Stability Theorems For Linear Motions: With An Introduction To Liapunov's Direct Method [1 ed.]

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STABILITY THEOREMS FOR LINEAR MOTIONS WITH AN INTRODUCTION TO LIAPUNOV'S DIRECT METHOD

PRENTICE-HALL INTERNATIONAL SERIES IN APPLIED MATHEMATICS

Introduction to ALGOL DEUTSCH, Nonlinear Transformations of Random Processes DRESHER, Games of Strategy: Theory and Applications HAHN, Theory and Application of Liapunov's Direct Method HARTMANIS AND STEARNS, Algebraic Structure Theory of Sequential Machines LEHNIGK, Stability Theorems for Linear Motions with an Introduction to Liapunov's Direct Method MORETTI, Functions of a Complex Variable PIPES, Matrix Methods for Engineering STANTON, Numerical Met/rods for Science and Engineering WACHSPRESS. lterati't'e Solution of Elliptic Systems and Applications to the Neutron Diffusion Equations of Reactor Physics BAUMANN, FELICIANO, BAUER, AND SAMELSON,

PRENTICE-HAll., INC. PRENTICE-HALL INTERNATIONAL, INC., UNITED KINGDOM AND EIRE PRENTICE-HALL OF CANADA, L TO., CANADA

STABILITY THEOREMS FOR LINEAR MOTIONS WITH AN INTRODUCTION TO LIAPUNOV'S DIRECT METHOD

SIEGFRIED H. LEHNIGK L/ni!ed Sillies Army .\fissile Command and Unirersily

ol Alahama in Html.'irille

PRENTICE-HALL, INC. E~GLE\VOOD

CLIFFS, N.J.

139176 London AUSTRALIA, PTY. LTD., Sydney CANADA, LTD., Toronto INDIA (PRIVATE), LTD., New De/hi JAPAN, INC., Tokyo

PRENTICE-HALL INTERNATIONAL, INC., PRENTICE-HALL OF PRENTICE-HALL OF PRENTICE-HALL OF PRENTICE-HALL OF

© 1966 by Prentice-Hall, Inc. Englewood Cliffs, N. J.

All rights reserved. No part of this book may be reproduced in any form, by mimeograph or any other means, without permission in writing from the publisher.

Current printing (last digit): 10 9

8

7 6

5 4 3 2

Library of Congress Catalog Card No. 66-14358

Printed in the United States of America 83995-C

PREFACE

This book deals \vith the basic theorems of LIAPUNov's direct method, with the theory of the first approximation (linearization), and with the stability criteria for linear motions. The concept of stability in the sense of LIAPUNOV and fundamental refinements of this concept are defined and illustrated in Chapter I. The core of Chapter 2 is a theorem correlating the various types of asymptotic behavior of the solutions of linear differential equations with constant coefficients and the location of the eigenvalues of the coefficient matrix, i.e., the zeros of the characteristic polynomial, in the complex plane. This theorem is derived from the explicit form of the solutions. Linear equations with constant coefficients are frequently used in many fields. Their solutions are well known. Various theorems on the asymptotic behavior of the solutions are available. However, in setting up mathematical models for dynamic systems, a pendulum, for example, or a missile, we end up with nonlinear equations more frequently than with linear equations. Why, then, do linear equations play such an important role? The answer is given in Section 3 of Chapter 3 by the theory of linearization. Under certain conditions a nonlinear equation with a linear part is equivalent with respect to its asymptotic properties to the equation of the linear part. To prove the theorems of Section 3, we need the main theorems of LIAPUNov's direct method. These theorems, expressed for nonautonon1ous equations, and generalizations for autonomous equations, are proved and illustrated in Section 1 of Chapter 3. The LIAPUNOV functions employed in the proofs of Section 3 are quadratic forms. The existence problem for such LJAPUNOV functions is treated in Section 2. v

vi

PREFACE

Chapters 4, 5, and 6 deal with stability problems of linear autonomous motions. The stability criteria of HERMITE, BILHARZ, HURWITZ, and ROUTH are established in Sections I through 4 of Chapter 4. They state necessary and sufficient conditions that a polynomial be H urwitzian. The starting point is a theorem of HERMITE for which an algebraic proof is given. It is possible to prove the algebraic stability criteria via LIAPUNov's direct method. The algebraic approach has been preferred here since we have to resort to algebraic tools in order to prove theorems on the number of zeros with nonnegative real parts of a polynomial. The stability criteria are not independent of each other. It is shown that they are special formulations of HERMITE's criterion. Section 5 paves the way for Section 6 and Chapter 5. It deals with EucLID's chain of two polynomials and with a theorem of STURM on the number of real zeros of a polynomial in a prescribed interval. A polynomial is not H urwitzian if the conditions of a stability criterion are violated. In general, an investigator is not satisfied with this negative result. In order to stabilize a system with the least effort, he wants to know the distribution of the zeros in the complex plane. The theorems of Section 6 provide methods for counting the numbers of zeros in each half-plane and on the imaginary axis in the regular case (no vanishing elements in the corresponding sequences) and in the singular cases. Geometric interpretations of the algebraic stability criteria for real polynomials are given in Chapter 5. The criterion of NYQUIST can be directly derived from the criterion of LEONHARD, which in turn can be proved via an algebraic criterion of L. CREMER. In Section 1, this theorem of CREMER is proved by means of the RouTH criterion. The interpretation of its conditions in terms of the LEONHARD locus leads then immediately to the LEONHARD stability criterion. Also in Section 1, the generalized LEONHARD criterion is proved which permits the number of zeros of a polynomial in each half-plane to be counted, provided that the locus does not pass through the origin (regular case). The proof is given independently of the corresponding theorems of Section 6 of Chapter 4. In Section 2, the generalized NYQUIST criterion is established by means of the generalized LEONHARD criterion. It contains NYQUIST's original stability criterion as a special case. Particular attention is paid to the NYQUIST criterion as an instrument in control engineering. Also in Section 2, KALMAN's concepts of controllability and observability and PoPov's theorem on the nondegeneracy of a transfer function are discussed. Sections 3 and 4 deal with the root-locus method and with the existence of stability domains for parameter dependent systems. The final chapter, Chapter 6, is concerned with the structural stability problem of AtZERMAN and GANTMACHER, i.e., technically speaking, with the problem of stabilizing a linear control system with unstable elements by a choice of suitable values for its parameters. This problem is certainly

PREFACE

VII

of practical interest in control engineering. However, little attention has been paid to it, probably because the general problem is still unsolved. The know·n theoren1s are presented in Chapter 6 and proved by means of the RouTH and the LEONHARD criteria. To avoid the ··snowball effect," only a relatively sn1all number of references have been given. One group of references is historical in nature; it refers the reader to the original publications in which the stability theorems presented in this book first appeared. The other group refers the reader to subjects w·hich he is very likely to be interested in after he has studied this book. The reader should be familiar \vith the initial value problem for ordinary differential equations, and he should definitely have a thorough knowledge of n1atrix theory. Matrices are used extensively throughout this book. I an1 very n1uch indebted to friends who, with criticism, suggestions, and encouragement, assisted n1e in preparing this book: RuDOLF K. FESTA, HANS J. SPERLING, WOLFGANG HAHN, and ROBERT C. CARSON. 1 wish to express heartfelt thanks to each of them. Also, 1 wish to thank Mr. JosEPH H. NoRDMAN, \vho assisted me in reading the proofs. Last, but not least, I wish to thank Mr. JOHN H. DAVIS, Assistant Vice President of PrenticeHall, for his understanding and patience, and Mr. JAMES E. RICKETSON, Production Editor, for making proofreading bearable. SIEGFRIED H. LEHNIGK

