Stability of Linear Systems: Some Aspects of Kinematice Similarity (Mathematics in Science and Engineering, 153)

STABILITY OF LINEAR SYSTEMS: Some Aspects of Kinematic Similarity

This is Volume 153 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.

STABILITY OF LINEAR SYSTEMS: Some Aspects of Kinematic Similarity c.j. HARRIS Department of Electrical and Electronic Engineering The Royal Military College ofScience Shriuenham; Sunndon, England.

and

J.P. MILES Super Proton Synchrotron Division European Organisation For Nuclear Research 1211 Geneve 23, Switzerland.

1980

@

ACADEMIC PRESS A Subsidiary ofHarcourt Brace Jovanovich, Publishers

London

New York

Toronto

Sydney

San Francisco

United States Edition published by ACADEMIC PRESS INC. 111 Fifth Avenue New York, New York 10003

Copyright © 1980 by ACADEMIC PRESS INC. (LONDON) LTO.

AJ/ Rights Reserved No part of this book may be reproduced in any form by photostat, microfilm, or any other means, without written permission from the publishers

British Library Cataloguing in Publication Data Harris, C J Srability of linear systems - (Marhernarics in science and engineering). 1. System analysis 2. Stability I. Title II. Miles, J F III. Series 003 QA402 78-75275 ISBN 0- 12-328250-0

Printed in Great Britain

Preface

In spite of the considerable development in the last two decades of the state space approach to stability theory for linear time invariant systems the corresponding status of time varying and nonlinear systems is comparatively retarded. This apparent lack of maturity in the theory of variable coefficient and nonlinear differential equations can be ascribed to the need to derive the solutions of such systems before the structural properties of stability, controllability and observability can be ascertained. However for line~r time invariant systems such properties can be determined directly (or indirectly through the algebraic approach of Laplace transforms) in terms of the coefficient matrices. It is the prime purpose of this book to identify classes of linear and nonlinear multivariable time varying coefficient differential systems whose stability can be characterised directly from their variable coefficient matrices by a suitable transformation, in much the same manner as linear time invariant systems. A secondary purpose of this book is to collect together and unify recent advances in linear stability theory and to highlight those results which are directly applicable to practical dynamic systems. The book is self-contained and in Chapter One a complete review of mathematical preliminaries and definitions necessary throughout the book is given; the mathematically mature reader may omit this chapter without loss. This chapter covers various elements of functional analysis including linear transformations; matrix measures and their applications in estimating the bounds of solution to linear ordinary differential equations (Coppels inequality); inner product spaces and Fourier series, including Bessels inequality and Parsevals equation; and Cesaro sums and their associated Fejer kernels used in the approximation of real valued functions on bounded intervals. As a prelude to the study of differential equations with almost periodic coefficients, the theory of almost periodic functions as a generalisation of pure periodicity is developed in Chapter Two. Properties such as Fourier series and Parsevals equation are established by analogy to the purely periodic case.

v~

PREFACE

It is shown in an approximation theorem that to any almost periodic function there corresponds a sequence of trigonometrical polynomials which are uniformly convergent to the function. As many dynamical systems have spatially varying coefficients as well as time varying coefficients, the continuity, algebraic properties and Fourier series of almost periodic functions dependent upon a parameter are developed at length for later use in the context of asymptotic Floquet theory in Chapter Six. Since the prime purpose of this book is the stability of linear dynamical systems, an introduction to ordinary linear differential equations and their properties is made in Chapter Three. Questions concerning the existence and uniqueness of solution are resolved via Picards method of successive approximations and the Gronwell-Bellman lemma which establishes bounds on solution. This latter result is important in stability studies since it yields an explicit inequality for the solution to an implicit integral inequality. Floquet theory describes linear ordinary differential equations with periodic coefficients; they occur in many theoretical and practical problems concerned with rotational or vibrational motion. It is shown that there exists a nonsingular periodic transformation of variables which transform linear periodic coefficient differential systems into constant coefficient systems; this form of Liapunov Reducibility or Kinematic Similarity is clearly important in stability studies. The question of structural invariants, such as stability, under Kinematic Similarity are discussed together with the necessary and sufficient conditions for Kinematic Similarity for a variety of coefficient matrices in Chapter Four. Special emphasis is given to systems whose coefficient matrices commute with their integral; for such systems it is shown that the state transition matrix and Liapunov transform are readily computed and that unstable time invariant systems can be stabilised by time varying control laws. Chapter Five is devoted entirely to the establishment of necessary and sufficient conditions for the stability of nonstationary differential equations with particular reference to linear systems with periodic and almost periodic coefficients. The theory of exponential dichotomy illustrates the danger of determining system stability based only on the characteristic values of time dependent coefficients. A more restrictive, but less conservative theory based upon the asymptotic behaviour of characteristic values for the stability of linear nonstationary systems is developed via matrix projection theory. The investigation of Kinematic Similarity is taken up again in Chapter Six in the context of linear differential equations with almost periodic coefficient matrices and those dependent upon a parameter. Analogues with Floquet theory are identified and conditions for Kinematic Similarity are established via the characteristic exponents of the almost periodic coefficient matrices and the characteristic values of the transformed system. By way of example, Chapter Seven contains a collection of

PREFACE practical applications of linear differential systems with var~ able coefficients; these demonstrations include a pendulum with moving support, parametric amplifiers, columns under periodic axial load, electrons in a periodic potential, spacecraft attitude control and a detailed study on the beam stabilisation of a proton beam in an alternating gradient proton synchrotron. This book is the result of a collaborative effort between the authors at the University of Manchester Institute of Science and Technology, Oxford University, European Organisation for Nuclear Research (CERN) and the Royal Military College of Science, and the authors wish to acknowledge their debt to these institutions for their support and the provision of facilities to carry out this work. Finally, personal thanks are given to Miss Lucy Brooks whose excellent typing turned an untidy manuscript into the final version of this book.

July 1980

C. J. Harris J. F. Miles

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CONTENTS

Preface

v

Mathematical Preliminaries

Chapter

1.1 1.2 1.3 1.4 1.5 1.6 1.7 Chapter 2 2.1

2.2 2.3

2.4 2.5 2.6

2.7 Chapter 3 3.1

3.2

3.3

3.4 3.5 3.6

Metric Spaces Normed Metric Spaces Contraction Mappings Linear Operators Linear Transformations and Matrices Inner Product Spaces and Fourier Series Notes References

1

7 13 16 19

26 33

34

Almost Periodic Functions Introduction Definitions and Elementary Properties of Almost Periodic Functions Mean Values of Almost Periodic Functions and their Fourier Series Almost Periodic Functions Depending Uniformly on a Parameter Bochner's Criterion Limiting Cases of Almost Periodic Functions Notes References

35 38

43 58

62 64 66 69

Properties of Ordinary Differential Equations Introduction Existence and Uniqueness of Solution Linear Ordinary Differential Equations Constant Coefficient Differential Equations Periodic Coefficients and Floquet Theory Notes References

70

71 79 85 90 92 93

x

CONTENTS LIST

Chapter 4 4.1 4.2 4.3 4.4

4.5

Chapter 5 5.1 5.2 5.3

5.4 5.5 5.6 5.7 5.8

Chapter 6 6.1 6.2 6.3 6.4

Chapter 7 7.1 7.2

Kinematic Similarity Introduction Liapunov Transformations and Kinematic Similarity Invariants and Canonical Forms Necessary and Sufficient Conditions for Kinematic Similarity Estimates for Characteristic Exponents References

95 96

98 101 104 124

Stability Theory for Nonstationary Systems Local Equilibrium Stability Conditions Asymptotic Stability Matrix Projections and Dichotomies of Linear Systems Asymptotic Characteristic Value Stability Theory Stability in the Large Total Stability and Stability under Disturbances Sufficient Conditions for Stability Notes and Input-Output Stability References

125 129 135 142 153 155 158 160 163

Asymptotic Floquet Theory Introduction The Coppel-Bohr Lemma and Linear Differential Equations with Almost Periodic Coefficients Coppel's Theorem Almost Periodic Matrices Containing a Parameter References

164 167 176 180 193

Linear Systems with Variable Coefficients Introduction and Survey of Applications Beam Stabilisation in an Alternating Gradient Proton Synchrotron References

194 199 206

Appendix

Existence of Solutions to Periodic and Almost Periodic Differential Systems

207

Appendix 2

Dichotomies and Kinematic Similarity

214

Appendix 3

Bibliography

220

Subject Index

233

Chapter I MATHEMATICAL PRELIMINARIES

I. I

Metric Spaces Metric spaces are fundamental in functional analysis since

they perform a function similar to the real line R in ordinary calculus.

A metric space is a set X with a metric defined on it.

The metric associates any pair of elements x,y of X with a distance function d(x,y) which is essentially a generalisation of the distance between two points in a Euclidean plane.

The metric

space is defined axiomatically by: Definition I. I: Metric space A metric space is a pair (X,d) where X is a set and d a metric on X, that is a function defined on the Cartesian product such that for all

x,y E X

ml.

d(x,y)

m2.

d(x,y)

0

m3.

d(x,y)

d(y,x)

m4.

d(x,y)

we have:

real valued, finite and non-negative

~s

S

X x X

if and only if

d(x,z) + d(z,y),

x

~

y

the triangle inequality

A subspace (Y,d) of (X,d) is obtained if we take a subset Y

C

X

and restrict d to

Y x Y,

that is

d = d/yxY

(which is

known as the induced metric on Y by d).

Example 1 a) On the real line R the metric ~s b) On the Euclidean plane

E2

=

R2 ,

d(x,y)

=

Ix-yl

the Euclidean metric is

STABILITY OF LINEAR SYSTEMS

2

d(x,y) = «a l-S l)2 + (U - S ) 2) ! where x = (a 2 2 l,u 2), y = (Sl,S2)' Alternatively d l (x,y) = lal-Sll + lu 2-S 2 1; this second metric illustrates that a given set X can have various metric spaces simply by choosing different metrices. 2 The generalisation of E to the complex n-Euclidean space or unin, tary space C is the space with the set of all ordered n-tuples of complex numbers ric

x

= [!a l-S l I 2

d(x,y)

Y

(ul, ... ,u),

n + ... + la

=

I

(Sl"" ,S)

with met-

n

I 2 ] 2. n-S n

c) Consider the set X of all real valued continuous functions x(t) on t over the closed interval

= max Ix(t)-y(t)/,

d(x,y)

I

= [a,b]

with metric

the space (X,d) in this case is called

tEl

the function space C[a,b]. d) Sequence spaces £P.

As a set X take all bounded sequences

of complex numbers

- x (a. )

J

00

la)P

L:

+ ... =

j=l

la·I J

the metric d(x,y) by

P < 00

for

= y(S.),

00

la.-S./p]p

j=1

x

=

x(a.) p =

= sup la.-S.

d(x,y)

jEN

J

converges for fixed p

J

the special case of by

J

for fixed p and

then in the metric space

j=1

each element

If we now define

d(x,y) = [ L:

L:

J

> p ? 1. 1

00

y

00

such that

J

J

I,

00,

For

(00)p?1).

the metric on this set X 1S given where

N = {1,2, ... }

and

y = y(S·). J

A particularly important example of the £P space is when 1n which case we have the Hilbert space with metric

p

=

d(x,y)

and the following so-called Cauchy-Schwartz 1ne-

( L:

j=l

quality holds 00

L:

j =1

( L:

la.S./ J J

~

+

~

=

( 1. 1)

k=l

A generalisation of this is possible for that

2

I,

p? 1

if a q 1S such

then we have Holder's inequality

I. MATHEMATICAL PRELIMINARIES 00

j=1

1

00

la.s·1 J J

L

3

00

laklP)p ( L k=1 m=1

(I. 2)

(L

Rather than use products of elements of X, if we use sums in the sequence spaces for (L

j= 1

la. J

+ s.IP)p J

x = x(a.) E £P,

for

p

we have Minkowski's inequality

~

1

1

J

~

(L

k=1

y = y(S.) E £P

(I. 3)

p ~ I.

and

J

ISkIP)P

Metric spaces are a special class of topological spaces which are characterised by open sets in a space X.

Since the important

analysis concepts of continuity of transformations and convergence of sequences can be defined for general spaces in terms of open sets completely independently of a metric. Consider a given metric space (X,d) we now discuss some of its topological properties: Defini tion 1.2 Given a

x

o

r > 0

E X and a real number

then we have the

following sets: B(x ,r)

{x E X: d(x,x ) > r }

is an open ball,

B(x ,r)

{x E X: d(x,x )

r }

is a closed ball,

Sex ,r)

{x E X: d(x,x )

r ]

~s

0 0 0

x

0

0 0

~

0

a sphere;

called the centre and r the radius.

~s

Sex ,r) o

Clearly

B - B.

An open ball of radius E is called an E-neighbourhood of x (E >

0).

o

Defini tion 1.3 A subset Y of X is said to be open if it contains an E-neighbourhood about each of its elements. be cZosed if its complement y ~s

C

A subset Y of X is said to

in X is open, that is

y

C

= X - Y

open. It is not difficult to show that the collection of all open

subsets of X, called J has the following properties: tl.

the null or empty set

8

E

J,

X

E

J,

4

STABILITY OF LINEAR SYSTEMS

U. X.t. n. X.

t2.

E

J,

for

X.

E

J,

E

J,

for

X.

E

J

~

t3.

i.

~

i.

t,

and i finite.

The space (X,J) is called a topological space with the set J a topology for X; clearly a metric space is a topological space. Open sets also play an important role

~n

the concept of con-

tinuous mappings on metric spaces. Definition 1.4: Continuous mappings X = (X,d)

Let f:X

-+

Y

be metric spaces.

said to be continuous at

~s

O(E) > 0

there is a that

Y = (Y,d)

and

such that

d(x,x) < O(E).

x

E

o

X

cl(fx,fx) < E o

A mapping

if for every

for all x such

f is said to be continuous if it

o

E >

~s

con-

tinuous at every point of X. The mapping definition

f:X

-+

is uniformly continuous if

Y

0 = O(E)

~n

the above

is independent of E.

