9) follows from (4.2.23) for x#-O and also holds (with equality) for x = O. This finishes the proof of (4.2.9). For

Ixl s

1, by applying (4.2.9) twice,

I(AB)x/

= /A(Bx)

I::; IAllBxl s IAllBllxl ::; IAIIBI,

(4.2.24)

68

4.

which maximized over

Linear Constant-Coefficient Equations: Theory

Ixl :5: 1 yields IABI :5: IAIIBI,

(4.2.25)

concluding the proof of (4.2.10). Equality (4.2.11) was already proved and (4.2.12) is a direct consequence of (4.2.11). This' concludes the proof of Proposition 4.1. The main reason for introducing the absolute-value function on /F is for convenience in talking about convergence and continuity. It is assumed that the reader is familiar with these ideas in the case of real or complex numbers; i.e., in dimension one. Norms are introduced in order to expand these notions to higher dimensions, where it soon becomes clear that formally there is little difference from the one-dimensional case.

Definition For A k (k = 1; 2, ...), a sequence of m x n matrices over /F, the sequence is said to converge to an m x n matrix Aoo as k - 00 if (4.2.26) as k - 00. The matrix Aoo is denoted by lim k _ oo Ak> and one writes A k as k- 00.

Aoo

Remark 4.2.2 Recall that a sequence of numbers ak is called Cauchy if for each real e > 0 there exists a real K. such that k ~ K. and p ~ K. implies that lak - api < s, The Cauchy criterion (theorem) says that a sequence of numbers is convergent if and only if it is Cauchy. The result is important because it can be used to prove a sequence convergent without the difficult task of finding the value of the limit. It is easily seen from formula (4.2.11) that a matrix sequence A k - Aoo as k - 00 if and only if (Ak)ij - (A oo )ij as k - 00 for each i, j. That is, convergence of a matrix sequence is equivalent to elementwise convergence. Thus it is clear that the Cauchy criterion is also valid for matrix sequences.

Definition the type

If A k is a sequence of m x n matrices arising as a matrix sum of

Ak =

k

L e,

(4.2.27)

i=O

and is convergent, then its limit is denoted by L~

0

B;.

The text follows the standard abuse of notation wherein the same symbol is used to denote the series, whether or not it converges. Series can be used to define matrix functions in much the same way as they are used to define scalar functions. The exponential function plays a particularly dominant role in this chapter.

4.2

69

Some Elementary Matrix Analysis

Definition For A any n x n matrix over of A is defined by the series

§,

the exponential function e A

(4.2.28)

The series is convergent for all A. This can be seen by computing the difference between two terms in the sequence of sums to get

Ai PAil IAli IAli "--"- B 2, ... , B s ), and accordingly

y

° --

[:~J

.

(4.6.27)

,

es

where in (4.6.27) e, denotes the first column of the identity matrix 1m " el

s,«,

B2e2 (y,° B y,0 ... , tr:' y 0) = e2 .. . . . . [

e,

s»,

B~-1ell B~-le2 ..

.

B:-

1

...

.

(4.6.28)

e.

The matrix in (4.6.28) was already proved invertible, and the inverse has unique polynomials associated with its columns via (4.6.24). Thus the polynomials associated with the first m1 columns of the inverse are precisely those satisfying [piBdet]k = bi pj(B;)e i = 0

(k = 1,2, (i = 2,3,

, md, , s; j = 1,2, ... , md.

(4.6.29) (4.6.30)

The proof reduces to showing that these polynomials are the ones given by (4.6.25) and (4.6.26). (The remaining columns of the inverse matrix can be dealt with by repetition of the same argument upon performance of the interchanges mentioned in the statement of Theorem 4.6.3.)

88

4.

Linear Constant-Coefficient Equations: Theory

Since each Bi is an mi x mimatrix of the elementary Jordan type

Bi =

0

*..

1 Ri 0 1

-..

0

-

9

-0

1 li 0 . . . 0 1 Ai

0

A

it follows (see Theorem 4.2.2) that (4.6.29) and (4.6.30) are equivalent to the equations (4.6.32) (4.6.33) i = 2, 3 , . . ., s; j = 1 , 2 , . . .)ml).Note that the poly(k = 1, 2 , . . .,mi; nomials of (4.6.25) and (4.6.26)are of degree n - 1 or less since (ml- 1) + m 2 + m 3 + . . . + m, = n - 1. Since the polynomials p j of degree n - 1 or less satisfying (4.6.29) and (4.6.30),and hence (4.6.32) and (4.6.33))exist and are unique (since the matrix in (4.6.28)has a unique inverse), it is sufficient to show that the polynomials of (4.6.25)and (4.6.26)satisfy (4.6.32)and (4.6.33). From formula (4.6.25) it is clear that those polynomials satisfy

(4.6.34)

(4.6.35) .-a

Thus (4.6.32)holds for k = 1. To perform a proof by induction, assume that (4.6.32) is valid for 1, 2, . . .) k c m l . The task is to show that it is likewise valid for k + 1. Rewriting (4.6.25) as mi-

1

1

ajXx -

r=j-l

n,)r

n (x S

=

pj(x)

(4.6.36)

-

i=2

and differentiating that equation k times relative to x produce the equation ml - 1

r=j-1

r(r - 1).. . (r - k k

=

1 C,kpjyx)

r=O

+ l)aj,(x - Aly-k

"n (x i=2

I' ( k -0

-

li)-m'

(4.6.37)

4.6

Explicit Formulas for GC) and

89

e'A

in which C~ = k !/r !(k - r)! is the binomial coefficient. The resultant equation, obtained by setting x = Al in (4.6.37) and invoking the induction hypothesis, is s

bl+lk!(ljk

=

pJkl(Al)

IT (AI

- ~i)-m,.

(4.6.38)

i=2

Hj t= k + 1, then certainly (4.6.38) implies that pjkl(Ad = O. For j = k formula (4.6.26) reduces to

+ 1,

(4.6.39) which when substituted into (4.6.38) gives p)kl(Ad = 1 in this case. Thus the inductive proof of (4.6.32) for polynomials (4.6.25) is achieved. The fact that those polynomials satisfy (4.6.33) is obvious because of the product term in (4.6.25). The proof of Theorem 4.6.3 is now complete. Since the minimal polynomial of a matrix A divides the characteristic polynomial, it is clear upon distribution of the multiplication in (4.6.25) that piA) = 0 for j ~ rl + 1, where rl denotes the multiplicity of the eigenvalue Al in the minimal polynomial of A. A similar remark applies to the polynomials associated with the remaining distinct eigenvalues. Each matrix A j in (4.6.22) is simply the matrix whose columns are thejth columns of p;(A) (i = 1, 2, ... , n). The net effect is that the active terms of yet) in (4.6.22) associated with a multiple eigenvalue Ak = (lk + iWk are limited to those containing maximal factors t", P ::;; r k - 1, where rk is the multiplicity of Ak (k = 1, 2, ... , s) in the minimal polynomial. Remark 4.6.4

Remark 4.6.5 The polynomials (4.6.25) and (4.6.26) provide generating formulas for the elements of the inverse matrix (4.6.28); i.e.,

pJil(O)

Po = -i!-

(4.6.40)

t. = 1,2, ... , md. The remaining elements of the inverse can be computed in similar fashion from the polynomials obtained via the specified interchanges.

(i = 1,2, ... , ml;

Example 4.6.1

The matrix A =

2 2 -2] 0 3 1 [ o 1 3

(4.6.41)

has characteristic polynomial PA(X) = (x - 2)2(X - 4), and thus the

90

4.

Linear Constant-Coefficient Equations: Theory

eigenvalues are Al = 2, A2 = 4, with multiplicities m, = 2, m2 respectively. Direct substitution into (4.6.25) and (4.6.26) provides

(4.6.42)

Pl(X) = - lx(x - 4),

= - t(x - 2)(x - 4):

P2(X)

The interchange ml remaining

+-+ m2,

Al

+-+

= 1,

(4.6.43)

A2 in (4.6.25) and (4.6.26) produces the

pix) = l(x - 2)2.

(4.6.44)

Substitution of A as given by (4.6.41) into (4.6.42)-(4.6.44) leads to p,(A)

~ [~

p,(A)

~

p,(A)

~ [~

j -tJ'

[H -~l

(46.45)

(4.6.46)

i ~,

(4.6.47)

from which can he read off Al

=

[~ ~ ~l, o

A2 = [

~J

0

t ~ j],

-2

A3 =

(4.6.48)

0

? -20

[!

-2

0

(4.6.49)

2

~]2

!

.

(4.6.50)

Substitution of (4.6.48)-(4.6.50) into (4.6.22) finishes the computation: 2t 2t e 2te2t _ 2te ~ 4t) 2t 4t etA = 0 t(e + e t(e _ e 2t) . (4.6.51) [ 4t) 2t 4t 2t) o !(e _ e !(e + e

4.6 Explicit Formulas for C(e) and

91

erA

EXERCISES

1.

In the representation elA

=

Z2~(t),

... , Zn~(t))

(Zn(t), Z!~(t),

... , Z:~(t))

(Z1~(t),

of Corollary 4.6.2, show that elA T

=

in which (i = 1, 2, ... , n).

2. As in Exercise 1, discuss the action induced upon the remaining representation formulas appearing in Corollary 4.6.2 and Theorems 4.6.1 and 4.6.3 under the transformation A __ AT. 3. Study the question dealt with in Exercises 1 and 2, but for the transformation A -- M AM -1 for M fixed. Can you say anything about A -- M A with M fixed?

Chapter

5

Linear Constant-Coefficient Equations:

Qualitative Behavior

In many applications of differential equations the dependent variables denote errors or displacement of the state of a system relative to some reference or equilibrium state. It is important in such situations that the errors converge to zero appropriately fast with increasing time. Sometimes it is extremely useful to know whether a given differential equation has one or more periodic solutions. The results in this chapter provide answers to such questions based upon somewhat less computational effort than is involved in finding closed-form solutions. 5.1 Stability and Periodic Oscillation

This section concentrates on the qualitative behavior of the solution to the Cauchy problem in £Jt",

x=

Ax,

(5.1.1) (5.1.2)

in which A is an n x n matrix over £Jt. All results can be extended to complex space in a rather obvious fashion. Definition Equation (5.1.1) is called stable if for each XO in £Jt" the solution x(t) to (5.1.1) and (5.1.2) has !x(t)! bounded on 0 :::; t < 00. If for some solution Ix(tk)l-+ 00 as tk -+ 00 for some sequence tk> then (5.1.1) is said to be unstable. 92

93

5.1 Stability and Periodic Oscillation

Definition Equation (5.1.1) is called asymptotically stable if each solution x(t) to (5.1.1) has x(t) - 0 as t - 00. The characteristic polynomial PA and minimal polynomial qA of A contain a great deal of information about the qualitative properties of the solutions to (5.1.1). Such results are summarized in the following theorem. Theorem 5.1.1

Equation (5.1.1) is asymptotically stable (Xj

< 0

(j = 1,2, ... , s) for all distinct eigenvalues satisfying (5.1.3) is called a stability matrix). Instability occurs if

if and

only

if (5.1.3)

Aj

=

(Xj

+

uo, of A. (A matrix

(5.1.4) for at least one j in {1, 2, ... , s}, No conclusion can be drawn if (Xj :-::;; 0 (j = 1,2, ... , s) without further information: If (Xj :-::;; 0 (j = 1,2, ... , s), then stability occurs if and only !f those Aj = (Xj + ico, with (Xj = 0 have multiplicity one in the minimal polynomial qA; i.e., dim "If/;.} equals the multiplicity of Aj in PA' A sufficient condition for stability is that (Xj :-::;; 0 (j = 1,2, ... , s) and that each Aj = (Xj + ico, with (Xj = 0 have multiplicity one in the characteristic polynomial pA of A. Theorem 5.1.2 Equation (5.1.1) will have a nonconstant (oscillating) periodic solution offrequency W if and only if at least one of the distinct eigenvalues A j = (Xj + ico, (j = 1,2, ... , s) of A has (Xj = 0 and Wj = w.

Both Theorems 5.1.1 and 5.1.2 are easily verified consequences of the analysis of the structure of etA that was made in Section 4.4, using the Jordan form of a matrix. The next theorem is useful in estimating the rate of decay of the solutions to an asymptotically stable linear system and provides an upper bound on the rate of growth of an unstable system. Theorem 5.1.3

For A an n x n matrix over

~

n

letAI:-::;;

L IpiA) IIYk(t) I

(5.1.5)

k= 1

for 0 :-::;; t < 00 in which the polynomials Pk(X) are those defined in Theorem 4.6.3 and the Yit) are the coordinate functions of yet).

94

5.

Proof.

linear Constant-Coefficient Equations: Qualitative Behavior

Definition (4.6.23) and elementary estimation give n

I t>

n

1

n

I I kI

I(AjY)il =

j= 1

= 1

n

I

n

s I I

j= 1 k = 1

n

=

(Pk(A»ijyit)

I(Pk(A»ijIIYit)1

n

I I I(piA»ij IIYit) I

k = 1 j= 1

n

I

=

k= 1

O::;t(t, ")1 ~ k~O IL~t, ")1 s exp (IS)A(a) 1do I) all (t, ..) E J x J (k = 1, 2, ...). lI> is continuous on J x J

(6.3.4) (6.3.5)

for and its series (6.3.3) is absolutely and uniformly convergent on compact subsets of J x J. Proof.

To minimize the number of absolute value signs, the proof is

109

6.3 The Fundamental Matrix Series Defining (t, r)

= In +

f

A(w)l1>(w, r) dco,

(6.3.20)

and moreover, because of (6.3.19), Eq. (6.3.20) can be differentiated to conclude that

o

ot l1>(t, r)

= A(t)l1>(t, r),

a.e. t e J

(6.3.21)

for each r ~ t in J. The case in which r > t is treated with the obvious minor change. The uniqueness claims are easy consequences of Gronwall's inequality (see Exercise 1 at the end of this section). Equation (6.3.12) can be proved as follows: Consider '¥(t) defined as '¥(t) = l1>(t, r)l1>(r, r),

t E J,

(6.3.22)

with arbitrary r E J fixed. Applying (6.3.8) and (6.3.10), Eq. (6.3.22) can be differentiated to get d

dt'¥(t) = A(t)'¥(t) - '¥(t)A(t),

a.e. t E J.

(6.3.23)

Thus '11(.) is a solution to (6.3.23) satisfying the initial condition '¥( r) = In as is clear from (6.3.9) and (6.3.22). But the matrix function '¥(t) = In, t E J, is another solution to the same initial-value problem. Application of Gronwall's inequality again shows that the solution must be unique, and the conclusion that 'P(t) = In, t E J, is inevitable. This finishes the proof of (6.3.12). If A( .) happens to be continuous, then by inspection of the proofjust given it is evident that (6.3.8) and (6.3.10) hold everywhere in J x J and moreover that l1>(t, r] and its first-order partial derivatives are continuous in J x J. It follows that l1> is thus once continuously differentiable on J x J. This concludes the proof of Theorem 6.3.2. Notation. Occasionally the matrix function l1> will be denoted by l1>A when its dependence upon A(' ) needs to be emphasized.

112

6.

General Linear Equations with Time-Varying Coefficients

Remark 6.3.1 From Theorem 6.3.2 and results derived in Chapter 4 it is apparent that A is just e(l- t)A in the case in which A( .) = A is independent of tEJ. EXERCISES

1. Prove the following form of Gronwall's theorem: Each continuous solution x( .) to the inequality x(t)

s

lX(t)

+

f

also satisfies the inequality x(t)

s

lX(t)

+

I

p(U)IX(U)

p(u)x(u) da,

exp(f

pew)

tE

[a, b),

dW) do,

t E [a, b),

where IX(') is continuous and 13(') is nonnegative and locally integrable on

[a, b).

2. Apply Gronwall's theorem to prove the uniqueness claims concerning Eqs. (6.3.8) and (6.3.9), (6.3.10) and (6.3.l.1), and (6.3.23) that were made in the proof of Theorem 6.3.2. 3. Prove the assertion concerning (6.3.17) that was made in the proof of Theorem 6.3.2.

6.4 Existence and Uniqueness of Solutions to Vector Differential Equations 'and Their Duals The results of Theorem 6.3.2 can be applied to vector differential equations. As often occurs in mathematics, it is natural to discuss some of the results in terms of the dual space of ~n rather than in terms of ~n itself. The appropriate definitions are recalled. Definition An tional on ~n if

~-valued

function I, defined on .

~n

is called a linear func(6.4.1)

for all Cl' Cz in ~ and all Vl, Vz in ~n. The collection of all linear functionals on ~n is denoted by ~". and can be turned into a vector space relative to the addition and scalar multiplication defined by

(11

+ Iz)(v) =

11(v)

+ Iz(v),

(c/ 1)(v) = c/1(v)

(6.4.2) (6.4.3)

113

6.4 Existence and Uniqueness of Solutions and Dual Systems

for all 11 ,12 e $'"*, v e $'", and c e $'. The resultant vector space, still denoted by $'"*, is called the dual space of $'".

Remark 6.4.1 $'"* has dimension n and its elements x* can be identified with I x n (row) vectors x over $' if x*(v) is defined to be the matrix product xv, for all v e $'". Remark 6.4.2 Any n x n matrix function A( .) defined on an interval J can have associated with it the differential equation in J x $'", x

= A(t)x,

(6.4.4)

as well as a differential equation in J x $'"*, (6.4.5)

x* = x*A(t).

Some interesting relationships between the solutions of the two equations appear in Theorem 6.4.1.

Theorem 6.4.1 Let A(·) be a locally integrable n x n matrix-valuedfunction (over $') defined on an open interval J c fit. Then there exists a unique solution x A to the initial-value problem on J x J x $'", x

= A(t)x,

x(t)/t=t =

(6.4.6)

a.e. te J,

e

(6.4.7)

namely,

(6.4.8) There also exists a unique solution x~ JxJx$'"* x* x*(t)lt=t

to the initial-value problem on

= -x*A(t), a.e. t e J, = e* for each (r, e*)eJ x

(6.4.9) $'"*,

(6.4.10)

namely,

(6.4.11) Moreover, the solutions to the two equations satisfy the invariance relation

e

x~(t,

r, e*)xA(t, r, e) = e*e

(6.4.12)

for all (t, r) e J x J, e $'", e* e $'"*. If A(·) is continuous on J, then x A and x~ are once continuously differentiable relative to their argument (t, r) e J x J, and Eqs. (6.4.6) and (6.4.9) hold for all (t, r) e J x J. Proof.

All statements are rather obvious consequences of Theorem 6.3.2.

114

6.

General linear Equations with Time-Varying Coefficients

Remark 6.4.3 Equation (6.4.9), transposed so as to appear as a differential equation in :Fn , is called the adjoint of (6.4.6) and appears frequently in control theory and other areas of mathematics involving differential equations or optimization. Remark 6.4.4 If A(') is assumed to be k-times continuously differentiable on J, then since A satisfies (6.3.8) and (6.3.10), it is clear that A will be (k + Il-times continuously differentiable relative to (t, T) E J x J (k = 0, 1, 2, ...). Thus the solutions (6.4.8) and (6.4.11) will likewise be (k + Il-times continuously differentiable on J x J.

6.5 Variation-of-Parameters Formulas for Vector and Matrix Differential Equations

Theorem 6.5.1 Let A(') and f(·) be locally integrable matrix-valued functions (over :F) of sizes n x nand n x 1, respectively, both defined on an open interval J c flt. Then there exists a unique solution x A to the initial-value problem on J x J x :Fn

x=

A(t)x

+

f(t),

a.e.

for each

x(t)lt=< = ~

(6.5.1)

tEJ, (T,

x J

~)EJ

(6.5.2)

given by the variation-of-parameters formula

xA(t, T,

~)

=



A(~' r)~ +

f

4> A(t, (1)f(a) da,

(6.5.3)

Proof. A proof can be based upon application of Theorem 6.4.1. The details as well as the proof of the following theorem are left to the reader.

Theorem 6.5.2 Let Al ('), A 2 ( ' ) , and F(') be locally integrable n x n matrixvalued functions on an open interval J with arbitrary r E J and n x n matrix Z fixed. Then there exists a unique solution to the initial-value problem

'P =

A1(t)'P - 'P A 2(t)

+ F(t),

a.e.

t EJ,

(6.5.5)

'Plt=< = Z, namely,

'P(t, r, Z)

Example 6.5.1

= 4>Al(t, r)Z4>A2(T, t) +

(6.5.4)

f

4>A/t, a)F(a)A,(a, t) da,

(6.5.6)

An important example of (6.5.4) is the equation a.e. t

E

J,

(6.5.7)

6.6

115

Estimates of Solution Norms

arising from A 1(t) = -AT(t), A 2(t) = A(t), and F(t) = In. Here A(') is assumed to be locally integrable on an interval J. In the special case in which A(') is independent of t, the (steady-state) equation, gotten by replacing P in (6.5.7) by zero, is Lyapunov's algebraic equation (5.2.19). Thus it should be not altogether surprising to find that in the next section (6.5.7) plays an important role in the problem of estimating rates of growth or decay of norms of solutions to differential equations associated with A(·). The solution to (6.5.7), as represented by the variation-of-parameters formula (6.5.6), is pet, r) =

0 on

['t,

00),

and if moreover

roo

J.

du f3(u)

=

00,

(6.6.9)

then the first inequality of (6.6.8) is likewise valid.

Remark 6.6.2 The proof of Theorem 6.6.1 relied on the assumption that iX(t) remains positive over the range of t being considered. (In applications, iX(t) is usually obtained, as a lower approximation to the smallest eigenvalue of P(t).) Examples demonstrate that the smallest eigenvalue of P(t) can become zero as t is increased from r to some T < 00. Theorem 6.6.1, for the given P('), will then not apply outside the interval ['t, T] since (6.6.3) must necessarily fail. One remedy for extending the interval determined by some approximating iX(·) would be to find a better solution P(·) to (6.5.7). Another would be to find a better lower approximation than iX(·). An effective procedure for accomplishing these objectives is provided in the next section.

DETERMINATION OF 01(') AND P(·)

To convert Theorem 6.6.1 into a result having practical as well as theoretical interest, two obstacles must be surmounted. The first is to single out solutions P(·) to (6.5.7) which are positive-definite over the interval of concern. The other is to find an effective procedure for approximating the extreme eigenvalues of such' P(·) to within any prescribed accuracy. The former is handled by restricting attention to any finite interval of the type r ~ t < b < 00, (r, b) E J x J, and using P(t)

=

i

b

cI>T(U, t)cI>(u, t) do

(6.6.10)

on the interval. The latter can then be approached through the series expansion appearing in the next theorem.

Theorem 6.6.2 Let A(') be locally integrable on J and consider any compact interval ['t, b] c: J. The integral (6.6.10) provides a symmetric positive-definite matrix solution to (6.5.7) on the interval ['t, b) and P(t)

=

r

Ao(t)

+ At(t) + ... + Ak(t)

+

cI>T(U, t)[AT(u)Ak(u)

+ Ak(u)A(u)]cI>(u, t) da,

(6.6.11)

118

6.

in which

General Linear Equations with Time-Varying Coefficients

r

(6.6.12)

Ao(t) = (b - t)I n , A k+ 1 i

::;;

=

+ Ak(a)A(a)] do,

(6.6.13)

[1'J~(~)]k

(6.6.14)

Ak(t),

(6.6.15)

[AT(a)Ak(o)

t ::;; b (k = 0, 1,2, ...). Moreover,

IAit) I ::;; (b -

t)

in which

In particular, co

pet) =

L

k=O

with the series uniformly convergent on

[i, b]

and

pet) ::;; (b - t)eqA(t)I n •

(6.6.16)

Proof. Routine calculations show that (6.6.10) is a matrix solution to (6.5.7) that is symmetric and positive-definite at each t E [i, b). Formula (6.6.11) with k finite follows by repeated integration by parts applied to (6.6.10). The estimate (6.6.14) is easily established using induction on (6.6.13). Equation (6.6.15), the uniformity of the convergence, and (6.6.16) are elementary consequences of(6.6.14) applied to (6.6.11). The reader is invited to check the details, and Theorem 6.6.2 is thereby proved.