CONTENTS

1

THE CONCEPT OF STABILITY OF MOTIONS

2

LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

16

3

LIAPUNOV'S THEOREMS AND THE THEORY OF THE FIRST APPROXIMATION

25

4

1

25

I The Theorems of LIAPUNOV 2 Existence of Quadratic Forms as LIAPUNOV Functions for Noncritical Linear Equations with Constant Coefficients 3 The Theory of the First Approximation

43 54

THE ALGEBRAIC STABILITY CRITERIA FOR LINEAR MOTIONS

72

I A Theorem of HERMITE 2 The HERMITE Criterion 3 The BILHARZ Criterion 4 The HuRwiTz and the RouTH Criteria 5 Eucuo's Algorithm and a Theorem of STURM 6 The Number of Zeros with Nonnegative Real Parts and Related Problems

viii

72 79 81 92 105

116

CONTENTS

5

6

.IX

THE GEOMETRIC STABILITY CRITERIA FOR LINEAR MOTIONS

152

1 The CREMER and the LEONHARD Criteria

152

2 The NYQUIST Criterion 3 The Root-Locus Method 4 Stability Domains in the Parameter Space

167 194 199

STRUCTURALLY HURWITZIAN POLYNOMIALS

210

BIBLIOGRAPHY

247

INDEX

249

GUIDE TO THEOREMS, DEFINITIONS, TABLES, AND ILLUSTRATIONS

THEOREMS, GENERALIZATIONS OF THEOREMS, COROLLARIES, AND LEMMAS CHAPTER Ill (cont.)

CHAPTER I

2 CHAPTER

12 13

II

1 2 3 4

17 17 20 23

1.1 1.2 Gen. 1.2 1.3 Gen. 1.3 1.4 Gen. 1.4 2.1 ,.,.........,.,

27 28 30 38 40 41 41 44 47 48 51 52

CHAPTER Ill

2.3 Gen. 2.2 2.4

2.5 3.1 3.2 3.3 3.4

CHAPTER IV (cont.)

53 55 61 62 62

Cor. Lem. Lem. Lem.

CHAPI ER IV

1.1 1.2 1.3 2.1 3.1 3.2 4.1 4.2 5.1 Cor. 5.1 5.2 5.3 5.4 5.5 6.1

72 76 76 80 86 89 I0 I 103 105 105 106 II 0 I I1 112 116

6.2 6.3 6.4 6.4 6.1 6.2 6.3 6.5 6.6 6.7 6.8 6.9

118 118 120 125 125 125 126 126 127 I3I 147 150

CHAPTER V

1.1 1.2 1.3 1.4 1.5 2.1 2.2 2.3

X

152 154 154 157 158 168 170 181

CHAPTER V (cont.)

2.4 2.5 2.6 2.7 4.1

181 183 190 191 200

CHAPTER VI

Lem. Lem. Lem. Lem.

I 2 3 4 1 2 3 4 5 6 7 8 9

10

212 213 213 215 215 218 219 220 221 224 225 228 236 245

THEOREMS, DEFINITIONS, TABLES, AND ILLUSTRATIONS

xi

DEFINITIONS CHAPTER I

CHAPTER Ill (cont.)

CHAPTER ll

2 3 4

3 4 4 8

5

11

6

13

1 ..,

-

16 20

CHAPTER Ill

1.1 1.2

1.3 1.4 1.5 3.1

33 38 43

CHAPTER IV

1.1 CHAPTER V

55

2.1 2.2

CHAPTER IV (cont.)

CHAPTER VI

27 28

72 190 193

TABLES CHAPTER IV (cont.)

CHAPTER IV

3.1 3.2 3.3 3.4

83 84 85 87

3.5 4.1 4.2 4.3

90 99 101 104

5.1 5.2 5.3 5.4

107 107 108 115

I

2

229 237

ILLUSTRATIONS CHAPTER Ill (cont.)

CHAPTER I

I

5

2

II

1.1

35

CHAPTER III

1.2 1.3 3.1 3.2

36 37 68 70

CHAPTER V (cont.)

CHAPTER V

1.1 1.2 1.3 2.1 2.2

158 159 161 170 171

2.3 3.1 4.1 4.2 4.3

178 198 204 205 207

REMARKS ON NOTATION

1. Cross References (i) If a chapter is subdivided into sections: 111.1.2 means Item 2 in Section 1 of Chapter 3. This item can be a theorem, a definition, a displayed expression, a table, or an illustration. (ii) If a chapter is not subdivided into sections: 1.1 means Item 1 in Chapter 1. (iii) Within a chapter we omit the chapter nu1nber: 3.1 in Chapter 3 means Item 1 in Section 3 of Chapter 3. 2. Complex numbers If a is a complex quantity, then a is its conjugate complex. The imaginary unit is i = ~. 3. Vectors Lower-case German letters denote column vectors. The superscript T denotes the transpose, 1 =II z1, .•. , Zn 11r. 111 = (jT t1) 1,.z is the absolute value (Euclidean length) of 3. 4. Matrices Capital German letters denote matrices. If W is an m x n matrix, (m rows, n columns) with elements a IlL'' we write ~t = II all" II'";.~~ t· If m = n: ~{ = II allv II ~.v=t or simply 2! = II a 1Lv 11. The superscripts T, I, and A denote the transpose, the inverse, and the adjoint, respectively. U is the unit matrix. There are four matrices not denoted by capital German letters, namely, V and V', in IV. I, ff., and .6. 1 and .6.2 in IV. 4. 5. Domains Jf'*(~; T): ~~~ < ~' T < f < 00 %(X; T): I~ I < X, T < t < co

X*(~):

I~ I < ~

f(X):

I~I 0. The bar denotes the conjugate complex.

Remark.

The continuity of t(J;, t) is essential for the investigations carried out in this book. We shall deal with consequences of the continuity condition in connection with Definition 3 and after Definition 6 (Remark 4 ).

All following considerations will therefore be based on differential equations of the type ~ =

t = f (~, 1) llx,, · · ·, Xnllr,

(f

E £)

f = 11/,, · · ·,./:zllr

(2)

In this notation we include, of course, the special case in which the righthand side of (2) does not explicitly depend on the independent variable t,

(f

E £)

(2*)

Although the right-hand sides of (2) and (2*) are different, we are going to use the same symbol f since there is no chance of confusion. (2) and (2*) bear different names. An equation of the form (2) in which f does explicitly depend on t is called nonautonomous (nonstationary), whereas an equation of the form (2*) in which f does not explicitly depend on t is called autonon1ous (stationary). In the theory of stability it is customary to talk about motions instead of solutions of an equation. Since we do not want to diverge from this usage, we introduce DEFINITION I. Let ~ = ~(1, ~ 0 , 10 ) be the solution of t = f(~, t) corresponding to arbitrarily given initial values ~o E Jt'~, 10 E (T, oo ). The set of the n + 1 variables {~, t} with t 0 < t < t" is called a motion oft = f(~, t ).

4

CHAP.

THE CONCEPT OF STABILITY OF MOTIONS

I

Remark. In the real case a motion may be interpreted as a space curve (integral curve, half-trajectory) in the (xh ... Xm t) space, the motion space. The solution t(l, ot, t 0 ) itself may then be interpreted as a space curve, the phase curre (halfcharacteriistic), in the (x 1, • •• , xn) space, the phase space. In this sense the phase curve is the projection of the motion onto the phase space, and ~ = t(l, ~o' to) is a parameter representation of the phase curve with the parameter t. Example. The motion of

determined by the initial values 10 = 0, x 1(0, ~o, 0) = x,o is {x" X:h t}, 0 < t < (X), with

x 1 = exp( -t)

+ exp 2t,

=

x 2 = -exp( -t)

2, x2(0, !o, 0) =

X2o

=

I,

+ 2 exp 2t

The motion {o, 1} oft = f(~, t }, (2), corresponding to the lril'ial solution ~ o is called the equilibrium of (2). Every other motion {~, t} of (2) corresponding to an initial point ~o 7=- o is called a perturbed motion of (2). Equation (2) is called the equation of the perturbed motions. DEFINITION

2.