Example 2 f:X

I f the mapping

{a .. } ~J

-+

Y

Y = Ax

such that

is represented by the matrix A = n, m with X = R Y = R real Euclidean

spaces, then m

t

d(y,yo )2

i=1 S

n L: a .. (x , - x oJ J j=1 ~J

.)1 2

m n [a .. 12)( L: Ix. - x .1 2 ) L: ( L: oJ J ~J j=1 i=1 j=1 (

Thus selecting

I

y2 d(x,x )2 o

L: i,j=1

yo

= E

(I. 4)

for any positive E, then by inequality

(1.4) and definition 1.4

the matrix mapping f is uniformly con-

tinuous. Continuity of a mapping

~n

terms of open sets is contained in

the following theorem whose proof utilises the above definition of open sets and continuity: Theorem 1.1 A mapping f of metric space X into a metric space Y

~s

0

1. MATHEMATICAL PRELIMINARIES

5

continuous if and only if the inverse image of any open set of Y is an open set of X. In a similar fashion open sets can be used to define convergence for a sequence in a topology (X,J). Definition 1.5: Convergence of sequences A sequence {x } n

~n

the topological space (X,J) converges to

x E X if and only if for large n, x

n

is in every open set that

contains x. We now consider two more related topological concepts. Y c X,

a metric space, then

x

o

E X

Let

(which mayor may not be

an element of Y) is called an accumulation or limit point of Y if every neighbourhood of Xo contains at least one point y E Y distinct from x. The set Y consisting of the points of Y and o

the accumulation points of Y is called the closure of Y. for the topological space (X,J),

n{c

Y

Y: C closed

~

~n

Y c X,

That

~s

the closure of Y is

(X,J)}.

Y is a closed subset of (X,J) containing Y (in fact the smallest); Y = Y

in addition if

then Y is closed in (X,J).

The concepts

of set closure and closed sets enables us to make the following equivalent statements about the mapping

f:X

+

Y

for (X,J) and

(Y,U) topological spaces; (i) f:(X,J)

+

(Y, U)

continuous.

~s

1

(ii) f- (C) is closed in (X,J) for all closed C in (Y,U), N c X.

(iii) feN) c feN) for all Definition 1.6

A subspace N of a metric space X is said to be dense

N

~n

X if

= X. This means that any

x

E

X can be approximated by some ele-

ment y of N with as small an error as we wish so that for arbitrary E.

d(x,y)

for the approximation it is useful if a countable dense subset can be found.

~

E

All linear normed space have dense subsets, but

6

STABILITY OF LINEAR SYSTEMS

Definition 1.7 A metric space X is separable if it has a countable subset which is dense in X. Obvious examples of separable metric spaces are the real line R, complex plane and the space tP(oo > p ~

I),

however the

tOO

space is not separable since it contains uncountably many sequences each contained within non-intersecting balls. Since metric spaces are special classes of topological spaces the definition of convergence in a metric space can be simplified to: Defini tion 1.8

{X }

~n the metric space n converge if there is a sequence x E X

A sequence

and x

n

-+

X

~s

=

(X,d)

is said to

such that

lim d(x ,x) n n-+oo

x.

Therefore if ~n

X

X

=

(X,d)

o

is a metric space a convergent sequence

bounded and its limit

~s

unique; also if

x

n as

-+

x

and

-+ y n -+ 00 The in X as n -+00 then d(x ,y ) -+ d(x,y) Yn n n convergence of sequences in a metric space is closely connected

with the continuity of a mapping between two metric spaces (X,d) and (Y,d), since the mapping

f:X

-+

Y

is continuous at a point

x

implies that fx -+ fx. We E X if and only if x -+ x o n 0 n 0 note that in ordinary calculus a sequence {x } converges if and n only if it satisfies a Cauchy convergence criterion, similarly for metric spaces we have:

Defini tion 1.9 {x }

A sequence if for every every

n

~

m,n > N.

E

X = (X,d)

is said to be a Cauchy sequence

d(x ,x ) < > for m n '7 Also if every Cauchy sequence in X converges

> 0

there is a N(~)

such that

then the metric space is complete. Whilst every convergent sequence

~n

a metric space is a Cauchy

sequence, not all metric spaces are complete.

This is unfortunate

since a large number of results in the theory of linear operators depend upon the completeness of the corresponding spaces.

1. MATHEMATICAL PRELIMINARIES

Example

7

:3

The real line R and complex plane are examples of complete metric spaces, other important metric spaces that when complete n, n, are R C tOO and £P. A particularly important complete metric space for our purposes

1S

the function space

C[a,b] for

[a,b]

R; in addition the convergence x + x in this metric space n uniform, and so the metric d(x,y) = max Ix(t)-y(t) I tE[a,b] called the uniform metric.

E

1S

Q

Examples of incomplete metric spaces are the rational line composed of all rational numbers and the set of all continuous valued functions with metric b d(x,y)

(f

Ix(t)-y(t) 12d t

J

defined on

[a,b]

E

R.

a We note that in this example the space of continuous valued functions defined on the int~rval

[a,b] has had two metrics de-

fined on it, however only one of the metric spaces is complete.

1.2

Normed Metric Spaces The most important metric spaces are vector spaces with metrics

defined by a norm which generalises the concept of the length of a vector in a three-dimensional space.

A mapping from a normed

space X into a normed space Y is called an

operator; also if Y

is a scalar field then this mapping is called a

functional.

Of

particular importance in the sequel are bounded linear operators and functionals since they are both continuous.

Indeed a linear

operator is continuous if and only if it is bounded. Consider the field K of scalar real or complex numbers: Definition 1.10:

Vector space

A vector space X (or linear space) over a field K is a nonempty set of elements

x,y, ... (vectors)

braic operations

vI.

x + y

v2.

x + (y+z)

y + x

(x+y) + z

which satisfy the alge-

8

STABILITY OF LINEAR SYSTEMS v3.

x + 0

x,

v4.

a(Bx)

(aB)x,

v5.

a (x+y)

o

x + (-x)

where a,B are scalars (a+B)x

ax + By,

ax + Bx.

Example 4 n, n Examples of linear vector spaces are R C the n-Euclidean real and complex spaces, the function space C[a,b], and £2. A linear subspace of a vector space X is a non-empty subset Yc X

such that for all

BY2 E Y.

and all scalars a,B,

aY1 +

Linear subspaces have the property that they all contain

the zero element.

space

Yl'Y2 E Y

A special subspace of X is the

imppopep sub-

Y = X.

A linear combination of vectors

space X is

a1x

+ a

1

any non-empty subset

+ 2x 2 N c X,

+ a x

m m

of a vector for all a. scalars. i.

For

the set of all combinations of vec-

tors N is called the span of N, which is also a subspace of X. The set of vectors

x1, ... ,x

be lineaPly independent if

r,-

ENe X

for

r

2

1

are said to

o 1 = 0. 2 A vector space X

only if i.e.

= a r = O.

0.

dim X

=

said to be of dimension n (and f i n i te) ,

~s

if X contains a linearly independent set of n-

n,

vectors whereas any other set of (n+l)-vectors in X are lineaply

dependent.

If

dim X

=

00

we say that the vector space X is infi-

nite dimensional. Clearly the vector spaces C[a,b] and £2 have dim X = 00 , n n n. If whereas R and C are finite dimensional with dim X dim X

=n

<

00,

then a set of n-linearly independent vectors in

X is called a basis for X and every vector

x E X has a unique

representation as a linear combination of the basis vectors. Clearly every linear vector space has a basis, and that all finite dimensional spaces are separable. To combine the algebraic concepts of linear vector spaces and the geometric concepts of a metric we need normed linear vector

1. MATHEMATICAL PRELIMINARIES

9

spaces or simply normed spaces:Definition 1.11: Normed spaces A norm on a (real or complex) vector space X is a real valued function on X whose value is denoted by /lxII, with the properties: nl.

Ilxll

n2.

II xii

n3.

Ilaxll

n4.

Ilx+yll

0,

:0,

°-

s;

x

= 0,

lal Ilxll, Ilxll + llvll .

I Ilyll - Ilxll I

x,y E X and a any scalar.

for

A norm defines on X a metric

d(x,y) = Ilx-yll

is called the metric induced by the norm. x f+ II xii

X = (X,

11·11)

i

E

and

X)

If the condition n2

does not hold then we call Ilxll a sem~-nopm. that

(x,y

Condition n4 implies

s a continuous mapping of the normed vector space A Banach space is a complete

into the real line R.

normed vector space. We have already shown that the Euclidean n n, spaces R and C spaces ~p (p=l, ... oo) and the function space C[a,b] are complete, in addition their respective metrics all satisfy the conditions of a norm and therefore they are all Banach spaces.

Example 5: The LP spaces We say that the function

f:R

~

R

~s

integrable if and only

if f is integrable over the bounded interval [a,b]. Consider the space LP of all (Riemann) integrable functions f:R ~ R such that fP ~s integrable on some interval [a,b] E R for any f in LP . We define the norm on this space by

II f

[(fCc)' d'P

II p

(1.5 )

a

So if

a E R

and

f,g E L P

then

lities

I , Ig I }

If+gl

s;

2

!f+gI P

<:

2P max{lfIP,lgI P}

max {I f

af E L P

and from the inequa-

STABILITY OF LINEAR SYSTEMS

10

(f+g) £ LP and therefore LP is a linear space. The norm (l.S) on LP satisfies all the conditions of a norm except it follows that

II f II

= 0 if and only if

0 almost everywhere. This P is because a Cauchy sequence in L does not have its limit in this space, so that LP is not complete and cannot therefore be a Banach that

space.

f

=

However it is well known that every metric space has a

completion which is unique up to an isometry. The completion of the LP space is achieved by using Lebesgue integration - a generalisation of Riemann integration.

(For simplicity, readers unfa-

miliar with Lebesgue theory should consider that all functions f:R ~ R

are piecewise continuous.)

We define a new space LP[a,bJ whose elements are equivalence classes f of functions in LP according to the equivalence relation, f

if and only if

g

'V

f = g

almost everywhere.

And define (1 .6)

co

f1g(t)I P dt <

with

o

equivalence class

Where g

co.

~s

any representative of the

£.

The linear space structure on LP[a,bJ is defined in terms of representations from LP such that

~

cf

(uf )

In this manner the norm on LP[a,bJ has the desired property that

11£11=0

if and only if

£=0.

It is not difficult to show that if

f,g E LP

Minkowski's inequality holds and in particular if g

E

Lq[a,bJ

then

fg

E L1

[a,bJ

(p?l) then f

E

LP[a,bJ

and Holder's inequality

11

1. MATHEMATICAL PRELIMINARIES

a

In LP[a,bJ spaces there are at least three kinds of convergence f € LP[a,bJ (we denote LP[a,bJ by L P

of a sequence {f }, with n

n

in the seque 1) • (i) If {f (r j } converges to f (t ) n ges pointwise to f.

(f:R

-+ R)

then {f } convern

(ii) If {f (t)} converges to f(t) for almost all t, then {f } n

n

converges pointwise almost everywhere to f.

The limit function in both these cases mayor may not belong to LP. (iii) I f {f (t)} converges to f in L P , if f € LP and 11 f -f II n n P as n -+ 00, this is called strong convergence or convergence

°

-+

in the mean (of order p). If {f } converges strongly to both f and g then £ = g almost n However in the case of L P spaces, pointwise conver-

everywhere.

gence does not imply strong convergence and strong convergence does not imply pointwise convergence.

For example take the sequ-

er:ce for t in (O,n f

n

-1

)

(t )

otherwise

the sequence converges pointwise to zero, however n

as

n

-+

00

for

p-1

p > J.

so that

II £n II p

n

(p-1/p)

-+

00

n However in the Euclidean R and en spaces

strong convergence and pointwise convergence are equivalent. Finally we note that for the LP[a,bJ space, the monotone convergence theorem shows that this metric space is complete and is therefore a Banach space. Finite dimensional normed spaces are much simpler than infinite dimensional spaces; for example every finite dimensional subspace Y of a normed space X is complete and closed in X.

12

STABILITY OF LINEAR SYSTEMS

Another important consequence of finite dimensional vector spaces X is that all norms on X lead to the same topology for X irrespective of the choice of norm on X.

This leads to:

Definition 1.12: Equivalent norms A norm Ilxll on a vector space X is said to be equivalent to a norm IIxlio on X if there are positive numbers a,S such that for all

x s X

allxll o

<;

Ilxll

<;

sllxll . 0

We note that the same a,S must work for all nite dimensional vector space X any norm Ilxll

x E X. ~s

r

On a fi-

equivalent to

any other norm Ilxll ; this result implies that convergence of a s

sequence in X does not depend upon the choice of norm on that space.

The same conclusion also holds for continuity and bounded-

ness, that is equivalent norms define identical topologies.

How-

ever in applications some norms are preferable since they give sharper results in say the estimation of eigenvalues (characteristic values) of matrix operators. An important topological concept

~s

that of compactness:-

Definition 1.13: Compact spaces A metric space X is said to be compact if every bounded sequence in X has a convergent subsequence. A general property of compact sets is that a compact subset N of a metric space X is closed and bounded (that

i

s

sup Ilx xsX

II

< (0).

In addition for a finite dimensional normed space X any subset N

C

X

is compact if and only if N is closed and bounded (this

not true for infinite dimensional spaces).

~s

An interesting con-

clusion can be made about a normed space X if it has a compact subset

N

=

{x: Ilxll < I},

in which case X is finite dimensional.

In connection with continuous mappings

f:X

+

Y

from a metric

space X to a metric space Y, the image of a compact subset N of X under the transformation f is compact and this mapping achieves a maximum and a minimum at some points of N if

Y

= R.

13

1. MATHEMATICAL PRELIMINARIES

For regular continuous valued functions defined on compact metric spaces an important theorem used in establishing the existence and uniqueness of solution to differential equations is the ArzelaAscoli theorem, but first we need to define equicontinuiuty: Definition 1.14: Equicontinuous sequences A sequence {x } in C[a, b] is equicontinuous if for every there is a for all x

o(~)

n

> !t1-tol > 0

and all

tl,t

such that

~ >

0

Ix (t )-x (t )/ < ~ n 1 n 0

E [a,b].

o n From this definition each x

n

is uniformly continuous on [a,b]

and 0 does not depend upon n. Theorem 1.2: (Arzela-Ascoli) A bounded equicontinuous sequence {x } in the compact metric n

space C[a,b] has a subsequence which converges uniformly in the norm of C[a, b] . Note that the sequence {x (t)} is uniformly bounded, that n

sup sup [x (t)[ n tE[a,b] n 1.3

<

1S

00.