Lemma 6.6.3 Let S(· ) be a real n x n symmetric matrix-valued function defined on an open interval J. The minimum and maximum eigenvalues of Sea) at each point a E J define real-valued functions on J which are continuous at each point in J at whicn S( .) is continuous. Proof. Since Sea) is symmetric at each a E J, its eigenvalues are real, and thus the smallest eigenvalue ac(a) of Sea) defines a function ac(') on J. Let t E J be a point at which S(·) is continuous, and in order to reach a contradiction suppose that ac(') were not continuous there. Then there would exist a sequence t k -- t as k -- 00 with t k E J such that ac(tk) -f+ ac(t) as k -- 00. Thus by extracting an appropriate subsequence, it can be assumed that either

(k = 1,2,

)

(6.6.17)

= 1,2,

)

(6.6.18)

or (k

6.6

119

Estimates of Solution Norms

for some number p. From the symmetry of S(·) it follows that there exist unit vectors ~k' ~* in [JItn such that Ct(tk) = ~k' S(tkRk and Ct(t) = ~*. S(t)~*. By extracting a further subsequence, it can be assumed that ~k ~ ~oo as k-r CXJ for some ~ 00 E [JItn that necessarily will have I~ 00 I = 1.Thus inequalities (6.6.17) and (6.6.18) could be written respectively as (k

= 1,2, ...)

(6.6.19)

(k

= 1,2, ...).

(6.6.20)

and By the continuity of S(·) at t, it follows upon taking the limit as k ~ (6.6.19) that

00

in

(6.6.21) over the collection of which is a contradiction since ~ * minimizes ~. S(t)~ unit vectors in [JItn. In the complementary case, (6.6.20) obtains and can be rewritten as - ~*'S(tk)~*

~k'S(tk)~k

+ ~*'[S(tk)

- S(t)]~*:2: (k

P> 0

= 1,2, ... ). (6.6.22)

By the continuity of S( .) at t, etc., I~*

.[S(t k) -

s(t)]~*1

~ IS(tk) - S(t)II~*12

= IS(tk) - S(t)1 ~O

(6.6.23)

as k ~ 00. However, for all sufficiently large k, inequalities (6.6.22) and (6.6.23) imply (6.6.24) over the collection which is again a contradiction since ~k minimizes ~. S(tk)~ of ~ E [JItn of unit length. This concludes the proof that the continuity of S(.) at a point t E J implies the continuity of Ct( •) at t. The proof of the remaining conclusion of Lemma 6.6.3 regarding the largest eigenvalue of S(· ) uses the same ideas and similar arguments and is therefore omitted. Lemma 6.6.4 Let Sk(-) and Soo(') be symmetric real n x n matrix-valued functions, continuous on a compact interval [a, b] (k = 1, 2, ...). Denote the minimum and maximum eigenvalues of these matrices by Cti'), CtooC) and Pi'), Poo('), respectively. If Sk(-) ~ Soo(-) on [a, b] as k ~ 00, then Ctk(') ~ Ct oo ( ' ) and Pk(') ~ Poo(') uniformly on [a, b] as k ~ 00.

Proof. To prove that Ctk(') ~ Ct oo ( ' ) uniformly, it is sufficient to prove only pointwise convergence since by Lemma 6.6.3 the Ctk(') and Ct oo ( ' ) are continuous real-valued functions on a compact interval. To effect a proof by contradiction, assume that, at some point t E [a, b], Ctit) fr Ctoo(t) as k ~ 00.

120

6.

General Linear Equations with Time-Varying Coefficients

By extracting an appropriate subsequence, there is no loss in assuming that either

= 1,2,

)

(6.6.25)

(k = 1,2,

)

(6.6.26)

(k

or for some number p > O. As in the proof of Lemma 6.6.3, inequality (6.6.25) can be rewritten as (k = 1,2, ...)

(6.6.27)

for appropriate unit vectors ~k' ~oo in fJtn. By assuming a further extraction of an appropriate subsequence, it can be assumed that ek -+ e* as k -+ 00 for some ~* E rJin of unit length. The same elementary manipulations as those that appeared in the proof of Lemma 6.6.3, along with the hypothesis that Sk( .) -+ S 00 ( • ) on [a, b] as k -+ 00, show that (6.6.27) leads to the contradiction eoo .Soo(t)e 00 >

e*· Soo(t)~*

(6.6.28)

and (6.6.26) leads to the contradiction of the fact that ~k minimizes ~. Sk(t)~ over the collection of unit vectors in fJtn (k = 1,2, ...). In any case, the appropriate contradiction is achieved, thus completing the proof of the statement that if Sk(') -+ Soo(') on [a, b] as k -+ 00, then OCk(') -+ oc oo(') uniformly on [a, b] as k -+ 00. The conclusion concerning Pk) and Poo(') can be argued in a similar manner and is thus left to the reader. Lemma 6.6.4is now established. The next theorem is a continuation of the results obtained in Theorem 6.6.2 and uses the same notation. It provides a means for approximating the extreme eigenvalues of P(·) without requiring the computation of cI>A' and in a form (6.6.3) needed for application of Theorem 6.6.1. The following string of definitions is involved:

I -t; a.e. a E ['t, b]}, = inf{s I AT(a)Ak(a) + Ak(a)A(a) ~ si; a.e. a E ['t, b]},

"Ik = sup{y AT(a)Ak(o) + Ak(a)A(a) ~ ~k

k

Ak(t)

= max{A I L

i=O

A;(t) ~ AIn},

k

f.Lk(t)

= min{f.L I L

i=O

(6.6.29)

A;(t) ~ f.LI n },

OCk(t) = Ak(t)/(l - Yk),

Pk(t) = f.Lk(t)/(l - ~k)'

ocoo(t) = max]« I P(t) ~ ocln }, Poo(t)

= min{PI P(t)

~ PIn},

r ~ t ~ b.

6.6

121

Estimates of Solution Norms

Remark 6.6.3

If

lim inf Yk

~

0,

k ....oo

then Yk -- 0 as k -- 00. To see this, for arbitrary e > 0 consider Yk = 1, 2, ...). For this choice of Y in the definition of Yb

= Yk - e

(k

AT(IT)Ak(IT)

+ Ak(IT)A(IT)

~ (Yk - £)In ,

(6.6.30)

a.e. IT E ["t, b], which implies that lim sup k-C(l

~. (b cIlT(IT, t)[ AT(IT) Ak(IT) + Ak(IT)A(IT)]cIl(lT, t) dIT ~ Jt

~ [lim sup Yk - £]~'P(t)~ k-+ co

(6.6.31)

.

for each nonzero ~ E f1tn, t E ["t, b). From (6.6.14) of Theorem 6.6.2 and the fact that pet) > 0, inequality (6.6.31) reduces to lim sup Yk ::s; e.

(6.6.32)

k ....oo

But

£

> 0 is arbitrary, and this is consistent with (6.6.32) only if lim sup Yk k ....oo

= O.

(6.6.33)

°

This concludes the demonstration that Yk -- as k -- 00. A similar argument shows that bk - - 0 as k -- 00 if lim sup, .... 00 bk ::s; O. The functions Ak(-), Jlk('), IXk('), Pk('), 1X 00 ( - ) ' and Poo(-) are continuous on ["t, b]-a consequence of Lemma 6.6.3. Both IX 00 ( . ) and Poo(') are strictly positive on ["t, b) since pet) > 0 on that intervaL They are the extreme eigenvalues of P(·).

Remark 6.6.4

Theorem 6.6.5 Let A(') be locally integrable on J and consider any compact interval ["t, b] c J. Assume that

lim infYk ~ 0,

(6.6.34)

k ....oo

lim sup k ....co

s, s

O.

(6.6.35)

Then for each e > 0 there exists a K. such that

o < IXk(t)In s 1X00(t)In s pet) s Poo(t)In ::s; Pit)In for

"t ::s; t ::s;

(6.6.36)

b - e, all k ~ K •. Moreover, assumptions (6.6.34) and (6.6.35)

122

6.

General Linear Equations with Time-Varying Coefficients

imply that

both uniformly on [""t", bJ as k -

(6.6.37)

and

IXk(-) - 1X 00(-) 00.

Proof. Lemmas 6.6.3 and 6.6.4and Theorem 6.6.2imply that Ak(' )-1X 00(') and Pi") - 1300(-) uniformly on [""t", bJ as k - 00. As discussed in Remark 6.6.3, assumptions (6.6.34) and (6.6.35) ensure that Yk - 0 and £>k - 0 as k - 00. The uniform convergences (6.6.37) thus follow from the equations defining lXi') and Pk(')' On applying the inequality defining Yk' it follows from (6.6.11) of Theorem 6.6.2 that for every f. > 0,

k

=

L

Ai(t)

i=O

+ (Yk

- f.)P(t),

(6.6.38)

r ~ t ~ b,

for all k sufficiently large that 1 - Yk > O. Since gives

f.

> 0 was arbitrary, (6.6.38)

""t" ~

t

s

b,

(6.6.39)

for all k large enough that 1 - Yk > O. In similar fashion it is easy to see that ""t" ~

t

s

b,

(6.6.40)

for all k so large that 1 - £>k > O. In summary, IXk(t)I" ~ P(t) ~ f3k(t) I" ,

r ~ t ~ b,

(6.6.41)

for all appropriately large k such that 1 - Yk > 0 and 1 - £>k > O. In view of Remark 6.6.4 and the uniform convergence (6.6.37), it follows immediately that for each f. > 0 there exists a K. such that (6.6.42) on

""t" ~ t ~

b -

f.

for all k

~

K •. This completes the proof of Theorem 6.6.5.

6.6

123

Estimates of Solution Norms

Remark 6.6.5 Theorem 6.6.5 and the definitions leading up to it were developed to provide computationally feasible approximations Q(i.), Pk(') to the extreme eigenvalues 01 00 ( -}, Poo(-) of P(·) needed in (6.6.3) for the application of Theorem 6.6.1 in computing upper and lower estimates of the integral norm of the solutions to the initial-value problem (6.6.1) and (6.6.2). The precise manner in which the theory accomplishes this goal deserves some reiteration and further comment. The extreme eigenvalues 01 00 ( ' ) , Poo(-) would provide the best possible choice of 01(') and P(·) in (6.6.3) were it not for the fact that their calculation would require the computation of A or some equivalent to get P(· ). Theorem 6.6.5 advocates the less accurate Q(k(' ), Pk(') on a subinterval T :::; t :::; b - e as appropriate choices of Q( .) and P(·). The computations of the Q(k(') and Pk( .) work more directly with A(') itself, and this of course vastly enhances the applicability of Theorem 6.6.1. The uniform convergence concluded by Theorem 6.6.5 ensures that any desired level of approximation can be achieved (at least in principle) by increasing k at the cost of the additional computation. Remark 6.6.6 In some applications the computational effort can be reduced by sacrificing accuracy. That is, in some cases, rather than compute 'Yk> Ok' Ak(-), and J.lk(· ) for some fixed (perhaps small) value of k, it might be easier to find numbers Yk' bk, E < b and continuous lk), f1.k(·) such that

AT(o)Ak(a)

+ Ak(a)A(a) ~

AT(u)Ak(u)

+ Ak.(u)A(u) :::; bkl n,

°< l

a.e.

Yk1n,

with 1 - Yk > 0, 1 -

I

i=O

(6.6.43)

EJ,

(6.6.44)

lJJ,

(6.6.45)

a.e. a E [T,

k

k(t)I n :::;

a E [T, EJ,

all

Ai(t):::; f1.k(t)I n,

t E [T,

bk > 0. Then application of(6.6.11) leads to for

t E [ r, lJ),

(6.6.46)

where iik(t) =

l k(t}/(l

- Yk),

(6.6.47)

Pit)

f1.it)/(1 - J'k)'

(6.6.48)

=

Thus (6.6.46) provides an estimate of the extreme values of P(·) of the type needed for application of Theorem 6.6.1. The acceptability of the estimates must be judged in the context of the application. If they are adequate, the question of arbitrary high degree of approximation is irrelevant.

124

6.

General Linear Equations with Time-Varying Coefficients

Remark 6.6.7 Hypotheses (6.6.34) and (6.6.35) are rather mild assumptions. For example, they are met if I AT(t) 1 + IA(t)1 is bound on [T, b] since, according to Theorem 6.6.2, A k ( • ) as k - 00 uniformly on [T, b].

1 1- °

The following corollary follows readily from Theorem 6.6.1 and Remark 6.6.6. Corollary 6.6.6 Let A(') be an n x n integrable real matrix-valued function on a compact interval [a, bJ. Assume that for some numbers y and b < 1, a.e.

t E [a, b

J.

(6.6.49)

Then x(t, ~), the unique absolutely continuous solution to the initial-value problem in [a, b] x PAn:

x = A(t)x, x(a, ~)

=

a.e.

t E [ a, b],

(6.6.50) (6.6.51)

~,

satisfies the inequalities

(6.6.53) Proof.

T

The conclusion follows directly from computing, using k =

= a in Theorem 6.6.1 and Remark 6.6.6.

Example 6.6.1

°

and

In the scalar equation

v + a(t)v =

a.e.

0,

t e [0,1],

(6.6.54)

[0, 1],

(6.6.55)

assume that a(' ) is measurable and satisfies

°s

aCt)

s

1,

tE

with strict inequality holding on a subset of [0, 1] of positive measure. The claim is that for each solution v(' ) to (6.6.54) on [0, 1], the norm (6.6.56)

125

6.6 Estimates of Solution Norms

satisfies the inequality

~(O)

[1 ~ f~

~

exp ( -

]

f~ ~:»)

Ix(O)J2

Ix(a)J2 da

13(0)

[1 -

f~ ~~:»)

exp ( -

]I

tE

x(O)J2,

[0, 1],

(6.6.57)

in which ~

1 - t - f/ (1 - a)[1 - a(a)] da 1 + 2 fA (1 - a)[1 - a(a)] do '

(t) -

(6.6.58)

~-::--;;-;----,------:-;>,=--------:--:-=i~

pet) = 1 - t + f: (1 - a)[1 - a(a)] da 1 - 2 fA (1 - a)[1 - a(a)] da

(6.6.59)

To verify this claim, write out (6.6.54) in first-order form to obtain the matrix coefficient A(t)

=

[-~(t)

~J.

(6.6.60)

With formulas (6.6.12) and (6.6.13), compute (6.6.61)

Ao(t) = (1 - t)12'

J1 (1 -

A 1(t) =

a)[1 - a(a)]

da[~

~J,

(6.6.62)

and hence A T(t)A 1(t)

+ A 1(t)A(t)

=

2

5,1 (1 -

a)[1 - a(a)] do [ -

~(t)

~J.

(6.6.63)

From (6.6.63) it is apparent that ")1112

s

A T(t)A 1(t)

")11

=

-2

+ A 1(t)A(t) ~

(6.6.64)

(;112'

where

(;1 =

fa1 (1 -

a)[1-a(a)] da,

(6.6.65)

2 fa1 (1 - a)[1 - a(a)] da

because of (6.6.55). That restriction on a( .) also ensures that 1 -

(6.6.66) ")I 1

> 0 and

126

6.

General Linear Equations with Time-Varying Coefficients

1 - 1 > O. The eigenvalues of 1

L

Ai(t) = Ao(t)

i=O

=

+ A 1(t)

[1

Jtr (l -

11

1- t

(l - 0')[1 - a(O')] dO'l

0')[1 - a(O')] do

~~

1- t

compute to be Al(t)

=

1- t

fll(t) = 1 - t

-11 + 11

(l - 0')[1 - a(O')] da,

(6.6.68)

(1 - 0')[1 - a(O')] da.

(6.6.69)

Now it is clear from (6.6.65), (6.6.66) and (6.6.68), (6.6.69) that (6.6.58), (6.6.59) arise from the choices and and the claim (6.6.57) is a direct consequence of Theorem 6.6.1. The preceding theory and technique for approximating the extreme eigenvalues of the solution PO to (6.5.7) can be applied to obtain estimates of Ix(tW as well as of its integral.

Theorem 6.6.7 Let PO be a symmetric solution to (6.5.7) satisfying (6.6.3) for all t in a subinterval of J with r ~ t. On that subinterval the solution x(' ) to (6.6.1), (6.6.2) satisfies

~~:~

exp( -

f~~:»)

1~12 ~

Ix(tW

~ ~~;;

fP~:»)

exp( -

1~12

(6.6.70)

for all t in the subinterval. Proof. Let vet) = x(t)· P(t)x(t). Since pet) satisfies (6.5.7), it is easy to check that

d~;t)

= -lx(tW.

(6.6.71)

Since

o < ~(t)In

~

pet)

~

P(t)I n

for

r

~

t,

(6.6.72)

6.6

127

Estimates of Solution Norms

successive applications of (6.6.72) to (6.6.71) imply

dv(t) vet) -

dt pet),

-- Y2, ... , Yn is a linearly independent collection ofabsolutely continuous n-vector functions defined on an open interval J, then its elements constitute an ordered basis of the vector space of solutions to the equation x = A(t)x for some n x n locally integrable matrix function A(') oti J,

3. Verify that $'n., the dual of $'n, is a vector space and show that its dimension is n by exhibiting an ordered basis. 4. Prove that cDA(t, 0) = cDA(t, 't) A(t, r) =

e sint - sinr

[c~S( cos

t - COS,) sm(cos t - cos r)

- sinecos t - cos r) J . (7.3.20) cos(cos t - cos r)

In particular, cI>A furnishes the solution to (7.3.11), satisfying Xl(') = x 2(r) = ~2: X1J = [ X2

esint-sint

= [~l ~l

~l'

c~s(cos t - cos r) - ~2 sin(cos t - cos r)J. (7.3.21) sm(cos t - cos r) + ~2 cos(cos t - cos r) (7.3.22)

144

7.

Commutative linear Differential Equations

EXERCISES

1. Verify that the following system is commutative on qj by application of Theorem 7.3.2, and determine a commuting antiderivative B(') of its coefficient matrix A(·). X2

=

+ (1 + 2t)X2'

-tx 1

2. Check the commutivity of the following system on qj and then compute its general solution by the, method discussed in Section 7.3.

3.

Consider the matrix function of qj,

A(t)

O1J {[1 0'

t

~ [~ ~J

irrational, rational.

Find an appropriate commuting antiderivative to demonstrate that A(') is commutative on qj. 4. The matrix function B(t) = sin 2t [

(0. ) sgn sin t

olJ

is absolutely continuous on qj. Why? Compute its derivative A( .) and confirm the commutivity of A(') on qj. Find those (t, r) E qj2 for which A(t)A(-r) - A(-r)A(t) "# 0.

5. Show that if A(') is commutative on J with commuting antiderivative B( '), then for each -r E J, B(-r)A(t) - A(t)B(-r) = A(t)

f

A(u) do -

f

A(u) da A(t),

a.e. t E J. Apply this result to conclude that the equality A(t)

t

A(lT) do

=

fo~(U)

do A(t),

a.e. t E qj, fails for the commutative system of Exercise 4.

145

7.4 Some Sufficient Conditions for Commutivity

7.4

Some Sufficient Conditions for Commutivity

The results of this section provide easy verification of commutivity where they apply.

Theorem 7.4.1 A locally integrable n x n matrix-valued function A(') on J is

if either of the following equivalent conditions are A(t)A(r) = A(r)A(t), a.e. t E J for a.e. r E J, A(t) J: A(O") da = J: A(O") da A(t) for a.e. t E J, all r E J.

commutative

(a) (b)

Proof.

met:

Clearly, (a) implies (b). Now assume (b), which is to say that A(t)

f

A(O") da -

f

A(O") do A(t)

=0

(7.4.1)

for a.e. t E J, all r E J. At each point t E J at which (7.4.1)holds, the left-hand side is an absolutely continuous function of r: therefore the equation can be differentiated relative to r and (a) results. Thus (a) and (b) are equivalent. If (b) is assumed, then, upon setting Bo = 0 in (7.3.6), Theorem 7.3.2 applies and A(') must be commutative. This concludes the proof.

Corollary 7.4.2 Let A o be any n x n matrix over CC and f(z) be a power series centered at the origin, as in Theorem 4.2.2, with radius of convergence p, If Ao is nilpotent, then the formula (7.4.2)

A(t) = f(tA o)

defines a matrix function A(') that is commutative on 9f. If P > 0, then (7.4.2) determines a commutative matrix function on the interval - plr < t < otr in which r is the maximum absolute value of the eigenvalues of A o.

Example 7.4.1 According to Corollary 7.4.2, A(t) =

ao

f'

L: I" A~ k.

= e tAo,

t

E

9f,

(7.4.3)

k=O

is commutative fo.r any n x n matrix Ao over CC. The commutivity of Example 7.2.1 is now transparent when it is recognized to be of type (7.4.3) with Ao the matrix [; ~ A].

Example 7.4.2 series is

Another commutative matrix function defined by a power ao

A(t) =

L:

k=O

t

E (

f' A~

= (In -

tA o)- t,

(7.4.4)

-llr, l/r), where Ao is any n x n matrix over CC whose eigenvalues

2 each satisfy 121

< r.

146

7.

Commutative Linear Differential Equations

If, in lieu of the series in (7.4.4), the inverse matrix in that equation is taken as the defining formula, then the interval can sometimes be extended. In fact, for A o having no real eigenvalues, commutivity occurs on (- 00, (0), and if OCl < OC2 < ... < oc, are the reciprocals of the real nonzero eigenvalues of a matrix A o over re, then intervals over which commutivity occurs can obviously be taken to be anyone of (- 00, ocd, (OCl' O(2), ••. , (OC'-l' oc,), (OC., (0). Although the inverse is defined for a.e. t E ~, the OCi must be excluded from the domain of A(') to maintain the local integrability. A more complete description of the manner in which analytic functions can be used in constructing commutative A(') is given in Section 7.5. Theorem 7.4.3 Let {Ak}O' be a commuting family ofn x n matrices over re and {OCk(')} 0' be a sequence of complex-valued measurable functions defined on an open interval J. If the series CXJ

L

(7.4.5)

locit)IIA k !

k~O

is convergent for a.e. t

E

J and is locally integrable, then the series

A(t)

converges for a.e. t

E

=

CXJ

L

k=O

(7.4.6)

ocit)A k

J and defines a commutative matrix function A(') on J.

Proof. The assumed convergence and local integrability of (7.4.5) imply the same for (7.4.6). Equation (a) of Theorem 7.4.1 follows from writing A(t) and A('l:) as series and applying the commutivity of the Aks. Thus the conclusion of Theorem 7.4.3 follows as a corollary to Theorem 7.4.1.

Series that are not necessarily power series serve as useful devices for producing commutative matrix functions. One example that illustrates the point is the series

Example 7.4.3

•

A(t) =

L CXJ

k~

(1 +-

e- kt [sin ktl I

1

k

)k

Ao ,

t E (0, (0), which clearly satisfies the hypothesis of Theorem 7.4.3. EXERCISES

1.

Consider any matrix function A(') of the type 00

A(t) =

L k~O

ocit)A k,

t

E

[0,

(0),

(7.4.7)

7.4

147

Some Sufficient Conditions for Commutivity

in which the n x n matrices A o' At> A 2 , .•. satisfy AiA j = AjA i for all i,j and the scalar-valued ak(') are locally integrable. (a) Show that if

k~O Loo lak(a) I do IAk ! 0 appropriately small the matrix function Bi«, t) = B(t) + el; will satisfy IB(f., t)1 #- 0 on some subinterval J, of J containing r. But a trivial calculation shows that B(e, t )

dB(f., t) _ dB(f., t) ( )-0 dt dt B f., t - ,

(7.5.18)

a.e. t E J" including the point t = r. It follows from what has already been proved that (7.5.19) a.e. t E J" including the point t = r, Now in view of the fact that the elements of B*(f., t) are polynomials in both f. and the elements of B(t) it is possible to let f. -+ 0 in (7.5.19), with the result that at least at t = r, (7.5.20) Since the argument applies to a.e. r

E

J, as restricted, the proof is complete.

Remark 7.5.1 A matrix function B(') on J is called analytic if it has a convergent power series

B(t) =

Lco k=O

(t

- rl B(k)(r) k!

(7.5.21)

about each point r ~ J. If B(') in Theorem 7.5.3 is analytic, then f(B(')) will likewise be analytic on J. Example 7.5.1 If fez) is any entire function such as eZ , sin z, any polynomial p(z), p(z)eZ , etc., then e", sin B, pCB), p(B)eB, etc., will be commuting antiderivatives of the consequent commutative matrix functions eBB, B cos B, p'(B)B, [pCB) + p'(B)]eBB, etc., for BO any commuting antiderivative. One allowable choice of B( .) would be B(t) = tAo, where A o is any n x n matrix. Theorem 7.5.5

Let B o be any n x n matrix over CC and f(t, z) be defined and continuous on J x E, where J is an open interval and E => a(B o) is an open

152

7.