We are now in position to define the subject matter of our investigations: We shall be concerned with the behavior for t > t 0 of the perturbed motions of autonomous or nonautonomous equations t = f(~, t ), f E E, which initiate at t = 10 from initial points !:o sufficiently close to o. According to the character of this behavior, lve shall attribute certain properties to the equilibriulns of these equations.

The equilibrium oft = f(~, t) is said to be stable if there exists a 10 E (T, oo) such that for every € > 0 there exists a o = o(€, t 0 ) > 0 •vith the property that for every ~o with l~ol < o the inequality l~ol < o implies 1~(1, lo, to)l < € in [t 0 , oo). DEFINITION

3.

' Remarks. 1. Definition 3 in1plies the existence of all solutions with l.ro I < in [t 0 , oo ). 2. Evidently, 8 < € since t(t 0 , !: 0 , t 0 ) = J-: 0 • 3. To be precise, we should say that the equilibrium is stable at the right of t 0 if all conditions of Definition 3 are satisfied. According to the restriction t > t 0 , we may omit the phrase "at the right of 10 ." However, if motions have to be considered also at the left-hand side of t 0 , the distinction between ··right" and ••teft" is essential.

o

The following question arises: If at some t 0 E (T, oo) the requirements of the stability definition are satisfied, can there exist a t ~ =F 10 , 1~ E ( T, oo ), at vlhich they cannot be satisfied? The answer is negative if f (~, 1) E E in ~-*(~; T). (It suffices to consider the real case.) Let a> 0 and "'> 0 be numbers such that T 0 in [1, co), we have to restrict x 0 to lxol < o = € inf exp t(t - 1) = € l€ [I,O 0, a number = e which satisfy the requirements of the definition of stability. Let us look more closely into the situation determined by Eq. (3). We shall show that the number o cannot be chosen independently of t 0 , i.e., o = o(e, to). For lx(t, x 0 , t 0 )1 to be less than every € > 0 in [t 0 , oo) with arbitrary to, we have to restrict x 0 to lxol < o = € inf exp {1 0 (1 - t 0 ) - t(l - t )} t

E

[lo, oo)

The exponential function at the right-hand side takes on a minimum at t = ~· Therefore, . { 1 if t 0 > ~ 1nf exp {10(1- to)- t(l- t)} = [ ( ')2} 1 1·f to< .J. lE [lo,oo) exp -to-~ < ~ Consequently, for lxol

lxol we have the < o(e,

lo)

=

{

€ €

bound if t 0 > exp [ - (t 0

!

1)2} < e

~

-

.f 1

to < 2•

2. We consider the equation

~~ = a

1

-1

a

(4)

(Capital German letters denote matrices.) The right-hand side of (4) belongs to the set E everywhere. The components of the solution for given initial values ~o = lix.o, X2ollr and to are x 1 (t, !"0 ,

t0)

x 2 (t, 60 , t 0 )

= [x 10 cos (t =

- t0 )

[x 10 sin (t -

10 )

-

x 20 sin (t - / 0 )} exp a (I - 10 )

+

X:w

cos (t - t 0 )} exp a(t - t 0 )

If a < 0, for every t 0 and every € > 0, the number o = € has the property that l!ol < = € implies l!(t, !:o, to)l = (xio + x~0)'i 2 cxp a(t - to) = l!ol exp a(t - t 0 ) < e in [t 0 , oo ). Consequently, the equilibrium is stable if a < 0.

o

3. ~ = 0 The right-hand side is in E everywhere. The solution for arbitrary initial values !o and / 0 is ~(t, ~ 0 , t 0 ) !'o; every solution of ~ = o is identically constant. The equilibrium is stable. For every t 0 and every € > 0 the number o = € has the property that l!:ol < o implies l~{t, ~o, / 0 )1 < e in [t 0 , oo). The initial point !o of an identically constant solution of an equation t = f(~, t) is called a singularity of the equation. If there is a neighborhood of a singularity !:o which contains no other singularity than 60 , then !o is called an isolated singularity. Observe that the definition of stability does not require the point ~ 0 = o to be an isolated singularity. For the equation t = o every point ~ 0 is a singularity; the point x = o is not an isolated singularity.

=

CHAP.

I

4. X=

THE CONCEPT OF STABILITY OF MOTIONS

{X- 2 0

if 2 0 be given. Then lx(t,X 0 ,

~)1=~2 lx0 lt- 2 +cost tov (v = 1, 2, ... ), we have g(lov) _ 2v + 2 g(tv) - (2v + 1)37r2

O -+

as

"

-+

oo

As in Example 1, the number o depends not only on € but also on 10 • However, although the equilibrium of x = xg(t)/g(t) is stable, it has a specific property not encountered in the equilibriums of the equations considered in Examples 1 through 4.

8

CHAP.

THE CONCEPT OF STABILITY OF MOTIONS

I

The length of the interval ( -o, o) for initial points Xo for which lxolg(t)./g(to) < € in [t 0, oo) decreases as the values of t 0 increase. This type of stability is called nonuniform stability. Suppose that the equilibrium of t = f(r, t) is stable; i.e., suppose that there exists a number t 0 E (T, oo) and, for every € > 0, a number o = o(f, to) such that l~(t, ~ 0 , 10 )1 < f in [1 0 , co) for every ~:o with lrol < o. The equilibrium is said to be uniformly stable if there exists a number o' < 0 such that 1~(1, ro, to,. )l < f in [lot. , 00) for every loa• > In and for every ro with Irol < o'. In Examples 2, 3, and 4, the numbers do not depend on 10 • The equilibriums are uniformly stable; we may take o' = o. In Example I, o does depend on I 0 • But this fact does not constitute nonuniformity of the stability of the equilibrium. For, if 10 is an arbitrary initial point, then o' = € exp {-(to - !} 2} is of such a nature that lxol < o' implies

o

lx(t, X0 , lo~. )l

=

lxol exp {t(l

- 1) - foa-·0 - lo1J)}


10 , i.e., the equilibrium is uniformly stable. There is no such o' in the case of nonuniform stability. We are not going to pursue this refinement of the stability concept any further. We refer the interested reader to the monograph of HAHN [II, Chap. 4, Sec. 17].

The equilibriun1 oft = f(~, t) is said to be quasi-asymptotically stable if there exists a 10 E (T, oo) and a o0 = o0 (t 0 ) > 0 such that l~ol < Do in1plies lim l~(t, ~o, fo)l = 0 as t ~ oo. DEFINITION

4.

Ren1arks. I. Note that, as in Definition 3, the existence of the solutions with lrol < o0 in [t 0 , oo) is implied. 2. Definition 4 implies that the point ~: = o is an isolated singularity of

t

= i(~, 1).

3. By means of arguments similar to those used in the case of stability, one can show that the requirements of Definition 4 are satisfied for every t~ =I= 10 , t~ E (,-,co), if they are satisfied for 10 • We refer again to the examples considered in connection with the definition of the concept of stability. The equilibriurn of (3) is quasi-asymptotically stable. For t 0 = 2, for example, and Oo < oo, lxol < On implies lin1 lx(t, Xn, 2)1 = 0 as 1 -+ oo. The equilibrium of (4) is also quasi-asymptotically stable if a < 0. For every to and every oo < CXJ, l~ol < o0 implies the limit relation of Definition 4. But the equilibrium of (4) is not quasi-asymptotically stable if a = 0 (although it is stable if a = 0). If a =· 0, the absolute value of ~:(t, ~: 0 , t 0 ) is a constant ,.e (t ' ~.o, t• I o) I I - (X 210 - : X 202 )1 ·'2 - I,. ·!- o ' lhe equilibrium oft = o is not quasi-asymptotically stable. I

From Example 2 for a = 0 and from Exan1ples 3 and 4 we conclude that stability does not imply quasi-asymptotic stability. Example l and

CHAP.