Contraction Mappings The contraction mapping (or Banach fixed point) theorem is a

very important result which is used in establishing the existence and uniqueness of solution to nonlinear differential equations; equally it has played an important role in developing practical stability criterion, such as the circle criterion, for multivariable nonlinear systems.

Definition 1.15: Contraction mapping For a metric space (X,d) the mapping

mapping if there is a real number k d(fx.fy)

~

kd(x,y)

for all

(0

f:X ~

~

X

k < 1)

is a contraction such that

x,y E X.

From definition 1.4 the above implies that the mapping f is uniformly continuous.

The operator f is called a contraction

since the images of any two elements x and yare nearer to each other than x and yare.

STABILITY OF LINEAR SYSTEMS

14

Theorem 1.3: Contraction Mapping Let (X,d) be a complete metric space and let

f:X

~

X be a

mapping such that d(fx,fy)

k d(x,y)

:S

holds for a fixed constant k Then there exists exactly one

X E X o

and for all

such that

fx

o

x,y E X. x

and

o

the sequence {x } (n = 1,2, ... ,00) defined n converges to X as n ~ 00; moreover d(x ,x ) :S fx n o o n

that for any by

k

(O:S k < 1)

X E X

n

(I-k) d(fxo'x o)' The unique point x

o

called a fixed point of the operator f,

~s

since it is fixed for every f applied to the space X.

The last

inequality of theorem 1.3 provides an estimate of the rate of convergence of the given sequence to the fixed point, and is particularly useful in evaluating numerical algorithms that compute the solution to integral and differential equations.

The theorem

holds globally since the contraction condition holds for all x,y in X; also it holds for all Banach spaces if (X,d) is a normed linear vector space. ing that the sequence

The proof of the theorem follows from show{x} (n n

=

is a Cauchy sequence

1,2, ... ,00)

and since X is complete, converges to a limit x

a

~n

X.

The defi-

nition of uniform continuity shows that the mapping f is uniformly continuous, therefore this limit x

o

~s

invariant under f, from

which the contraction condition of the theorem shows that x unique in X.

a

lS

A corollary to theorem 1.3 is:-

Cora llary I. 3 If (X,d) is a complete metric space and if the mapping is such that fr has a contraction for some

r > 0,

f:X

~

X

then the map-

ping f has a fixed point.

Example 6: Iterated solution to linear equations To apply Banach's theorem 1.3 we require a complete metric space.

Take the set X of all ordered n-tuples of real numbers

x = (x 1,x2, ... ,xn),

y = (Yl'Y2""'Yn)'

Z

= (zl,z2,,,,,zn)' etc.

I. MATHEMATICAL PRELIMINARIES

15

On X the norms £00, £1 and £2 define respectively d oo ' d , and d 1

metrics such that the space f:X

the transformation fx

Y

or

~r

r=l

where

A = {a .. }

€

i j

2

Define

X by

-+

+ b

r n

(1.7)

(the set of constant nxn matrices) and

M

e X

(b , ••• , b ) 1 n

b

x

a.

I:

t

is a Banach space.

Ax + b n

y.

X = (X,d)

is a constant vector.

For the d 00 metric and

fw,

z

max ly.-z·1 J

j

max J

I

n L

a.

r=1

Jr

(x -w )

r

r

J

I

n

doo(x,w) max L la. I Jr j r=1 ~

where

k doo (x,w),

n max L j r=l

k

x = Ax + b

la.Jr I·

x (k + 1)

L

r=1

la. j

r

I

n L

r=1

la. I < 1 Jr

for all j then

(0)

,x

(l)

, ... to

Ax (k ) + b,

k = 0,1, ...

n

So if

has a unique solution x which is given by the limit

of the iterated sequence x

for

(1.8)

< 1

( I •9)

and x(o) arbitrary.

for the £00 norm ~s

The contraction condition

a row sum, if instead the £1

norm was used then the contraction condition would be

n

L

j=J

la. F

I

for all r (column sum condition), and similarly if the ,£2 norm

<

STABILITY OF LINEAR SYSTEMS

16

n

n

L

was used the contraction condition would be

L lajkl2 <

1.

j=1 k=1

The iteration algorithm (1.9) can be used to solve the linear equation

Cx = g,

det C '"

° by

setting

C = H-G with

det H '" 0,

so that on rearrangement -1

H Gx

x

+

-1

H g

Clearly if we put

A

(1.10) -1

H G and

-1

b = H g

can be used directly to solve for x.

the algorithm (1.9)

Two iteration schemes based

upon this decomposition idea are Jacobi and Gauss-Seidal iteration. Banach's fixed point theorem has many other applications including existence and uniqueness theorems for ordinary differential equations and for Fredholm and Volterra integral equations. Linear Operators

1.4

Definition 1.16: Linear operator f:X ..... Y

A linear operator

is such that

(i) the domain, D(f) , of f is a vector space X and the range, R(f) , of f lies in a vector space Y over the same field F. (ii) For all

x,z

€

D(f)

f(x+z)

fx + fz

f(ax)

a f (x)

and scalars

a

F

€

( I. 1 1)

A generalisation of the above definition

~s

if

n

x.

i,

€

D(f) ,

for

i = J ,2, ... ,n,

then

f( L a. x.) ~ i. i=l

a. ~

€

F,

and

n L a.fx .. r, ~ i=l

This property of linear operators

~s

useful in matrix analysis

and in differential equations and

~s

called the principle of

superposition.

Example 7 The following are examples of linear operators: (i) The vector space X consisting of all polynomials on the interval [a,b]; then

1. MATHEMATICAL PRELIMINARIES fx(t)

for every

x(t)

17

X

X €

is the linear differential operator and maps X into itself. (ii) A linear operator f from C[a,b] into itself can be defined by

t

f

fx(t)

x(s)ds.

a

(iii) The real (nxr ) matrix A = {a .. } ~J r n f:R ~ R by means of the equation y If the mapping

f :D(f)

~

R(f)

Y =

exists the linear inverse mapping f

-1

fx

=

x

for all

X

E D(f)

f

Ax. is one to one then there

-1

f-

and

defines a linear operator

:R(f)

1fy

~

Y

D(f).

Clearly

for all

Y E R(f) .

The inverse of a linear operator exists if and only if the null space of the operator consists only of the zero vector, that is fx

= 0 "* x =

o.

If we now consider linear

vector spaces we have:

~ormed

Definition 1.17: Bounded linear operators Let X and Y be linear normed spaces and D(f) E X.

linear operator, where

bounded if there [l f x l]

<;

c Ilxll

~

Y be a

The operator f is said to be

a real number

~s

f:D(f)

c > 0

for all

such that

xED(f).

Clearly there is a smallest c such that the above inequality ~s

satisfied and we call this the induced norm of the operator f.

Corresponding to the vector norm ated with the linear operator f,

11£11.

min(c)

sup xED(f)

~

Ilxll,

the induced norm associ-

~s

} {~ ll x l]

(I. 12)

xlO or equivalently

IIf II . ~

sup [l f x ]] XED (f) Ilxll=1

(1.13)

the above definitions imply that II fxll

<;

II f II. Ilx II ~

( 1 • 14)

STABILITY OF LINEAR SYSTEMS

18

Clearly

II f. II

ping f.

We note that the induced norm satisfies conditions (nl-

~

can be interpreted as the maximum gain of the map-

n4) of definition 1.11, and that an induced norm is topologically equivalent to another non-induced norm.

A special property of

induced norms for linear transformations defined by matrices is that they are submultiplicative, that is IIABII. s IIAII. IIBII. ~ ~ i, n xn. A,B E R Finally, if the space Y is a Banach

for all matrices

space then it can be shown that the space of all bounded linear operators

f:X

7

Y with the induced norm is also a Banach space.

Example 8 (i) In example 7(i) the differential operator was defined on the space X, of all polynomials t I = [0,1)

Ilxll = max l x Ct ) l , tEl

with norm

but not bounded, since on setting

x

= x(t)

fx(t)

on the interval

We see that f is linear t

n

n

n

for

n

E:

the

N,

IIx II = I and Ilfx II = n so that the induced n n norm for the differential operator II f II = n (which ~s arbitra-

norm of x a s rily large).

(ii) The integral operator fx I

where

=

yet)

=

fl

f:C[O,I)

k(t,T)X(T)dT

7

and k

C[O,I]

~s

y

continuous on lXI,

o

is both linear and bounded. n r, (iii) Also the operator f:R 7 R which ~s

tion

defined by

[0,1),

y

bounded.

= Ax with

A

= {a .. } ~J

defined by the equa-

a (nxr) matrix, is linear and

A functional is a special operator in that its range lies on the real line R or complex plane C; it obviously satisfies the above conclusions concerning norms, continuity and boundedness. A general result for finite dimensional systems with normed space X, is that every linear operator on X is bounded.

Further

for any linear operator f boundedness and continuity are equivalent and if f is continuous at a single point it is continuous everywhere in X. Matrices are the most important tool for studying linear operators on finite dimensional vector spaces, since

19

I. MATHEMATICAL PRELIMINARIES

the operators can always be respresented by matrices.

1.5

Linear Transformations and Matrices Let X and Y be finite dimensional vector spaces over the same

field and and

G

f:X

+

Y be a linear operator.

= {gl.g2 •... ,gr}

=

... ,h l.h 2, n} basis vectors for X and Y res-

act as

Let

H

{h

pectively. with the respective vectors in fixed order.

Thus each

x E X has a unique representation x

+

where a. are scalars. i.

a h

(1.15)

n n

Since f is linear, x has the image

fx

y

(1.16)

again this representation is unique so f is uniquely determined if the images

prescribed.

Yk = f h Since

k Y'Y

of the n basis vectors k

Y,

E

are n they have unique representations r

B.g. ,

Y

hI"" ,h

L:

J J

j= I

a' g., Jk J

where each g. form a linearly independent set. J

(1.17) Therefore equa-

ting (1.16) and (1.17) gives, (I. 18)

Y

Therefore,

13.

J

j

1,2, ...

,r.

(I. 19)

The coefficients {a

form a (rxn) matrix A = {a for j k} j k} HG fixed H,G, which is uniquely determined by the linear operator f.

This matrix represents the operator f with respect to the bases H,G.

However the matrix A is not unique, it depends upon the HG basis vectors chosen in X and Y. We shall see that the operator is uniquely represented by an equivalence class of similar matrices

STABILITY OF LINEAR SYSTEMS

20

{~G,AH'G'}

n X ~ Y ~ R

for

which are such that

for e a (nxn) nonsingular matrix. So consider the linear operator

f:X

+

X,

on a normed space X

and h ~ {hI" .. ,h be n} n} two row vectors which act as different bases for the vector space

of dimension n.

Let

h ~ {hl, ... ,h

X, which by definition of vector basis each h. is a linear combination of the

and conversely, that is

~'s

where e is a nonsingular (nxn) matrix. 1S

J~

h

~

he

For every

or h'

x

E

X

~

C'h'

there

a unique representation with respect to these bases n x

hX

L:

l

j~l

a.h.

J J

'*

n L:

hx2.

j~1

a.h.

x

1

Cx 2.

J J

fx hy2. '* Yl ~ Cy2.' Consequently if hYl the matrices which represent f with respect to Al and A2. denote hand h respectively then Similarly for

y

~

and

A x

2. 2.

hence

or (1.20)

which is the definition of similar matrices Al,A2.' characteristic determinants of this equality gives Det(C

-1

AIC - AC

-1

Also the

IC)

Det (AI - AI)

(1 .21 )

which establishes: Theorem 1.4

All matrices representing a linear operator

f:X

+

X on a

finite dimensional normed space X, relative to various bases for

21

1. MATHEMATICAL PRELIMINARIES

X have the same characteristic values.

Also two matrices repre-

senting the same linear operator f are similar, each with the same characteristic values. Since the characteristic equation of the operator f is of order n, f has at least one characteristic value (and at most n) for

X oF {a}.

An important aspect of similarity transformations is the possibility of reducing a matrix to block diagonal form to ease computation.

Suppose that X is a n-dimensional vector space over

some field F.

Then any non-zero vector

e(i) s X

a characteristic vector of the linear operator \

I\.e

(i)

f:X

is said to be ~

X

if (1.22)

t.

where A is a (nxn) matrix associated with f and Ai s F is a l characteristic value of A given by solution of Det(AI-AI) = O. Define the matrix

C

(e

(l) ,

, ... ,e

(n)

)

(the so-called modal mat-

rix) then from (1.22) CDiag (A. ) ~

or

Diag(A.) r,

(1.23)

so that Al is similar to a diagonal matrix that contains the characteristic values of Al along its diagonal.

In this case we

have assumed that the transforming matrix C has n linearly independent column vectors each associated with n distinct characteristic values of the matrix AI'

It is important to note that al-

though similar matrices have the same characteristic values the corresponding characteristic vectors are not necessarily the same.

In the more general case when Al has multiple characteris-

tic values, it can be reduced via a similarity transformation to a unique block diagonal form, -1

CAlC where

J

(1.24)

STABILITY OF LINEAR SYSTEMS

22

A.

0

1

A.1

1

J=

h

0

0

J2

0

J

0

0

1

a

A.1

0

1

o

m 0

0 lS

0

0

J. =

and

o

A.

0

1

(r.xr.) matrix with r. the multiplicity of the ith charac1

1

1

m

teristic value of A1 and such that

I:

i=l

r.

n.

1

This canonicaZ

form is called the Jordan form for A1 , in this case the characm teristic equation for Al can be written as

II

i=1

r. (A-A.)

1

=

O.

1

Consider matrices over the field of complex numbers, and denote the complex conjugate of matrix A by A*. said to be normaZ when

AA* = A*A,

The matrix A is

special cases of normal mat-

rices are Hermitian matrices and Unitary matrices which satisfy A* = A

and

AA* = A*A = I

respectively.

All normal matrices

can be diagonalised by using unitary matrices in the similarity transformation, therefore any (nxn) normal matrix has n orthonormal characteristic vectors.

A consequence of this is that all

unitary matrices have characteristic values that lie on the unit circle.

The location of the characteristic values of diagonal

and triangular matrices clearly lie on the main diagonal.