Commutative linear Differential Equations

subset of~. Further assume that f(t, z) is absolutely continuous relative to t for each fixed z E E and that bothf(t, z) andfl'/, z) are analytic in zfor each fixed t E J. Then the matrix function B( .) defined as B(t) = f(t, B o) is a commuting antiderivative on J and moreover, BoB(t) - B(t)B o = 0, Proof.

t

E

(7.5.22)

J.

The assumptions allow application of the Cauchy integral formula 1 B(t) = -2. tu

1.

jll"',

f(t, z)(zI - BO)-l dz,

t E J.

(7.5.23)

The details are left as an exercise. Corollary 7.5.6 Iff(z) is analytic on an open set nand A o is an n x n matrix over ~ for which a(tA o) c: for all t

E

n

(7.5.24)

J, J an open interval, then the matrix function

A(t) = f(tA o)

(7.5.25)

as well as its derivatives A(k)(t) =

Pk)(tAo)A~

(7.5.26)

(k = 1, 2, ...) are analytic commutative matrix functions on J. Proof.

Apply Theorem 7.5.3 to verify condition (a) of Theorem 7.4.1.

Theorem 7.5.7 Let A o be any n -x n matrix over ~ and f(t, z) be defined on J x E where J is an open interval and E::J a(A o) is an open subset of~. Assume thatf(t, z) is locally integrable relative to t for fixed z and analytic in z for a.e. fixed t E J. Then the matrix function A('), defined as A(t) = f(t, A o), is commutative on J. Proof.

The assumptions allow application of the Cauchy integral formula A(t)

= -2. 1 1

1tl

jll"',

f(t, z)(zI - AO)-l dz,

(7.5.27)

The details are omitted. The next theorem is another example of the many interrelations between linear and nonlinear differential equations and linear algebra.

Theorem 7.5.8 Let B( .) be an n x n matrix-valued commuting antiderivative over

~

on an open interval J. If the characteristic polynomial of B(t),

(7.5.28)

153

7.5 Matrix Functions and Operations Preserving Commutivity

has distinct roots at a.e. t

E

J, then B(') satisfies the differential equation

a.e.

t

E

J.

(7.5.29)

(Subscripts denote partial derivatives.) Conversely, for every monic polynomial pet, 2) - over ((5 with absolutely continuous coefficients, each n x n matrix solution B(') to (7.5.29) will be a commuting antiderivative. If pet, 2) annihilates B(t) and has distinct roots at a.e. t E J, then the roots of the minimal polynomial of B(t) are also distinct at a.e. t E J and the eigenvalues of B(t) are roots of pet, 2) for a.e. t E J. Proof. If B(') is an n x n absolutely continuous matrix function over ((5, then its characteristic polynomial (7.5.28) will obviously have absolutely continuous coefficients. By the Cayley-Hamilton theorem, pet, B(t))

=

t

0,

E

(7.5.30)

J,

which can be differentiated, and when B(') is a commuting antiderivative, that differentiated equation will be of the form pit, B)B

+ pit, B)

=

0,

a.e.

t E J.

(7.5.31)

By a standard theorem of algebra the assumption that p(t,2) has distinct roots implies that pit, 2) #- 0 at those 2 in ((5 where pet, 2) = O. Hence since a(p(t, B)) = p(a(t, B)), the matrix pit, B(t)) is nonsingular at a.e. t E J. This allows (7.5.31) to be rewritten as .

B

=

-pit, B)

-1

a.e.

pit, B),

t

E

J.

(7.5.32)

To prove the last half of the theorem, let pet, 2) be a monic polynomial over absolutely continuous coefficients. Then the right-hand side of Eq. (7.5.29) is defined. Now let B(') be an absolutely continuous solution to that equation. (The existence and uniqueness problems for such nonlinear equations are treated later in the text and are of no direct concern here.) Since B(t) commutes with the right-hand side of(7.5.29), it satisfies (7.5.1) at a.e. t E J and hence is a commuting antiderivative. Suppose that ((5 with

per, B(r)) = 0

at some r

E

(7.5.33)

J. Since B(') is assumed to satisfy (7.5.29),differentiation leads to

d dt pet, B(t))

=

pit, B)

.

+ pit, B)B

= p,(t, B) - pit, B)p;: let, B)p,(t, B) a.e. t E J. = 0,

(7.5.34)

154

7.

Commutative Linear Differential Equations

The only continuous function satisfying (7.5.33), (7.5.34) is pet, B(t)) = 0,

all

t

E

(7.5.35)

J.

If (7.5.35) holds and pet, A) has distinct roots at a.e. t E J, then the minimal polynomial of B(t) divides pet, A) at each t E J, and hence that minimal polynomial must have distinct roots at a.e. t E J. Obviously the eigenvalues of B(t) are roots of pet, A) (with not necessarily the same multiplicity) a.e. t E J. This completes the proof. Corollary 7.5.9 Let B(') be an n x n matrix-valued commuting antiderivative over on an open interval J. If the characteristic polynomial pet, A) of B(t) has n distinct roots Ai(t) (i = 1,2, ... , n), a.e. t E J, they satisfy the scalar differential equation

rc

Z = - Pr(t, z)j pz(t, z), The eigenvalues of A(t) = B(t) are ~.lt)

a.e.

t

E

J.

(i = 1,2, ... , n), a.e. t

(7.5.36) E

J.

Proof. The conclusion that the A;(t) satisfy (7.5.36) on the intervals where they are distinct follows from differentiation of the equations pet, A;{t))

= 0.

(7.5.37)

Since B(') will satisfy (7.5.29), A(t) = -p;,(t,B)-lPrCt,B),

(7.5.38)

which in tum implies that the eigenvalues of A(t) are - p;,(t, ..1;)-1 PrCt, A;) (i

=

i;

(7.5.39)

= 1,2, ... , n), a.e. t E J. This proves Corollary 7.5.9.

Remark 7.5.2 Commuting antiderivatives exhibit a great deal of geometric regularity; i.e., they show a proclivity to be solutions to differential equations. Theorem 7.5.8 concludes this to be true even for those whose eigenvalues collide (but do not stick together). The differential equations are quite nonlinear.

The next theorem provides a clear picture of what is happening during the intervals between collisions. Its proof must borrow some knowledge about nonlinear differential equations from subsequent chapters. In this instance the maintenance of subject continuity seems to warrant the break in the traditional order of writing. Theorem 7.5.10 Let A(') be a continuous commutative n x n matrix-valued function over C(j on an open interval J and suppose that A( .) has a commuting

155

7.5 Matrix Functions and Operations Preserving Commutivity

antiderivative B(' )for which B(t) has n distinct eigenvalues at each t E J. Then B(t)B(I:) - B(I:)B(t) = 0, Proof.

all (r, 1:) E J x J.

(7.5.40)

Assume the hypothesis and consider the characteristic polynomial pet, z) = det[ zI - B(t)J,

t

E

(7.5.41)

J.

The proof revolves around the nonlinear differential equation in J x i

= - plt, z)/pit, z).

~,

(7.5.42)

As stated by Corollary 7.5.9 the eigenvalues A.;{t) of B(t) are each solutions to = 1,2, ... , n). The argument advanced in the proofof Theorem 7.5.8 explained that the denominator pit, z) in (7.5.42) is not zero along the paths {(t, Ai(t)) It E J} (i = 1, 2, ... , n). From the once continuous differentiability of B(') it is readily shown that for each compact subinterval [a, bJ c J the corresponding segments of those paths {(t, Ai(t)) It E [a, bJ} (i = 1,2, ... , n) are contained in open tubular neighborhoods on which the right-hand side of (7.5.42) is continuous in (t, z) and analytic in z for fixed t. The existence of the solutions Ai(') to (7.5.42) and this analytic behavior ofthe differential equation in their neighborhoods is enough to allow the classical theory of analytic nonlinear differential equations to apply (see Theorem 11.4.6). This theory states that (7.5.42) has a unique general solution ntt, 1:, z), regarded as a complex-valued scalar function of the initial parameters (I:, z) as well as t satisfying 11(1:,1:, z) = z. In the situation being considered I: is regarded as fixed in J. The classical theory concludes that for each fixed t E J the function l1(t, 1:, z) is analytic. in z on some open neighborhood of the eigenvalues Ai(I:)(i = 1, 2, ... , n) of B(')' in the complex plane. The uniqueness aspect of the theory ensures that l1(t, 1:, Ai(I:)) = A;(t), t E J (i = 1,2, ... , n). More could be said, but it is primarily the existence of the general solution l1(t, 1:, z) and its analyticity relative to z that are germane in the present proof of Theorem 7.5.10. They allow application of the Cauchy integral formula. The claim is that (7.5.42)onJ(i

B(t) =

~J:

2mjow,

l1(t,1:, z)[zI - B(I:)]-l dz,

t

E

J.

(7.5.43)

In (7.5.43) the open set co, is centered about the eigenvalues Ai(1:) of B(1:) = 1,2, ... , n). Note that the integral in (7.5.43) does define a matrix function of B(I:) for t E J fixed and that (7.5.43) is valid at t = I: since 11(r, r, z) = z. As a consequence of the fact that (as a function of t) l1(t, r, z) satisfies (7.5.42), it is easily verified that the integral (7.5.43) does indeed provide a matrix solution to the differential equation (7.5.29). But the anti(i

156

7.

Commutative linear Differential Equations

derivative of A(') hypothesized does satisfy (7.5.29)-one of the conclusions of Theorem 7.5.8-and the classical theory moreover ensures the uniqueness of the solution to (7.5.29), taking on the value B(7:) at t = 7:. This proves the equality (7.5.43). With (7.5.43) established, the commutivity claimed in (7.5.40) is now transparent. Since initially 7: E J was arbitrary, the proof is complete.

Remark 7.5.3 For those A(') satisfying the hypothesis of Theorem 7.5.10, Eq. (7.5.40) can be differentiated relative to t and 7: to get (t,7:) E J x J.

A(t)A(7:) - A(7:)A(t) = 0,

(7.5.44)

Thus at least for continuous A(') with commuting antiderivatives B(') satisfying the conditions imposed in Theorem 7.5.10, it is now clear what is happening on the open intervals between collisions of the eigenvalues of B(.).

EXERCISES

1. Show that the matrix function A(') on

- 2t A(t) =

[

~

defined by

2t]

1

o1

2t

1 -2t

1

is analytic and commutative. 2. Verify that B(') defined on ~ by

m,) ~ [,;

'~1

:;]

is a commuting antiderivative. Let A(') = (B*)'(') where * denotes the classical adjoint. Why is A(') commutative on ~? Compute A(·). 3. Show by direct computation that if B(') is a commuting antiderivative on J with B(t) nonsingular for each t E J, then B- 1( .) is also a commuting antiderivative. 4. Explain what is wrong with the following reasoning: For a scalar z, j(z) = det(z) is analytic in the entire plane. Hence for any n x n matrix A, 1

.i.

det(A) = -2

mj"w

det(z)[zl - AJ-l dz.

7.6 The Special Case n

5.

=

157

2

Without doing any extensive computing, show that the matrix function A(t) =

[2~2

-4t

3

~

t2

2t

~

t ] 3 _t 2

is commutative on &l and that on those intervals in which it is invertible, its inverse A - 1(.) is likewise commutative.

7.6 The Special Case n - 2

Clearly, all absolutely continuous scalar functions are commutative. This section analyzes 2 x 2 systems in detail.

Theorem 7.6.1 If a commutative 2 x 2 matrix function A(') on an open interval J is analytic, then A(t)A(-r) - A(T)A(t) = 0

for all

(r, T) E J x J.

(7.6.1)

On the other hand, if A(') is but Coo, then (7.6.1) can fail on a subset of J x J of positive measure. Proof. Let A(') be analytic on J. If A(') is assumed to be commutative, then there exists a commuting antiderivative B(') of A(') and B(t) = B(T)

+

f

A(s) ds

(7.6.2)

for each T E J and all t E J. Since A(') is analytic on J, it follows from (7.6.2) that B(') is likewise analytic there and that the equation (7.6.3)

B(t)B(t) - B(t)B(t) = 0

must hold for all t

E

J. Write out B(t) in terms of its elements as

B(t) - [bit) - bz(t)

b 1(t) ] bit) - bit) .

(7.6.4)

It follows upon substitution of(7.6.4) into (7.6.3) that the latter is equivalent to the system b 1(t)bz(t) - b 1(t)bz(t) = 0,

(7.6.5)

bz(t)b 3(t) - b 2(t)b 3(t) = 0,

(7.6.6)

b 3(t)b 1(t) - bit)b 1(t) = 0

for all

t

E

J.

(7.6.1)

158

7.

Commutative Linear Differential Equations

If btU, bi'), and bk) are all identically zero on J, then

= bit)Iz,

B(t)

t

E

(7.6.8)

J,

and hence

= bit)Iz.

A(t)

(7.6.9)

Certainly in this case A(t) satisfies (7.6.1). Now consider the remaining case in which at least one of btC-), bi'), bi') is nonzero on some open subinterval (ex, f3) c J. Suppose that it is b l (- ) that is nonzero on (ex, f3). Multiplying (7.6.5) and (7.6.7) by b I Z(t) and integrating produce the equations bz(t) = c Zb l(t),

(7.6.10)

bit) = C3bt(t)

(7.6.11)

for some constants Cz, C3 for all t E (ex, f3). But since b t('), bi'), and bi') are analytic on (ex, f3), it follows that (7.6.10) and (7.6.11) hold for all t E J. Substituting (7.6.10) and (7.6.11) into (7.6.4) and differentiating show that A(t)

. [1° OJ + . [0 1J

= bit)

1

bt(t) Cz -C3 '

(7.6.12)

and again it is a trivial matter to check that (7.6.1) holds. A similar argument applies in the two remaining possible cases for which it is b z or b 3 that is nonzero on a subinterval of J. This concludes the proof of the first part of Theorem 7.6.1. To establish the last part, consider the matrix function A(t)

=

a(t{~

~J

+ a( -t)[~

~J

(7.6.13)

on J = (- 1, 1), in which a( .) is the classical example of aero but nonanalytic function .

{

aCt) =

°,

2

e -1/t ,

0< t < 1, -1 < t ::::;; O.

(7.6.14)

Note that

_ { A(t) -

aCt)

[~ ~l

[0 IJ

a( - t) 1 0 '

0::::;; t < 1,

(7.6.15) -1 < t ::::;; O.

159

7.6 The Special Case n = 2

The next task is to verify that A(') is commutative. Consider B(t) =

J:

t

A(s) ds,

E

(7.6.16)

(-1,1).

Certainly B(t) = A(t)

(7.6.17)

for all t E( -1, 1), and B(') is not only absolutely continuous but Coo. By looking at the two cases, first that in which 0 ~ t < 1 and second that in which -1 < t ~ 0, it follows readily from (7.6.15) and (7.6.16) that A(t)B(t) - B(t)A(t) = A(t)

J:

A(s) ds -

J:

A(s) ds A(t) = 0 (7.6.18)

for all t E ( -1, 1). The commutivity of A(') is thus established. The last item of concern is calculation of the term involving A(') in (7.6.1). Restrict attention to -1 < , < 0 < t < 1. Applying (7.6.15), compute A(t)A(,) - A(,)A(t) =

a(t)[~

~Ja( -,{~

- a(

-,{~

~Ja(t{~

= a(t)a( -,{~ = exp( -

~J

t~

-

-

,~)[~

~J

~J -

~J

# O.

(7.6.19)

The proof is complete. Theorem 7.6.2 Every 2 x 2 commutative matrix function A(' ) on an interval J has a commuting antiderivative of the form B(t) , =

et(t{~

~J

+

ezet)[C2~t)

~:~:~J

(7.6.20)

in which etC'), ek) are absolutely continuous on J and CtC'), ck), ck) are constant over the open intervals at which ezet) # 0 (i.e., the c i ( · ) are step functions). Every such matrix function is a commuting antiderivative. Hence the 2 x 2 commutative matrix functions are those of the form a.e.

t

E

J.

(7.6.21)

160

7.

Commutative Linear Differential Equations

Remark 7.6.1 Not every absolutely continuous cl(.), c2(.) and step functions cl(-), cz(-), c3(*)in (7.6.21) will produce a commutative A(.). The switching must be synchronous with the zeros of tz(t) to maintain the continuity of B ( - ) .The proof of Theorem 7.6.2 is left as an exercise for the reader.

7.7

A Criterion for Exponential Decay of Solutions

For applications it is useful to know conditions on the coefficient matrix A( of (7.2.3) which will ensure that all solutions to the differential equation approach zero as t + 03. One such criterion for the variable-coefficient case is provided by the following theorem. a )

Theorem 7.7.1 Let A( *)be a locally integrable, commutative, complex n x n matrix-valued function on an interval (a, co) with commuting antiderivative B( .). If for some positive z E (a, co) (7.7.1) and the set (7.7.2) is contained in the half-plane Re z c v, then p t ) I 5 ce"t-" z

(7.7.3)

< t < co,for some c E a. Moreover, c can be chosen so that Jx(t)lI cIxoJeu(t-.F),

(7.7.4)

< co,for all solutions of (7.2.3) with x(z) = xo. In particular, ifu < 0, then each solution x ( - )of (7.2.3) will approach zero as t + co.

z 5 t

Proof. Assume the hypothesis. It is sufficient to prove (7.7.3) since each solution to (7.2.3) can be represented as x(t) =

eB(t)e-B(r)XO.

(7.7.5)

Since B ( * )is assumed to satisfy (7.7.1), it follows that the set K defined by (7.7.2) will be compact. Thus K lies in the interior of the closed half-disk with center at z = u + Oi, side on the line Re z = u and radius p for some p > 0 appropriately large. With the boundary ao of this half-disk oriented in the

161

7.7 A Criterion for Exponential Decay of Solutions

counterclockwise direction, the Cauchy formula provides the representation

eB(tl = -1. 2m

t. [ tJ(fJ

e'" zI - -B(t}]-l dz, t

(7.7.6)

t E [T, co). For fixed t, estimation of (7.7.6) gives

B(t)]-l/ ds

< -etut. /[ zI - - 2n tJ(fJ t

'

(7.7.7)

in which ds is the differential element of arc length. From the fact that zl - B(t}jt is invertible for each z E OW, t E [T, (0), the compactness of OW, and the boundedness (7.7.1), it follows that

/[zI for some b e flt, all z E OW, t is the inequality

B(t)]-l/ 5, b
0 called the period. To save words, a periodic function with period p is called simply p-periodic. The period is not unique; if p is a period, then so is kp (k = 1,2, ...), and if A(') is constant, then it is p-periodic with p > 0, P otherwise arbitrary. As intuition might suggest, a certain aspect of constant coefficient behavior is prominent in tile solutions to all linear periodic equations. A precise account of this phenomenon is described by the Floquet theorem and is amplified in the applications to commutative systems. Lemma 8.1.1 For each nonsingular n x n complex matrix M there exist complex n x n matrix solutions C to the equation

M

=

eC •

(8.1.3)

Proof. Since M is nonsingular, there exists a simple closed curve ow bounding an open set to that contains the eigenvalues of M with the origin in the complement of its closure w. It follows that each branch In(z) of the 162

163

8.1 Periodic Homogeneous Equations

logarithm function is analytic on an open set containing 6.Consequently Theorem 7.5.1 can be applied to determine a matrix (8.1.4)

In(z)[zZ - MI-' d z .

The matrix obtained in (8.1.4) depends on the branch of the logarithm chosen. However, irrespective of the choice, eln(') = z and a second application of Theorem 7.5.1 then gives

This proves Lemma 8.1.1.

Lemma 8.1.2 For each real nonsingular n x n matrix M there exists a real n x n matrix solution R to the equation (8.1.6)

M Z = eR.

Proof. According to Lemma 8.1.1 there exists a complex matrix C = h(M) that satifies (8.1.3). The claim is that CC = CC. To show that this is true, compute r r (8.1.7) Thus it is sufficient to show that C M a real matrix it is clear that

=

MC. From the assumption that M is

(8.1.8)

ln(z)[zZ - MI-' dz M = M c and the equation

cc = Cc

(8.1.9)

is established. The commutivity (8.1.9) and reality of the matrix M facilitate the final calculations MZ

where R = C

= M R = [eC][Tq =

[ e c ] [ g ] = ec+T

=

,

(8.1.10)

+ C is a real matrix. This concludes the proof of Lemma 8.1.2.

Theorem 8.1.3 Floquet. Zf A ( * )is a locally integrable n x n matrix function over F on Se and moreover is p-periodic, then each fundamental matrix @( -) of A( .) can be factored as @(t) = P(t)efc

(8.1.1 1)

164

8.

Periodic Linear Equations

in which the n x n matrix function P(·) is absolutely continuous and p-periodic and C is an appropriate complex n x n matrix. Proof.

Let cD( .) be a fundamental matrix of A(·). Hence (p)

f: ~-l(O")~-T(O")

do

e.

(8.3.12)

The last integral in (8.3.12) is noted to be a symmetric matrix and is moreover, nonsingular. To demonstrate the nonsingularity, compute the quadratic form

e fP c:I>-l(O")c:I>-T(O") do ~ = fP 1~-T(0")eI2 °

.

°

°

da 2: 0,

(8.3.13)

which by continuity is zero if and only if ~-T(O")e = 0, 0" E [0, p]. This of course is possible precisely when ~ = since fundamental matrices are nonsingular. This shows that the integral of (8.3.12) under consideration is in (8.3.12) is also positive-definite and therefore nonsingular. Since ~(p) nonsingular and is free to vary in :Fn , it is now obvious that the right-hand side of (8.3.6) takes on all values of :Fn under all choices of p-periodic and locally integrable f(·). Consequently for each such f(·) (8.3.6) will have a solution x? only if(8.3.?) obtains. This completes the proof of Theorem 8.3.2.

e

Remark 8.3.1 With A(') p-periodic it is possible for (8.1.1) to have a nontrivial p-periodic solution and yet for (8.3.1) to have nontrivial p-periodic solutions for some p-periodic nontrivial f(·) (see Exercise 1). EXERCISES

1. Why does the equation ii

+ u = sin 2t

(8.3.14)

fail to produce a counterexample to Theorem 8.3.2? 2. Consider the differential equation

x=

A(t)x

+ B(t)u(t)

(8.3.15)

in which A(') and B(') are continuous matrix-valued functions of size n x n and n x m, respectively. If A(') and B(') are defined on an interval [to, t 1 ] and for each choice of'(x", x') E:Fn x :Fn there exists a continuous m vectorvalued function u(') (called a control function) on [to, t 1 ] such that the resultant nonhomogeneous equation (8.3.15) has a solution x(·) on [to, t 1 ]

175

8.4 Periodic Commutative Equations

that satisfies the boundary conditions x(t o) = x",

(8.3.16)

x(td = x ',

(8.3.17)

then (8.3.15) is called controllable on [to, t 1l In particular, suppose that A(') and B(') are p-periodic on f7t. Call (8.3.15) p-controllable if for each to E f7t and x? E /Fn there exists a p-periodic control function u(·) on f7t such that the resultant solution x(·) of (8.3.15) satisfying (8.3.16) is likewise p-periodic. Prove that (8.3.15) is then p-controllable if and only if it is controllable on [0, pJ.

8.4

Periodic Commutative Equations

The classical Floquet theory of Sections 8.1 and 8.2 establishes a close tie between the characteristic roots of a p-periodic matrix function and the qualitative behavior of the solutions to the associated linear differential equation. This elegant theory fails to dispense with one important obstacle to its practical applicability, namely, the problem of computing (or at least estimating the absolute values of) the roots. The hitch is that their definition involves knowledge of the generally unknown fundamental matrix at two points. (Recall Remark 8.1.4.) In this section we present one significant class of time-varying periodic systems for which this obstacle can be overcome-commutative periodic equations.

Theorem 8.4.1 Let A(') be a locally integrable, commutative, p-periodic n x n matrix function on f7t and let B(') be one of its commuting antiderivatives. Then the characteristic roots of A(') are the eigenvalues of the matrix M

=

(8.4.1)

e-B(OleB(pl.

If B(') satisfies the additional condition B(p)B(O) - B(O)B(p)

= 0,

(8.4.2)

then

(8.4.3) in which C is the average matrix

C = -1

p

fP A(u) da. °

(8.4.4)

176

8.

Proof.