I

THE CONCEPT OF STABILITY OF MOTIONS

9

Example 2 for a < 0 might lead to the conjecture that quasi-asymptotic stability implies stability. This conjecture turns out to be correct only in special cases, but not in general. A special case (represented by Example I) in which quasi-asymptotic stability implies stability is that of a scalar differential equation x = f(x, t) (f E £). let to and Oo :-: ooUo) be numbers such that lx(t, X 0 , t 0 )j --• 0 as t ----J> oo ·whenever lxo I . - ._: Oo, and let .Xo > 0 be an initial point with .x 0 < 0 • Then x(t, .Xu, lo) - -~ +0 and x(t, -.Xo, 10 ) ----J> -0 as t- • oo. Let e > 0 be given. Then there exists a number tf > t 0 such that

o

-e




0, cp 0

E

0, q; 0 , t0)

(-oo, +co), t 0 Y

( t,

Y0,

cp 0, t 0)

E

=

=0 (0, oo), it has the solution h(t, q;0 )

Y0 h(l

o' e2" > ... > ed"" > 1 be the (nonvanishing) SEGRE numbers of the characteristic matrix sU - ~ for S", i.e., the exponents of the elementary divisors of sU - ~ for S". The number d" is the defect of S"U - ~. The numbers eP" and 1t" are connected by the relation (~e=1,

... ,k)

The SEGRE number e 1" shall be called the highest matrix ~ can now be written as ~

The ~K are

1rK

x

1t"

SEGRE

number of S". The

= diag II 3,, ... , ~k II

(6)

matrices of the form ~" = diag

II ~ '/" ... , ~d " II

(7)

" and the ~PIC are ePK x eP" matrices which can be written as ~PIC = SICU

+ (ff'PIC

(p = 1, ... , d";

IC

=

1, ... , k)

U is the ePK x eP" unit matrix, and, if ePIC ..> 1, tfeP" is that eP" X ePIC matrix the rows of which are (i) for tL = 1, ... , ePIC -

I:

II 0,

... , 0, I, 0, ... , 0

------- ------II o, ... , o II f.l

epiC-~.L-l

II (8)

If eP" = I, ~. reduces to the element 0, and hence ~PIC reduces to the scalar quantity SIC.

CHAP.

II

LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

The matrices SJl and

etc

piC

19

are commutative. Hence

exp (t - to)~PIC = (exp (t - t 0 )SJ1) exp (t - t 0 ){.tc =--c

(

jJK

exp (t - t 0 )SK) exp ( t - to)CfePK

Here exp (t - t 0 )~ t'piC

=

U

-+- ~ ~

l·'=i

(t - ftoYrc,. ~t' V •

PIC

From (8) it follows that the rows of~~VIC are ( i) f o r tL

=

1, . . . ,

II 0,

v :

ePIC -

. . . , 0, 1, 0, . . . , 0

II

~~

11 +I•

+A.

(ii) for J1 ~-·-.!PIC- v and that Cf~pK

=

0 if v >

f'piC-11- L•-l

II o, ... , o II

(A.= I, ... , v):

epiC· Therefore

and exp (t-

to)~PK =

(exp (t- t,)S,)( U

+ ':~.· (I-;; ('>"(f:,,.)

(9)

The relations (6), (7), and (9) give us the desired insight into the structure of exp (t - t 0 )~. It is now easy to see that erery component y 14 (t, t) 0 , t 0 ) (/l = 1, ... , n) of the general solution

t)(t,

t) 0 , t 0 )

= (exp (t -

t 0 )~)t) 0

(10)

of (4) is a linear con1bination of a finite nu1nber of ter~ns of the form

( 11)

YaoPa exp (t - to)SK

in which Yao is a component of t) 0 , p,r is a polynon1ial in t or a constant, and S" is one o.f the distinct eigenvalues of~- Since the matrix 9R in (5) is constant, the components of the general solution of (3) are also of the form (II). We turn now to the stability properties of the equilibrium of a linear homogeneous equation with constant coefficients, (t real) The right-hand side of such an equation is in E everywhere, i.e., in ~*(+(X); -(X)). Let sL. = uv + iwp (v = 1, ... , n) be the (not necessarily distinct) eigenvalues of the complex n x n matrix iJ(, i.e., the zeros of the characteristic polynomial n

f(s) =

~ GvSn-v = G0

det (sU -

~()

1'=0

of~(.

The determinant of~( is n

det ~( = (-t)n an/Go

=IT Sv V-l

(12)

20

LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

CHAP.

If~( is real, we associate with f(s) the n x n HURWITZ matrix .~) II H 11 ,. II ~·''=' (cf. TV. 4), the general element of which is H#J~·

= a 2 ,,_1!

(aa = 0

if a


n)

II

=

(13)

The reader is urged here (and on various other similar occasions) to write out this matrix for some specific value of n to get a better feeling for its actual form. As will be shown in IV. 6, the next to the last main principal minor Hn- 1 = det II Hll,.11 ~.~ . ·=, of·\) is of the form Hn-l

'' = ( -J)n(n-l)/2a~l-l II

(sll

+ s,.)

(14)

J.l,l'= l

(formula of ORLANDO [23], cf. (IV. 6.4)). H,_ 1 is also called the next to the last HuRWITZ determinant off(s). From ( 13) it follows that Hp.rl =:: 0 if 11 :::; n - 1 and H,w = an. Hence det ~

=

or, since, by (12), a11 = (-1)na 0 det

H"

Hn

=

a11Hn-1

~(,

= ( -l)naoHn- 1

det 9(

(15)

DEFINITION 2. 1. The complex matrix ~~ is said to be critical if a,. < 0 (v = I, ... , n) and if at least one a,. = 0. 2. The complex 1t1atrix 9( is said to be noncritical either ({a,. < 0 (v = I, ... , n) or if at least one av > 0. Relation (15) in connection with (12) and (14) shows that the realntatrix 91 is critical if a < 0 (v = 1, ... , n) and Hn = 0. 1,

Remark. The words ..critical'' and .. noncritical" have been chosen in view of the theory of the first approximation (cf. 111.3).

For equations i = ~{~ we prove now a theorem which forms the basis for the considerations of several of the following chapters. THEOREM 3. Let SIC = ~IC + ill" (K = I, ... , k < n) be the distinct eigenvalues of9L 1. Let 9( be noncritical. 1(a) The equilibrium of i = 9{~ is asymptotically stable if ~" < 0 (K = I, ... , k). (By Theoretn 1.2 and by the Remark added to that theorem. the equilibriwn is even asyn1ptotical/y stable in the whole.) 1(b) The equilibriun1 is unstable ({ there is at least one eigenvalue with positive real part. 1(c) The equilibriu1n is contpletely unstable if all ~IC > 0 (K = 1, ... , k). 2. Let 9( be critical. Let S,, ... , Sl (I< k) be those distinct eigenvalues

CHAP.

II

LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

21

with ~., = 0 (A. = I, ... , 1), and let e 11 , • • • , ell be the highest corresponding SEGRE nurnbers. 2(a) The equilibriun1 of~ = ~{6 is stable, but not asyn1ptotica/ly stable, if e •• = . · . = e.t = I. 2(b) The equilibriunz is unstable, but not con1pletely unstable, ({ there is at least one SEGRE nu1nber e~.\ > I.