In the

case of more generalised matrices characteristic value location is given by Gershgorin's theorem: Theorem 1.5

An arbitrary (nxn) matrix

A

= {a .. } over the complex field

lJ has its characteristic values located in the union of circles in the complex plane z defined by

Iz-a .. 1 11

where

~

r., 1

i = 1,2, .•. ,n

(1.25)

I. MATHEMATICAL PRELIMINARIES

n i.

n

·1 ~J

L:

r.

or

la.

j=1 j,ii

23

L:

i=1 i,ij

1a.~J ·1·

Clearly the more diagonally dominant A is, the smaller the Gershgorin circles and the tighter the bounds on the location of the characteristic values of A. In the remainder of this section we introduce the concept of a measure of a matrix,

this has the advantage of allowing

~(A);

tight estimates to be made upon the location of characteristic values of a matrix (dependent upon the norm used) since \leA) un-

II All i

like the induced norm

can have negative values. n; X = C from the norm

Consider the complex Euclidean space condition tors

x,y

1

E

ll y ll

Ilxll -

1

s Ilxll- ll y ll it follows for any two vecu Cll x ll ) = lim

that the limit

X

Ct-+O+

{II x+ay II a

exists, since the function inside the limit is a nondecreasing n function of a and bounded by ±llyll. The function \l:C -+ R is a one-sided derivative of the norm function. in the definition of

If the vector norm

is replaced by the induced norm for a

~(.)

matrix A, we have the definition of the measure of a matrix, ~(A).

lim (11I+aAIi. - l)aa-+O+ ~

~

1

(1.26)

It is easily seen from this definition and the algebraic pron perties of vector norms that for any A,B E CnxC and all a € R+ that

fl(aA).r,

afl(A).

Ifl(A+B)·1

s

~

~

fleA). +fl(B). ~

Ifl(A). - fl(B)·1

s

jjAII·t,

?

~

~

Also if

?

fleA).~

~

IIA - BII· ~

-fl(-A). ~

A. (A)

(j=I,2, ••• ,n)

J

?

-IIAII· ~

(1.27)

is a characteristic value of

matrix A and if its associated normalised characteristic vector

.

~s

(')

e J , then as

a

-+ 0

+

STABILITY OF LINEAR SYSTEMS

24

a- 1 ( 11 +CJ.A • (A) ) J

and

a -1 (

II (I+aA) e (j ) II.1. - II e (j) II)

1- I)

Re (A. (A))

->-

J

(1.28)

s a -1 (11(I+aA)II.-I) 1.

j..1(A).

->-

1.

(1.29)

so that j..1(A).

Re(A.(A)),

2:

1.

for

J

i = 1,2, .•• ,n

(1.30)

The computation of the measure of a (nxn) matrix

A

X = en

relatively straightforward; consider the space

{a .. } is 1.J

with the

vector norms £1, £2 and £00, their respective matrix induced norms n

are

Ia.1.J. I)

sup( L j i=1 n

sup( L i j=1

Ia.1.J·1)

I

[max A. (A*A) ] :1

(column sum),

1.

i

JJ

n L

i=1 Uj

and

(row sum); similarly the matrix measure j..1(A) asso-

ciated with these induced norms are respectively (Re(a .. ) +

,

Ia 1.J .. I), ri

sup(Re(a .. ) + L i 1.1. j=1 j#i

j..1(A)2 = max A. 1. i

n (A+A*»,

j..1(A)l = sup j

and

j..1(A)oo =

!a..I). 1.J

Example 9 Consider the (2x2) real matrix

A

=

(~

~J

'

its characteris-

tic values are 2,3, and its associated £1, £2 and £00 matrix measures are

j..1(A) 1

3,

).l(-A) 1

-1,

12

IJ(A)2

-25

12

+-

2 '

1J(-A)2 =

5 +-2. So that the characteristic 4, ).l(-A)oo 2 2 ' ).l(A)oo value for bounds for these respective matrix measures are given

-

by inequality (30) as £1: 2 £ : £00:

s Re(A) s 3, 5

2

12 s Re(A) s

- -

2

2 s Re(A) s 4.

-5

2

12

+-

2 '

I. MATHEMATICAL PRELIMINARIES

25

Note that the upper bound matrix measure on the £1 norm and the lower bound on the £00 norm give the exact characteristic values of the matrix A.

Whilst the matrix measure associated with

the £2 norm gives the narrowest bandwidth for the location of the characteristic values of A. The above example shows that matrix measures can have negative values and therefore cannot be a norm Cunlike the induced norm II-All. = IIAII.),

for which

~

~

however this very property is parti-

cularly useful in estimating the bounds of solutions to ordinary differential equations.

x

Consider the homogeneous differential equation where ACt) is a (nxn) continuous matrix defined for II x I t

)

II

ACt)x, t? to; then

i.s a solution to this differential equation with right

handed CDini) derivative lim cllx+axlL - Ilxll)aa-+O+

1

lim CII (I+aA) x II - II x II ) a-I

a-+O+

lim {( II (I+CiA) II. - 1) II xii }a

a-+O+

-1

~

C1.31)

fl(A)·llxll ~

fleA) ? -fl(-A)

Remembering that

integrating (1.31) over [t ,t] o

and utilising the properties C1.27) of the matrix measure we get

Coppel's inequality, t

Ilx(t o) II ex p{-

f flC-A(T»dT} t

<;

Ilx(t)

o

II

<;

flx(t )11 x o

t

(1.32)

exp{f flCACT»dT} t

From which it follows that if

x

Ct ) = 0 0

the unique solution (if it exists) to x(t ) f 0, 0

x Ct j x ,t ) o 0

o

the null solution is

x = ACt)x.

is a unique solution to

Further, if

x=

A(t)x,

s~nce

26

STABILITY OF LINEAR SYSTEMS

the difference of two solutions to also a solution (if it exists).

x

=

A(t)x

is by linearity

The general existence and unique-

ness conditions for solutions to differential equations are given by the contraction mapping theorem 1.3 since -].l(A). ~

2

IIAII.

~

2].l(A). ~

2

the measure of a matrix in Coppel's inequality

-IIAII. , ~

can be replaced by the induced norm of the matrix, but clearly the resulting bounds on solution will not be as tight. 1.6

Inner Product Spaces and Fourier Series The concepts of dot product and orthogonality of vector algebra

can be generalised to arbitrary normed vector spaces; this leads to inner product spaces and complete inner product spaces which are called Hilbert spaces.

These spaces retain many of the qua-

lities of Euclidean spaces and geometry, in particular that of orthogonality and p r o j e c t i.ons

v.

Defini tion I. 18 An inner product space X is a vector space with an inner product <x,Y> defined on it, that is for every pair of vectors x,y there is a scalar <x,Y> such that pl.

<x,z> +

<x+y, z>

p2.



p3.

<x,Y>

p4.

<x,x>

pS.

<x,x>

a<x,y> 0

2

0

<=*

x

=

0

An inner product on X defines a norm a metric

d(x,y)

normed spaces.

=

Ilx-yll,

Ilxll

I

=

<x,x>':!

on X and

hence an inner product spaces are

A complete inner product space is called a Hilbert

space, and also a Banach space.

Also since the normed linear

space (X, 11·11) has a uniformly continuous norm 11·11 on X, then !

similarly for the inner product space (X,<·,·>2) the function x~x,y>

is uniformly continuous for each

y E X.

All norms on inner product spaces satisfy the additional socalled parallelogram equality,

1. MATHEMATICAL PRELIMINARIES

27

<x+Y,x+y> + <x-y,x-y> = 2«x,x>+
p6.

This equality does not hold for other norms and clearly not all normed spaces are inner product spaces. Utilising the properties of norms and inner product spaces we have the following additional special properties for inner product spaces, 2

p7.



p8.

I<x,y>!

p9.

<x+Y,x+y>

lal <x,x> 1

1

<X,X>2 2

s

<x,x> +

S

(Triangle inequality)

Property p8 is Schwartz inequality which becomes an equality if {x,y} is a linearly independent set. concept of inner product spaces is if

A final and most important

x,y E X are such that

0,

piO. <x,y>

then vectors x and y are

orthogona~.

10

Examp~e

(a) The Euclidean spaces

kn

n and C are both Hilbert spaces

with inner products defined by <x .y>

for

x

=

(al,C1.2,'" ,a) and y = (Sl,S2" .. ,S ) n n L 2 [a , b ] . The vector space of all continuous

(b) The space

valued functions on [a,b] forms a normed space X with norm Ilxll

{t

1

<X,X>2

X ( t ) 2d t

}!

x c L 2 [a,b].

a

x,y E L 2 [a , b ]

Also for any

which are complex valued, their in-

ner product is defined by <x,y>

f

b

x(t)y(t)*dt

a

and norm is given by II

xii

{f a

b

1

!x(t )

*

1

2

dt

} Z

But s i.nce x(t)x(t) = Ix(t) I , then L2[a,bJ i s a Hilbert space. We also note that for any x,y E L2[a,bJ Holder's 2

28

STABILITY OF LINEAR SYSTEMS

inequality becomes Schwartz inequality and Minkowski's inequality becomes the triangle inequality. (c) The sequence space £2 with ~nner

product

<x,y>

= L:. a.S· * 00

~

~

~

is a Hilbert space, but £P (p#2) is not an inner product space nor a Hilbert space (it is however a Banach space since £P is complete). (d) Similarly the function space C[a,b] is a Banach space but not an inner product space nor a Hilbert space. Consider now the Euclidean space R3 with basis vectors h3 then any x E R3 has the unique representation

h l,h 2,

Taking inner products with hI' h 2, and h 3 enables us to calculate the unknown coefficients

k 1 = ,

where

al,a2,a3

and

as

= = 0

by the ortho-

If k = 1 then we say that the basis set i is orthonormal otherwise it is orthogonal. In general for

gonality property. {h.} ~

n x E R

and {h

an orthonormal sequence in an inner product k} space X, we have the unique representation

x

(1.33)

where the coefficients <x,h

are independent of n. Clearly an k> orthonormal set {h.} is linearly independent, conversely any arJ

bitrary linearly independent {gk} in (X,<','»

can be orthonor-

malised into a sequence {h.} in (X,<','» ... ,h) n

=

by noting that span(h l, J span(gl, ... ,g ); the resulting procedure (Gram-Schmidt n

process) gives the relationship between the sequences {h

k}

and

{gk} as (1.34)

where

MATHEMATICAL PRELIMINARIES

I.

29

k-I 2:

h

r=1

r

-k

r

ExarrrpLe 11

Consider the inner product space (X,<','» continuous functions on

f

TI

[TI,-TIl

for all real valued

with inner product

<x,y>

x(t)y(t)dt.

-TI An orthogonal sequence in X is u

n

cosnt ,

n = 0,1,2, ...

sinnt

n

or v

n

1,2, ...

TI since

f -TI

m n

0

m#n

TI

m=n=I,2, ...

2TI

m=n=O

{

cosmt cosnt d t

Hence an orthonormal sequence in X is, u

m

u

m

II u

m

cos mt

11- 1

lIT

for

m

1,2, ... ;

u

1

o

VzTI

Similarly for v , its orthogonal sequence 1n X is n

v

sin nt

n

n

1,2, ...

Returning to the unique representation (1.33) of respect to the orthonormal sequence are called FOUPier coefficients.

~,

x

E

n

R

the coefficients

with <x,~>

By taking norms of expression

(1.33) and utilizing the Pythagorean relation then for every x

E

X n 2:

k=1

l<x,~>12

~

Ilx11

2

(1.35)

,

which is known as Bessels inequaZity. to show that for the space (X,<','»

Also it is not difficult and any

x,y E X

that,

STABILITY OF LINEAR SYSTEMS

30 n

1::

k=l

I<x,hk>1

II xii.

<:;

Clearly Bessel's inequality

II vl]

(1.36)

a special case of the above ine-

~s

quality. In the remainder of this section we take

X

=

L2.

Let

Nc X

be the linear space spanned by an orthonormal sequence {hI' ... , h ... } ~n L 2 and if we consider the smallest subset N (the cIon sure of N) of L2 containing N, then if f E L2 the Fourier series of f converges in the mean (L 2 norm) to the orthogonal projection g of f on N (g EN); in particular the Fourier series of f converges in the mean to f if and only if fEN. If in addition N is dense in L2 (more generally in the Hilbert space (X,<·,·»), that is N = L2 (more generally N = X) we say that the orthonormal set {h.} is complete or total in L2 (respectively in (X, J <.,.», and for any f E L2 (f s (X,<·,·») the Fourier series of f with respect to {h.} converges in the mean (in the norm <., .»

J

to f. x

That is the partial sum ah

+ .•. +ah

lIn n

n

converges to x in the mean, or

lim Ilx -xii n--

v

n

=

O.

In examp Le 11

u

the sequences and form a complete orthonormal sequence. m m For complete orthonormal sequences Bessels inequality (1.35) becomes Parseval's equation,

(1.37) and inequality (1.36) becomes 1:: I<x,hk>1

k for

k

x,y

S

L2

(or

Ilxll

(1.38)

Ilyll

x,y s (X,<·,·»).

Parseval's equation is a

necessary and sufficient condition for convergence in the norm of the generalised Fourier series (1.33). We know that the classical Fourier series for a real valued function x:R ~ R which is periodic with period 2rr (i.e. x s

MATHEMATICAL PRELIMINARIES

I.

x(t)

!a

+

o

IT

~

with

IT

f

r

1.

xcosktdt,

IT } -IT

-IT

exist since they are dominated by Ix!. v

k

(I .39)

(a cos kt + b sin k t ) k k

L:

k=1

31

xsinktdt,

which both

By setting

uk = coskt,

= sinkt in the above definitions of a

b multiplying by u k, k, j respectively by v.) and integrating with respect to t over

(~nd

[-IT,IT]

J

we obtain

a.