Periodic Linear Equations

Theorem 7.3.1 implies that

=

(t» E B. for T, S t < co , . Thus suppose that (Tz , 4>(Tz » E qj X qjn - B. for some T z satisfying T, < T z < «i , . By continuity of 4> there exists a largest T < «i., such that (t, 4>(t» E Bn all t for which T, S t S T; call it T 3 . Thus

If(t, 4>(t» I S b

(10.3.7)

for T, S t S T 3 < T z < w+. Since (T l , 4>(Td) E B./ z by the choice of T l , (T3 , 4>(T3» E B. by the choice of T 3 , and by (10.3.4)and (10.3.7) it follows that

~

15: f'ia, 3

-

4>(Tl )! =

3 -

Tli S

< !4>(T3 ) S

bjT

4>(lT»

dlTl

b(;b) =~,

(I0.3.8)

which is a contradiction. This proves (10.3.6). Combining (10.3.5)and (10.3.6) shows that

14>(t) - ~<Xli

(10.3.9)

Sf,

and since e was arbitrary, we conclude that (10.3.10) Since (w+, ~<Xl) E D, by the continuity of f on D, the Cauchy-Peano theorem ensures the existence of a solution (1 co s 4><Xl) of(1O.1.1)through (w+, ~<Xl). The claim now made is that the curve

co : < t < w+, «i.,

S tE1<Xl'

(10.3.11)

extends (1, 4». Certainly $ is continuous because of (10.3.10) and satisfies (10.1.1) for co : < t, < w+ and w+ < t e I <Xl' Since 4><Xl satisfies (10.1.1) at w+' all that remains to be shown is that (10.3.12) as fLO. Because of(10.3.1O) and the facts that 4> satisfies (10.1.1) on (aL, w+) and that $(o» +) = ~ co s it follows that $(w+ - f) - $(w+) = 4>(w+ - f) - ¢<Xl = -

f:++-. felT, 4>(lT» da;

(10.3.13)

194

10.

Global Solutions

and hence (10.3.12) reduces to showing that 1

£"

t+"co+

-r

c

feu, O. 6. Assume that

forall

XE~,

I and g are continuous on ~ and that all

XE~,

in which the integers r, s as well as the real coefficients /1' gl are positive with

10.3

199

The Behavior of Solutions on Maximal Intervals

for some f3 E fjl and all x E fjl. Show that for each bounded and continuous forcing function e on [0, 00) X fjl2, Eq. (10.3.22) will then have all of its solutions extendable to co , = 00. 7. A classical example of a nonlinear equation satisfying the conditions imposed in the result of Exercise 6 is the Van der Pol equation it + (u 2 - 1) u + u = e(t, u, u). Exercise 6 expands this example to include, in particular, all polynomial equations with polynomial coefficients f(u)

=

+ fm um,

fo + flU + f2U2 +

+ gnun,

g(u) = go + glU + g2u2 +

with m even, n odd, and fm' gn positive coefficients. Verify these claims. To see that the conditions imposed upon the polynomials f(u), g(u) are not superfluous, verify that the equation it - u

u-

u

3

=

°

has a solution u(t) = 1/(1 - t) for which w+ < 8. Consider the equation it

+ [(hg)'(u)] u + g(u) =

00.

e(t, u, u)

(in which the prime denotes the derivative relative to u). Assume that e is continuous and bounded on (a, 00) x fjl2, both hand 9 are once continuously differentiable, h is nonnegative, and (hg)(O) = 0. Show that the conclusion of Theorem 10.3.4 applies when 9 moreover satisfies condition (a) of that theorem. 9. Show that all equations of the type it

+ f(u)u + h(u)

f:

f«(J) da = e(t, u, u)

have the immortality property if f and h are continuous on fjl, h is nonnegative, g(u).= h(u) So f«(J) do satisfies (a) of Theorem 10.3.4, and e is continuous and bounded on (a, 00) x fjl2. 10. Verify that the equation it

+ g'(u) u + g(u) = 0,

in which the prime denotes derivative and g(u) = u cos u + sin u, is an example of an immortal equation that is capable of taking on negative potential energy of arbitrarily large magnitude. Compute the potential energy function. Will the immortality persist if the equation is forced with any continuous e bounded on [0, 00) x fjl2? Is w_ = - 00 for each maximal solution to the unforced equation?

Chapter

11

The General Solution-Dependence of Solutions on Parameters

In previous chapters we established the existence of a maximal solution curve through each initial point (r, ~) in an open subset D of fJIl x fJIl" on which is defined the differential equation x = f(t, x), with f: D -- fJIl" continuous. In this chapter we topologize the collection of maximal solution curves eJ>/D) in such a manner that for those equations with unique solutions, the map, called the general solution, that assigns to an initial point (r, ~) E D the maximal solution curve through it, turns out to be continuous relative to the topology imposed on eJ>f(D). From this geometric approach such classical results as the semicontinuous dependence of the maximal interval of existence on the initial point, openness of domain, and the uniformly continuous dependence of solutions over compact time sets relative to parameters are derived as corollaries. The differential dependence of solutions on parameters is developed in the same framework and the results are used to derive related variational equations. As an application of the main theorems it is shown that eJ>/D) is an n-dimensional (topological) manifold; moreover, it has the richer structure of orientable C'-manifold when f is r times continuously differentiable relative to x for (t, x) in D (r = 1, 2, ...). 11.1

Mathematical Preliminaries

The notions of vector space and linear transformation provide a natural context within which the theory of linear differential equations can be 200

11.1

Mathematical Preliminaries

201

developed. In deference to the student, in previous chapters we held the level of abstraction to a minimum by the extensive use of matrices. One would expect that the transition to nonlinear differential equations would require a more general mathematical framework. The generalization adopted here is topological space and continuous map. A topological space is a nonempty set 8 (called the underlying set) together with a distinguished family !F of subsets of 8, satisfying the following conditions: !F contains the empty set, (2) every collection from !F has its union in !F, (3) the intersection of any two elements in !F is again in !F, (4) the union of all elements of!F is 8. (1)

The elements of !F are called open sets. Thus a topological space is the pair 8, !F. (This text frequently follows the standard abuse of notation in which 8 is used to denote the topological space as well as the underlying set.) A map f : 8 1 - - 8 z from one topological space into another is called continuous at p E 8 1 if for each open set Oz containing f(p), f(Ol) c Oz for some open set 0 1 containing p. If f is continuous at each point in 8 1 , then it is simply called continuous on 8 1 , A collection fJ6 of subsets of a nonempty set 8 is called a basis if the union of its members is 8; and whenever p is a point in the intersection B 1 (\ B z of two elements B 1 , B z offJ6, then p E B 3 C B 1 (\ B z for some B 3 E/!J. Note that all possible unions of elements drawn from fJ6 produce a family of open sets for 8. fJ6 is called a basis for the topological space defined by that family. Familiar examples of topological spaces are 8 = PAn (n = 1, 2 ... fixed), in which the open sets are generated by the basis fJ6 = {B~ I x E PAn, r > O} where B~ = {y IY EPAn, Ix - yl < r}, called the open ball of radius r centered at x. Similarly, cc n is a topological space. A standard way of producing a topological space 8 1 , !F1 from a given topological space 8, !F is to let 8 1 be any non empty subset of 8 and choose !F1 to be the family of subsets of 8 1 obtained by all intersections of elements of !F with 8 i- The topological space 8 1 , !F1 is called a subspace of 8, !F and 8 1 is said to be given the induced or subspace topology. A subset K c 8 of a topological space 8 is called closed if the complement of K in 8 is open. Note that this generalization agrees with the use of the term in PAn. The axioms of a topological space strip away the irrelevancies that occur in the first appearance of the notions of continuity, connectedness, compactness, etc., in PAn. Many important topological spaces are function spaces. For example, the topological space 8 = {c/J I c/J: [a, b] -- PAn continuous} with basis fJ6 = {B~ I c/J E 8, r > O}, where B~ = {t/J I t/J E 8, max1a,bllc/J(t) - t/J(t) I < r},

202

11. The General Solution-Dependence of Solutions on Parameters

already provides the setting for the proof of the Picard-Lindeloff theorem (Theorem 9.2.4 in Chapter 9). It is hoped that the treatment of differential equations appearing in the remainder of this chapter will encourage the student who is not yet acquainted with the basic notions and theorems of topology to engage in some outside reading.

11.2 A Space of Curves

For the moment, we suspend direct concern with differential equations in order to develop an ambient space within which to view the collection of solution curves of differential equations. Let D continue to denote a nonempty open subset of PA x PAn and define (D) to be the collection of ordered pairs (1, ¢) with I an open subinterval of PA, ¢: I - PAn continuous, and (r, ¢(t» E D for all t e I, The space of curves (D) will be topologized, using as a base for the topology the family of subsets of (D),

N'f:!) = {(J,t/!)/(J,l/J)e(D),J:::l K,!¢(t) -

t/!(t)! < e for all r s K}, (11.2.1)

where (1, ¢) E (D), K c I is compact, and e > O. Although mathematicians frequently topologize collections of maps from one space into another, the topic under consideration involves maps that have individual domains.

Lemma 11.2.1 The family of sets (11.2.1) is the base for a topology of (D) relative to which a sequence Uk' ¢k) converges to a point (1, ¢) in (D) if and only if both I k :::l K

for all large

00,

(11.2.2) (11.2.3)

¢klK -¢IK'

uniformly as k -

k,

all compact K c I.

Proof. It is clear from the definition (11.2.1) that the subsets cover (D) since (1, ¢) e N'f:t J• Now consider any

N'f:t J

(11.2.4) It is necessary to show that

(11.2.5) for some (13' ¢3) E (D), compact K 3

c

13 , and £3 > O. Consider the

11.2

203

A Space of Cu rves

tentative choices

= (1, cP), (11.2.6) K3 = K 1 U K2, (11.2.7) £3 = min ls, - maxlcP(t) - cP1(t)l, £2 - maxJcP(t) - cPit)I}. (11.2.8)

(13' cP3)

K,

K2

Observe that (13' cP3) E cJ)(D) since (1, cP) E cJ)(D), and K 3 c 13 since 13 = 1, and due to (11.2.4)both 1 =:> K 1 and 1 =:> K 2 • Moreover, £3 > 0 since t

E

K; (i = 1, 2),

(11.2.9)

which by the continuity of the functions involved and the compactness of the K; implies via the extreme-value theorem that maxlcP(t) - cP,{t) I < £; K,

(i = 1,2).

(11.2.10)

This shows that at least N~;:t.3) is a subset of type (11.2.1). The first part of (11.2.5) follows from the choice (11.2.6). Now the last part will be verified. Thus select arbitrary (14' cP4) E N~;:t.3). Consequently, (14' cP4) E cJ)(D), 14 =:> K 3 =:> K 1 because of (11.2.7), and obviously

cPit) - cP1(t)

= [cPit) - cPi t)] + [cPit) - cP1(t)] = [cPit) - cP3(t)] + [cP(t) - cP1(t)],

t E K 1, (11.2.11)

in view of (11.2.6). From (11.2.11) and the definition (11.2.8) it follows that

IcPit) - cP1(t)1 ~ IcPit) - cPit)1

< £3

+ IcP(t) -

+ maxlcP(t) K,

cP1(t)!

cP1(t)1

= min{£l - maxlcP(t) - cP1(t)l, £2 - maxlcP(t) - cPit)j} K,

+ maxlcP(t) .K,

K2

cP1(t)1 ~ £1'

(11.2.12)

all t E K 1 • Note that (11.2.12) uses the fact that K 3 =:> K 1 • Thus it has been verified that (h, cP4) E N~::~tl and since the choice of (14' cP4) in N~;:t3) was arbitrary, the conclusion (11.2.13) follows. By symmetry it is clear that by repeating essentially the same argument one gets (11.2.14)

204

11. The General Solution-Dependence of Solutions on Parameters

and thus verification of (11.2.5) is complete. This proves that the subsets (11.2.1) constitute a base for a topology on (D). Now consider any convergent sequence (Ik' c/Jk) - (I, c/J) as k - 00 in (D) relative to the topology established. Select arbitrary compact K c I. Then for each e > 0, (Ik> c/Jd E N~:t) for all large k. But then (11.2.2) and (11.2.3) follow directly from definition (11.2.1). Conversely, if(11.2.2)and(11.2.3) hold for a sequence {(Ik' c/Jd (k = 1,2, ...) in (D) and (I, c/J) E (D), then (Ik' c/Jk) eventually enters and remains in each basic open set centered on (I, c/J) for large k and thus does likewise for all open sets containing (1, c/J). This completes the proof of Lemma 11.2.1. With the topology on (D) vividly in mind, we shall close this section with a lemma that will be needed later in this chapter. Lemma 11.2.2

If the map D - 9l"I(t, x) -

(11.2.15)

f(t, x)

is continuous, then the induced map

(11.2.16) is likewise continuous. Proof. The proposed map (11.2.16) is defined and maps into (91 x 9l") since f is assumed continuous on D, (t, c/J(t» E D for all t e I, and c/J: I _ 9l" is continuous for each (1, c/J) E (D). Select arbitrary (Io, c/Jo) E (D) and denote fe-, c/JoC-» by fo. Consider arbitrary e > 0 and compact K c ID" It is sufficient to show that the proposed map carries the basic open set

into

(11.2.17)

N(Io.!o) K,f

for some compact eel0 and 1J > O. Consider the candidate C = K. The problem then reduces to showing that there exists a 0 such that for each (I, c/J) E (D) with I ::::J K and Ic/J(t), - c/Jo(t) I < 1J(£),

t E K,

(11.2.18)

it necessarily follows that I!(t, c/J(t» - f(t, c/Jo(t»

I
(D) to fail to be in a given basic open set of the type defined in (11.2.1), one of the two remaining conditions in that definition must be violated. In particular, (11.3.7) can occur if and only if I k ::J K fails or IrP(t) - rPk(t) I < e, t E K, fails. But from Theorem 10.3.1 it follows that IrP(t) - rPk(t) I < £, t e K n I k, must fail for each k E {I, 2, ... }. By the continuity of each rPka!1d the compactness of the boundary aT of T, it follows that for each k E {I, 2, ...} there exist unique points (IXk' xd, (13k> Yk) in aT such that (IXk' 13k) Elk, (11.3.10)

rPiIXk) = xi, rPk(f3k) = Yk' and (r, rPk(t» E T for IXk < t < 13k' Once again, by extracting a subsequence, it can be assumed that (IXk' Xk) - (IX, x) and (13k> Yk) - (13, Y) as k - 00. Note that IX < r < 13 (see Exercise 10.2.1). Since IX < 13, certainly IX + lin < 13 - lin for all large n. On the other hand, since IXk - IX and 13k - 13 as k - 00, IXk < IX + lin and 13 - lin < 13k for each n E {I, 2, ... } for all large k. Thus it is apparent that IXk < IX

+ lin
2(13 - IX),

{rPk I[a + I/n.p -lin]}

(11.3.12)

for all large k, is an equiuniformly continuous sequence and is uniformly bounded. Consequently the Arzela-Ascoli theorem [19] applies, thus assuring that each subsequence of {rPd, k E {I, 2, ... }, has in turn a subsequence that is uniformly convergent on [IX + lin, 13 - lin]' In particular, starting with the smallest integer value n > 2(13 - IX) and working inductively, one can extract for each n > 2(13 - IX) a subsequence {rPn.k}, k E {I, 2, ... }, of {rPk} that is uniformly convergent on [IX + lin, 13 - lin] such that {rPn.d, kE{1,2, ... }, is a subsequence of {rPn-l.d, kE{1,2, ... }. Now consider the diagonal sequence t/fk = rPk.k' It follows that {t/fd is a sequence of continuous functions, uniformly convergent on each compact subset of (IX, 13), and its pointwise limit, denoted by t/f, is defined and continuous on (IX, 13). From the definition of t/fk observe that E

(IX

Since IX < r < 13, as already shown, r

E

t

+

11k, 13 - 11k).

(IX

+ 11k, 13 - 11k) for all large k.

(11.3.13)

208

11.

The General Solution-Dependence of Solutions on Parameters

Integration of (11.3.13) shows that

!/tk(t) - !/tk(T:)

f

=

Ito, !/tlu») de,

(11.3.14)

r

t E (a + 11k, fJ - 11k), for all large k. By the uniform continuity of Jon and the uniform convergence of!/tk on compact subsets of («, /3), it follows on taking limits as k --* 00 in (11.3.14) that

f

!/t(t) ., !/t(T:) =

f'to, !/t(u» do,

t

E

(a, fJ).

(11.3.15)

It was proved that !/t(t) = lim i --+ oo 4Jk.(t), t E (a, fJ), for some subsequence k, --* 00 as i --* 00. Since the convergence is uniform on compact subsets of (iX, fJ), 4Jk,(T:kJ = ~k" and T:kl --* T: as i --* 00, it therefore follows that

!/t(T:) = lim 4Jk,(T:k.) = lim ~k, i-r o:

i-v co

=

~'

(11.3.16)

Thus by differentiation of(11.3.15), along with (11.3.16), it has been shown that

~ = J(t, !/t), !/t(r)

=

t E

(ex, /3),

e,

(11.3.17) (11.3.18)

and, in particular, by the (assumed) uniqueness of the solutions to the Cauchy problem (D, f),

!/t(t) = 4J(t),

t

E

(ex, fJ).

(11.3.19)

For each i E {I, 2, ... }, by definition of the aks and fJkS, at least one of l4Jk,(akJ - 4J(iXk,) I, l4Jk,(fJk.) - 4J(/3kJ Imust equal £. Hence it can be assumed,

by extraction of a subsequence, that one of the following holds for all large i:

l4Jk'(ak.) - 4J(iXk.) I = e,

(11.3.20)

l4Jki(/3k,) - 4J(/3kJI =

(11.3.21)

£.

Suppose that (11.3.20) were the case. For each tJ satisfying a < ex

+ tJ < /3,

+ tJ) - 4J(i/.k, + tJ») = !4Jk,(ak, + tJ) - !/t(exk, + tJ)!--* 0

(11.3.22)

!4Jk,(exk,

as i --* 00 because of the convergence of 4Jk, to !/t on (a, /3). Since 4Jki and 4> satisfy the same differential equation

¢ki

=

J(t,4JkJ,

(11.3.23)

¢

=

J(t,4J),

(11.3.24)

209

11.3 The General Solution and Continuous Dependence

subtraction and integration give

cPki(ak,) - cP(ak,)

=

cPk,(aki + £5) - cP(rxk,

-f:~'

+0

+ b)

[J(O", cPk,(O"» - f(O", cP(O"»] da. (11.3.25)

Application of (11.3.19) and (11.3.20) to (11.3.25) shows that

e = IcPk;(rxk,) - cP(ak,) 1 ::;; IcPk;(ak; + b) - cP(ak, + 15)/ +

r.

IbJc

for all large i, where IJ(t, x)1 ::;; c for (r,x) E Thus letting i -+ and taking into account (11.3.22), it can be concluded that

(11.3.26) 00

in (11.3.26) (11.3.27)

which provides a contradiction since 1151 could be chosen arbitrarily small independently of e and c. The case in which it is (11.3.21) that obtains can be treated similarly, and thus the long-sought contradiction establishing the claimed continuity of the map (11.3.5) is reached. Since 1'/lt, T,~) = f(t,1'/(t, T, ~», t E l( T, ~), by definition of 1'/ the map defined in (11.3.6) is the same as the map of D into (~ x ,!~") given by (T,

~) -+

(I( T,

~),

f(· , 1'/(., T, ~»).

(11.3.28)

But the latter map is the composition of the map D -+ (D) that was just shown to be continuous, followed by the continuous map of (D) into (~ x ~") given by (11.2.16). Thus (11.3.6) is continuous and Theorem 11.3.1 is proved. . Theorem 11.3.1 encompasses a number of results of the classical theory of differential equations. Recall some of that terminology: An extended-real-valued function b(x) is called lower semicontinuous at X o if for each e > 0, b(xo) - e < b(x) for all x in a neighborhood of Xo' Similarly, a(x) is called upper semicontinuous at Xo if for each e > 0, a(x) < a(x o) + e for all x in a neighborhood of Xo' For such functions, if a(x) < b(x), then the interval-valued function lex) = (a(x), b(x» is said to be semicontinuous at Xo' Amongst other consequences, Theorem 11.3.1 implies that for each compact K c I(TO' ~o), I(T,~) ::;) K for all (T,~) in a neighborhood of (To, ~o) E D. This translates into the statement that 1(', .) is semicontinuous at each (T,~) E D, i.e., on D. In particular CO+(T,~) is lower semicontinuous and CO_(T,~) is upper semicontinuous on D. As a final note about classical terminology, it should be pointed out that a lower semicontinuous function is simply a continuous function if its range is retopologized by the topology generated by the base {(a, 00] 1 a E ~}. A similar statement applies to an upper semicontinuous function. Remark 11.3.2

210

11.

The General Solution-Dependence of Solutions on Parameters

Coronary 11.3.2 Let (D,17) denote the general solution to a continuous Cauchy problem (D, f). Then the subset P)

=

{(t, r, ~)I('t,~)

ED,

t E I('t,~)}

c 9t x D

(11.3.29)

is open. Furthermore the maps £1) -

9t n l(t, 't,~)

-17(t, r, ~),

(11.3.30)

p) -

9t n l(t, 't,~)

-17,(t, 't,~)

(11.3.31)

are continuous and Ie· ,..) is semicontinuous on D.

To prove £1) open, choose arbitrary (to, 'to, ~o) E £1). Then which is open, and hence toE(a,b)c[a,b] cI('to,~o) for some a, bE 9t. By the continuity of (D,17) proved in Theorem 11.3.1, I('t,~) =' [a, b], all ('t,~) in a neighborhood of ('to, ~o)E D. Since (to, 'to, ~o) E P) was arbitrary, it follows that p) is open. To prove the continuity of (11.3.30) - ('to, ~o), so that by the continuity let (t, 't,~) - (to, 'to, ~o) in p). Then ('t,~) of (D, 17) provided by Theorem 11.3.1, 11(', r, ~) - 11(', 'to, ~o) uniformly on compact subsets of I('t o, ~o). Hence 17(t, r, ~) - 11(to, 'to, ~o) as (t, 't,~) (to, 'to, ~o). This proves the continuity of (11.3.30). In tum it follows that £1) - DI(t, 't,~) - (t, 11(t, r, ~» is continuous. But since f is continuous on D and 11lt, 't,~) = f(t, 11(t, r, ~», it is clear that (11.3.31) is continuous because it is the composition of continuous maps. The semicontinuity of 1(', .) was already pointed out in Remark 11.3.2 to be a corollary of Theorem 11.3.1. This completes the proof of Corollary 11.3.2. Proof,

toEI('to,~o),

Coronary 11.3.3

Assume the same hypothesis and notation as in Corollary 11.3.2. Then for each (t, r, ~o) E p), (t, 11(t, r, ~o»

r

E

E

~o»

11('t, t, 11(t, r, ~o»

(11.3.32) (11.3.33)

I(t, 11(t, r, ~o»,

(r, t, 11(t, r, In particular, h(~) with inverse h-l(~)

D,

E £1),

= ~o.

(11.3.34) (11.3.35)

= 11(t, 't,~) defines a local homeomorphism h about ~o = 11('t, t, ~).

E

9t n

Proof. Statements (11.3.32)-(11.3.35) follow successively from the definitions of £1), 11, and I and from the assumed uniqueness of maximal solutions to the initial-value problem. For fixed t and r the semicontinuity of I('t, .) implies that (t, r, ~) E p) for all ~ in a neighborhood of ~o. Hence (11.3.32) remains valid for ~ in a neighborhood of ~o and (11.3.33)-(11.3.35)

11.3

211

The General Solution and Continuous Dependence

remain valid as well. The continuity of h( .) and h- 1( . ) is provided by Corollary 11.3.2 and the proof is complete.

Remark 11.3.3 Theorem 11.3.1 and Corollaries 11.3.2 and 11.3.3 illuminate the nature of the dependence of the solution curves of (10.1.1), (10.1.2) on the initial parameters Cr,~) E D. In both the theory and the applications of differential equations one frequently encounters differential equations with additional parameters, say Z E Bim, entering the function! directly, and the nature of the dependence of the solutions on z is likewise of interest. In most instances the following device reduces z to the status of initial coordinates of ~ and therebyplaces the questions within the purview of Theorem 11.3.1 and Corollaries 11.3.2 and 11.3.3: Regarding f as a function of (t, z, x) E 15 c Bi x Bim x Bin, pass to the Cauchy problem (15, f) in which f(t, z, x) = [!(t,Oz, X)]

(0

E

(11.3.36)

Bim).