Proo_f

We shall prove the statements of this theorem by investigating the properties of the solutions of the equation t) = ~t) in which ~ is the JORDAN normal form of~( and by carrying over the results to the original equation ~ = ~(~. To do this we use the relation det 9J1 =f.:. 0

( 16)

(which follows from (5)) and a relation between the initial points !:o and t) 0 • Let 9J1* = ~1Jlr9J1 =II n1!1· 11. Then

I~0 I = or, if \Ve set

/3

=

max

(ijf9Jl*tJo) 1 '


ln 1t~titf!:o exp{t- to)(~.' +ill,)

Since at least one of the mu 1 (J-L = I, ... , n) is different from 0 (911 is nonsingular), it follows that I~(t, !: 0 , 10 ) I -> oo as t _,. oo. For every given > 0

o

22

CHAP.

LINEAR DIFFERENTIAL EQUATIONS WITH CONSTANT COEFFICIENTS

II

there are initial points ~o with Ito I < o such that this limit relation holds. We only have to take for to the unique solution of 9JFro = t}o for given t}o with y 10 -::1=- 0 and, according to (17), with It} 0 I < (n/3) _,,.. 2 o. Consequently, according to Remark 3 to Definition 1.6, the equilibrium of t = ~(~ is unstable. However, the equilibrium is not completely unstable if there is at )east one eigenvalue with nonpositive real part. Without loss of generality we may assume that S 2 is one of the eigenvalues with nonpositive real parts and that the JoRDAN blocks of~ are arranged as in (6). Then the component y.,..+ 1(t, t} 0 , t 0 ) of t)(t, t) 0 , t 0 ) contains the term (18)

*

We set now y,.• +,.o 0 and Y1-1o = 0 for 11 =I=- 1r 1 + 1. Then all components of t}(t, t)o, to) are identically equal to 0 except y.,..+ 1(t, t) 0 , t 0 ), which is equal to (18). Hence, by ( 16), if we denote the (1r 1 + I )th column vector of 9R by m'~'•+'' ~(t, ~o, to)= nt~r.+tY~r.+t,o exp (t- / 0 )(~ 2 + i!l 2 ) and, consequently, since

~2

< 0,

l~(t,~o,lo)l < ltn~r.+II·IY'~'•+t,ol

Now let e if

> 0 be given. Then I~(t,

~ 0 , 10 )

in[to, oo)

I < e in [t 0 , oo) for every

10

certainly

or, if we use ( 17), if I~o I < e(n/3)'· 2/ Int 0 (< 0) in f(X). The function cp(~) 0 in ~(X) may be interpreted as positive or negative semi-

=

definite in .X'"( X). A function cp(~) is said to be positive (negative) definite in %(X) if cp(o) = 0 and if cp(~) > 0 ( < 0) for every~ =1=- o in %(X). A function (~, t) is said to be positive (negative) se1nidejinite in a half0 fort> Tand ifO ( T and if there exists a continuous function cp(r) which is positive definite in .Yt"(X) such that (~, t) > q>(~) (< -q>(!:)) in X'"(X; T). A not identically vanishing positive (negative) semidefinite function which is not positive (negative) definite shall be called strictly positive

=

(negative) semidefinite.

If cp(~) is a (semi) definite quadratic form, we simply call its coefficient matrix (semi) definite. The derivative of v(~, t) with respect to the independent variable t in terms of a scalar vector product is given by v. ( ~' t ) =

ov

~, u.\ 1

ov ... , uX ~' 11

ov ~

ut

. Ill r · II x. 1 ~ • • • ,x,,

Substituting into this expression for x" (fl = I, ... , n) the corresponding component of the right-hand side of ( 1.1 ), we obtain

.

v(r, t) =

ov

ov ·11/,(7;, t), ... ,f~~(~, t), 111r. ut

(tv ~' · · ·' ~, ~

uX 1

uX,1

(1.2)

SEC.

II I. 1

LIAPUNov·s

THEOREMS AND THEORY OF FIRST APPROXIMATION

27

1.1. The expression ( 1.2) is called the total derivative of v(!:, t ) for Eq. ( I. I ). If we have to consider simultaneously more than one differential equation, we shall \\'rite the number of the equation for \vhich N~:, t) is formed as a subscript toN!:, t)~ e.g., l\ 11 ,(~:.t). Occasionally, \vhen no confusion is possible, we shall omit the arguments~~ and t. In the following. unless othenvise indicated, ~(1. ~~h / 0 ) shall always stand for a nontrivial solution of ( 1.1 ). DEFINITION

A. Liapunol' 's Theorems on Stability

1.1. The equilibriun1 of the equation t. = f(~, t) is stable ~f there exists a function r(!:, t) lrhich is posith·e definite in._~( X~ T) and whose total derivatire for the given equation is negatire sen1ide[lnite in _x-·( X; T) (X< ~. T > T). THEOREM

Remarks. 1. This and the fo11owing theorems of this section ren1ain valid if .. positive" is replaced by "'negative" and vice versa. 2. We made a c1ear distinction between the numbers ~ and T on the one hand and X and Ton the other. This is the situation: (i) We consider a real function l(L I) which has certain properties in a domain .Jt"*(~; T), i.e., in a semicylindrical domain 1~1 < t 1 > T (open with respect to ~ and 1). (ii) We consider a real function v(~, 1) which has certain properties in a domain f(X; T), i.e., in a semicylindrical domain 1~1 < X, 1 ~> T (dosed with respect to ~ and 1). The inequalities X < ~ and T > T introduced in Theorem 1.1 guarantee that {(L 1) and r(L t) have the required properties in the common domain f(X; T) c f*(~; T).

3. In Theorem I .1 and all other theorems of this section, T may be replaced by every number greater than -co if the given differential equation is autonomous, i. = fOJ, and if we employ functions r which are functions of [ only, v ~ v(O. In other words, Tis immaterial in the autonomous case.

Let v(7-:, t) be positive definite in .f( X; T). Then there exists a real continuous function (p(~) which is positive definite in the spherical domain $"(X), such that the inequality Proof of Theorem 1.1.

v(~, t)

holds. Furthermore, let we have the inequality

i'(l.l)

> qJ(J;)

in f(X; T)

be negative semidefinite an ff(X; T). Then

1\ .. n(~, 1) < 0 · in Let

€

( 1.3)

( 1.4)

Jf'(X; T)

be a positive number not greater than X, 0
0

as t --• (X)

v(~,

t)

( 1.8)

We shaJI show that a = 0 as a consequence of the decrescentness of v(r, t ). Suppose a were positive. Under this assumption the inequality V(6(1, }; 0 , t 0 ), f) >

a in [1 0 ,

CXJ)

(which follows from (1.8) and from the fact that v(~(t, !:o· to), t) is decreasing) would imply the existence of a positive number ry < e such that 1!:(1, !: 0 , lo)l > 'Y

in [1 0 , (X))

For, if !:(1, 60 , 10 ) would approach o, the relation lim v(r(t, !:o, lu), t) == 0 as 1 --+ oo would follow as a consequence of the decrescentness of 'V(!:, t ). But this would contradict the assumption that a in ( 1.8) is positive. Now let {3 = inf 1fr(r) > 0 on the annular domain 0 < ry < l!:l < e. Since {r(t, !: 0 , 111 ), 1 J remains in the subdomain 0 < 'Y ~ 161 < e, I > lu, of ..Jf.'(X; T), it follows from (1.7) that 1\J.I)(!:(f, 60 , 10 ), t) < -{3

in [to, (X))

Integration of this relation leads to the inequality 'V(6(1, !: 0 , 10 ), 1)

=

V(!:(to, !:o• lo), to) -l-

Jt ;\~.n(};(u, 1-:o, to), u)da lo

< V(~(tO, ~IH fo), 1o) - (I - 1u)f3

(f3

>

0)

30

LIAPUNOV'S THEOREMS AND THEORY OF FIRST APPROXIMATION

Ill. 1

SEC.