<x,u·>llu·11 J

J

-1

J

-

b. = <x,v.>llv.11

,

J

J

where u., v. are defined in example 11. J

J

-1

J

(1.40)

So that the Fourier

series (1.39) can now be rewritten as 00

x( t )

<x,u >u + L: «X,R >u + <x,vk>v k} o 0 k=1 k k

(1.41)

which justifies our earlier terminology of Fourier coefficients for <x,h

k>. Consider now the partial sum S

n

the Fourier series (1.39), as S (t) n

!a + o

n L:

k=1

of the first (n+l) terms of

kt + bksin k t

(~cos

)

n

L:

k=1

(cos k.r cos k r + sin kr sin kT)dT}

IT

~f

x(T)D (T-t)dT

(1.42)

n

-IT

{X(t+T) + x(t-T)}D (T)dT n

(1.43)

STABILITY OF LINEAR SYSTEMS

32

D (T) n

where

sin(n+D 2 sin T /2

=

for

T f 2nr,

r any integer,

is an

even function and is called the Dirichlet kernel. Setting x(t) = in (1.42) gives 1 = -1 fn 2n (t)dt, multiplying this result non by x(t) and subtracting from (1.42) gives tr

~ fo

{X(t+T) + X(t-T) - x(t)}n (T)dT

S (t) - x(t)

n

so that

S (t)

zero as

n

7

n

7

00.

x(t)

n

(1.44)

pointwise if the above integral tends to

This sufficient condition for convergence is

called Dini's condition and is certainly satisfied if x(t) is differentiable.

Surprisingly the same result on the convergence of

{S } can be governed by an arbitrary small interval n

[-n,n],

for

0 < a

I,

$

[-an,an]

in spite of the fact that the Fourier

series depends upon the whole of the interval

[-n,n].

If in ad-

dition x is of bounded variation on the restricted interval an]

of

then by considering the upper bounds of

fn (T)dT, n

[-an,

the

Fourier series of x at a discontinuity at points t converges to this is known as Jordans condition.

+ x(t-O)},

~{x(t+O)

The fact that the Fourier series of continuous functions need not converge everywhere endangered the whole theory of representation of a function by its Fourier series.

This situation was

salvaged by Fejer who showed that the Fourier series of a continuous function x(t) is summable to x(t) by the method of arithme-

tic means (or Cesaro sums).

Essentially this averaging process

smooths out the oscillations caused by the method of partial sums which utilise Dirichlet kernels.

o (t) n

Let

(S (t ) + ... + S (t»)(n+l) o n

-1

for n=O,I,2, .. (1.45)

which is clearly the arithmetic mean of the first n partial sums S (t) of the continuous function x(t). n

Also from equation (1.42)

1T

o (t) n

1 f (n+l)n)

-n

x(T)F (T-t)dT n

(1.46)

I. MATHEMATICAL PRELIMINARIES

F (t ) = (D (t ) + ... + D (t)) = sin 2(n+ (2sin 2¥)-1 > 0, n o n 21)t

where if

33

2nn

f t.

The kernels F (t) are the well known Fejer kernels n

which converge positively to zero as

n

~

00.

Since the Fejer

kernels are positive, of period 2n and satisfy the same integral type equations as the Dirichlet kernels then it is not difficult to see that if as

n

~

00

S (t) ~ x(t) then also 0 (t) ~ x(t) pointwise n n Fejer's important result, that parallels that of

Jordons condition, is that if

x E L 1 [-n,n],

then for any dis-

continuity t at which the limits x(t+O), x(t-O) exist the Fourier series of x(t) is Cesaro surnrnable to

!{x(t+O)+x(t-O)}.

Essen-

tially this result shows that the Fourier series of a continuous function x of period 2n is Cesaro surnrnable at every point t to the function, in addition the series {o } converges uniformly to n

x.

Also since the power series for the sine and cosine are

un~-

formly convergent on a bounded interval, the Fejer approach can be used to approximate any real valued continuous function x(t) on a bounded closed interval [a,b] by a polynomial pet) such that Ix(t)-p(t) I < E

for any

E > 0

and for all

t E [a,b] - this is

the classical Weierstrass approximation theory.

1.7

Notes Throughout this chapter results mainly germane to the

rema~n

der of this text have been presented without formal proof.

The

style and approach adopted is similar to that of Curtain and Pritchard (1977) which appears in the same series. introductory text in functional analysis ~s

A suitable

that of Naylor and

Sell (1971), whilst a more advanced text by Bachman and Narici (1966) provides the majority of the omitted proofs of this chapter.

The classical text of Dunford and Schwartz (1963) provides

the necessary background 1n topology and linear operators.

Re-

sults on linear transformations and linear algebra can be found in the very readable text of Hadley (1961), whilst Luenberger

34

STABILITY OF LINEAR SYSTEMS

(1969) provides an excellent introduction to vector space methods for readers with an engineering mathematics background.

The re-

sults on inner product spaces and Hilbert spaces can be found in the specialist text of Halmos (1957).

Finally, the material on

matrix induced norms and measure of a matrix is summarised in Coppel's (1965) text on stability theory. References Bachman, G. and Narici, L. (1966). "Functional analysis", Academic Press, New York Coppel, W.A. (1965). "Stability and asymptotic behaviour of differential equations", Heath, Boston Curtain, R.F. and Pritchard, A.J. (1977). "Functional analysis in modern applied mathematics", Academic Press, New York Dunford, N. and Schwartz, J. (1963). "Linear operators", Vols.I, II, J. Wiley, Interscience, New York Hadley, G. (1961). "Linear algebra", Addison Wesley, New York Halmos, P. (1957). "Introduction to Hilbert spaces", Chelsea, New York Luenberger, D.G. (1969). "Optimization by vector space methods", J. Wiley, New York Naylor, A.W. and Sell, G.R. (1971). "Linear operators in engineering and science", Holt, Rinehart and Winson, New York

Chapter 2

ALMOST PERIODIC FUNCTIONS

2.1

Introduction

The theory of almost periodic functions was created and developed in its main features by Bohr (1924) as a generalisation of pure periodicity.

The general property can be illustrated by

means of the particular example f(t)

s1n 2TIt + sin 2TIt

/2

This continuous function is not periodic: that is there exists no value of T which satisfies the equation all values of t.

=

f(t+T)

f(t)

for

However, we can establish the existence of num-

bers for which this equation is approximately satisfied with an arbitrary degree of accuracy.

For given any

n > 0

as small as

we please we can always find an integer T such that T/2 differs from another integer by less than n/2TI.

It can be shown that

there exist infinitely many such numbers T, and that the difference between two consecutive numbers is bounded.

This property

of the T'S defines almost periodicity in general (Fink, 1974). Almost periodicity is a structural property of functions which is invariant with respect to the operations of addition and multiplication, and also in some cases with respect to divison, differentiation, integration and other limiting processes.

To the

structural affinity between almost periodic functions and purely

36

STABILITY OF LINEAR SYSTEMS

periodic functions may be added an analytical similarity.

To any

almost periodic function there corresponds a Fourier series in the form of a general trigonometric series f(t)

Ak being real numbers and ~ real or complex. The serles lS obtained from the function by the same formal process as in the

with

case of purely periodic functions, that is, by the method of undetermined coefficients and term by term integration.

As in the

purely periodic case, the Fourier series need not converge to the almost periodic function for all values of t.

Nevertheless,

there is still a very close connection between the series and the function.

In the first place Parseval's equation holds; 00

where the mean value M is defined by t

~

Mt{g}

Lim T-7=

~

T

f g(t)dt. o

The uniqueness theorem, according to which there exists at most one almost periodic function having a given trigonometric series for its Fourier series follow from Parseval's theorem. Further, the series is summable to f(t) in the sense that there exists a sequence of polynomials (m = 1,2, .... )

where

d

k

S

[0,1]

and only a finite number of the d

k

differ from

zero for each m, such that (i) the sequence of polynomials converges to f(t) uniformly in t; (ii) the sequence of polynomials converges to the Fourier series associated with f(t), by which is meant that for each k, d(m) k

7

I

as

m

7

00•

Conversely, any trigonometric polynomial is an almost periodic

2. ALMOST PERIODIC FUNCTIONS

37

function, and so is the uniform limit of a sequence of trigonometric polynomials.

It is easily proved that the Fourier series

of such a limit function is the formal limit of the sequence of trigonometric polynomials (Cordeneanu, 1968).

Thus the class of

Fourier series of almost periodic functions consists of all trigonometric

L

series of the general type

~

exp(iAkt)

to which

there corresponds a uniformly convergent sequence of polynomials of the type

L d~m)

~

exp(iAkt),

(m

= 1,2, .... ) which formally

converge to the series. The first investigations of trigonometric series, other than purely periodic series, were carried out by P. Bohl (1906).

He

considered the class of functions represented by series of the form

where

wl,w2, ... ,w n

are arbitrary real numbers and

"

are real or complex numbers.

A k

K1 2'"

k

The theory of these functions,

!L

however, follows in a more or less natural way from existing theories on purely periodic functions, rather than the theory generated by Bohr.

A quite new way of studying trigonometric

series

~s

and

the sequel we will use the strong correspondence between

~n

opened up by Bohr's theory of almost periodic functions

the two to develop methods of studying differential equations with almost periodic coefficients. Our two main aims in this chapter are to review the development of the Fourier series theory of almost periodic functions and to consider the question of approximating almost periodic functions by trigonometric polynomials.

Material is taken from

five main sources, namely Bohr (op.eit.), Bochner (1927), Besicovich (1932), Corduneanu (1968), and Fink (1974).

Proofs

of standard theorems are given only when some clarification is necessary, otherwise they are omitted for brevity.

The final

section is devoted to almost periodic functions depending

38

STABILITY OF LINEAR SYSTEMS

uniformly on a parameter.

This section

~s

a prelude to the ma-

terial contained in Chapter Six on almost periodic differential equations dependent upon a parameter.

Section 2.6 briefly con-

siders limiting cases of almost periodic functions. 2.2

Definitions and Elementary Properties of Almost Periodic Functions Bohr developed the theory of almost periodic functions in di-

rect analogy with the theory of purely periodic functions, although theorems which are decidedly trivial for purely periodic functions are no longer trivial for almost periodic functions.

Theorems for

purely periodic functions are trivial because the investigation of such functions can be restricted to a finite interval, say the period itself.

For the almost periodic case, similar theorems

can be deduced by effectively restricting interest to a finite interval called the inclusion interval.

The numbers, corresponding

to periods in the purely periodic case, which characterise the almost periodicity of an almost periodic function, are chosen from the inclusion interval and are called almost periods.

We

clarify the situation by considering the class of continuous functions which have the following property: for every there exists a translation number !f(tH) - f Ct;) I < n Note that the numbers

,en)

~,

of

f(t)

for all t

,en)

n > 0 such that (2. l)

are arbitrarily large and this is not,

however, a satisfactory situation.

Examples may be constructed

to show that the class of functions we have chosen does not even remain invariant under the operation of addition.

Clearly some-

thing more must be said about the L(n) to produce a class of functions which are well behaved.

To this end Bohr introduced

the concept of relative density. Definition 2.1: Relatively dense set (Bohr, op.cit.) A set T of real numbers is said to be relatively dense if there exists a number

£ > 0

such that any interval of length £

2. ALMOST PERIODIC FUNCTIONS contains at least one member of T.

39

Any such number is called an

inclusion interval of the set T. The number T(n) is called an n-translation number of a function f (t ) (Bochner, 1927), and we denote the set of all translation numbers n of f(t) by

T(n,f(t».

It is easily verified that the

following properties hold: (i) T(n' ,f(t»

~

T(n,f(t»)

for any

n' > n.

(ii) If n is an n-translation number then so is -no (iii) If n 1 , n 2 are (n 1 and n 2)-translation numbers respectively, then n

1±n 2

is an (n

1±n 2)-translation

number.

Thus we are led to the Bohr definition of an almost periodic function f(t): Definition 2.2: Bohr almost periodic function (Bohr, op.cit.) A continuous f:R n > 0

the set

T(n,f(t))

Henceforth let £ Each

E is called Bohr almost periodic if for any

~

T E T(n,f(t))

n

it is clear that as

denote 'an inclusion interval of

T(n,f(t)).

is now called an n-almost period of f(t) and n

whereas, in general,

is relatively dense.

~

£ n

0, ~

the set +00.

T(n,f(t»

becomes rarefied,

From the definition it follows

that any continuous purely periodic function f(t) is Bohr almost periodic, since for any n the set T(n,f(t»

contains all numbers

kw (w a period of f(t) and k an integer) and thus is relatively dense.

In the sequel we shall be concerned mainly with uniformly

almost periodic functions, although generalisations do exist (see section 2.6) and choose to drop the qualifier "uniformly": almost periodic will imply uniformly almost periodic unless otherwise specified.

Several theorems now follow, which establish the ele-

mentary properties of almost periodic functions. Theorem 2.1: (Bohr, op.cit.)

Let f:R

E be an almost periodic function, then f(t) is boun-

~

ded on E.

Proof: Put

n

=

and denote by M the maximum of

If(t)1

in an

interval [0'£1]' It can be easily seen that corresponding to any t we can define a number T E T(l,f(t», such that t + T belongs

STABILITY OF LINEAR SYSTEMS

40

to

and consequently that

[O,~l],

If (t+T)

I

M

<

but If(t+T)-f(t)

I

<

thus If (t )

I

for all values of t,

M + 1

<

which proves the theorem. The next theorem is an important statement about the continuity of almost periodic functions.

(Bohr, op.cit.)

Theorem 2.2:

Let

f:R

~

E

be an almost periodic function then f(t) is uni-

formly continuous on R.

Proof: Given an n > 0, take an ber such that if only It'-t"l <

If(tl)-f(t2)

o.

Itl-t21 <

o.

I

<

~

/

1n 3

3n

and let for any

0(0<0<1) be a num-

tl,t 2

[0'£n/ 3+ 1 ]

Let now-t' ,t" be any two numbers such that

There exists a number T of the set

that both the numbers

€

t'+T, t"+T

1

T("3 ll,f(t))

belong to the interval [0,£ / +

n

We have then

1].

If(t'+T)-f(t"+T)1

such

<

Also If(t+T)-f(t)

I

<

for any t.

Thus

If( t ' ) - f ( t") I <: If ( t ' ) - f ( t ' +T) I + If ( t" +T) - f ( t") I < n

+

If ( t ' +T)- f ( t " +T) I

which proves the theorem. The above proof exemplifies the idea of almost periodic function theory that everything that we need to know about almost periodic functions happens on some finite interval. The following theorems 2.3-2.6 establish the algebraic and topological properties of the Bohr definition of almost periodicity.

3

2. ALMOST PERIODIC FUNCTIONS Theorem 2.3:

41

(Bohr, op.cit.)

If the complex valued function f(t) is almost periodic then for any constant complex number a and real number S, af(t), f * (t), f(t+S) and f(St) are also almost periodic.