If 15 is open and! is continuous, thereby making f continuous on 15, then Theorem 11.3.1 and Corollaries 11.3.2 and 11.3.3 apply to (15, f). The conclusions concerning ft, i, and 15 then translate directly into statements about the original problem since 1](t, t, z,~) consists of the last n coordinate functions of ft(t, t, Z, ~); e.g., 15 - 0,

So

E S,

constitutes a base for a topology on S. Show that the resultant topology on S is Hausdorff.

212

11. The General Solution-Dependence of Solutions on Parameters

3. Show that there is no distance function generating the topology that is defined on ~)

= exp

(f2 fX(U''1(U,T1>~))dU)

>

tr

° (11.4.48)

and the orientability now is an immediate consequence of(I1.4.47), (11.4.48). This concludes the proof of Theorem 11.4.7.

EXERCISES

In the following exercises, unless otherwise specified, let (D, '1) denote the general solution to the Cauchy problem (D, f) having [f, fx] continuous at all (t, x) E D.

1.

Derive the formulas tr fx(u, ni«, T, ~») dos, (a) det '1~(t, T,~) = exp(J~ (b) '1~(t, T, ~)-1 = '13(T, t, '1(t, T, ~)), all (t, T,~) E ~. 2. (a) Show that '1(t, T, ~) is a solution to the problem '1ft

+ '1f~'1Z

l'1t = 0,

'1(T, T,

0 =

~,

(t, T,

0

E

~.

(b) In the case in which (D, f) is autonomous, let '1(t,~) = '1(t, 0, ~) and w±(~) = w±(O, ~), (T,~) E D = [)l x E. Show that the equations of part (a) then reduce to tl« -

'1f~'1

~-1 '1f

=

°,

3. Fix any (T, ~)'E (a)

D. Derive the equation

z(t) - '1(t,

T,~)

=

f

'1z(t, a, z(u»

[d~~)

- feu, Z(U»] da,

t E I( T, ~), for any C 1 curve z: I( T, ~) -+ [)In, satisfying both (1) Z(T) = ~ and (2) (t, a, z(u») E~, all a in the interval with ends T, t. (Hint: (11.4.27), replacing (T,~) by (e, z(u»).)

Apply

222

11. The General Solution-Dependence of Solutions on Parameters

(b) Verify that the derived equation of (a) reduces to (11.4.26) with the special choice of z being z(t) = ~, all t E I(r, ~). (c) Let (D, z) be the general solution to the "perturbed" Cauchy problem (D, f + h). Apply the derived formula of(a) to obtain the Alekseev"variationof-parameters formula" z(t,

r,~)

= rt(t, r,~)

+

f

rtzCt, (J, z((J, r,

~))h( J,

z((J, r,

~))

da.

For which hand (t, r, 0 is it valid? 4. Show that if D = (to, tr) x PJtn and f satisfies

fit, x) + f':(t,

x) :s:; yIn

for some y E PJt, all (r, x) E D, then w+(r,~) = t 1 for each (r,~) ED. Further such that prove that there exists abE PJt (generally (r, ~)-dependent) Irt~(t, r, ~)I :s:; b for r :s:; t < t 1 • (Hint: Derive the inequality

Irt~(t,

r,

~W

s

1

+y

f

Irtl(J, r,

~W

da,

and apply (11.4.26) and Corollary 10.3.2.) 5. As a consequence of Exercise 4, show that if II, I is bounded on PJt x then w±(r,~) = ± 00 for all (r, ~). 6. Prove that if(D, f) satisfies the hypothesis of Exercise 4 and if [h, hxJ is continuous with Ihxl bounded on D, then (D, f + h) will likewise have w+(r,~) = t 1 for all (r,~) in D. 7. An autonomous Cauchy problem (D, f) has solution curves given by the formula

ee:

rt(t,~)

= (1-

t~d

1

+ t2~~

[~1

- t(~i~2

+ ~DJ

n

(~1'~2)EPJt.

2

Determine f and the general solution rt(t, r, Compute the 2 x 2 matrix function A(t, r, ~) that appears as the coefficient in the variational equations (11.4.18) and (11.4.19).

Chapter

12

Limit Properties of Solutions

Although the results of Chapters 9-11 have the virtue of applying to a broad class of equations, by themselves they generally do not provide the detailed description of solution behavior needed in applications. By building upon those foundations the material covered in this chapter leads to a more complete structure.

12.1

Limit Sets and Invariance

We shall restrict attention to those autonomous Cauchy problems (D, f) that have a general solution (D, ,,). That is, D = f1l x E, E an open subset of f1l etc. (See Corollary 11.4.5 for a reminder about the notation.) Hence ,,(t, ~) is uniquely determined by the equations ft

,

~ ,,(O,~)

= 1(,,),

(12.1.1)

= ~

(12.1.2)

for ~ E E, t E «(JL(~), w+(~». The definition of a general solution given in Chapter 11 assumes local existence and uniqueness of solutions to the initial-value problem (12.1.1), (12.1.2). As was shown in Chapter 9, a sufficient condition for ensuring the existence of a general solution in the autonomous case is that I be locally Lipschitz on E. Throughout this chapter I is understood to be continuous onE. For (E E define (12.1.3) 223

224

the union over 0

12. Limit Properties of Solutions

~

t < w+W. Similarly,

(12.1.4)

the union over w-W < t ~ O. The sets y+W, negative semiorbits of ~, respectively, and

y-:-(~)

are called positive and (12.1.5)

is called simply the orbit of ~. The notation is shortened to y when there is no need to distinguish a particular point on the orbit and to y+(y-) when any respective semiorbit of y will do. A subset SeE is called positively (negatively) invariant if it contains y+(~)(y-W) for each ~ E S. S is called invariant ifit contains y(~) for each ~ E S. The general solution Yf to an autonomous Cauchy problem satisfies the equation

Remark 12.1.1

Yf(t, Yf((J,

~))

= Yf(t + (J,

~)

(12.1.6)

when both (J and t + (J are in (w_(~), w+(~)). This relation, sometimes called the semigroup property, is a direct consequence of Corollary 11.4.3. Note that the semigroup property substantiates Y+W, y-W, and y(~) as examples of positively invariant, negatively invariant, and invariant sets, respectively. Further observe that a point ~ E E at which fW = 0, called a rest point, equilibrium point, or critical point, is its own orbit. More generally, a point ~ E E is called periodic if Yf(', ~) is periodic on (- 00, (0). Then y(~) is called a periodic orbit and is invariant. The positive limit set n+(y) of an orbit y is the set of points in E of the type lim, .... co Yf(t k , ~), some ~ E y, some t k - 00 as k - 00. The negative limit set n-(y) is similarly defined as those points in E of the type lim, .... co Yf(t k , ~), some ~ E y, some tk - - 00 as k - 00. The limit set n(y) consists ofthe points in E that are limits of Yf(tb ~) for some ~ E y, where either tk - 00 as k - 00 or tk - - 00 as k:+ 00. Remark 12.1.2 Some authors, at the outset, restrict the notions of orbit, limit set, and invariance to those differential equations having Iw±WI = 00 for all ~ E E. This condition can always be achieved by a reparametrization of the solution curves; for example, by replacing f(x) with [1 + If(xWJ- 1f(x). The approach taken in this book avoids that restriction and seems more natural. Thus the differential equations considered can have an orbit y(~) with w-W > - 00 or w+W < 00, but of course it then follows from Corollary 10.3.2 that n-(y) = 0 or n+(y) = 0, respectively.

12.1

225

Limit Sets and Invariance

Proposition 12.1.1 n+(y) =

ny+W,

(12.1.7)

~EY

n-(y)

=

n

(12.1.8)

y-W,

~EY

(12.1.9) (The overbars in (12.1.7) and (12.1.8) denote closure relative to E.) Proof. To prove (12.1.7), consider arbitrary x E n+(y). Thus x = limk--+oo ,,(tb '0) for some '0 E Y and t k - - 00 as k -- 00. Select arbitrary, E y. Then '0 = "(0', e) for some 0' E (w-(e), w+(e)), and according to (12.1.6),

(12.1.10) with t k + 0' E [0, w+W) for all large k since 00 = w+('o) = w+W - 0' and tk - - 00 as k -- 00. This shows that x E Y+W, and since E y was arbitrary, the conclusion reached is that

e

n+(y) c

n y+W.

(12.1.11)

~EY

To establish the reverse inclusion, suppose that some x E E satisfies x E y + (e) for all eE y. Thus for some fixed eE y, ,,(tk , e) -- x as k -- 00 for tk E [0, w+W)· Ify is periodic, say with period p > 0, then ,,(Tk, e) = ,,(tb ,), where T k = t k + kp -- 00 as k -- 00; this says that x E n+(y). The same conclusion follows trivially if y is a rest point. Suppose, on the other hand, that 't is not periodic. If tk can be selected so that tk - - w+(e) as k -- 00, then clearly w+W = 00 and x E n+(y), as required. Now consider the complementary case which would imply that for some e > 0 and {} > 0, I,,(t, ,) - xl > e for all t E [{}, w+W). In this situation, by first extracting a convergent subsequence, it could be assumed that tk - - too as k -- 00 for some too E [0, (}). Then x = limk--o oo ,,(tk , , ) = ,,(too, ,) E y. But f(x) i= 0 since y is not periodic, and consequently x ¢ y+(,,({}, ')). The preceding argument shows that x ¢ y+("({}, ')), the desired contradiction of the assumption that x belongs to the closure of every positive semiorbit of y. This completes the proof of (12.1.7). The remainder of the proof of Proposition 12.1.1 proceeds similarly and therefore is omitted. Amongst other uses, the next lemma provides a precise statement of the sense in which the general solution will be shown to carry semiorbits to limit sets.

226

12. Limit Properties of Solutions

Lemma 12.1.2 Let (M, d) be a metric space. (Recall Exercise 2 following Section 11.3.) g is taken to be the collection of nonempty compact subsets 5 of M. Denote the open ball in M of radius e and center m e M by B'tm). Then g together with the formula c5(5 1 , 52)

= inf{£.IB£(sd n

52 #

0, B£(S2) n

51 #

0, s, E 5 i (i = 1,2)} (12.1.12)

for (5 1,52) E g x g defines a metric space (g, c5). (The distance function c5 is called the Hausdorff metric.) Proof. The function c5 is real-valued since compact subsets of a metric space are bounded. Suppose that c5(5 1 , 52) = O. Fix SI E 51 and let 0 < £.k - 0 as k - 00. There exists a sequence ~ E B£k(SI) n 52 and ~ - SI as k - 00. But 52 is compact and hence closed, which puts lim, .... co s~ E 52' Therefore SI E 52' and since SI E 51 was arbitrary, 51 c 52' By symmetry 52 c 51 and thus 51 = 52' Conversely, if 51 = 52 = 5 E g, then B£(s) n 5 # 0, all s E 5, all £. > 0, which says that c5(5 1 , 52) = o. To continue the proof, define d(m, 5)

= inf{d(m, s) Is E 5}

(12.1.13)

for mE M, 5 c M. As a first step in showing that c5 satisfies the triangle inequality, it will be shown that c5(5 1, 52) = J(5 1 , 52) where 8(5 1,52)

= max{max des!> 52), max d(s2' 51)} 81

$2

(12.1.14)

in which s, E S, E g (i = 1, 2). By compactness there is an s! E 52 with d(SI' s!) = d(SI' 52) ~ max" d(SI' 52) ~ $(5 1, 52), which implies that BP(sdn5 2 # 0,all SIE51' where p =3(5 1,52), By symmetry BP(s2)n5 1 # 0, all S2E 52' which proves that c5(5 1 , 52) ~ J(5 1 , 52)' Clearly the reverse inequality likewise holds and the proof reduces to showing that 3 satisfies the triangle inequality. The inequalities (12.1.15) imply that

d(SI' 52) ~ d(SI' S3)

+ d(s3' 52)

~

d(sl' S3)

+ max '3

d(s3' 52),

(12.1.16)

and consequently max d(SI' 52) ~ max d(SI' 53) $1

for

Si E S, E

81

+ max d(s3' 52)

(12.1.17)

+ max

(12.1.18)

83

g (i = 1, 2, 3). By symmetry,

max d(s2' 5d ~ max d(s2' 53) b

82

83

d(S3' 5d·

12.1

227

Limit Sets and Invariance

As a direct consequence of (12.1.15) and (12.1.17), (12.1.18) it follows that S(SI, Sz) ~ max{max d(sl' S3), max d(s3' SI)} 81

83

+ max{max d(S3' Sz), max..d(sz, S3)} 83

=

82

J(Sl' S3) + J(S3, Sz)

(12.1.19)

and Lemma 12.1.2is proved.

Remark 12.1.3 It can be shown that if (M, d) is a complete metric space, then the metric space (i/, b) of compact subsets is complete. If (M, d) is compact, then (i/, b) is likewise compact. Theorem 12.1.3 The sets n-(y), n+(y), n(y) are invariant and closed(relative to E). If y-, y+, or y has compact closure in E, then, respectively, n-(y), n+(y), or n(y) is nonempty, compact, and connected. Compactness of y+(e) implies that 17(t, y+W) - n+(y) in the sense of the Hausdorff metric as t - 00. Likewise 17(t, y-W) - n-(y) as t - - 00 if y-W is compact in E. Proof. The sets n-(y), n+(y), and n(y) are closed since, according to Proposition 12.1.1, they are the intersections of closed sets. In proving the invariance ofn+(y) there is nothing to do ifn+(y) = 0. Now suppose there is an element x E n+(y). Then by the definition of positive limit set,

x = lim 17(tk> e) k .... co

e

(12.1.20)

for some E y, tk E [0, w+W) with t k - W+W as k - 00. By openness of domain, continuity, and the semigroup property it follows from (12.1.20) that for each t E (w_(x), w+(x)), 17(t, x) = lim 17(t, 17(tk> e)) = lim 17(t + tk , e)· k-oo

k-oo

(12.1.21)

Since x E E, (12.1.20) along with Corollary 10.3.2 ensure that W+W = 00 and consequently t + tk - W+W as k - 00 in (12.1.21). This proves that 17(t, x) E n+(y), and the invariance of n+(y) is established. The similar proof of the invariance of n-(y), and hence of n(y), is omitted. Now assume that y+(e) is a compact subset of E. Obviously n+(y) is then not empty, and n+(y) is compact since it is a closed subset of y+W. Since n+(y) is compact, it follows readily that for any open set S (of E) containing n+(y), 17(t, e)E S for all large t e [0, (0). The conclusion that 17(t, y+(e)) c S for all large t now follows from (12.1.6). In particular, since n+(y) is compact, for arbitrary £ > 0 the open set S can be taken to be the union of a finite number of open balls of radius £ centered on points in n+(y). But 17(t, y+(~))

228

12. Limit Properties of Solutions

is compact since it is the continuous image of a compact set, and consequently it too will be contained in the union of the finite collection of e-balls for all large t. Then the relation B£(s) n g+(y) '1= 0, SE 11(t, y+(~» obviously holds for all large t, and moreover, B£(s) n 11(t, y+(~» '1= 0, s E g+(y), for otherwise the defining property of g+(y) would be violated. Since £ was arbitrary, we con-- g+(y) relative to the Hausdorff metric as t -- 00. clude that 11(t, y+(~» The proof that 11(t, y-W) -- g-(y) as t -- - 00 when Y-W is compact is similar and therefore is omitted. Recall that a set is connected if it is not the union of two nonempty disjoint open sets. Again consider the situation in which y+(~) is a compact subset of E. To reach a contradiction, suppose that g+(y) = Sl U S2 with S 1, S2 nonempty open subsets of E with S 1 n S2 = 0. From the first part of the proof, 11(t, y+(~» c Sl U S2 for all large i. But 11(t, y+(~)) is a connected set and hence must lie wholly within one of Sl or S2' This is impossible since both Sl' S2 are nonempty and consist of points of the type lim k -+ oo 11(tk , ~), tk E [0, w+W), t k - - W+W as k -- 00. Thus the proof that g+(y) is connected is complete. The proofs of the corresponding statements concerning g-(y) and g(y) proceed in a similar fashion and thus are omitted. The next theorem takes a Lyapunov approach to deriving an upper estimate of the sets g+(y). The examples following its proof demonstrate how the theorem can be applied to obtain detailed information about the limiting behavior of solutions to some nonlinear differential equations.

Theorem 12.1.4 Let f: E--~" and v: E-fYi be once continuously differentiable on a nonempty open subset E of fYi". For fixed c E fYi define V = {XIXEE,v(x) < c},

(12.1.22)

= {x IXE V, f(x)'vx(x) = O}, Z* = {the largest invariant subset of Z for (D Z

Assume that V is bounded and f(x)'vix) positive-invariantfor (D; f), W+W = 00, c

g+(y(~»

and 11(t, y+W) - g+(yW) as t -

00

~

°

= E x fYi,

fn.

(12.1.24)

throughout that set. Then V is

Z1 c Z,

for each ~

(12.1.23)

E

(12.1.25)

V.

Proof. Assume the hypothesis. To prove the positive invariance, consider arbitrary ~ E V. Then the unique solution (D, 11) to (D, f) has 11(t, ~) E V on some open interval containing t = since V is open. On that interval,

°

(12.1.26)

12.1

229

Limit Sets and Invariance

and hence v(I'/(t, ~» is monotonically nonincreasing relative to increasing t and v(I'/(t, ~» v(~) < c. Suppose that v(I'/(t, ~» < c were violated at some t 1 E [0, w+(~». Let t* be the infimum of those t E (0, w+(~» at which v(I'/(t, W ~ c. By continuity it would follow that v(I'/(t*,~)) = c. Since v(I'/(t, ~» E V for 0 ::;; t < t*, v(I'/(t, ~» is nonincreasing on that interval; hence v(I'/(t, ~» v(~) < c on 0::;; t < t*, which contradicts the continuity of v(I'/(t, ~». This establishes the positive invariance of V. Moreover, it shows that y+<e) c V, and by Corollary 10.3.2 we conclude that w+<e) = 00. Since V is bounded, y+<e) must be compact in E and Theorem 12.1.3 applies. The next task is to show that the now compact and nonempty n+(y(~» c z. By the boundedness of V, its positive invariance, and the continuity of v(·), the monotone nonincreasing v(I'/(t, ~» is bounded below and hence has a lim, 00 v(I'/(t, = Co < c. By continuity of v(·) it follows that vex) = Co for all x E n+(y(~». But according to Theorem 12.1.3, n+(y(~» is invariant, and thus

:;

:;

m

V(I'/(t, x)

= f(I'/(t, x»· v~(I'/(t,

x)

=0

(12.1.27)

which says that n+(y(e» c Z. However, since n+(y<e) is for all x E n+(y(~)), invariant, n+(y(~» c Z* as well.The remaining conclusion that I'/(t, y+(~» n+(y(~» is a direct consequence of Theorem 12.1.3, and the proof of Theorem 12.1.4 is now complete. The function f(x)'vix) that appears in (12.1.23) is sometimes called a Lyapunov derivative or directional derivative ofv in the direction f. Its sign at a point x E E gives some indication of the direction in which the solution curve is moving across the level set of v through x. Theorem 12.1.4 clarifies the solution behavior in certain regions that contain points at which the Lyapunov derivative is zero. The significance of Theorem 12.1.4 is that it provides a useful device for in applications. (This will be brought out in the helping to locate n+(y(~» example following this remark.) It should be noted that the proof of the as being inside Z* but also actually shows theorem not only locates n+(y(~» that n+(y<e) lies inside some level set of v. The theorem imposes no requirement on the sign of u; however, any v satisfying the hypothesis could be converted into one that is positive by adding an appropriately large constant to v. The literature sometimes imposes the condition that vex) - 00 as x - 8E. That condition would ensure that V is bounded, but it is not necessary. Remark 12.1.4

Example 12.1.1

The pair of coupled nonlinear second-order scalar

equations ii

+ (1 - v2 ) U + U =

0,

jj

+ (1 - u2 ) v +

V =

0

(12.1.28)

230

12. limit Properties of Solutions

has the first-order form in [)f4 (12.1.29)

X4 = -X 3 + (xi - 1)x 4.

X3 = X4'

The function v(x) = t[xi + x~ + x~ the direction of f determined by

+ xi]

has the Lyapunov derivative in

+ x 2[ -Xl + (x~ - 1)x 2] + X3X4 + X4[ -X3 + (xi - 1)x4] - [(1 - x~)x~ + (1 - xi )xi]

f(x)'vx(x) = XlX2

::;;0

(12.1.30)

on the set

v

I

= {x v(x)


I + x~ + x~ + xi < I}.

= {x xi

(12.1.31)

By noting that 1 - x~ > 0 and 1 - xi > 0 for x E V, it is apparent from

(12.1.30) that

Z = {x Ixi

+ x~
0, it follows from (12.2.6) that p = p(f.) can be chosen appropriately small so that vex) < p(f.) implies that Ixl < c5(I/4I p j), Ixl < ~, and x E E. For arbitrary r E (a, 00) consider any ~ E fYt" that satisfies v(~) < P and let c/J(.) be any solution curve of (12.2.4) on a right-maximal interval [r, co ,) with c/J(r) = ~. Then by continuity v(f/J(t» < P on some open interval containing r, and hence by (5.2.19), (12.2.4), and the choice of p it follows that d

dt v(f/J(t»

= 2f/J'P[Ac/J +

h(t, c/J)J

= f/J'(ATp +PA)f/J + 2f/J'Ph(t, f/J) = -1f/J1 2 + 2f/J' Ph(t, f/J)

s

-1f/J1

2

+ 21pllc/J1 (41~1)

:$;

-1f/J12

+ 2Ipl/f/Jllh(t, f/J)I

If/JI = -

~ 1f/J1 2 s O.

(12.2.7)

By continuity it can be concluded from (12.2.7) that v(f/J(t» :$; v(~) < p for all t E [r, w+), and this inequality along with the first in (12.2.6) allows application of Corollary 10.3.2, which shows that ce, = 00. Now it can be concluded that (12.2.7) is valid for all t E [r, 00). By the same argument found in the proof of Theorem 5.2.4, it follows from (12.2.6) and (12.2.7) that

If/J(t) I s

~

(12.2.8)

e-(t-t)/2PI~I,

for t E [r, 00), vW < p. Since ex, P depend only on A and since p = p(f.) is independent of r, inequality (12.2.8) provides the subsequent inequalities

Ic/J(t) I :$; e-(t-t)/2Pf. :$; f. (12.2.9) I s c5 o(f.) where c5 o(f.) = min{c5(I/4I p l),J p(f.)/P}. This establishes

for If/J(r) I= I~ the claimed uniform and asymptotic stability since r E (a,

00)

was arbitrary.

Remark 12.2.1 The definition of Lyapunov asymptotic stability requires more than just the solutions f/J(-) with f/J(r) = ~ having 1f/J(t)1-0 as t - 00 when I~I is small. There are examples of nonlinear equations with this property that are nevertheless not Lyapunov asymptotically stable since their maximum value If/J(t) I over r :$; t < 00 does not approach zero as the initial ~ - 0 (cf. [7J). Remark 12.2.2 Note, from the estimate (12.2.9) establishing the uniform and asymptotic stability of the zero solution of (12.2.4), that for solutions

234

12. limit Properties of Solutions

with initial ~ satisfying I~I ~ c5 o(£) there is a uniform length of time during which I O. In particular, the steady-state analysis of(12.3.19) translates into the study of(12.3.24) with c set to zero. Thus (12.3.24) can be taken as a starting point and looked at afresh from the point of view of the notation and theory developed in preceding sections of this chapter. Since for m > 1 the generalization (12.3.24) of (12.3.20) to a higherdimensional space variable is no longer autonomous but clearly is asymptotically autonomous, the embedding device introduced in Example 12.3.2 is in order. Thus, introduction of an additional coordinate variable, say y = r", embeds the equations into the autonomous system

X2

= - [(m - 1)y1n + cD-1JX2 - D-hvF(xd,

y=

(12.3.25)

_y2

with (t, Xl' X2, y) E 9f X 9f2n+ 1. The original problem is concerned with the solutions in the subregion in which t > 0 and y > 0 with the initial values of those variables related by the equation y = t- 1 • The study of the limit sets n+(y) of the orbits of those solutions provides a geometric approach to investigation of the limit behavior of solutions to (12.3.24). In the context of Theorem 12.3.1 it is interesting to note that the limit system of (12.3.25) is just (12.3.20)! In other words the theorem provides a way to use information about nonlinear diffusion along one space dimension to study diffusion through higher-dimensional regions of space. For detection of bounded positive sem'iorbits of(12.3.25) that are needed for application of Theorem 12.3.1, it is useful to work with the Lyapunov function (12.3.21), regarded as a function defined on the domain of (12.3.25). Its derivative relative to the latter system can be checked to be (12.3.26) Detailed analysis of examples is left to the reader. Similar topics can be pursued for other classes of partial differential equations.