This inequality implies that v(~(t, ~ 0 , t 0 ), t) < 0 for sufficiently large values of t in contradiction to the assumption that v(~, t) is positive definite in S(X; T), i.e., in contradiction to relation (1.8). Consequently, the assumption of a positive limit value a for v(~(l, ~ 0 , 10 ), t) as t---+- oo cannot be correct; a must be equal to 0. But, if Jim v(~(t, ~o, 10 ), I) = 0 as t --+ oo, then lim

v(l.l)

(~(t,

!: 0 , t 0 ), t)

=

0

as

I

---+-

oo

and, because of ( l. 7), lim 'o/(~{1, ~ 0 , t 0 ))

=

0

as t

-~ oo

This limit relation implies lim !:(1, [ 0 , t 0 )

=

o

as I---+-

oo

since 1/J'(~) is positive definite. Therefore, there exist numbers to E (T. oo) and which meet the requirements of Definition 1.4, and, since !: (t, !:o, to) stands in this proof for an arbitrary solution of ( 1.1) with I!:o I < the equilibrium of ( 1.1) is quasi-asymptotically stable. Since it is also stable, it is asymptotically stable.

o

o,

For autonomous differential equations we have the following 1.2. The equilibrium oft = f(~) is asymptotically stable {{there exists a function v(~) lvhich is positive definite in S(X) (X < ') and whose total derivative v (!:) for the given equation is negative semidefinite in f(X) and of such a nature that v (~(t, !: 0 , 10 )) :fo 0 for every nontrivial solution !:(1, ~ 0 , t 0 ) oft = f 0 the existence of a positive = o(e) < € such that l!:ol < implies l!:(t, !:o, _to) I < e in [to, oo ). Furthermore, if !:o =t= o is an arbitrary, but from now on fixed, initial point with l!:ol < o < e < X, the assumptions about v and i'; imply that v(!:(t, !:o, to)) is decreasing and that

o

o

lim V(!:(l, !: 0 , 10 )) = a> 0

as t

-·)>

oo

(1.9)

which means that ( 1.1 0) Suppose a were positive. Then, since v is continuous (basic property 2 of v) and since v(o) = 0, there would exist a positive number ry such that

0 < "/ < l!:(t, !:o, t 0 )l
0 there exists a positive number N = N(7J 1) such that

(PONTRYAGIN

(1.12)

If L?' E .9', then, according to (1.11 ), 0 < ry ~ lr.~l < e. Consequently, for every 1-:.P E f!iJ and for every / 0 there exists exactly one solution ~(t, [_.p, / 0 ) oft = f 10 such that ( 1.13)

We use now the continuous dependence of the solutions of t = f(~) on their initial values. For every 7] 2 > 0 there exists a number 1]1 > 0 such that (1.14)

if I[? - ~(tL., ~ 0 , t 0 ) I < 7J 1 ; in other words, if we observe ( 1.12), inequality (1.14) holds if v is sufficiently large. Furthermore, r.,(t, r.,(t,., ~ 0 , to), to)= !:(1 + 1,.- / 0 , 1;, 0 , 10 ) in [1 0 , oo). Hence, by (1.13) and (1.14),

v(,;(t* _:_

tl'- lo,

1-:o, lo))

fo

in contradiction to ( 1.10). Consequently, the assumption that a in (1.9) is positive cannot be correct. We have a = 0. But then v(~(t, ~o, to))~ 0 as 1 _, oo in1plies r..(t, 7;, 0 , 10 ) ~ o as t ~ oo. Since 1;, 0 with 11-:ol < is arbitrary, the equilibrium is asymptotically stable.

o

As an elementary example for the application of Theorems 1.1 and 1.2, we consider the scalar equation ( 1.15) i = (l - 2t)x Its right-hand side belongs to the set E everywhere. The function v(l l(x, 1) = tx 2 and its total derivative for (1.15) satisfy the conditions of Theorem 1.1. First of all, v( 1 J has the three properties mentioned at the beginning of this secti·.Jn (with !~ arbitrarily large, Tarbitrarily sma11). Secondly, vc 11 is positive definite for arbitrarily large X, and, e.g., for T = I, i.e., in Jf"(X; 1), since tx 2 > x 2 in X"(X; 1). The total derivative of v< 1 J for ( 1.15) is vH~I5) =

-x 2 (4t 2

-

21 - I)

32

LIAPUNOV'S THEOREMS AND THEORY OF FIRST APPROXIMATION

SEC.

III. 1

It is negative semidefinite in .Jf"(X; 1) (X < oo ). Hence, v( 1 > and vU~•s> satisfy the conditions of Theorem 1.1. The equilibrium of (1.15) is stable. The function v< 1'(x, t) = tx 2 cannot be employed to make a statement in the sense of Theoren1 1.2; tx 2 is not decrescent. By Definition 1.2, a function v(~, t) is decrescent if and only if ]v(~:, t)] < f for every~ with 1~1 < p, where p is a function off only, and not oft. For 7} = tx 2 and t > I, however, we find tx 2 < f if

lxl < (

f)

12 /

=

p(E, t)

We choose next the function v< 2 )(x, t) = x 2

This function (which has the three basic properties mentioned earlier) and its total derivative for (1.15) satisfy the conditions of Theorem 1.2. The function v< 2 > is positive definite everywhere, i.e., in particular, in .Jf"(X; I), X < oo. It is also decrescent; x2 < f if ,xl < £ 1 2 = p(€). The total derivative vU?,a> is negative definite in f(X; 1) since

v(i:, 5)

= -2x2(2t- I)< -2x 2

in f(X; I)

All conditions of Theorem 1.2 are satisfied; the equilibrium of Eq. ( 1.15) is asynlptotically stable.

Next, let us consider an example for the application of the Generalization of Theoretn 1.2. Let the equation t = ~-l}; with ~( -

be given. ~(~:

E

0

1

-a:?

-a,

a,

·-.

0,

.-·

E everywhere. The function v([)

==

a:? > 0

[r~~

with

'

and its total derivative for the given equation satisfy the requirements of the Generalization of Theoren1 1.2. 1.• is positive definite everywhere, and v.~ 1 !

=

~:r(9F'~~

+ ~Wl)~ =

-2af.;d

is strictly negative semidefinite everywhere. Now, v~lt vanishes identically if and only if x'!.(t, ru, 10 ) -- 0. But, if x 2(t, ~: 0 , 10 ) :-: 0 with ~o o, then, as the given differential equation shows, x 1(1, ~: 0 , 10 ) = 0 which can be only if ~n =-= o. Thus, the equilibrium of the given equation is asymptotically stable (in the whole).

.*

If the function v(~, t) satisfies the conditions of Theoretn 1.2 for an Eq. ( 1.1) in a dotnain .%(X; T), we can be sure that all phase curves of the equation, initiating from points of a domain f"*(c) with sufficiently small 8, satisfy the. limit relation lim]~{t,ro, fo)] = 0

as t __.

00

(]rol 0 is

JxJ o

x

(

1' Xo, 10

)

=

x0

+ (310

-

3x0 1 x 0 )exp

3(1 2 -

d)/2

A sketch of the corresponding integral curves for 1 ~ I is shown in Figure 1.2. From the explicit form of the solutions it follows that, as 1 ~ co,

.

l1m x(t, x 0 • to)

=

{0 oo

if X 0 < 31 0 'f I X 0 = 3to

36

LJAPUNOV'S THEOREMS AND THEORY OF FIRST APPROXIMATION

SEC.

I II. 1

)(

-2~-4---.,r--+~------~--------~

I

I I

I

Fig. 1.2

lim x(t, x 0 , 10 ) = co

as

I~ 11

if x 0 > 31 0

'

with the finite escape time - (

11 -

~

lo

2 I

+J

,.