Similarly f(t)2 and

f(t)f * (t) are almost periodic. The next theorem tells something of the convergence properties of a sequence of almost periodic functions and provides the key to many results obtained in the theory of almost periodic functions, particularly with respect to polynomial approximation. Theorem 2.4:

(Bohr, op.cit.)

If a sequence of almost periodic functions {fk(t)} converges uniformly on R to a function f(t), then f(t) is also almost periodic.

Proof: Given n, there exists a function f k (t) such that o !f(t) - fk (t) o

I

:Q

<

for all values of t.

3

Let now T be a number of

Then

If

( t +T) -

f (t )

I +

[f k (t.) - f(t)[ o

<

n

which shows that T(n,f(t» and thus

~

1

T("3n , f

T(n,f(t»

ko ( t »

is relatively dense.

This being true for

any n we conclude that f(t) is almost periodic. The following lemma tells us something about the properties of almost periods of almost periodic functions and from this we can investigate the arithmetic properties of the functions themselves. Lemma 2.1: (Bohr, op.cit.) For any

n > 0

and for any two almost periodic functions

f 1 (t) and f 2(t) the set dense.

T(n,f 1 (t»

n T(n,f 2(t»

is relatively

42

STABILITY OF LINEAR SYSTEMS Now the arithmetic properties can be deduced.

Theorem 2.5: (Bohr, op.cit.) The sum of two almost periodic functions f 1(t) and f 2(t) is an almost periodic function.

Proof: Taking an arbitrary

n > 0,

let T be any number of the

set T(~n,fl

(t»

n

T(~n,f

2

then

(t»

which shows that T belongs to the set T(n,f 1 (t) + f 2(t» so that

~

T(n,f 1 (t) + f

2(t»

T(~n,fl

(t»

T(n,f

n

1(t)

+ f

2(t».

Thus

T(~n,f2(t»,

is relatively dense, which proves the

theorem. The theorem can be generalised immediately to the case of the sum of any finite number of almost periodic functions. result holds for subtraction.

A similar

The next theorem is almost trivial.

Theorem 2.6 The product of two almost periodic functions f (t) and f 2(t) 1 is an almost periodic function.

Proof: Using Theorems 2.5 and 2.3 together with the relation

proves the theorem. The corresponding result for the ratio of two almost periodic functions f 1 (t) and f 2(t) requires that inflf2(t) I is positive, t since we write the ratio f (t)/f 2(t) as the product of two al1 most periodic functions f1(t) and ( t ) . The last two elementary 2 properties of almost periodic functions presented concern the

i

almost periodicity of the derivative and integral of an almost periodic function.

The result for the derivative is quite

straightforward, while that for the integral is a little more complex and will receive considerably more attention in later chapters in connection with the solution of differential equations.

2. ALMOST PERIODIC FUNCTIONS

43

(Bohr, op.cit.)

Theorem 2.7:

If the derivative of an almost periodic function is uniformly continuous, then it is almost periodic. Theorem 2.8: (Bohr, op.cit.) If an indefinite integral of an almost periodic function f(t) is bounded, then it is almost periodic.

Froof: is postponed until Section two, Chapter Six.

A proof

based on establishing the relative denseness of a set of almost periods for the integral (bounded) is rather involved and no useful purpose will be served by reproducing such a proof here. Much simpler proofs are developed from the Fourier series theory of almost periodic functions and this approach is used in Chapter Six.

Theorem 2.8 is essentially a result about the solution of

the differential equation

x

= f(t),

f(t)

~

almost periodic.

In the next section we examine the Fourier series theory of almost periodic functions which is developed analogously to the corresponding theory for the purely periodic case. 2.3

Mean Values of Almost Periodic Functions and their Fourier Series We begin by recalling some of the main aspects of the Fourier

series theory of periodic functions.

Let f(t) be periodic with

period w, then there is a formal relation f(t)

L

~

Ikl
~

exp(iAkt)

where A k

~

2kn

w

~

and

~s

M

w

f f(t)

1

is given by

exp(-iAkt)dt

(2.2)

0

Questions of convergence aside, the main results are: (i) the mapping from f(t) to the numbers (ii) Parseval's equation holds, that is

{~,Ak}

is one-to-one,

44

STABILITY OF LINEAR SYSTEMS

(iii) there are altered partial sums which approximate f(t). To extend the above results to almost periodic functions we need to replace A by a more general real number and replace the k formula (2.2) by something more appropriate. No fixed w will give a satisfactory theory since many almost periodic functions would have the same Fourier series. On the other hand, the Fourier transform would not exist in general.

Bohr introduced a compromise and took an average of

the Fourier transform, that is, applied a limiting process to (2.2) as w .....

00

w

~ wf f(t)

A(f ,A)

(2.3)

exp(-iAt)dt

o for

f(t)

AP(C), where the space

€

AP(C)

= {f:f is an almost

periodic complex valued function of the real variable t}

II f II

the uniform norm each fixed (f,A). a real number}.

=

sup If (t ) t

I.

Note that

A(f ,A)

Here we have used the notation

€

A

R

with for

{AlA

~s

The next important question concerns the exis-

tence of the Bohr transforms of any almost periodic function and this is answered as follows: Theorem 2.9: For any

op.cit.)

(Bohr~

A

€

A,

A(f,A)

exists.

Proof: We first prove the theorem for by Mt{f}. ger.

= 0 and denote A(f,O)

Let w,n be two positive numbers and m a positive inte-

Write mw

--!.mw

A

f

m-I l:

f (t j d t

k=O mw

o Denote as usual by

~

'k be a number of

T(n,f(t»

n

(k+\ )w

f

f(t)dt

kw

an inclusion interval of T(n,f(t», and let included in the interval

(kw,kw+~

).

n

2. ALMOST PERIODIC FUNCTIONS

45

Then, (k+l)w

J

(k + l )W-Tk

fkW--T

f(t)dt

kw

f(t+Tk)dt k W

W

f

f

f(t)dt +

o

[f ( t

+Tk) - f ( t ) ] d t

o

f(t+Tk)dt +

+

f

(k+l )W-T k f(t+Tk)dt

W

I

1121

Evidently

<

rxo,

1

+ I

2

+ I

3

Now writing

+ I

(2.4)

4

M = IIf(t) II

and observing

that the length of the range of integration in 1 3 and 1 4 is less than Zn' we have 1 1 31 < MZ 11 41 < MZn . Hence n, (k+ I)w

fkw with

fW f(t)dt + 8 (nw+2MZ

f(t)dt

o

181

:0;

I.

Therefore

mw mw

W

f

~ fo

f(t)dt

0

where

e

n

<

f(t)dt + 8(n +

2M£ W

n)

'

(2.6)

has changed its value but is still less than 1 in abso-

lute value. please.

(2.5)

n),

Now let

~

be a positive number as small as we

In the above formula set

~

"8 '

and this gives

W

16MZ > ----!l ~

.

46

STABILITY OF LINEAR SYSTEMS w

mw

f

mw

f(t)dt

W

+e~4

f f(t)dt

(2.7)

o

o

Corresponding to any positive number T, define the integer m by the condition

mw

T

S

(m+l)w.

S

From the boundedness of f(t)

we conclude that T

~~ [~

m f(t)dt -

J

f

mw

0

O.

fCt)dt]

0

Consequently there exists a number

I

f

T

f(t)dt -

mw

0

T > T .

for all

o

m

T 1

exists as T

+

r

) 0

f(t)dt

> 0

T <

such that

~

4'

This argument shows that the limit of

o 00, and if we define

lr T)

T

f (t l d t

o

T

lim

T->oo

t f f(t)dt

(2.8)

o

the first part of the theorem is proved.

To solve the case of

the Bohr transform, we need only consider the function f(t)exp (-iAt)

which, for real A, is the product of two almost periodic

functions and according to Theorem 2.6 is therefore almost peri-· odic.

Thus the mean value

defined to be

A(f,A).

Mt{f(t)exp(-iAt)}

exists and is

This proves the assertion.

It is worth noting at this stage that the mean value M of t almost periodic functions f 1 (t), f 2(t) has several simple algebraic properties:(i) Mt(f * (t»)

(ii) Mt(f(t» (iii) M ( f +f

t (iv) I f

1

::> 2)

{f (t)} n

* Mt(f(t» o

if

Mt(f 1 ) + Mt(f2.) is a uniform convergent sequence of almost

periodic functions such that then

f(t)::> 0

lim f (t)

n->=

lim M (f (t») = M (f(t». n->OO t n t

n

=

f(t),

f(t) E AP(C),

47

2. ALMOST PERIODIC FUNCTIONS

Following Bohr's development of the theory of almost periodic functions by analogy with the purely periodic case, we next look at Bessel's inequality as a first step in getting Parseval's equation.

This is important because it is used to prove unique-

ness for the Fourier series of an almost periodic function.

The

first result of interest concerns polynomial approximation to almost periodic functions. Theorem 2.10: (Bohr, op.cit.J Let f(t) be an almost periodic function; A ... ,A be m 1,A 2, m distinct arbitrary real numbers and B ,B , ... ,B be m arbitrary 12m real or complex numbers. Then m

Mt{!f(t) -

=

k=1

2

Bk exp(iAkt) I }

2 m 2 m 2 Mt{lfCt)1 }- = IA(f,A k)!, + = IB - A(f,Ak)1 k k=1 k=1 with

A(f,A

Proof: Write

k) m

Mt{lf(t) -

= M

(2.9)

=

k=1

2.

B exp(iAkt) I } k

rU(t) -

t ~

(where the asterisk denotes the complex conjugate) then the above,

48

STABILITY OF LINEAR SYSTEMS

As

Mt{exp[i(~

only for

)t]}

-~

k

1

Thus

differs from zero (and is equal to I)

1

2

k2

the last sum reduces to the sum

m

Mt{lf(t) -

2

L:

B exp(iAkt)! } k

k=l

m

2

Mt{lf(t)1 }-

L:

k=1

B

*

k

m

m

2

L:

Mt{lf(t)1 }-

k=1 m

+

L:

k=1 m L:

2

Mt{lf(t)I}-

k=1

-

A(f,~)

A(f,~)A

L:

k=1

+ B A* (f,A k) k

m

BkBk*

L:

k=1

* (f,Ak) HB * - A* (f,~)} k

{B - A(f ,~) k 2

IA(f,Ak)1

+

and equation (2.9) is called the equation of approximation In the mean. From the last theorem it is clear that the polynomial m L:

Bkexp(i~t) with fixed exponents A gives the best approxik k=1 mation in the mean to f(t) if B = A(f,A for all k, in which k) k case we have

m

Mt{lf(t) -

A(f,A

L:

k=1

k)

exp(i~t)1

2

}

m

2

Mt{lf(t)1 } -

L:

k=l

The left-hand side of this equation being nonnegative, it follows that m

L: k- l

IA(f,A

2

)! k

This inequality

lS

(2. 10)

known as Bessel's inequality and

lS

true for

2. ALMOST PERIODIC FUNCTIONS

49

an arbitrary number m of real numbers A It appears that to any k. positive n there corresponds at most a finite number of values of A for which

!A(f,A)1 > n.

This leads immediately to:

Theorem 2.11: (Bohr, op . ci.t . ) There are at most a countably infinite set of values of A for which A(f,A) differs from zero. Denote these values of A by

A

1,A 2

and write the set of

, ••• ,

these numbers as A. Definition 2.3: Set of exponents (Bohr, op.cit.) A(f,A) #

The set A for which

a

with

A s A is called the

set of exponents of f(t) and may be written Af• Definition 2.4: ModuZe of f (Fink, 1974) The set of all real numbers which are a linear combination of the elements of A with integer coefficients is called the moduZe

off, mod (f), i . e . N { L:

mod Cf)

j=!

That is for

n. A.; n., N ~ 1, integer} J

J

f s AP(C)

J

the module of f is the smallest additive

group which contains the exponents of f.

The relationship bet-

ween the exponents of two almost periodic functions f and AP(C)

g s

are contained in the following equivalent relationships

(Favard, 1933; Fink, 1974):(i) mod(f)

~

mod(g)

(ii) for every T(n' ,get))

n>O

(module containment) a

n'>O

exists such that

T(n,f(t)) c

(translation set containment)

= f implies Thg = g and that there is a h' c h so Th,g = g (assuming that Thf exists). Convergence here

(iii) Thf that

is either uniform, uniform on compact sets, pointwise or in the mean sense, since they are all equivalent for

g,f s AP(C).

Definition 2.5: Fourier coefficients (Bohr, op.cit.) The numbers fi~ients

A(f,A)

for

A s A are called the Fourier coef-

of f(t).

Since A is countable, it can be enumerated by the positive

STABILITY OF LINEAR SYSTEMS

50

integers and one can write the Fourier series associated with f(t) as f(t)

Z A(f,A) exp(iAt)

~

(2.11 )

Given the Fourier series (2. II) of f*(t)

~

f(t+S)

a,S

then also

Z A*(f,A)exp(-iAt) ~

Z A(f,A)exp(iAt)exp(iAS)

exp(iat)f(t) for

f(t) E AP(C),

Z A(f,A)exp(i(A+a)t)

~

real numbers.

If f(t) is a periodic function, then the Fourier serLes (2.11) reduces to the usual Fourier series (2.2) for periodic functions. Of course no convergence is implied by the Fourier series representation (2. II), although since Theorem 2.10 holds for any finite set of numbers in A, the sum over A converges, that is Bessel's inequality (2.10) holds.

In fact Bessel's inequality

can be replaced by an equality and this gives Parseval's equation. The details of the procedure for demonstrating this are omitted (see Jensen, 1949). Theorem 2.12: Parseval's Equation (Jensen, 1949) For any

f(t) E AP(C) N L: K=I

IA(f, AK) I

2

If the two almost periodic functions (f 1 (t) - f 2(t» > 0 then from Parseval's

f1(t), f

with 2(t) for all t, have the same Fourier series equality applied to

f (t ) - f :: Set), 2(t) 1 2 Set) E AP(C), it would follow that M I} O. However, t{18(t) since S(t) is a non-negative and non-vanishing almost periodic 2} function it has a positive mean and therefore Mt{18(t)1 # 0, and we conclude that two distinct almost periodic functions have distinct Fourier series - the so-called Uniqueness theorem. The final aspect of the Fourier series theory of almost

2. ALMOST PERIODIC FUNCTIONS

51

periodic functions concerns their convergence, that is we ask whether or not one can actually compute the function from its corresponding Fourier series.