243

12.4 local Behavior and linearization about Rest Points EXERCISES

1. Find a simple example of an autonomous differential equation in 9f2 that substantiates Remark 12.3.1. 2. Analyze the limiting behavior of the solutions to the following system for increasing t: Xl =

X2 = X3

=

+ X 2X3, - X l - X2 + X3 + x2x I + xt

X2

[sin

Xl -

2]x 3 .

Investigate the limiting behavior of the system

3.

• Xl

=

3 X 2,

4. Consider the diffusion-reaction equation in one space variable

au a2u at

=

ax2

+ 4(u

2

- l)u.

Derive the first-order system of ordinary differential equations governing the traveling wave solutions u = u(x - ct) and verify that the results agree with (12.3.20) if D = 1 and F(u) = (u 2 - 1)2. Note that the system obtained with c = 1 is (12.1.35) of Example 12.1.2. Describe the behavior of the bounded forward traveling wave solutions u(x - t) of speed c = 1 as Itl- 00.

12.4

Local Behavior and Linearization about RestPoints

In this section we shall concentrate on autonomous equations of the form X = Ax

+ h(x)

(12.4.1)

in which A is an n x n matrix and the n-vector function h is once continuously differentiable on a neighborhood of the origin and of higher order in a sense to be made precise. Such an equation arises from an autonomous equation with a rest point when the rest point is translated to the origin under a suitable change of coordinates. The rest point-which is to say, the origin for (12.4.1)-is called hyperbolic if all the eigenvalues of A (regarded as a matrix over the complex field) have nonzero real parts. Such a matrix will likewise be called hyperbolic. For example, a stability matrix is an extreme example of a hyperbolic matrix. Throughout the literature an extensive and successfuleffort has been made to relate the solution behavior about the origin of(12.4.1) to that of the linear equation resulting from deletion of the perturbation term h. An example of

244

12. Limit Properties of Solutions

one such result is Corollary 12.2.2. The rather remarkable Theorem 12.4.1 accounts for much of this success. (See [9] for a historical account of its development and for related results.)

Theorem 12.4.1 Suppose that in (12.4.1) the n x n matrix A is hyperbolic and that h is of class C 1 on a neighborhood of the origin in :)In. If as

Ih(xWlxl-O

Ixl-O,

(12.4.2)

or, equivalently, h(O) = 0

(12.4.3)

and

then there exist open sets U, V in :)In that contain the origin and a homeomorphism g: U - V such that the general solution '7 of (12.4.1) factors as

(12.4.4) for each ~ E U and corresponding t in the open interval about the origin in over which etAg(~) E V.

:)l

Proof. By a preliminary nonsingular linear change of coordinates, it can be assumed that A is of a block diagonal form A = diag[ A 1 , A 2 ] in which the nl x nl matrix A 1 has eigenvalues all with negative real parts and the n2 x n 2 matrix A 2 has eigenvalues all with positive real parts. (One of A 1 , A 2 might not occur.) Then there exist positive numbers c, 0(1' and 0(2 such that

(12.4.5)

and - 00

< t s; O.

(12.4.6)

Moreover there exist matrix solutions Pi > 0 (i = 1, 2) to the Lyapunov equations (12.4.7) (12.4.8) (Recall Theorem 5.2.3.) The associated quadratic forms are Vi(X i ) = Xi' Pix, = 1,2) and vex) = v 1(x d - V2(X2) for x = (Xl' X2) E :)In! X :)ln2 • Let = diag[ P 1> P 2] and with the aid of (12.4.2) choose f. > 0 sufficiently small so that

(i P

2I p llh(x)//lxl < i for all x

E

BE' the open ball of radius

f.

(12.4.9)

centered about the origin. Note that (12.4.10)

245

12.4 Local Behavior and Linearization about Rest Points

for all xEB•. Define H = {xllv(x)1 < c5} n B. and 8H = {x I xEH, Iv(x)1 = c5}. It is left as an exercise to show that c5 > 0 and r > 0 can be chosen appropriately small so that B zr c H c B. and, moreover, so that for each ~E U = B" y+W n 8H:F 0 ify+W n(,gfR - B.):F 0, and y-Wn 8H:F 0 ify-W n (,gfR - B.) :F 0. (In other words the seniiorbits must exit B. across 8H only.) The above choices of £,15, and r remain valid if hex) is replaced by A.(x)h(x), where A.(x) is any function on ,gfR with 0 ~ A.(x) ~ 1. By a classical construction such a A.(x) can be selected to satisfy the further requirements that A.(x) = Ion V and A.(x) = 0 on,gfR - B zr . Hence there is no loss in assuming that the domain of hex) is ,gfR and that hex) = 0 on 8H. The general solution l1(t, ~) of (12.4.1) with 11(0, ~) = ~ is then defined for all t E,gf and ~ E ,gfR. The required map g(~) = (glW, gzW) on the set U defined above is to be determined by the formulas

Coo

(12.4.11)

(12.4.12)

in which h(~) = (h1W, hzW)E,gfRl X ,gfR2 and ~ = (~l' ~Z)E,gfRl X ,gfR2. Inequalities (12.4.5) and (12.4.6) and the boundedness of Ih(x)1 ensure the uniform convergence of the integrals on ,gfR. It follows that g is continuous on ,gfR. Note that g(O) = O.: The next step is to establish the validity of the equation (12.4.13)

By referring to (12.4.11) and (12.4.12), applying the semigroup property of 11, and then making a simple change in the variable of integration, it is easy to check that g(l1(t,~))

=

- (f~oo

(111(t, ~), e(t-a)

m h (I1(u, en da,

I1z(t, A1

1

f:

e(t-a)

A2

hz(I1(U,

e» dU).

(12.4.14)

Since (12.4.15)

246

12.

Limit Properties of Solutions

Eq. (12.4.9) can be differentiated, with the result that d dt g(y/(t, ~»

=

(Algl(y/(t,

m, Azgz(Y/(t, ~»),

(12.4.16)

which immediately implies (12.4.13). Completion ofthe proof of Theorem 12.4.1 requires only the demonstration that the restriction of g to U is one-to-one. Because of(12.4.10), each nonzero element of U lies in the level set A(t) =:= {x Ix = y/(t, ~), ~

(12.4.17) E oH} n U Z for some t E &t. Select any nonzero x! and X in U. Then Xl = 1](t l, ~l) and XZ = Y/(tz, ~Z) for some t i E &t, ~i E oH (i = 1,2). Observe that Iv(~l)1 = Iv(e)1 = b. Since h = 0 on 8H, it is clear from (12.4.11) and (12.4.12) that the restriction of g to oH is the identity. From this and (12.4.13) we have ~l = gee) = e-t1Ag(y/(tl, ~1» = e-ttAg(x l), (12.4.18) ~z = g(~Z) = e- t2Ag(1](tz, = e- 12Ag(xZ). (12.4.19)

e»

If g(x l)

= g(XZ), then certainly (12.4.20)

But v is strictly monotone along the solutions of x = Ax. Equation (12.4.20) then violates the earlier observation that Iv(e)1 = Iv(e)1 unless t l = Vz» which is to say that ~l = ~z and hence that Xl = xZ. Our calculations further show that x#-O implies g(x) #- O. This shows that g is one-to-one on U, and Theorem 12.4.1 is established. Remark 12.4.1 Examples show that it is not always possible to choose the function g of Theorem 12.4.1 to be C l .

The conclusion of Theorem 12.4.1 says that the solution curves of (12.4.1) near the origin are mapped onto the solution curves of the linear system y = Ay by the coordinate transformation y = g(x) with the parametrization preserved. The conclusion of Theorem 12.4.1 can fail if the matrix A has an eigenvalue with zero real part. (See the exercises following this section.) Although an extensive treatment of the marginal case wherein A is not hyperbolic will not be presented, the following result gives some indication of its delicate nature. Remark 12.4.2

Theorem 12.4.2

In the scalar differential equation

i = iwz + h(z),

(12.4.21)

12.4

247

Local Behavior and Linearization about Rest Points

assume that the real co #- 0 and that h is a complex analytic function on a neighborhood of the origin in the complex plane with h(O) = h'(O) = O. Then there exists an open set U containing the origin and a bijective, analytic map g: U - V = {z Ilzl < p}, some p > 0, for which the general solution 1/ to (12.4.21) can be written as 1/(t,z)=g-l(e i wtg(z))

forall

ZEU

and tE81.

(12.4.22)

Proof. Let Uo be an open disk about the origin in which h(z) is analytic. Since h(O) = h'(O) = 0, then h(z) = Z2 fez) for some fez) analytic in a neighborhood of the origin, and thus by making the radius of U 0 small it can be assumed that 1

1

icoz

1

h(z)

fez)

+ h(z) - -itoz - iwz[iwz + h(z)] = iwz - iw[iw + zf(z)] (12.4.23)

is analytic on the complement of the origin in U 0 • Fix any Zo #- 0 in Uo . For z #- 0 in Uo let C be a path from that does not meet the origin. Now define g(z)

=

(i

Jiz exp(iW)j~iww

Zo to

dW) dW) exp(F(z)) + hew) = exp Jzo -;-

z in Uo

Z

(12.4.24)

where

_i few) dw . Jzo uo + wf(w) z

F(z) =

But independent of C,

(f

exp

Z

-dW) = -z ,

Zo, W

(12.4.25)

Zo

and it is further noted that the integral defining F(z) is independent of C. Thus g(z)\ =

-z

(12.4.26)

eF(z)

Zo

is an analytic function on U 0 witn g(O) = 0 and g'(O) #- O. By the inverse function theorem, g is therefore bijective on a neighborhood of the origin with g-l likewise analytic. Thus it is possible to choose a disk V c g(Uo ) centered about z = 0 and an open set U c U o containing the origin such that the restriction g: U - V is a bijection with analytic inverse. For any z E U, g(z) E V. Now restrict t to the open interval about t = 0 where 1/(t, z) E U. Then

(f

g(1/(t, z)) = exp u»

q (t .Z)

zo

.

dw

lWW +

) h( ) w

(12.4.27)

248

12.

Limit Properties of Solutions

in which C is a path connecting Zo to I'/(t, z) and can be taken to be C = C I + C2 where C I is a path from Zo to z and C 2 is a path from z to I'/(t, z) that will be taken to be 1'/(-, z). Therefore (12.4.27) can be rewritten as g(I'/(t, z)) = eXP(iW

f~

iww

() (. rt

a;

= g z exp IW Jo iwI'/(u,~) =

heW)) eXP(iW

f; such that any solution x(t) of (12.5.1) with x(o) E N exists for I tl :$; f. and crosses T exactly once for 1 tl :$; e. Proof.

Let

1] be

the general solution to (12.5.1), and consider the equation 1](t, e) - (q

+ O"V)

=

°

(12.5.2)

in which v is a unit vector in [H2 parallel to T. The scalar t and 0" are regarded as unknowns with a vector parameter in [H2. Certainly the equation has the solution t = 0, 0" = for = q. Computation of the Jacobian of the term composing the left-hand side of (12.5.2) relative to t and 0" at t = 0, 0" = 0, = q gives

e

°

e

e

de{;:~:~

=~:J

= _[Jl(q)V

2

-

j2(q)v

1

]

#

°

(12.5.3)

252

12. Limit Properties of Solutions

since T is a transversal to f at q. By the implicit function theorem there exists a unique solution t = t(~), (J = (J(~) to (12.5.2) for ~ in a neighborhood of q. This completes the proof.

Lemma 12.5.3 If T q is a transversal of(12.5.1) containing a point q E n+(y), then either the orbit y is periodic or else y + crosses T q at a sequence of times t 1 < t 2 < .. " and the corresponding crossing points ~k converge monotonically to q along T q as k -

00.

Moreover, q is the only point of n+(y) on T q •

Proof. If y is not periodic, then certainly Lemma 12.5.2 applies. Hence, since q E n+(y), the semiorbit y+ intersects Tq at times forming a sequence t 1 < t 2 < " " with corresponding crossing points ~ 1, ~ 2, .• '. Certainly ~ 1 , ~2' ... and q are distinct, for otherwise y would be periodic. Consider the Jordan curve determined by the arc of y" connecting ~1 to ~2' together with the segment of Tq connecting ~2 to ~1' There are three cases to consider: The point q lies on, outside, or inside the Jordan curve. If q lies inside, then since all crossings occur in the same direction, ~2 must be between ~1 and q, for otherwise the tail of y+ would be outside and would not be able to reenter the inside, which, however, it must do since q E n+(y) (see Fig. 12.5.1). There is also the possibility that q lies outside the Jordan curve, in which case ~2 would again have to be between ~1 and q, for otherwise the tail of y+ would enter the inside and would remain trapped away from q. Finally, q cannot lie on y+; consequently, the only place it might occur on the Jordan curve would be on the segment of Tq between ~1 and ~2' But that too is impossible since all crossings of Tq are in the same direction, and once the tail of y+ enters the inside or outside it cannot get back to the outside or inside, respectively, to reapproaeh q. In conclusion, ~2 always lies between q and ~1' The argument can be repeated to show that ~3lies between ~2 and q, etc. Since Tq can be identified as a compact subinterval of Yl, ~k converges as k - 00 because it is a monotone bounded sequence. Moreover, the limit of ~k is q. This follows directly from the fact that q E n+(y) and from Lemma Tq

Fig. 12.5.1 Two cases in the argument leading to monotonicity of crossing points.

12.5 Planar Systems

253

12.5.2. The only point of g+(y) on Tq is q, for any other would produce crossing points other than ~ 1 , ~ 2' •... Theorem 12.5.4 Poincare-Bendixson Theorem. If y + is a bounded positive semiorbit of(12.5.1) and g+(y) does not contain a rest point, then either

(a) g+(y) = g(y) = v or (b) g+(y) = y+ - y+. In either case, g+(y) is a periodic orbit. Proof. Let g+ satisfy the hypothesis of Theorem 12.5.4. Since g+(y) is not empty, it is possible to select one of its points q 1, which necessarily is not a rest point. Denote the orbit of q1 by Y1' Then g+(Y1) c g+(y) since g+(y) is closed. Select any q E g+(Y1) and let T q be a transversal of(12.5.1) containing q. As a consequence of Lemma 12.5.3, Y1 must be periodic. Otherwise Y1 would intersect T q at a sequence of distinct points which necessarily are in g+(y); but that is ruled out by the uniqueness statement in Lemma 12.5.3. Thus g+(y) contains at least one periodic orbit Yl' The remaining task is to show that g+(y) = Y1' Suppose that g+(y) - Y1 were not empty. Since g+(y) is connected, it would be possible to choose a point q2 E Yl which is a limit point of g+(y) - Y1' Let Tq be a transversal through q2' Any small disk about q2 contains a point q3 ~ g+(y) - Y1' For such a q3' (12.5.1) has a solution x(t) such that x(O) = q3 and the orbit Y3 of q3 is contained in g+(y). If q3 is sufficiently close to q2' then Y3 crosses Tq • By the uniqueness statement of Lemma 12.5.3 the crossing point is ne~essarily q2' However, since q3 ¢ Y1' a contradiction of the uniqueness of solutions to (12.5.1) is achieved. Thus g+(y) - Y1 is empty and the proof is complete.

The conclusion of Theorem 12.5.4 remains valid under slightly weaker assumptions. The differential equation need only have f defined on an open subset E of 911 2 and then y+ is required to have compact closure in E. The once continuous differentiability of f can be relaxed to f continuous as long as the initial-value problem has. unique solutions. For example, it is sufficient that f be locally Lipschitz on E. The more general result, using essentially the same proof, along with an extensive treatment of many related subjects, can be found in Hartman [9]. Those interested in a comprehensive treatment and references to the extensive literature should consult Hale [8] as well as Hartman [9] and Lefschetz [14]' Remark 12.5.1

The following corollary gives a detailed description of the manner in which the tail of y+ approaches the periodic orbit shown to exist in Theorem 12.5.4 (see Fig. 12.5.2).

254

12. Limit Properties of Solutions

Fig. 12.5.2 The approach of the tail of y'

to R+(y)

Corollary 12.5.5 Let y satisfy the assumptions of Theorem 12.5.4 for the case in which y is not periodic. Fix any q on the periodic orbit R+(y) and let T, be a transversal at q. 7hen.T, n R+(y) = q. y + crosses T, at a sequence of points ( k ( k = 1, 2, . . .), converging strictly monotonically to q as k ---t co. The respective crossing times t , < t , < * * * o fthe solution x(t) to (12.5.1) for which x ( t l ) = have (12.5.4) the smallest period of the solution x,(t) of (12.5.1) satisfying x,(O) = q. The sequence of functionsgk(t)= x(tk + t )converges uniformly to x,(t) on 0 I tI p as k + c o . Proof. All but (12.5.4) and the uniform convergence have already been proved. The g k s are all solutions of (12.5.1) with the same orbit y, and since y + is bounded they are all defined on 0 I t < co.Because of the convergence of the initial values gk(0) = tk+ q as k + co, the uniform convergence of &(t)to x,(t) on 0 I tI p as k--+ 03 is ensured by Theorem 11.3.1. In particular, g k ( p ) x,(p) = q as k + co. The proof is completed on noting that with Lemma 12.5.2 the negation of (12.5.4) would violate the uniform convergence of gk@ to x,(t) for 0 It Ip. --+

THE INDEX OF A CLOSED PATH

There is a very useful formula that computes how many times a curve y winds around a given point zo. This number is called the index of y with respect to zo (see Fig. 12.5.3). The definition is given in terms of complex variables, from which it naturally arises.

Dehition Let y be a closed curve in the complex plane V and zo be a point not on y. Then the index of y with respect to z,, (also called the winding

255

12.5 PlanarSystems

I

(Y

,zo) " 0

Fig. 12.5.3 The index of a curve about a point.

number) is defined by I(y,zo) =

~

2m

r ~.

Jy Z

-

Zo

It is relatively easy to show that I(y, zo) is an integer.

Rather than work with y directly, a complex number (vector) f = fl + if2 might be attached to each point along y to get a new curve yf = f 0 y and select Zo = O. The latter is the situation of interest in the study of (12.5.1). Since solutions to differential equations ordinarily do not cross, y will be a Jordan curve J and thus the index of J f will be denoted as IiJ). Thus I f(J) counts the net rotation of f as a point moves around J in counterclockwise fashion. The following basic properties of I f(J) are stated without proof. For more details and derivations see [16].

Theorem 12.5.6 If J is a Jordan curve surrounding a finite number of rest points PI' P2"'" Pn of f, then If(J) =

n

I

If(p;}·

;=1

If f has no rest points inside or on J, then I f(J) = O.

Theorem 12.5.7 If J is a C 1 Jordan curve and .f is a continuous unit tangent vector field along J, then I f(J) = 1. Corollary 12.5.8 . If Y is the orbit of a periodic nonconstant solution to (12.5.1), then Iiy)

=

1.

Corollary 12.5.9 A periodic orbit contains at least one rest point in its interior.

APPLICATION TO L1ENARD EQUATIONS

The Poincare-Bendixson theorem finds many uses in the study of periodic solutions in the plane. A demonstration of this is provided by the proof of

256

12. Limit Properties of Solutions

the next theorem, taken from Hale [8]. It considers the Lienard equation

u + f(u) u + u =

0,

(12.5.5)

written in the equivalent first-order-system form

x= yin which F(x) =

Y = -x

F(x),

(12.5.6)

H f(u) da. (See the proof of Theorem 10.3.4.)

Theorem 12.5.10 The planar system (12.5.6) has a periodic solution circling the origin if the continuous function f has F satisfying the conditions (a) F(x) is an odd function of the variable x. (b) F(x) - 00 as x - 00, and for some p > 0 it is positive, monotone increasing for x > p. (c) F(x) < 0 on 0 < x < ex, and F(ex) = 0 for some ex > O. Proof.

It will be useful to consider the scalar equation dy dx

-x

y - F(x)

(12.5.7)

for the orbits of (12.5.6) (refer to Fig. 12.5.4). The approach taken will be to show that for A = (0, a) on the y axis, with a sufficiently large, the positive semiorbit initiating at A enters the right half-plane and returns to the y axis, as indicated, to a point D = (0, d) with 0 < -d < a. Since F(x) is an odd function, any solution (x(t), y(t» to (12.5.6) will also have ( - x(t), - y(t» as a solution. Consequently, any positive semiorbit initiating at a point on the y axis strictly between the origin and D will be bounded. Examination of (12.5.6) and (12.5.7) shows that in the right half-plane solution orbits have negative slope above the curve y = F(x) and positive slope below it, and they cross the curve at vertical slope. Moreover, they cross the y axis horizontally from left to right above the x axis and from A

o Fig. 12.5.4 Starting a bounded positive semiorbit.

257

12.5 Planar Systems

right to left below the axis for t increasing. Hence, with assumption (b) there is no problem in concluding that for a sufficiently large an orbit initiating at A = (0, a) will return to the y axis at some point D = (0, d) on the negative y axis. The problem now is to show that 0 < - d < a for a appropriately large. . Consider the function vex, y) = t(x 2 + l). Along solution curves of (12.5.6) and (12.5.7), dv dt = -xF(x),

(12.5.8)

dv dx

(12.5.9)

-xF(x) - F(x)'

=y

dv dy = F(x).

(12.5.10)

From these three equations it follows that v(D) - v(A)

=

f

=

(f

dv

ABECD

+

AB

r )[y--

J +r

xF(x) dx F(x)

JCD

F(x) dy

(12.5.11)

JBEC

along orbits of (12.5.6). Clearly, the integrals in (12.5.11) over AB and CD remain bounded as a - 00.. By fixing any G on the x axis between (P, 0) and E, assumption (b) can be invoked to estimate the integral over BEC as

-r

JBEC

F(x) dy =

r

JCEB

F(x) dy >

r

JEK

F(x) dy >

IGHI x IGKI,

(12.5.12)

where IGHI and IGKI are the lengths of the line segments indicated in Fig. 12.5.4. As a- 00, IGHI remains fixed, whereas it is apparent from (12.5.7) that IGKI- 00. 'rhus for a sufficiently large, the inequality v(D) - veAl < 0 is implied by(12.5.11), which establishes the required inequality 0 < -d < a. Thus there exists a bounded positive semiorbit y+ of (12.5.6) that crosses the negative y axis. The system (12.5.6) has the origin as its sole rest point. However, y remains bounded away from the origin since (c) and (12.5.8) imply that v > 0 on a deleted neighborhood of the origin in al 2 • Thus the Poincare-Bendixson theorem ensures the existence of a periodic solution. Its periodic orbit must circle the origin-a consequence of Corollary 12.5.9 and of the fact that the origin is the only rest point. This concludes the proof of Theorem 12.5.10.

258

12. limit Properties of Solutions

Remark 12.5.2 Further results concerning the uniqueness and location aspects of the periodic solutions to the Lienard equation (12.5.5) exist (see [8]). Example 12.5.1 The Van der Pol equation U + k(u 2

1) Ii

-

+ u = 0,

(12.5.13)

k a positive constant, has a periodic solution circling the origin. (See Exercise 1 at the end of this section.) Remark 12.5.3 At this point the global and topological nature of the Poincare-Bendixson theory should be rather apparent. The subject of differential equations naturally extends to equations on two-dimensional surfaces and, more generally, on differential manifolds. There, under the title of differential equations on manifolds, topological dynamics, and differential topology, the constraints imposed on the solutions by the underlying manifold are an active element of the theory. We end this discussion with the remark that there exist periodic-free differential equations on the two-dimensional torus for which each orbit has the entire torus as its positive limit set. Thus there is no simple generalization of the Poincare-Bendixson theorem to manifolds in general. A natural generalization of a periodic orbit is a minimal set (a closed invariant set containing no proper such subset). A famous result of Schwartz [23] proves that minimal sets are either rest points, periodic orbits, or tori. Thus the search resumes.