"'0

og Xo- 3to

)1,:!

----..

_.- fo

The domain of attraction of the equilibrium is !1 to: depends on 10 •

-

CX)

< x 0 < 31 0 • It

Before we turn to LIAPUNov's instability theorems, we introduce a geometric interpretation of Theorems 1.1 and 1.2 for equations t = H!:, t) for which there exist functions v in the sense of these theorems which are functions of! only, v = v(~:). This interpretation will show that the restriction to spherical domains S(X) was adopted in the proofs of Theorems 1.1 and 1.2 only for the sake of a convenient notation. Let K 1(~) c 1 = constant define a bounded closed surface in the phase space which contains the origin in its interior :31. (Figure 1.3 illustrates the case for two dimensions.) Let K(~:) = c = constant define another closed surface in the phase space which contains the origin in its interior .!2. and which is a subdomain of.?!. Let v(~:) be single valued, continuous, and posi:_c:

SEC.

II I. l

LIAPUNOV'S THEOREMS AND THEORY OF FIRST APPROXIMATION

37

V =(/JK v=V

x, K=c

7 f2

Fig. 1.3

tive definite on .!! . Furthermore, let cp ,..- = inf v(J.:) on K(~J = c. Since v(J.:) is positive definite, CfJn: is positive, and, as a consequence of the compactness of K(J.:) ::-:-: c, the function r(J.:) takes on the value (PI\ some\vhere on K(J.:) = c. We shov.. now that the surfaces v(~J = V = constant are closed and bounded ifO < V< CfJ~t·· Let x 1(p), ... , x~~(p) (0 < p< I) be the parameter representation of a continuous path connecting the origin of the phase space (x 1(0) = · · · = Xn(O) = 0) with an arbitrary point of K(~:) == c (K(x 1(1), ... , x (1)) = c) without intersecting the surface K(J.:) = c. We consider v(!:) along this path. We have v(x 1(0), ... , x ~~ T if there are points !:o -=/=- o in every neighborhood of o such that v(6 0 , t) < 0 for all t > T. DEFINITION

1.3. The equilibrium of the equation t = f T THEOREM

SEC.

III. I

LIAPUNOV'S THEOREMS AND "fHEORY OF FIRST APPROXIMATION

39

and whose total deril'ative for the given equation is negative definite in .%(X; T) (X

T).

Proof. We have to show that for every t 0 E (T, oo) there exist numbers e > 0 and i = i(l o, ~o, e) > 0 such that for every o :=s::: e there is at least one solution !;(t, ro, to) of (I. I) with l~ol < o for which l~(to+i,~o,to)l=e

(1.17)

Suppose the equilibrium of ( 1.1) were stable. Then there would exist a to E [T, oo) such that for every e > 0 (which we assume to be not greater than X) there exists a positive number < e with the property that the inequality lrol < implies l~(t, ~ 0 , t 0 )l < e in [t 0 , oo). Now assume that a function v(,;, 1) with properties required by the theorem exists. Let ~(I, ~ 0 , t 0 ) be a solution whose phase curve remains in the domain ff*(e), and let its initial point ~0 (0 < lrol < o) be so chosen that

o

o

v(~o, to)
T. Furthermore, there exists a function 1fr(r) which is continuous and positive definite in .X"(X) such that i'(l.l)

< -t(r)

(1.19)

in f(X; T)

Consequently, since {~(1, ~ 0 , t 0 ), t J remains in ff(e; T) from ( 1.18) and ( 1.19) that

,-.~(X;

T), it follows

( 1.20) In analogy to the corresponding part of the proof of Theore1n 1.2, the decrescentness of v(~, 1) and inequality (I .20) imply the existence of a positive number ry such that l~(t, ~o,

lo)l > "/

in [t 0 , oo)

Let f3 =-= inf 'o/(!:) on the annular domain 0 < ry < 1~1 < e. Since {r(l, ro, t 0 ), t J remains in the subdomain 0 < 'Y < 1~1 < e, t > to, of f(X; T), it follows from ( 1.19) that 1)

0)

from which we conclude that v(1;(t, ~ 0 , 10 ), t) decreases unboundedly as t increases, although lr(t, !:o, 10 )1 < e in [to, (X)). But this contradicts the assumption that v(~, t) is decrescent. A decrescent function is bounded for all values of t if its argument r is bounded. Consequently, the assumption

40

LIAPUNOV'S THEOREMS AND THEORY OF FIRST APPROXIMATlON

SEC.

Ill. l

that the equilibrium of (1.1) is stable cannot be correct. The phase curve corresponding to the solution ~(t, ~ 0 , 10 ) with l~ol < and ~o so that (1.18) holds cannot remain in f*(e) for all t > 10 • Since < e, there must exist a number i = i(t 0 , ~ 0 , e) > 0 such that ( 1.17) is satisfied.

o o

If ( 1.1) is autonomous, we have the following 1.3. The equilibrium oft = f(~) is unstable if there exists a function v(~) lvith v(o) = 0 which has a domain v < 0 and whose total derivative b,(~) for the given equation is negative sen1ide.finite in f(X) (X< ~)and of such a nature that Vr(~(t, ~ 0 , 10 )) =F 0 for every nontrivial solution ~(t, ~ 0 , to) oft = f(~) with l~ol < X. GENERALIZATION OF THEOREM

Remark. This theorem is a special case of a more general theorem of KRASOVSKII [16, Chap. 3, Sec. 15] for equations t. = p(~, 1) with p(~, 1 + f-)) = p(~, 1).

Proof Suppose the equilibrium were stable, i.e., that for given E > 0 there would exist a positive number = < e such that l~ol < implies l!:(t, ~ 0 , 10 )1 < e in [t 0 , oo ). Let 0 < e < X and Jet ~o =1=- o be an arbitrary, but from now on fixed, initial point with l~ol < and v(~(t 0 , ~ 0 , t 0 )) < 0. There are such points ~ 0 in every neighborhood of o which satisfy the last condition since v has a domain v < 0. Then l~(t, ~ 0 , 10 )1 < e < X in [1 0 , oo) and, as a consequence of the assumptions about i: ,

o o(e)

o

o

v(~(t, ~o, to))< v(~(t 0 , ~ 0 , t 0 ))

Since v is continuous, and since v(o) of a. positive number ry such that 0

< "/
... > ed"" > 1, eh: > 1 (~e

k

ePK = 7CK,

p~1

L

1r/C =

n,

«=1

diC = defect of S"U - ~(, d" < n, we define @5 by setting those elements of® equal to 1 whose subscripts are 2, e 11 +I ; 2, ir 1 +e 12 + I;

3,e 11 +e:n+1; ... ; 3, if 1 + e 12 + e22 + 1 ; . . . ;

d1,e11+ ... +ed,-1.1+1 d 2, ir 1 +e 12 + · · · +ect 2 -1.2+ I

. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ... 1, n-ifk+ 1; 2, n-ifk+e1k+ I; 3, n-ifk+eJt+e2k+ 1; ... ; dt, n-ifk+etk+ · · · +ed~-1.k+ 1

and by setting all other elements of 8 equal to 0. We obtain the same result as in the special case k = 1. We can now consider the general case as a combination of (i) and (ii). The result is that, if 9l =1= aU, there exists a singular matrix ~ such that ttl( I) ¥= o unless to = o, i.e., there exists a strictly positive semidefinite matrix (£such that !r(t, to, 1 0 )(£~(t, ~ 0 , 10 ) =F 0 for every nontrivial solution ~(t, ~ 0 , t 0 )

oft

=

~(~.