The answer to this question has

been provided by several authors, in particular Bochner (1927), by recourse to the classical approximation theorem due to Weierstrass.

It is interesting to note that two problems are

contained in the above discussion: one is concerned with convergence questions, while the other involves polynomial approximation.

Although the two problems are closely related, they may

be and have been on occasions, treated as separate.

Indeed Bohr

tackled the approximation problem by developing the theory of purely periodic functions of infinitely many variables - functions that can be written as:

r(k)w t + .•. )] m mm with the r.(k ) J

.

rat~ona

I numb ers (J'

1,2, ..• ) .

A convergent

sequence of such functions converges to a function which Bohr called a limit periodic function of infinitely many variables. Associated with each limit periodic function there exists a diagonal function f(t) formed by setting each of the variables t. t(j

=

1,2, ... ),

f(t)

J

that is:

f(t ,t , ... ,t , ... ) 12m

and by means of a theorem due to Kronecker on Diophantine approximations, Bohr was able to show that the aggregate of all the values of the diagonal function f(t) is everywhere dense in the aggregate of all the values of

fCt,t , ... ,t , ... ). 12m

Further-

more, he was able to show that fCt) is almost periodic. Unfortunately an answer to the convergence question does not come easily from Bohr's work.

In fact it comes in two parts.

The first stems from the classical theory of Fourier series of purely periodic functions which tells us that in the purely

52

STABILITY OF LINEAR SYSTEMS

periodic case we should consider a more general concept than convergence, namely that of summability, because there are examples of Fourier series known to diverge at a point.

Therefore, instead

of considering the convergence of sequences of partial sums, for summability we consider the convergence of sequences of arithmetic means of the partial sums.

Summability

~n

this form is called

Fejer summability and not only does it tell us something about the convergence properties of Fourier series of periodic functions but also it contains the classical theorem of Weierstrass on trigonometric polynomial approximation (see Chapter One).

The second

part of the answer we seek stems from the fundamental dissimilarity between purely periodic and almost periodic functions that prevents the development of the theory of the latter by direct analogy with the former.

The problem of summation by partial

sums of the Fourier series of a purely periodic function has already been examined and similar difficulties are to be expected with diagonal functions of limit periodic functions of several or an infinite number of variables.

However, from the same point

of view we may expect summation by arithmetic means to be applicable to the general case of almost periodic functions.

Bochner

(op.cit.) realised that the continuity of an almost periodic function implies the continuity of the corresponding limit periodic function of many variables and that this is a sufficient condition for the uniform convergence of the Fejer sums of the latter functions.

He then took the diagonal functions of these

Fejer sums to obtain the Fejer sums of the almost periodic function itself.

The implication of Bochner's approach is the fol-

lowing theorem:Theorem 2.13: Approximation theorem (Bochner, op.cit.) To any almost periodic function there corresponds a sequence of trigonometric polynomials - Bochner-Fejer polynomials - uniformly convergent to the function.

Proof: The proof of the Approximation Theorem is based on the proof for the same theorem for periodic functions.

The proof for

2. ALMOST PERIODIC FUNCTIONS

53

periodic functions is to show that the Cesaro-means of the partial sums converge uniformly to the continuous function.

To formulate

the central idea, let A exp[ivwt]

L:

f (t )

IV I<00

V

then the Cesaro means (Fejer sums) are given by the formula A exp [ivwt] V

Since

A

V

M {f(s) exp[-ivws]}, s

it follows that

M {f(s) exp[-ivw(s-t)]}

A exp[ivwt]

s

V

M {f(s+t) exp[-ivws]} s

whence MS{f(s+t)TIk(WS)} where

is the Fejer kernel and is defined by the equation

TI (s ) k

L:

Ivi
1 -

t~l]

exp[-ivs]

fn~'r k

. s Sln "2

The main properties of TIk(s) are (i) TIk(s) 2 0 and (ii) M(TI

k)

1.

Property (i) is clear from the second representation and (ii) from the first since M(TI is as follows, If(t) - 0k(t) I

k)

is the constant term.

The proof then

Ms{f(t)ITk(ws) - f(S+t)ITk(ws)} S

Ms{lf(t) - f(s+t)!ITk(ws)}

and the idea is that for lsi small the first factor is small and for lsi bounded away from 0,

TIk(ws)+O

almost periodic case, products of

~'s

as k~.

To solve the

are taken, one for every

STABILITY OF LINEAR SYSTEMS

54

periodic component of the almost periodic function.

That is, a

finite product of the kernels is TIkI (WIS)TIk2(W2S) ... TIk (Wms) m

where

Ivi
< k

and is taken, for

m, independent numbers.

IVI!
wI,w2""'w certain real linearly m, This composite kernel has the same charac-

teristic properties as the Fejer kernel - it is never negative and its mean value is equal to 1.

This kernel is called the

Bochner-Fejer kernel and using it we can form a Bochner-Fejer polynomial by direct analogy with the purely periodic case, that is,

where f(t) is now an almost periodic function, f (t )

~

L

A

V

Thus we can write

I~I

L
exp[i\ tJ V

as

2. ALMOST PERIODIC FUNCTIONS

55

where as usual A(f.A) and each

A E A is written as V

A

V

In this representation for A the numbers v1.V2 •...• Vm are V rationals and the linearly independent set of numbers w1.w2 •...• w are called a base for A It follows from (2.12) that the f. m exponents of BF k k (t) are contained in the set of exk 1. 2 ... ·• m W1. W2.··· 'W

m

ponents of f(t). Introducing a simpler notation. the Bochner-Fejer polynomials are written in the form

Z d(k) A exp[i~ V

V

t]. V

where the d(k) satisfy the inequality V

0 s d(k)

S

V

finite number of them are different from zero. d

(k )

depend only on

klok 2 ..... k

V

and only a

We observe that

w1.w2 ..... w

v m• not on the values of the coefficients A.

1

m

and on A .but V

Bochner has shown

(Bochner. 1927) that the set of all Bochner-Fejer polynomials together with the set of f(t) are uniformly continuous and uniformly almost periodic.

Under these conditions. our main problem of

finding a sequence of Bochner-Fejer polynomials uniformly convergent to the function f(t) is equivalent to finding a sequence convergent in mean.

Thus the problem is given an

a Bochner-Fejer polynomials

Recall that the Fourier series of f(t) is 00

F (t )

Z v=\

Avexp [iAvtJ

~

> 0

to find such that

56

STABILITY OF LINEAR

and let

be a base of A f.

00

Z

V=V +1

IA I

2

o

Define v

o

so that

S.

<

v

~YSTEMS

(2.13)

2

Let m be the largest index of a's in the linear expressions of

A1,A 2 ,

•••

A v

,A ' so that we can write vo

r

(v)

(v= 1,2, ... , \) )

1

o

where all r's are rational. all the numbers

r~v)

(j

J

=

Let q be the common denominator of 1,2, .•. ,m;

R~v)

are integers.

J

Define numbers v ¢2

o

Z

v=1

IA I v

¢ > 0 and 2

<

=

1,2, ... ,v). 0

We write

am

.•• +

where all

V

q

Let R be the maximum of all N > 0

by the conditions

S.

IR~v)

J

I.

(2.14)

2

(2. 15)

Take N.

+

w. J

a·

= -l q

(j = 1,2, .•. , m)

and all

k,k, ... ,k

12m

greater than

By (2. 12) ,

L: V=V +1

d A

v v

exp riA t] v

o

where d differs from zero only for a finite number of values of v v and 0 ~ d v ~ 1 for all v > vo.

2. ALMOST PERIODIC FUNCTIONS

57

We have

00

+

L: (l-d ) v v=v + 1

2

o

By (2.13) , (2.14) and (2.15 ) V

2

s

Mt{lf(t)-BFk(t)I }

Therefore BFk(t) is a

¢2

polynom~al

o

L: v=1

00

2

+

IA)

L: IAV I v=v +1

2

<

i;

0

of the kind required and this

concludes the proof of Theorem 2.13. Note that Bochner's procedure not only proves the existence of trigonometric polynomials BFk(t) such that <

n

for some

n

> 0

I

s~plf(t)-BFk(t)

as small as we please, but also it gives a

definite algorithm for finding the BFk(t).

Furthermore, the

polynomials BFk(t) have exponents which are Fourier exponents of f (t.) .

An obvious corollary to theorem 2.13

~s

that almost periodic

functions are precisely those functions that can be uniformly approximated by trigonometrical polynomials.

Since we are

~n

the

main concerned with almost periodic differential equations, questions of differentiability and integration of Fourier series of almost periodic functions are of special interest. that f(t) and AP(C)

~s

f' (t) E AP(C),

the Fourier series of

Suppose f' (t) E

just the formal derivative of the Fourier series of

f(t), r.v e . A(f',,\) Given

iAA(f,A

f(t) E AP(C),

K),

i

R.

the simplest condition (Meisters, 1958)

STABILITY OF LINEAR SYSTEMS

58

t

on the Fourier series of f(t) which yields

f f(s)ds E AP(C)

~s

o

since t

F(t)

f f(s)ds

A

o

o

Ak

Although the numbers

occur

~n

the denominator of the Fourier

series of F(t), it does not effect the validity of the series since

A ~ O.

That is for

F(t) EAP(C)

it is necessary (but

not sufficient) that the Fourier exponents zero. In anticipation of the

ma~n

A of f(t) are nonk

theme in later chapters concerning

differential equations whose coefficients are almost periodic functions containing a parameter, in the next section we examine the fundamental properties of this class of functions. 2.4

Almost Periodic Functions Depending Uniformly on a Parameter In the study of vector differential equations with almost

periodic coefficients, the properties of vector continuous complex valued functions dependent upon a parameter vector x most important.

~s

Fortunately almost all of the properties of

almost periodic functions discussed in sections 2.2 and 2.3 can easily be extended to almost periodic functions dependent upon a parameter vector. f(t,x) (f:RxD

7

Consider the n-vector continuous functions

En)

where D is an open subset in En (more gene-

rally a separable Banach space) and

XED.

Definition 2.6: Almost periodic functions dependent upon a para-

meter

A function f(t,x) is called almost periodic with respect to

XED

if for any

n > 0

~n

t uniformly

and compact set FeD

there exists a positive number £n(F) such that any interval of

2. ALMOST PERIODIC FUNCTIONS

59

the real line of length £ (F) contains a T for which

n

- f(t,x)1 < n

t E R

for all n). f E AP(E

we denote

and for all

x E F.

If(t+T,X) In this case

Similarly following definition 2.1, the number n is called a n-translation number of f(t,x) and we denote by

T(n,f,F)

the

set of all n-translation of functions dependent upon a parameter which are identical to those discussed in section 2.2.

Note that

the translation numbers (almost periods) are again selected from a relatively dense set. Almost periodic functions dependent upon a parameter have a variety of continuity and algebraic properties which are readily n) For example if f E AP(E

derived from the above definition.

then f(t,x) is bounded and uniformly continuous on compact subset

D (x E D).

~n

functions E AP(E),

s.n

£(~'

f(t,x)

=

f)

n =

I

and selecting

Also for vector valued

for

{i.(t,x)}

each component

~

and conversely.

F any

These boundedness and continuity

conditions can be established by setting an interval

R x F,

f. (t,x) i.

Similar algebraic results to those of

theorems 2.3-2.6 also hold for the components f of f(t,x) E i n): AP(E i.e. if each f.(t,x) E AP(E) and for any g(t,x) E i. n) AP(E then o.f . (t,x), f. 2(t,x), f. (t ;») + g(t,x) and f. (t; ,x) i.

x g(t,x)

i.

i.

are all almost periodic in t uniformly for

some constants a and for all i. Inf Ig(t,x)

I

>

0,

i.

xED,

for

Moreover if

FeD,

tER xEF then xED.

f.(t,x)g(t,x) i.

-)

is almost periodic in t uniformly for

Some of the above results are given more formly in

Theorems 2.14-2.16 since they are of vital importance in the development of polynomial approximations of almost periodic functions dependent upon a parameter. The question of integrability n) of functions f(t,x) E AP(E will be dealt with in Chapters Four and Six.

60

STABILITY OF LINEAR SYSTEMS The next two theorems are of vital importance to the develop-

ment presented in Chapter six.

They define properties of almost

periodic functions containing a parameter which are essential in the problem of polynomial approximation.

By analogy with Theorems

2.1 and 2.2 we have: Theorem 2.14: (Corduneanu, 1968) If D is a compact set in En, then the function

f(t,x) sAP(E

n)

is almost periodic in t uniformly with respect to x, is bounded on RxD.

Proof: uses the same argument as in Theorem 2.1.

The details are

omitted. Theorem 2.15: (Corduneanu, 1968) Under the same hypothesis as ~n Theorem 2.18, it follows that n) is uniformly continuous on the set RXD.

f(t,x) s AP(E

Proof: Essentially the same as in Theorem 2.2.

The details are

omitted. The next theorem we present in this section is analogous to Theorem 2.4 for almost periodic functions without parameters, and tells something of the properties of convergent sequences of almost periodic functions containing a parameter. Theorem 2.16: (Corduneanu, 1968) If a sequence of almost periodic functions formly dependent on the parameter

x

S

{fk(t,x)}

uni-

D is uniformly convergent

on RxD to the function f(t,x), then f(t,x) is also almost periodic in t uniformly with respect to

Proof: Given

n,

!f(t,x) - f

ko

x s D.

there exists a function (t,x)

I

<

n 3 '

f

for all

ko

(t,x)

such that

(x,t) s DXR.

Let now x be an almost period from the set of translation numbers 1

T(3 n,f

ko

(t,x».

Then

If(t+T,x) - f(t,x)1

s

If(t+T,X) - f

ko

(t+T,X)

+ If k (t+T,X) - f (t,x)1 + If (t,x) - f(t,x)1 k0 ko o

I <

61

2. ALMOST PERIODIC FUNCTIONS which shows that T(ll,f(t,x) and this proves the theorem.

The relationship between the exponents of two almost periodic functions

f(t,x),

g(t,x)

dependent upon a parameter vector x,

is given in the following theorem due to Favard (1933): Theorem 2.17: Module containment Let f(t,x) and g(t,x) be almost periodic in t uniformly for x s D,

then for any compact set

implies

mod(f,D)

~

mod(f,F)

~

mod(g,F)

mod(g,D).