EXERCISES

1. Verify that the Van der Pol equation (12.5.13) satisfies the hypothesis of Theorem 12.5.10. 2. Let y be an orbit of (12.5.1) in fJt2. Suppose that there is a continuous curve g: [a, b] - n+(y) which is onto and that Q+(y) contains no rest points. Show that n+(y) is a periodic orbit. 3. Let y: [a, b] - f!l.2 be a Jordan curve. Suppose that (12.5.1) has a solution which enters and remains inside y for all large t. Show that (12.5.1) has at least one rest point inside or on y. 4. Show that it is impossible for the positive limit set of each orbit initiating inside a given Jordan curve y to be outside y. 5. Starting with the definition of index of a closed curve, derive the formula I (J) = f

~

211:

f J

11 dI2 - 12 dI1

Ii + I~

259

12.5 Planar Systems

for the index of a real C 1 vector field f = (fl' f2) in ~2. (J is a C 1 Jordan curve oriented in the counterclockwise direction.) 6. Prove Corollaries 12.5.8 and 12.5.9 by applying Theorem 12.5.7. 7. Let J: [a, b] -- ~2 be a Jordan curve and let f(t), g(t) be two continuous vector fields on J that can be continuously deformed into one another without vanishing; that is, there exists a continuous function H: [a, b] x [0, 1] __ ~2 such that H(t, 0) = f(t), Hit, 1) = g(t), H(a, s) = H(b, s), and H(t, s) #- 0 for all t and s. Show that IiJ) = IiJ). 8. Show that IiJo) = If(J 1 ) if J o and J 1 are C 1 Jordan curves that can be deformed into one another without crossing rest points of f. 9. Prove that if f has div(f) of one sign not zero inside a Jordan curve J, then (12.5.1) has no periodic solution other than possible rest points. (Hint: Use Green's theorem.) 10. The problem of determining rest points is the problem of computing the zeros of f(x). A periodic solution is just a rest point in another dynamics. That is, a Jordan curve y in ~2 is carried along by the general solution to get the trajectory of Jordan curves ~(t, y(. )): [a, b] -- ~2 definded by ~(t,

y(s))

= l1(t, y(s)), S E [a, b]'

(a) Show that the rest point set of (12.5.1) inside y is invariant along the trajectory ij. (b) Denoting the inside of a Jordan curve J as I J for y a Jordan curve, derive the area formula

and apply it to obtain the result that ify is a periodic solution of(12.5.1),then

which says that

r div(f)(O d~l

Jry

u, =

O.

(Note the agreement with Exercise 9.) (c) Let y: [0, A] -- ~2 be a C 1 Jordan curve parametrized by arc length.

260

12. Limit Properties of Solutions

Derive the curve length formula

to, y) =

fij

ds

=

f"l7uCt , y(O'»/ da

and apply it to show that if y is a periodic solution, 'then

fy T'[Jh) + f~r(y)]T

ds

= 0,

where T is the unit tangent vector field to y and ds is the element of arc length.

Chapter

13

Appl ications in Control Theory

This chapter is a brief introduction to a few of the important mathematical notions that appear in the control of differential equations. It demonstrates how some of the results of previous chapters bear on the subject. 13.1 The Controlled and Uncontrolled Equations

The system in !J.ln

x = dx,

(13.1.1)

in which d is an n x n matrix, hereafter taken to be real, has been treated extensively. The results revolved about the solution formula x(t) = [XO, dxo, ... , d

n

-

1xO]

[yO, Byo, ... , B n - 1 yO] - l y(t), (13.1.2)

in which the n x n matrix Band n-vector function yet) are derived entirely from the eigenvalues of d, and yO = yeO). This formula proves useful in the theoretical and numerical calculations that follow. Control problem situations in which the appropriate model is the control differential equation

x=

dx

+ (J4u,

(13.1.3)

were described in Chapter 1. The term with the n x p matrix coefficient (J4 accounts for control variable u E!J.lP entering the system. (Note that in conformity with contemporary notational usage the letters d and (J4 are used for the coefficients in (13.1.3). Script letters are employed to avoid confusing the coefficient (J4 of u in (13.1.3) with the matrix B in (13.1.2), which also occurs in a number of previous chapters.) 261

262

13. Applications in Control Theory

When the control variable u in (13.1.3) is replaced by a control function u = u(t), the resulting nonhomogeneous equation can be written in the solved forms provided by the variation-of-parameters formula x(t) = etdxO

+ f~

e(t-a)dBlu(a)da = etdxO

+ {e"'dBlU(t

- a)da. (13.1.4)

13.2 Transient and Steady-State Partsof the Solution

Suppose that d is a stability matrix and that the control input function u(t) is bounded; that is, it satisfies /u(t)1 s b for some b e 9f for - 00 < t S O. (For example, it might be zero there.) Then the integral terms in (13.1.4) can be rewritten to obtain the decomposition (13.2.1 ) where x'(r) = etdxO,

1 00

x 2(t)

=

-

XS(t) =

f~

(13.2.2) ead Blu(t - a) do,

and 00

e(t-a:dBlu(a) do =

r

(13.2.3)

ead Blu(t - a) do.

(13.2.4)

In (13.2.1) the first part, x 1(t), is the solution to the uncontrolled equation and is a transient term associated with the initial value of the output x". Methods for computing x 1(t) in closed form, e.g., formula (13.1.2), methods for estimating the decay rate of Ix 1(t)I, and the possible qualitative behavior of that part of the output were covered earlier in the text. The second part, x 2 (t ), is likewise a transient term but it arises from the input u(t). The techniques developed also provide estimates of its decay rate, Ix 2(t)1

s

1 00

blBlI

leadl da,

(13.2.5)

for example, with leadl estimatable by methods applicable to x 1(t). The last term, XS(t), in (13.2.1) is called the steady-state solution. (Note that it is a solution to the differential control equation, although XS(O) i= xo.) It is the part of the solution that persists after the transients have died out. Equation (13.1.4) defines a linear integral transformation !I', the steady-state

263

13.3 The Frequency Response Function

~

u (t)

0--------

xs(t)

Fig. 13.2.1 The steady-state operator.

operator, which maps whatever linear space of control inputs it might be

allowed to act on onto the resultant linear space of steady-state responses (see Fig. 13.2.1). The relative importance of the three parts of the outputs x(t) to (13.1.3) depends on the application at hand. In some applications all three need to be considered to some extent. Most of the important information about the steady-state response of interest in the control of(13.1.3) is readily extractable from the frequency response function-the next item up for discussion.

13.3

The Frequency Response Function

In the typical situation (13.1.3) is composed of connected subsystems, some to be controlled and others built in by the engineer to do the controlling. A major question of interest concerns the modification of the amplitude of a control input as it passes through the system. In some situations it is desirable for a small-amplitude control input u i( . ) to produce a large value xit) of one of the output variables. An example would be a thrust control on an airplane in which the pilot inputs a low-energy command to increase the thrust of the engines as needed in takeoff. On the other hand, some of the inputs might carry high frequency noise from the equipment or other sources which must be attenuated (filtered out) as it passes through the system. Intuition would suggest that delays exhibited by the system in responding to the inputs would also influence important matters such as stability. These concerns lead to the notion of a frequency response function In (13.2.4) let j E {I, 2, ... , p} be fixed and consider the special control input u(t) = eirotej, with co real and ej the jth column of the p x p identity matrix. Since d is' assumed to be a stability matrix, the response is the n-vector function xj(t)

=

foro e4t et>.l4< e1,

I

jq+ 1 jq ... ,el.>4t e2• .>4< ez, ... ,ez ,..

I j q ep'.l4<j q - l ep, ... ,ep'}

'.>4 del' el' dez, ez, the rows of the left end matrix P of (13.7.33) are rearranged accordingly to get 0 0 P= 1 0 1

0 0 0 1 1 0 0 0 0 -1

0 1 3 1 1

1 1 1 0 0

(13.7.35)

The cycle lengths r l = 3, rz = 2, obtained in step 4, dictate thatthe canonical matrix fZ in formula (13.7.10) is

fZ=

0 1 0:0 0 0 0 1 : 0 0 0 0 oio 0 --------4---0 0 o :0 1 0 0 o:0 0

(13.7.36)

which, along with the original .91, Bl, and P found in (13.7.35), is substituted into (13.7.10) to finally compute

K= [-1 -1 0-4 -lJ. -1

0

0

1

0

(13.7.37)

It can be verified that the computed P, Q, and K satisfy the equations imposed by the conclusion of Theorem 13.7.2, P(d

+ BlK)P- l

o o PBlQ

=

(13.7.38)

0 0

= 1 0 o 0

o

fZ,

(13.7.39)

1

Remark 13.7.3 The decoupling algorithm makes all the control variables work in breaking up the system into the maximal number of decoupled (independent) controllable subsystems by inserting appropriate feedback.

296

13. Applications in Control Theory

In the system U 1 and U2 control the same two-dimensional subspace of fJl3. Theorem 13.7.2 says that with proper feedback the system can be- decoupled into three independent subsystems with each input variable controlling one of the subsystems. In some problem situations an intermediate mode is appropriate whereby distinct control variables control different subspaces but those subspaces are allowed to intersect In a nontrivial fashion. In other words the subsystems need not be totally decoupled. In Exercise 4 at the end of this section, a simple formula is derived for computing a universal control function that achieves this mode. It has the advantage of involving no change of state variables. Conceivably, it might be taken as the first step in numerous design procedures since it provides another way of strongly "tying in" all the control variables into the system.

The General Case. The first and last segments of the algorithm for solving the general problem are the same as those in the diagonal case. Thus, every problem is started by writing out the augmented controllability matrix (13.7.6) and then row reducing it to get (13.7.7). As in the diagonal case, a finite sequence of row, column, and feedback steps is then performed. For the nondiagonal case it is the choice of this sequence that remains to be described. At any rate the sequence leaves the augmented matrix in a form in which the computations can be completed by following the same procedure as that in the diagonal case, beginning with step 4. The ultimate goal of the sequence of matrix operations is to get the cycles to produce an ordered cycle basis. This is the requirement that all cycles be closed and that the vectors of each cycle preceding the first vector in the cycle with zero foot, taken together, constitute a permutation of the columns of the identity matrix In. A vector in a cycle can be pictured as moving through the row in which it has a 1 as the cycle is traversed. Those elements, with the order that they are given by the choice of basis in the controllability matrix (in the diagonal case, the natural order from left to right) are defined as the ordered cycle basis. In the diagonal case the ordered cycle basis arose from the columns containing the diagonal of leading ones in the row-reduced form, and all cycles were closed by a single sequence of feedback insertions that forced the feet of the p columns at the end of the diagonal (the G matrix) to be zero. The matrix (13.7.14) was just that of the state coefficient matrix (viewed as an operator on fJln) relative to the standard unit basis sitting in the first n columns of the controllability matrix. It was the diagonal of leading ones in that matrix that carried the feedback matrices inserted above the foot down

297

13.7 Decoupling and Recoupling-Noninteractive Design

along the diagonals to close the cycles in G under the repeated multiplication by the closed-loop matrix that occurs in regeneration of the controllability matrix. In the non diagonal case the (ordered) basis cannot be made to consist entirely of columns containing leading ones because there are not enough of them. Fortunately, other columns of the controllability matrix are always available to complete the basis due to the assumed controllability. This alternative choice of basis implies that the procedure for reading off the open-loop matrix into which the feedbacks are to be inserted must be altered accordingly. The riondiagonal case allows additional coupling of the cycles that does not occur in the simpler case. This is no insurmountable problem. It means that only one cycle at a time can be cut loose, as opposed to the diagonal case in which all are closed simultaneously by a single sequence of feedback insertions and the disconnecting accompanies that closing of the cycles. The complications that must be dealt with in the non diagonal case will be discussed in terms of an example rather than by working abstractly. The system is one with p = 2 control variables and n = 6 state variables. The augmented controllability matrix is

Example 13.7.2

1 0 0

I

0 0 0 0 0 0 0

1 0

0

0

I

0 0 0 0

0 0 0 0

0 I 0 0 0 0 0 I 0 0 0 0 0 I 0 0 0 0 0 I

0 0 0 0

0 0 0 0

I 0 0 0

0 1 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

0 0 1 0

0 0 0 1

0 I 0 1 0 I I I

0 I 0 I 0 I I 2

I I

0 0

I 1 I 0

I 0 0 I

(13.7. 41)

For a good reason soon to be discussed, the state coefficient matrix ,r;I is not given explicitly. (Neither is !!4, but of course that is just the second block.) The system is not diagonal because of the bottom zero element of the sixth column of the controllability matrix (not to be confused with the augmented controllability matrix). The diagonal of ones that does not reach the bottom of the matrix will be called a short diagonal (of ones). The short diagonal in (13.7.41) prevents the annihilation of the foot of the fifth block by insertion of feedback using the technique of the diagonal case. Notice that a short diagonal allows nonzero terms to remain, after row reduction, in the foot of the matrix above the diagonal line. (In (13.7.41) they are the two Is in the sixth column ofthe controllability matrix.) These nonzero terms "tie" or "connect" the cycles together and must be eliminated by row and column operations and feedback if the cycles are to satisfy the definition of an ordered cycle basis.

298

13. Applications in Control Theory

The strategy of the algorithm is to start untying the cycles at the first point at which they come together, using feedback. To do this the open-loop matrix must be made to appear so that the feedback can be inserted. As already explained, in the diagonal case this matrix is already found sitting, after row reduction, in the controllability matrix by shifting over p columns from the left end. The reason why this matrix (13.7.14) is the open-loop matrix should be clear from the facts that the standard unit basis for f!Jln occupies the first n columns of the controllability matrix in the diagonal case and that the latter is generated by multiplication by .91. However, if a short diagonal occurs, say oflength s, then only the first s columns of the open-loop matrix will be found by shifting p units over. The remaining columns must be recovered from other columns of the controllability matrix. Thus in (13.7.41) the first five columns of the open-loop matrix can be read off by shifting two columns to the right, and it is the seventh column that can easily be made to produce the remaining column, namely, the ninth column, since the last base vector appears in the seventh. The matrix just read off is then

0 0

o o

0 0

1 0 o 1

o o

1 1

o

o o o

0 0

1 0

o

0

o

o o o o

0 0 0 0 0

(13.7.42)

1 1

To help locate the feedback that needs to be inserted to eliminate the Is in the sixth column of the controllability matrix that tie the cycles together, drop the row beneath the short diagonal of Is. Since there are two control variables, G consists of the foot of the two columns that follow the end of the short diagonal. Thus the problem looks like a diagonal one with p = 2, n = 5. Proceeding with the method for treating diagonal problems, we find that the two zeros allow insertion of the second stage of feedback into (13.7.42) with no column reduction required, and hence the closed-loop form of (13.7.42), following from insertion of feedback, is

o -1 o -1 1

o o o

o

0 0

o o

0 0

o

1 1

o o

0 0 0

o

0 1

o

0 0

o

1 0

0

o

1 1

(13.7.43)

299

13.7 Decoupling and Recoupling-Noninteractive Design

Regeneration of the controllability matrix, by using this matrix, and row reduction produce the matrix I 0 0 I 0 I 0 I

0 0 0 0

0 0 0 0

0 I 0 0 0 0 I 0 0 0 0 I 0 0 0 0

0 0 0 I

0

0 I

0 0 0 0

0 0 0 0

0 0 0 ·0

0 0 0 0

0 0 0 0

I 0 0 I

0 0 0 0

0 0 0 0

I 0 0 0

0 0 I 0

0 I 0 I 0 0 I 0

0 0 0 I

0 0 I 0

0 I 0 I 0 0 I I

(13.7. 44)

I

0 I

0 0

0 0 0 0

(After the row reduction that recorded the insertion offeedback, the contents of the second through the sixth blocks ofthe controllability matrix above the foot in (13.7.44) were set to zero. This is permissible since the direct feedbacks are taken into account in the formula that computes K.) Notice that the insertion untied the cycles in the third block and closed the second cycle. This disconnected closed cycle must be "gotten out of the way" so that the remaining cycle can be worked on. The process, to be called casting out closed (and disconnected) cycles, simply relocates the rows through which the cycle passes to the bottom positions of the matrix as I

0 0

I

0 0

0

0 0

0 0

0 0

0 0

0 0

I

0 0 I 0 0 0 0 0 0 0 I 0 0 0 0 0 0 I

0 0 0 0 0 0

I 0 0 0 0 0

0 0 I 0 0 0

0 I 0 0 I 0

0 0 0 I I 0

0 I 0 0 I I

0 I

0 0 0 0

0 I 0 0

0 0 0 I

0 0 0 0

0 0 0 I

0 0 0 0

0 0 0 I

0 I 0 I 0 0 0 I

I

0

(13.7.45)

The remaining (first) variable is in control in the problem with p = 1. n = 4 above the bold-face line in (13.7.45). Note that this problem is of diagonal type and that the cycle is not yet closed. The basis to use to make the insertion of feedback work in the top problem is el' del' dle l, d3el> el' del' Relative to this choice the open-loop matrix is 0 1 0 0 0 0

0 0 1 0 0

0 0 0 1 0 0 0

0 0 0 1 0 0

0 0 0 0 0 1

0 0 0 0 0 0

(13.7.46)

300

13. Applications in Control Theory

To close the first cycle of(13.7.45) (see the ninth column), the insertion of the first stage of feedback in (13.7.46) gives the closed-loop matrix -1 1 0 0 0 0

0 0 1 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 0 0 0 0 0 0 0 1 0

(13.7.47)

The resulting row-reduced augmented controllability matrix is then

o

0

1 0

o

0

0 0

0 0

o

0 0 1 0 1 0 0 0 0 0 1 1 0 0 0 0 0 1

0 0 0 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 1 0

0 1 0 1 0 0 0 0 0 1 0 0

0 1 0 0

0 0 0 1

0 0 0 0

0 0 0 0

1 0 1 1

o

0

1 0

0 0 1 0 0 0

0 0 0 0 1 0

0 1

0 0 0 0

0 0 0 0

0

(13.7. 48)

Relative to the same basis el> del> d2el> d3el> ez, dez, the open-loop matrix read off (13.7.48) is 0 1 0 0 0 0

0 0 1 0 0 0

0 0 0 1 0 0

0 0 1 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0

(13.7.49)

To close the first cycle, inspection of (13.7.48) shows that feedback can be inserted at the second level with the resulting closed-loop matrix being

o -1 1 0 0 0 0

0 1 0 0 0

0 0 0 1 0 0

0 0 1 0 0 0

0 0 0 0 0 1

0 0 0 0 0 0

(13.7.50)

301

13.7 Decoupling and Recoupling-Noninteractive Design

Its row-reduced controllability matrix computes to be 1 0 1 1 1 1

1 0

o

0

0 0

o

0

1 0

0 0

1 0

0 0 1 0 1 1 0 0 0 0 1 1 0 0 0 0 0 1

0 0 0 0 0 0

1 0 0 0 0 0

0 0 1 0 0 0

0 0 0 0 1 0

0 0 0 0 0 0

1 0 0 0 0 0

0 1

0 1 0 1 0 0 0 0 0 1 0 0

0 1 0 0

0 0 0 1

0 0 0 0

0 0 0 0

0 0 0 0

0 0 0 0

(13.7.5 1)

Inspection of the ninth column of the controllability matrix in (13.7.51) shows that the first cycle is now closed. (The 1 in that column is in a line of direct control; hence it can be set to zero but it need not be.) With the cycles closed and disconnected, the algorithm now finishes up with steps 4 and 5. Thus an ordered cycle basis is to be read off. In the diagonal case the basis elements occurred in the natural order. In (13.7.51) they were selected in the order et> del' dZel, d3el' ez, dez, and this then is the ordered cycle basis. The cycle lengths are r l = 4, rz = 2. Continuation with step 5 permutes the rows of Pin (13.7.51) in accordance with the ordered cycle basis to give 0 0 0 p= 1 0

a

0 0 0 0 0

0 0 1 1 0 1 0

0 a 1 0 1 1 0 1 1 1 1 1 1 0 0

(13.7.52)

a a

1

Q is read off(13.7.51)to be 1z . The cycle lengths determine the canonical form 0 1 0 0 0 0 1 0 0 0 0 1 !!Z= 0 0 a 0 -----------~-----

0 0 0 0 0 0 0 0

I I I I I I I I I I

I I I I I

0 0 0 0

0 0 0 0

(13.7.53)

0 1 0 0

Formula (13.7.10) is applied to compute the decoupling feedback matrix

302

13. Applications in Control Theory

coefficient K

=

[-1o -1 -1-2-1-IJ -1

0 -1

0

(13.7.54)

0'

and the problem is solved. Remark 13.7.4 In a high-dimensional problem that is not diagonal, a large number of cycles can occur. Each time one is closed and disconnected it is cast to the bottom of the. controllability matrix by row operations and the remaining cycles are worked on until the ordered cycle basis is achieved. This puts the coefficient matrix in block diagonal form. The Mathematical Basis of the Algorithm. The reasons for a number of steps in the algorithm become quite transparent when nonsingular changes of state and control coordinates y = Px, v = Q- 1U are made and the induced transformation of the controllability matrix is examined. The coefficients of the differential equation undergo the transformation .91 ---. PdP- 1, fJI---. PfJlQ and it follows that [fJI, .91fJI, ... , d

n

-

1

fJI] ---. p[fJlQ, .91fJlQ, ... , d

n

-

1

fJlQ]

(13.7.55)

is the induced transformation on the controllability matrix. A standard elementary theorem of matrix theory says that elementary row operations on a matrix can be realized by performing those operations on an identity matrix and then doing left multiplication by the resulting matrix. Hence the n x n matrix attached to the left end of the controllability matrix in (13.7.6) records the operations that are later (in step 5) converted back into a change of state coordinates. Similarly, elementary column operations on a matrix can be realized by performing those operations on an identity matrix and then doing right multiplication by the resulting matrix. Thus the right end matrix of (13.7.6) records column operations on fJI. In short, the augmented controllability matrix, along with row and restricted column operations, provides a systematic and convenient device for changing state and control coordinates (which, alternatively, could be viewed as changing bases in the column spaces of the controllability matrix and of fJI, respectively). The full rank assumed of f!J and induced in the controllability matrix by the assumed controllability is of fundamental importance. One significant consequence is that the basis used to read off the open-loop matrix needed for insertion of feedback can always be taken to be the columns of fJI together with other columns of the controllability matrix. This ensures that the numbers in the associated p rows of the row-reduced controllability matrix can be chosen arbitrarily. In the diagonal case the remaining leading Is in the diagonal are picked up by the open-loop matrix, and under regeneration

303

13.7 Decoupling and Recoupling-Noninteractive Design

of the controllability matrix they translate the inserted feedbacks downward along the diagonals to close the cycles in the G matrix. Under repeated matrix multiplication, mixing can occur, and this is prevented by the column reduction and by the insertion of feedback in stages. The verification that this happens is left to the reader as an easy exercise. In the nondiagonal case, by the full rank available, a short diagonal of Is is always present, which allows one cycle at a time to be closed and disconnected. The row reductions that follow the insertion of feedback are not similarity transformations since they are guided by new numbers that have been inserted in the matrix. The decoupling matrix does not have to be explicitly computed and updated during the process because of the uniqueness of the solution to the equation that it must satisfy (see Exercise 9). By selecting a basis from the controllability matrix and writing out any solution to (13.5.1) in terms of coordinates relative to that basis, it is easy to see that the closed property of all the cycles prevents the coordinates of the solution associated with one cycle from interacting with those of another. Finally, the permutation of the matrix P in (13.7.20) to get P amounts to reordering the ordered cycle basis in conformity with the order needed for the matrices of the decoupled subsystems to come out in the prescribed canonical form.

EXERCISES

1. Find a feedback controller u = Kx coefficients

+

v for the system (13.5.1) with

0 1 OJ [

.5#=000, 000 such that the Jordan form of the closed-loop coefficient matrix .5# as indicated. (a)

2.

[-~ _.~

~l,

° o-d

(b)

[-

+ fJlK

is

~ _ ~ ~l. ° ° -dJ

Consider the pair of controllable decoupled systems

X2 =

AtX2

+

u,

Y2 = A2Y2

Under what conditions on the parameter values

At

+

v.

and A2 does there exist a

304

13. Applications in Control Theory

control function of the type u = clp, u = czp for some scalar cl, c2 such that the joint system is controllable relative to the control variable p? 3. Apply the decoupling algorithm to compute the matrices P, Q, and K of Theorem 13.7.2 for each of the diagonal systems (13.5.1) determined by the following coefficients.

1, j, 1, [;d9

0 1 0 0 (c) .=[l 0 0 10 0

0 0

a=k

0 0 1 0

1-1 2 1 (d) d = [ 2 0 1 1 2 1-1

0 -1 a=[!-; Jl

1 1

[.