Remark. We observe that for given ~-l =t= all not every singular matrix the property that m(t) = ffi(exp ~lt)r 0 =I= o for every ro =t= o. Let

~~ =

I I

I I '

9Jl =

I -1

1I I 1'

~

-

mhas

0 0 0 2

Then 9Jl(exp .~t)lJ

= Jly. + Y2 exp 2t, -y. + Y2 exp 2tllr

With ~u

_ m

Jl -

_

i.I\J -

1 1 1 1

we obtain lt(t) = 2y 2(exp 21)111, lllr, and lV(t) = o for Y1 ro = Y.lll' -Ill T n. However, if we take I 0

*

m= m2 =

it follows that lU(t) = I!Yt and only if ~ = ro = 0.

+ Y2 exp 2t, our,

* 0, Y2 =

0, t.e., for

0 0 and this vector vanishes identically if

If the equilibrium of t = ~~~ with ~( =1= aU is asymptotically stable and (f ffi is a singular matrix such that t-u(t) = ffi(exp ~(t )~ 0 ¥= o for every ~ 0 =1= o, then there exists a positive definite solution GENERALIZATION OF THEOREM

matrix~ of~lr~

+ ~~( =

-ffi 7'ffi.

2.2.

52

LIAPUNOV'S THEOREMS AND THEORY OF FIRST APPROXIMATION

SEC.

III. 2

Proof I. If the equilibrium oft = 91~ is asymptotically stable, then a unique real symmetriC SOlUtiOn ~ Of 2( T~ + ~9( = eXiStS. 2. We prove the positive definiteness of t1 indirectly by assuming that the quadratic form v(~) = {'~~~ is not positive definite and by constructing contradictions. Suppose that v has a domain v < 0. Since v(~) = 0 if ~ = o, i.e., since v is decrescent, v with v~1!(~) = ~r(9tr~~ + ~9()~ = -~rmrm~ would be a LIAPUNOV function for t = 9(~ in the sense of the Generalization of Theorem 1.3. The equilibrium of t = 91~ would be unstable in contradiction to our assumption. Suppose next that ~ is strictly positive semidefinite. Then in every neighborhood of o there would exist initial points !:o -=F o such that v(r(to, ~ 0 , t 0 )) = v(r 0 ) = 0. For a corresponding solution r(t, ro, to) we have, by the assumptions about tu(t) and since - ffF"ffi is strictly negative semidefinite, the following relations

mrm

1) }I! (

r ( t, r o, to)) ¥=

o,

l) ~ -r (

Hence, for sufficiently large values oft v(r(t, ro, lo)) =

ft

to

i,lls

>

r ( t, r o, to))
10 in contradiction to the assumption that v is strictly positive sen1idefinite. Consequently, v must take on negative values in every neighborhood of o, and, since v is independent oft, it has a domain v < 0 for all t. We consider now the case in which\>( is noncritical, but in which Hn = 0. In this case ~( has at least one eigenvalue with positive real part, i.e., the equilibrium of the corresponding differential equation is unstable. As we know from Theorem 2.1, Eq. (2.3) has no unique solution ~ for arbitrarily given (£ if H, = 0. The following theorem will help us bypass this difficulty. 2.5. Let the equilibriunz of the equation t, = ~~~ be unstable, and let H, = 0. Then, given a real synunetric positive definite, but otherwise arbitrary, quadratic form !:r(.f!:, there exists uniquely a quadratic fonn v(~) = !:r~!: whose total derivative for the gil•en equation is 1\ 2_1) = A.v - ~rli!:. v(~) is a LIAPUNOY function for t = ~(~ in the sense of Theorem 1.4. THEOREM 1

Proof We consider the characteristic polynomial det (sU - ~(*) of the matrix

i>t*

=

9l- ~u 2

s:

(~e = 1, ... , k < n) denote the disin which A. is a parameter. If S" and tinct eigenvalues of~( and ~(*, respectively, then

s: =SIC-

~

We choose now for the parameter A. a positive number such that the determinant of~(* and the next to the last HuRWITZ determinant H~-~ of the characteristic polynomial of ~( * are different from 0. Such a choice of i\ is always possible since \>( has at least one eigenvalue with positive real part and since det \>( and H n-l are continuous functions of the eigenvalues of~(. Then the equation

t

=

~~*~

= ( ~( -

~ u)~

(2.8)

satisfies the assumptions of Theorem 2.4. Consequently, there exists for every given (£ with the properties mentioned in Theorem 2.4 a quadratic form v(~) = !:r~3~ whose matrix ~3 is uniquely determined by the equation

54

LIAPUNOV'S THEOREMS AND THEORY OF FIRST APPROXIMATION

~~*T~

+ ~2(* ==

SEC.

Ill. 2

(2.9)

-~

such that v is a LIAPUNOV function for (2.8) in the sense of Theorem 1.3. This means, in particular, that v(~) = ~rm~ has a domain v < 0 (for all 1 ). Furthermore, v is bounded in every domain .Jr(X) since it is continuous. This implies that vis bounded in .Jt'"(X; T) since it is independent oft. Hence, v(~) = ~r~~ satisfies condition I of Theorem 1.4. We consider now the total derivative of v(~) = ~r~~ for the original Eq. (2.1 ), 1\2.1) =

~T(2(T~

+ ~2{)~

+ A.U/2. Then + 2t*r'B +- ~21*)~ =

We replace here 2l by 2(*

vc2.1)

=

~T(A.m

A.~T~~

+

~r

=

A.v(~) -

~r~~

Furthermore, we denote -~r(£~ by w(~, 1). Then, since A. > 0 and since w(~, t) = - ~rQ:~ is negative definite everywhere, v< 2. 1> is of the form required in condition 2 of Theorem 1.4. Hence, v(~) = ~r~~ is a LIAPUNOV function for (2.1) in the sense of Theorem 1.4. From the way of proving Theorem 2.4, it follows that, if ~( has at least one eigenvalue with positive real part, there always exists a quadratic form which is a LIAPUNOV function for (2.1) in the sense of Theorem 1.4, regardless of whether or not the inequality Hn =1=- 0 is satisfied. Remark.

'

3. THE THEORY OF THE FIRST APPROXIMATION We are going to investigate the stability property of the equilibrium of a real differential equation

t

= f(~,t)

(fEE in .Jt'"*(~;7))

which either is of or can be brought into the form

t=

2(~

+ g(~, t)

(3.1)

with a real constant n x n matrix 21 ::/=. 0 and with a not identically vanishing, "strictly" nonlinear term g(~, 1 ). Under a strictly nonlinear function g(~, 1) we understand a function which neither is of nor can be expanded into the form g(~, 1) = 2('~ + g'(~, t) with 21' 0.

*

The condition imposed on O(~:, t) in (3.1) of being strictly nonlinear rules out addition and subtraction of a linear term. For exan1ple, the equation x = x 2 can be brought into the form (3.1) by adding and subtracting ax, a =1=- 0: i = ax + (x 2 - ax). But here g(x, t) = x 2 - ax is not strictly nonlinear. Remark.

SEC.

Ill. 3

LIAPUNOV'S THEOREMS AND THEORY OF FIRST APPROXIMATION

55

Since f E E in%*(~; T), we have g(~, t) E E in ..*-"*(~; T), i.e., in particular, g(o, t) o.

=

3.1. The linear part t = ~~~ of t = ~(~ + g([, t) is called the equation of the first approxitnation of t = ~~~ + g(~, t ). The equation t = ~~~ + g(~, t) is called the complete equation. DEFINITION

The linear part 9(~ of Eq. (3.1) is uniquely determined under the condition that g(~, t) be strictly nonlinear. The concept of the equation of the first approximation has been introduced by LIAPUNOV [18]. In his original theory, LIAPUNOV assumed that the components of g([, t) have power series expansions in some neighborhood of o and t = 0 starting with terms of at least second order. In the following we shall present the more general theory of the first approximation based on the condition that g([, t) satisfies the inequality lg(~, t )I


0 and suitable X and T lg(~, t )I