0 0

0 1 0 0

&=

0 0 1 0 0 0 1]’ .

a=

0 0 0 0

0 0

(f) d

=

1’ j-1. 1

0 0 (g) d = 0 0 0

’

1 0 0 0 0

0 1 0 0 0

0 0 1 1 0 0 0

a=b

0

0

1-1

Assume that in (13.5.1) the columns of B are linearly independent. This exercise does not require d,B to be controllable. (a) Show that the equation d29 = 0, in which d , = d BK, has a unique solution. Find an explicit formula for the solution K in terms of

4.

+

13.7 Decoupling and Recoupling-Noninteractive Design

305

.91 and fJI. For the solution show that .91K =

[I -

fJI(fJI*fJI)-lfJ1*Jd.

(b) Show that the process in (a) that takes .91, fJI - .91K takes .91K' fJI- .91K' What control-theoretic interpretation can you give this property? (c) What simplification occurs in the above formulas for K and .91K if the columns of fJI are orthonormal? (d) Show that bi'(dK)kb j = 0 (k = 1, 2, ... , n - 1; i = 1, 2, ... , p; j = 1,2, ... , p) if K is the solution of (a). Note that the relationship also holds for k = 0, i '" j if the columns of fJI are orthogonal. (e) Assume that the columns of fJI are not only linearly independent but also orthogonal. Let no be the dimension of the controllable subspace of .91, fJI. Show that there exist positive integers SI' S2' ..• , sp with SI + S2 + ... + sp = no such that the column-vectors of the n x no matrix [b 1, dKb b ... , dr1bl/" '!bp, dKb p, ... , d~p-lbpJ constitute a basis for being a linear combination of the controllable subspace of .91, fJI with d~bi the column-vectors d~bj' 0 :s; k :s; Sj - 1, 1 :s; j :s; i for i = 1,2, ... , p. (The point here is that all of the columns of fJI appear in the basis.) 5. Application of Exercise 4. Compute the controllable subspaces of the individual control variables in the system

to show that Ul and U2 control the same subspace. Then compute the matrix K of Exercise 4, substitute in the feed-controller u = Kx + v, and examine the subspaces controlled by the individual variables VI' V 2, V3' Investigate the extent to which this controller decouples the system by finding the basis discussed in part (e) of Exercise 4 and rewriting the state vector in terms of that basis. 6. A simple geometric interpretation underlies the controller discussed in Exercise 4. Let .A.P be the vector space of real n x p matrices endowed with the inner product (M1IM2) = tr(Mt M 2 ) for M 1 , M 2 in .A.p • In terms of the columns of fJI :;= [bb b2 , ••• , bpJ, define fJI(i,j) to be the matrix in .A.P with jth column b, and the other columns zero. Then fJI = L fJI(i, i), and if the b, are orthonormal (or orthogonal), which is to say that fJI *fJI = I p(or diagonal), then the fJI(i,j) constitute an orthonormal (orthogonal) system in .A.p' That system is of interest since in terms of it one can write fJlK = Li,j KijP(i,j). It is natural to ask whether or not some K in .91K = .91 + fJlK makes (.91 KIfJI(i, j)) = 0 for all i, j. The unique K that solves these equations is the one found in Exercise 4, and the equation drfJI = 0 is simply the statement that .91K .1 fJI(i, j) for all i, j. Verify the statements made. 7. Consider the situation described in Exercise 6 under the assumption that the columns of fJI are linearly independent and orthogonal. From the

306

13. Applications in Control Theory

facts that the g ( i , j ) constitute an orthogonal set and that B K lies in the spanned by that subset for each K , it follows from the subspace of A,," general properties of inner product spaces (see, for example, [111) that the solution K of Exercise 4 is the unique solution to the minimization problem minld, 1, where the minimization is over K E Mp,,and the norm is the one induced by the inner product, i.e., IdK = tr(d;dK). Using the formula for K found in Exercise 4,show that

l2

rninId,l

=

Jtrd*[z

-B(~*B)-'B*]~

and note the simplification that occurs if g has orthonormal columns. 8. Apply the decoupling algorithm to the following systems:

j7

0 1 0 0 0 0 1 0 (a) & = [o 0 0 0 0 0 0

(b) d =

[%j

1 '

:

1,

0 0 & = [l 0

1 0 0 1

0 0 1 0 0 0 0

0 0 (d) d = 0 0 0

1 0 0 0 0

0 1 0 0 0

0 0 1 0 0

0 0 0 , 1 0

B=

0 0 (e) d = 0 0 0

0 0 0 1 0

0 0 0 0 1

0 0 0 1 1

0 0 0 , 0 0

g=

1 "1 1 0

0

9. Assume that d,B, P,and the canonical matrix 2 in Theorem 13.7.2 are known. Derive formula (13.7.10) for the decoupling feedback matrix K.

307

13.8 Controllability of Nonlinear Equations

13.8 Controllability of Nonlinear Equations

The significance of the controllability hypothesis is evident in the study of linear control systems that appeared in the preceding sections of this chapter. Generally, controllability theory attempts to define and isolate the theoretical limits to which various types of dynamical systems can be controlled. It turns out to have a fundamental bearing on many aspects of control (see, for example, [13,21]). The discussion here focuses on nonlinear equations of the type

x=

f(t, x, u) = .9Ix

+

fJlu

+ h(t, x, u)

(13.8.1)

with the standing assumption that the term h(t, x, u)is defined and continuous at all (t, x, u) E PAl X PAn X PAP. The real matrices .91, fJI are of respective sizes n x n, n x p. The following theorem is a generalization of Theorem 13.5.1, which was first proved by the author in [15]. Theorem 13.8.1 In Eq. (13.8.1) let h(t, x, u) be continuous and Ih(t, x, u)1 be bounded on PAl X PAn X PAP. Further assume that h(t, x, u) is periodic in t with period T > 0 for each fixed x, u. Finally assume that

Then for each ()(, control function u(·)

p E PAP . croCCO,

E

X o , Xl E PAn . and 0 < t I < T there exists a T], PAP) such that

(a) u(O) = u(T) = ()(, u(t I) = p, and (b) a corresponding solution xC) of(13.8.1) for which x(O) = both X(tl) = x., x(T) = ·Xo.

Xo

satisfies

T his control function extends to a continuous periodic function on ( - 00, 00) with period T and with a response agreeing with the periodic extension of x(t). Remark 13.8.1 In Theorem 13.8.1, T need not be a minimal period. Hence that theorem applies to the special case in which h = hex, u) is independent of t (autonomous systems). In particular, it applies to the case in which h is the zero function (linear autonomous systems). It says not only that the response can be steered through any two states but also that the values of the control function can be prescribed as well. In particular, the response can be steered through the prescribed states with any attainable velocities. Remark 13.8.2 If the periodicity requirement on h(t, x, u) is dropped altogether, then T > 0 can be taken to be arbitrary and all conclusions of Theorem 13.8.1 remain valid, with the exception of the last sentence concerning the periodicity, which is lost.

The proof of Theorem 13.8.1 will appeal to the following lemmas.

308

13. Applications in Control Theory

Lemma 13.8.2

Define O. If

then the matrix ST = f: Ofor each fixed x, u and,finally, that rank [ !JI, d!JI, ... , dn-1!JIJ = n. Then the conclusion of Theorem 13.8.1 and the remarks following it are valid for all e in a neighborhood of £ = O. Proof. As in the proof of Theorem 13.8.1, the problem reduces to one of showing the existence of a solution to Eq. (13.8.18) in the Banach space E, with the difference that (13.8.2,0) is now replaced by N(x, u)(t)

=

e

r

e(t-W),,",'ii(w, x(w), u(w)) do:

(13.8.23)

since the previous role of h is now played by (h. It is easy to see that N does define a map of E into itself. From application of (13.8.22) to (13.8.19) and (13.8.23) and from routine estimation, it follows that there is aCE [}l such that

IIN(x,~)

- N(y, v)11 ~ !£lcLII(x, u) - (y,

v)11

(13.8.24)

for all (x, u), (y, v) in E and all e in some neighborhood of e = O. Thus for 1£/ < (CL)-l the map N: E- E is a contractive operator on a complete metric space E. Thus by the standard theorem which says that such an operator has a unique fixed point (see, for example, [19J), it follows that (13.8.18) has a solution. Application of Lemma 13.8.3 concludes the proof of Theorem 13.8.4. Remark 13.8.4

The controllability of a linear system can be destroyed by

313

13.9 The Domain C of Null Controllability

addition of a term h satisfying (13.8.22). An example is

x=

f.(x, u) = u

+ c J u2 +

x2.

(13.8.25)

J

For it h(t, x, u) = x 2 + u 2 does satisfy (13.822) for some L and is continuous. However, if 1£1 ~ 1, then £ x(t) ~ 0 for all t. This shows that the requirement in Theorem 13.8.4 that 1£1 be sufficiently small is not superfluous. It is worth noting that if the square root in (13.8.25)is dropped, the resultant equation (13.8.26) fails to satisfy the conclusion of Theorem 13.8.4 for all e but e =

o.

13.9 The Domain C of Null Controllability

In Section 13.8 we were concerned with finding conditions on .91, PA, and h in (13.8.1) which assure that it is possible to steer the system from any state to any other in finite time. However, in applications in which x denotes the error in some controlled physical system, the point of interest is the capability for steering all initial states to one fixed state, the origin. This leads to the definition of the domain of null controllability that follows. The discussion is restricted to autonomous systems in !?An of the type

x=

f(x, u) = dx

+ PAu +

h(x, u),

(13.9.1)

with the standing assumption that h and its first-order partial derivatives are continuous in !?An X ~P and are zero at the origin in !?An X !?Ap.

Definition The domain C of null controllability for (13.9.1) consists of all initial states in !?An that can be steered to the origin in finite time using continuous controls. Theorem 13.9.1 1f(13.9.1) has . rank [ PA, dPA, ... , dn-1PA] = d ~ n,

(13.9.2)

then C contains ad-dimensional submanifold of !?An that contains the origin. Proof.

The approach taken is to consider the equation

x=

- f(x, u)

(13.9.3)

and to study the collection of states that can be attained by steering from the origin as the initial state. This set is contained in C. By an appropriate nonsingular linear change of state coordinates, (13.9.3)

314

13.

Applications in Control Theory

is transformed into the system

+ !?IIU + d 12 x 2 + X2 = d 22X2 + h2(Xl , X2, u) in which dimtx.) = d, dim(x2) = n - d, and = d

Xl

11 X l

rank [!?II' d

11 fJI1> ••• ,

h l ( X l , X 2 , U),

(13.9.4) (13.9.5)

dft 1!?II J = d.

(13.9.6)

Now introduce the d-parameter family of control functions u(~,

=

t)

~

1

UI(t)

+

+ '" -:t-

~2U2(t)

~duit).

(13.9.7)

For I~I restricted to small values there exists the corresponding unique solution of (13.9.4), (13.9.5) defined on 0 :s; t :s; 1:

for which '1t(u(~, ·),0) (13.9.5) with respect to

°

ot (Ortl) ~ =

(13.9.8)

'12(U( ~, .), t)

(13.9.9)

= rtl(U(~,

X2

=

= 0, 112(U(~, ~

'),t),

Xl

·),0)

it follows that

(art ~ 1) + d (art2) ~

d

11

+

!?II [Ul (r),

e

+ o~.

= O. By differentiation of (13.9.4), 12

hl(rtl,

U2(t), ... , uaCt)J

rt2, u(~,

t))

(13.9.10)

(13.9.11)

(These are the variational equations discussed in Chapter 11.) In particular, at ~ = 0, (13.9.12)

(13.9.13)

In view of (13.9.6) the

U I (r), U2(t),

... , ud(t) can be chosen so that (13.9.14)

315

13.9 The Domain C of Null Controllability

By the implicit function theorem, it follows that (13.9.8) at

t

= 1, (13.9.15)

can be solved for ~ = ~(xd for all Xl near the origin with tion of ~(Xl) at t = 1 into (13.9.9) shows that

~(O)

=

O. Substitu(13.9.16)

which defines the required d-dimensional submanifold of attainable states. Generally, the manifold discussed in the proof of Theorem 13.9.1 is a proper subset of C. Furthermore, the manifold is not always unique since the control function selected to satisfy (13.9.14) is not unique. This remark is further illuminated by Exercise 2 at the end of this section. Remark 13.9.1

Remark 13.9.2 Any equation satisfying the hypothesis of Theorem 13.8.1 or Theorem 13.8.4 clearly has domain of null controllability C = ~n.

Controllability criteria for the nonautonomous equation that arises when d, Pi in (13.5.1) are allowed to be time-varying can be derived in terms of the fundamental matrix of the uncontrolled system (see, for example, [13]). Likewise, the coefficients d, Pi in (13.8.1) can be taken to be functions of t satisfying such criteria, and then analogous results to those of Theorems 13.8.1 and 13.8.4can be derived by using similar methods. Remark 13.9.3

Chapter 13 was written with the intention of giving some indication of the directions that control theory has imparted to the classical theory of differential equations. The material selected lies very much at the foundation of the subject and the treatment avoids elaboration. For example, it is often assumed that not all the state information is available for generating the feedback control functions. This gives rise to the notion of observability, which was not covered. To keep this introductory glimpse of the subject simple, less emphasis was placed on nonautonomous equations than is found in texts devoted solely to control theory. In situations in which the mathematical model describes the dynamics of the system to be controlled to a high degree of accuracy, various notions of optimality can be defined to determine values of free parameters in the controllers. Sometimes an optimization problem is formulated and solved only to see what level of improvement is available in the design of an existing control system. The topic of optimization has received a great deal of attention in the literature, particularly by the mathematically inclined, to whom the notion of optimization is an old and familiar one in mathematics. For coverage of this topic the reader is referred to [13] and [14] as a starter.

316

13. Applications in Control Theory

EXERCISES

1. Verify the claims made in Remark 13.8.4. 2. For the system compute explicitly the solutions (13.9.8), (13.9.9) and apply them to examine the manifold (13.9.16). By varying the choice of the control function u('), show that this family sweeps out C = f7t2. 3. For the system

show that the manifold determined by (13.9.16) is X2 = txi, which is precisely C. 4. An initial state Xo of(13.5.1) is called returnable if there exists a continuous control function steering x(O) = Xo to x(T) = Xo for some T > O. Find necessary and sufficient conditions on .91, f!4 for the collection of returnable points to be a linear subspace of f7tn. What is its dimension? 5. Show that there exists a periodic and continuous forcing function J(t) for which the nonlinear scalar differential equation

x + [sin(x)] x + e-

x2

=

J(t)

has a periodic solution x(t) that satisfies the boundary conditions x(O) = 0,

x(O) = 1,

x(l) = 1,

x(l) = O.

References

I. 2.

3. 4. 5. 6. 7. 8. 9. 10. I I. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

Boraiko, A. A. (1980). The pesticide dilemma. Natl. Geogr. 157, 145-183. Breiman, L. (1969). "Probability and Stochastic Processes." Houghton, Boston, Massachusetts. Brunovsky, P. (1970). A classification of linear controllable systems. Kybernetika (Prague) 173-188. Coddington, E. A., and Levinson, N. (1955). "Theory ofOrdinary Differential Equations." McGraw-Hill, New York. Conte, S. D., and de Boor, C. (1980). "Elementary Numerical Analysis." McGraw-Hill, New York. Dawkins, R. (1976). "The Selfish Gene." Oxford Univ. Press, London and New York. Hahn, W. (1967). "Stability of Motion." Springer-Verlag, Berlin and New York. Hale, J. K. (1980). "Ordinary Differential Equations." Krieger, Huntington, New York. Hartman, P. (1964). "Ordinary Differential Equations." Wiley, New York. Heymann, M. (1968). On pole assignment in multi-input controllable linear systems. IEEE Trans. Autom. Control, AC-l3, 748-749. Hoffmann, K., and Kunze, R. (1971). "Linear Algebra." Prentice-Hall, Englewood Cliffs, New Jersey. Kolman, B. (1976). "Introduction to Linear Algebra." Macmillan, New York. Lee, E. B., and Markus, L. (1967). "Foundations of Optimal Control Theory." Wiley, New York. Lefschetz, S. (1977). "Differential Equations: Geometric Theory." Dover, New York. Lukes, D. L. (1972). Global controllability of nonlinear systems. SIAM J. Control, 10, 112-125; errata 186. 11, (1973). Marsden, J. E. (1973). "Basic Complex Analysis." Freeman, San Francisco, California. Maxwell, J. C. (1868). On governors. Proc. Roy. Soc. 16,270-283. Ortega, J., and Poole, W. (1981). "An Introduction to Numerical Methods for Differential Equations." Pitman, Marshfield, Massachusetts. Phillips, E. R. (1971). "An Introduction to Analysis and Integration Theory." Intext Educational, Scranton, Pennsylvania. Russell, D. L. (1978). Controllability and stability theory for linear partial differential equations: recent progress and open questions. SIAM ReI'. 20,639-739. Russell, D. L. (1979). "Mathematics of Finite-Dimensional Control Systems." Dekker, New York. 317

318

References

22. Schechter, M. (1971). "Principles of Functional Analysis." Academic Press, New York. 23. Schwartz, A. J. (1963). A generalization of a Poincare-Bendixson theorem to closed two dimensional manifolds. Amer. J. Math. 85, 453-458; errata (1963), 85, 753. 24. Smale, S. (1967). Differentiable dynamical systems. Bull. Amer. Math. Soc. 73,747-817. 25. Warren, M., and Eckberg, A. (1975). On the dimensions of controllability subspaces. SIAM J. Controll3, 434-445. 26. Watson, L. (1979). "Lifetide: The Biology of the Unconscious." Simon & Schuster, New York. 27. Whyburn, G. T. (1964). "Topological Analysis." Princeton Univ. Press, Princeton, New Jersey. 28. Wonham, W. M., and Morse, A. S. (1972). Feedback invariants of linear multivariable Systems. Automatica-J. lFAC 8, 93-100.

Index

A

function, 174 history, 3 variable, 15 Controllability, 16,269,271,279 nonlinear systems, 307. 312 null,313 Controllability matrix, See Matrix subspace, 283 Controllable part, 274 Controller, decoupling, 283, 285 feedback, 275, 279 recoupling, 285 regulator, 17 universal, 296, 305 Cycle, 287, 299

ABC algorithm, 47, 48, 78 in computation of matrices, 266 and correspondence rule, 48, 52, 85 formula, 59, 84, 135, 261 matrix equation, 54 Absolute convergence, 69 Adjoint equation, 114 Airy equation, 127 Alekseev formula, 222 Amplitude response, 264 Analytic matrix function, 151

B Boundary, 192

c

Caratheodory, 182 Cauchy criterion, 68, 69 Cauchy integral formula, 148 Cauchy-Peano existence, 184 Cauchy problem, 12,47, 190,205 Cauchy sequence, 68 Cayley-Hamilton, 40 Characteristic roots, exponents, 165 Closed cycle, 287, 29&, 300 Closed loop poles, 275 Closed loop system, 275 Closed set, 201 Column space, 27 Commutative equation, see Differential equation Commuting antiderivative, 139 Continuous map, 201 Control differential equation, 261 feedback, 15

D Decoupling algorithm, 286, 288 Dependence, on parameters, 200 analytic, 219 continuous, 205 differential, 212 Differential equation analytic, 219, 246 asymptotically autonomous, 239 autonomous, 14, 218, 223 commutative linear, 139 constant-coefficient linear, 9, 47,64,92 control, 261 diffusion-reaction, 240 dual, 112, 131 general solution, 200, 205 history, 1 linearization, 244 matrix, 76, 109, 114, 153 nth order, 130

319

320

Index

periodic, 93, 162, 172, 307 periodic commutative, 175 sources, 4 time- varying linear, 8, 11, 106 variational, 212, 215, 216

E EISPACK,31 Electrical circuits, 7, 11 Embedding, 53 Epidemic model, 133 Estimates, solution norms Cetaev, 95, 101, 115 commutative equations, 160 constant-coefficient equations, 12 exponential function, 93, 95 variable coefficient equations, 115 Euler equation, 131, 138 Existence, local, 182, 184, 186 Exponential decay, 93, 160, 168 Extension of solution, 191

F Feedback, 15 Field,I9 Floquet, 163 Flow and mixing, 5 Fundamental matrix, 49, 59, 75, 77, 108, 128, 129, 130, 131, 140, 163, 165, 166, 167, 176

G Gain,264 Gauss-Jordan elimination, 30, 31, 33 Generalized eigenvectors, 54, 55 Gershgorin's theorem, 105 Gronwall's inequality, 74, 111, 112

H Hausdorff metric, 226, 227 Hausdorff space, 211, 212 Homeomorphism, 210, 244 Homogeneous system, 25, 27 Hyperbolic rest point, 243

I Index (winding number), 254 Infmitesimal transition scheme, 134 Initial-value problem, 47 Insertion of feedback, 290, 291, 298

J Jacobian matrices, 212 Jordan curve, 251

L Laplace transform operator, 264 Laplacian, 240 Levinson-Smith equation, 9 Lienard equations, 9, 195 Limit set, 224, 236 Linear algebraic systems, 25 Linear functional, 112 Linear independence, 51, 79, 81 Linear transformation, 81 Linearization, 244 UNPACK,31 Lipschitz continuity, 185 Locally integrable, see Matrix Lyapunov approach, 95, 228 asymptotic stability, 168, 232, 234 derivative, 229 energy function, 196 equation, 99 function, 241 partial stability, 236 stability, 92, 232 uniform stability, 232

M Manifold, 200, 220 Markov process, 132 Matrix 19 algebra, 20 characteristic polynomial, 39,45, 89, 93, 276 classical adjoint, 150 computing, 31 congruence, 98 controllability, 269, 271 convergence, 68 determinant, 35, 36, 37, 38, 46 diagonalization, 44, 99 differential equations, 114, 153 eigenspace, 39, 43, 45 eigenvalue, 39 eigenvector, 39 equation, 49,51,54,59, 82, 84,99, 101, 104, 105 exponential, 69, 75, 77, 78, 84, 85, 86, 91 feedback, 275

Index

321

frequency response, 264 function, 70, 148 hyperbolic, 243 inverse, 29 Jordan form, 43, 44, 45, 64, 77 locally integrable, 107 logarithm, 162, 163 minimal polynomial, 40, 44, 45, 89, 93 norm, 65, 66 positive definite, 92, 102 power series, 70 row-reduced echelon form, 26 similarity, 98 stability, 93, 99, 104, 232, 234 symmetric, 98 trace, 46 transfer function, 265, 266 transpose, 27, 30, 37 triangular factorization, 32 Maximal interval, solution, 191 Maxwell, J. C., 3 Mechanics, 6, 10, 11 Metric space, 211 Method of eigenvectors, 61

N Newton, I., 1,9 Norm equivalent, 72 Euclidean, 72, 99 matrix, see Matrix vector, 65 n-vectors, 23

o Open set, 201 Optimization, 315 Ordered cycle basis, 291, 292, 296

p. Periodic equation, See Differential equation Periodic orbit, 253, 255 Phase shift, 264 Picard-Lindelof theorem, 186 Pole relocation, 277, 281 Poincare-Bendixson theorem, 253, 255 Probability, 132

Q Quadratic form, 98, 99

R Reachable set, 271 Reduction, scalar control, 279, 281, 284 Regulator, 3, 17 Rest point, 224 Restricted column operation, 289 Routh-Hurwicz criterion, 95, 103 Row reduction, 286 Row space, 27

s Semicontinuity, 209 Semigroup property, 224 Semiorbit, 224 Short diagonal, 297 Solution, 47, 107, 182 general, 200, 205 global, 190 local, 182 maximal extension, 190, 191 periodic, 93, 168, 172, 173, 175, 176, 177, 253, 254, 255, 256, 258, 259 steady-state, 240, 262 transient, 262 Stability, 92, see also Lyapunov Stabilizability, 277 Stabilization, 18 Steady-state solution, 240, 262 Submanifold, 315 Subspace, 24 Successive approximations, 188 Systems linear algebraic equations, 25

T Time-varying coefficients, 106, 128 Topological space, 201 basis, 201 subspace, 201 Transfer function, see Matrix Transient solution, 262 Transversal, 251 Traveling wave, 240

u Undetermined coefficients, 52 Uniqueness, 185

v Van der Pol equation, 9, 199, 258

322

Index

Variation of parameters, 50, 52, 74, 114, 173, 222,262 Vector field, 205 Vector space, 81

w Watt, J., 3

z Zorn's lemma, 191