Thf(t,x) = f(x,t)

Froof: Let

FeD,

uniformly on RXD then by the equi-

valent statements on module containment following definition 2.6. Thg = g

Thf(g,t) = f(g,t)

uniformly, and also

mod f(g,t) c mod(f)

or equivalently

mod(f,D) c mod(g,D).

The fourier series theory'for functions pendent upon a parameter

x

S

D

so that

n) f(t,x) S AP(E

de-

En, is in general a direct ex-

c

tension of the Bohr transform theory of section 2.3.

The parallel

Bohr transform of an almost periodic function to that of (2.3) n) for f(t,x) S AP(E uniformly for x S D, a compact subset of En ~s w lim

A(f,A,x)

w-+<'O

where for each A(f,A,x) ous in x. A(f,A,x)

1

w

f

(2.16)

f(t,x) exp(-iAt)dt

0

(A,f,x), A(f,A,x) S En,

exists uniformly for x

~n

and for any

A s R,

compact sets and is continu-

This latter condition on the fourier coefficient follows directly from Theorem 2. II.

Hence we have a

formal correspondence between an almost periodic function

f(t,x)

and its fourier series 00

f(t,x)

~

Z A(f,AK,x) exp(iAKt)

(2.17)

K=I

The definitions for sets of exponents and the module of f(t,x) s AP(En) follow identically to those of definitions 2.3 and 2.4.

62

STABILITY OF LINEAR SYSTEMS

Finally we note that both BesseVs inequality and Parseval's equation hold in the mean for any f(t,x) E AP(En). 2.5

Bochner's Criterion In certain applications, notably in the study of differential

equations, Bochner (l927) found it useful to characterise almost periodic functions by means of a compactness criterion.

This

definition plays an essential role in the general theory of almost periodic functions and leads to results which are not easily derived from the Bohr definition.

The starting point consists of

considering together with a given continuous function the set of its translates

{fT:fT(t) = f(t+T),

f:R

~

C

T E R} = S(f)

and its closure S(f) in the topology of uniform convergence. Definition 2.7: Almost periodic function (Bochner, 1927) The continuous function

f:R

C

~

is almost periodic if from

every sequence of real numbers {h } one can extract a subsequence {h

nk

} such that

lim f(t+h

k~

n

nk

) = get)

exists uniformly on R.

To facilitate further discussion and remove the necessity of writing double subscripts, we introduce the following notation

{S }, then a. + S = {a.n +Sn }. a. = {a } and S n n } is a subsequence of {a}. If will mean that {S n n Se S' , we say that ex and S are common subsequences

for sequences: if Also

S ea

ex e a.'

and

of ex' and S' if

{o.k} = {a. } and {Sk} = {S } for a cornmon nk nk Finally we define the translation operator

set of indices {~}. Thf = lim to:

f(t+~)

which notationally simplifies definition 2.7

k~

Definition 2.8: Almost periodic function (Fink, 1974) The continuous function

f:R

~

every sequence h' there exists an

C

is almost periodic if for

h e h'

such that

Thf

exists

uniformly. Not only is this definition equivalent to Bohr's definition of almost periodic functions, it is also the definition of f(t) being normal (Bochner, 1927) and each of these definitions implies

2. ALMOST PERIODIC FUNCTIONS the characteristic properties of the others.

63 The equivalence of

the Bochner definition of almost periodicity (given essentially by the space AP(C»

and the Bohr definition is summarised in the

following theorem:Theorem 2. 18:

1974)

(Fink~

For any continuous complex valued function E AP(C)

then

f:R~C,

f

if and only if f(t) is Bohr almost periodic.

It is clear that through the use of the translation operator we are able to generate new almost periodic functions from f(t). The collection of all such functions g such that there is an h for which

Thf = g

noted by R(f). f(t)

For general almost periodic functions such as

sin t + sin

=

uniformly is called the hull of f and is de-

translate of f.

12

t

~n

R(f), g can be shown not to be a

It is not difficult to show that R(f) is a metric

space which is compact in the uniform norm if and only if E AP(C), this is equivalent °to the set of translates

fT(t)

=

f(t+T), T E R}

Neumann, 1935). if

f E AP(C)

f(t)

S(f) = {f

T

being totally bounded (Bochner and Von

Clearly from the definition of the hull, R(f), then for any

g E R(f),

R(g)

R(f).

The space AP(C) is an algebra over C, since for f and g both E AP(C),

thence

Th(f+g) + Thg = Th,f + Th,g (f+g)

E

AP(C),

also

Th(fg)

uniformly for

= Th(f)Th(g)

ly so that AP(C) is closed under products.

he h'

and

exists uniform-

Then clearly finite

sums of periodic functions are almost periodic functions and we may conclude that AP(C) is a complete metric space (Banach space) that contains all periodic functions.

It is in fact the smallest

completely normed space with this property and most importantly for our purposes, AP(C) is closed under differentiation and integration (Corduneanu, 1968). simply from the fact that all

The differentiation property follows f E AP(C)

are bounded and uni-

formly continuous, then if the derivative f' exists everywhere then

f' E AP(C).

To complete this section we present a further definition of almost periodicity which relies only on pointwise convergence~

64

STABILITY OF LINEAR SYSTEMS

although uniform convergence is implied. Definition 2.9: Almost periodic function (Bochner. 1962) The continuous function

f:R7C

every pair of sequences hI" quences

hI

C

hI'

and

h

2

h C

h

2'

is almost periodic if from one can extract common subse-

2'

such that

f

pointwise. The above condition is both a necessary and sufficient condition for convergence whether it be pointwise or uniform on compact sets.

The utility of Bochner's pointwise definition for almost

periodicity is revealed in Chapter Six in the context of differential equations with almost periodic coefficients. 2.6

Limiting Cases of Almost Periodic Functions The concept of asymptotically almost periodic functions f(t)

defined on the half real line R+ = [0,00) with values in En n) (f:R++C was first introduced by Frechet (1941) and dealt with in some depth by Corduneanu (1968).

For completeness we intro-

duce its main properties. Definition 2.10: Asymptotically almost periodic functions A continuous valued function

n

f:R+ + C

is said to be asymp-

totically almost periodic if it is the sum of a continuous almost periodic function

pet)

En

S

which tends to zero as t+oo. f (r ) if

p (t )

P (t.) + q(t) S

n), AP(C

This decomposition

S

and a continuous function

+

n + C

i.e. AAP(C

n q:R+ + C ~s

q:R

n) and

unique if

lim q(t) t+oo f(t)

€

AAP(C

0. n)

and f(t) is

bounded and uniformly continuous on R+ [0,00). Equally, if f n) and its derivative f' are both AAP(C then the decomposition of the derivative f',

65

2. ALMOST PERIODIC FUNCTIONS p'(t)+q'(t)

f' (t )

is also unique with

p'(t)

€

AP(C

n).

We can relate the above definition of asymptotically almost periodicity to the Bohr and Bochner definitions of almost periodicity through the following definitions: Definition 2. II: Property P (Corduneanu~ 1968) n f:R+ ~ C has property P if given an n > 0

£ > 0 n

and a

T(n) 2 0

there is a

such that every interval of length £

on R+ contains a L such that If(t+L) - f(L)1

n

<

for

2

t

n

T(n).

Definition 2.12: Property L (Corduneanu~ 1968) n f:R+ ~ C has property L if for any real sequence {h'k} such that

h

k' quence {h

> 0

k}

C

and~'

{h

a

~

k'}

as

k

~

00,

such that f(t+h

k)

we can select a subseconverges uniformly on

R+, i.e. for

get)

Definitions 2.10-2.12 are equivalent (Yoshizawa, 1975) so that n any f:R ~ C which has either property P or L is also asymp+

totically almost periodic. A special case of almost periodic functions dependent upon a n (f:RXE ~ En) parameter are quasi-periodic functions f(t,x) k (Nakajima, 1972). Let I. be a unit vector in D such that the J

jth component is k

E

and the others are zero.

such that all of the components are one (k

Let I be a vector S

~n

n).

Definition 2.13: Quasi-periodic functions n The function f(t,x) (f:RxE ~ En) is said to be quasi-periodic ~n

t if there is a finite number of non-zero real numbers

... w and a function F(z,x) where F(z,x) k' such that F(z+w.I.,x)

F(z,x)

J J

xED,

and

j

=

J,2, ... n

for all

Z E

w ,w ~s

66

STABILITY OF LINEAR SYSTEMS

and F(tI,x)

f(t,x)

t

for all

R

€

and

x

€

D where

D is an open subset in En. The most important result on quasi-periodic functions for our purposes is the existence of a fourier series:Theorem 2.19: (Nakajima, 1972) n Let f(t,x) (f:RXE + En) for

x

D an open subset in En.

€

The function f(t,x) is quasi-periodic in t if and only if it is almost periodic in t uniformly for mod(f) f (t , x)

E

x

€

D

and its module,

{2Tf , ... ,2Tf} has a finite integrable base. That is w1 w k n) QP(E is almost periodic with fourier series

L

f(t,x)

m

A

m (x,A)

eXP{2Tfit{;~

+ •.. + ::}},

where

w .,w are real pos{tive numbers, k' 1'" for integer m., and ~

Am(x,A)

2.7

~

i

=

FT

m

w

lim w-+ro

~

J

f(t,x) exp(-At)dt

# O.

o

Notes The functions studied by Bohl, the so-called quasi-periodic

functions, are a special class of almost periodic functions whose structure closely resembles the limit periodic functions introduced by Bohr.

Perhaps this observation conditioned Bohr's ori-

ginal thinking and led to his investigation of diagonal functions of limit periodic functions of many variables.

Work similar to

Bohl's was performed by M. E. Esclangon and is reported in two readily accessible papers: "Sur une extension de la notion de periodicite", Comptes Rendus Acad. Sci., Paris, 1902, and "Sur les fonctions quasi-periodiques moyennes deduites d'une fonction quas Lr-per i od i qua'", Comptes Rendus Acad. Sci., Paris, 1913. The early papers by Bohr, "Sur les fonctions presque periodiques"

2. ALMOST PERIODIC FUNCTIONS

67

and "Sur l'approximation des fonctions presque periodiques par des sommes trigonometriques", both in Comptes Rendus Acad. Sci., Paris, 1925, seem to substantiate the view that he was inspired by Bohl's and Esclangon's work; the following extract is taken from the second paper ... les recherches tres interessantes de ces geometres (Bohl and Esclangon - our brackets), les fonctions quasi periodiques peuvent se representer par la forme f(t)

= f(t,t, ... ,t),

est une , •.• ,t 1,t 2 m) fonction continue des variables t 1,t 2 , ... ,t qui m, est rigoureusment periodique par rapport a chacune de ses m variables.

ou

f(t

C'est

a

l'aide de cette repre-

sentation que Bohl a obtenu son resultat principal, que les fonctions quasi periodiques sont identiques aux fonctions f(t), qu'on peut approcher d'une maniere uniforme par des sommes trigonometriques de B exp!i(k1W 1 + k 2 w2 + ... + kmWm)t!. k Pour les fonctions presque periodiques dont les

la forme

L

exposants

Ak possedent une base entiere quelconque,

on peut generaliser Ie resultat de Bohl et Esclangon de la maniere suivante: Tout fonction presque periodique f(t) appartiennent des exposants

a

laquelle

A avec une base entiere,

k peut se representer sous la forme f tt )

ou

f(t

f(t,t,

,t, ... )

, ... ,t

) est une fonction uniforme1,t 2 m, ment continue d'une infinite de variables reeles, qui est rigoureusement periodique par rapport

a

toutes les variables. The Fourier series theory of almost periodic functions was developed soon after Bohr's original work, and apart from Bohr's own efforts in this direction, a valuable paper is S. Bochner, "Properties of Fourier series of almost periodic. functions", Proc.

68

STABILITY OF LINEAR SYSTEMS

Lond. Math. Soc. (2), 26, 433 (1927)

~n

which the tests for con-

vergence and summability, arithmetic properties and polynomial approximation has been dealt with by many authors.

In his paper

"Localisation of best approximation", Ann. Math. Stud. 25, Princeton, 1950, S. Bochner investigates polynomial approximation by considering the convergence and summability of Fourier

ser~es.

Throughout this chapter little has been said about the almost periodicity of integrals of almost periodic functions since, in general terms, little is known about the problem.

Certainly

~n

Euclidean spaces the boundedness of the integral is necessary and sufficient to ensure its almost periodicity, although extension to general metric spaces is difficult.

For more on this see the

book by L. Amerio and G. Prouse, "Almost periodic functions and functional equations", Van Nostrand Reinhold, New York, 1971. For our purposes the most easily interpreted conditions for the almost periodicity of such an integral are derived from the Fourier series theory.

The conditions of such an integral are

derived from the Fourier series theory. simple and there are many of them.

The conditions are not

A recent comment on this is

by G. H. Meisters, "On the almost periodicity of the integral of an almost periodic function", Am. Math. Soc. Notices, 5, 683 (1958), which may be consulted with profit.

An earlier paper

which is also useful is by S. Bochner, "Remark on the integration of almost periodic functions", J. Lond. Math. Soc., 8, 250 (1933), which is really a comment on the differential equation f(t)

~(t)

=

with f(t) almost periodic.

Generalisations of Bohr's theory have appeared and different forms of convergence.

ar~se

from

A good review of this develop-

ment is A. M. Fink, "Almost periodic functions invented for specific purposes", SIAM Review, 14, 572 (1972).

To conclude this

section it is certainly worth noting that one of the richest sources of material on almost periodic functions is "Collected Mathematical Works" of H. Bohr, Copenhagen, 1952.

2. ALMOST PERIODIC FUNCTIONS

69

References Amerio, L. and Prouse, G. (1971). "Almost Periodic Functions and Functional Differential Equations", Van Nostrand and Reinhold, New York Besicovich, A.S. (1932). "Almost Periodic Functions", Cambridge Univ. Press Bochner, S. (1927). Math.Ann.~ 96,119 Bochner, S. (1927). Proc.London Math.Soc.Ser.2, 26, 43 Bochner, S. (1933). J.London Math. Soc. , 8, 250 Bochner, S. (1950). Ann. Math. Stud. , 25, Princeton Univ. Press Bochner, S. (1962). Froc. Nat. Acad. Sci. , 48, 2039-2043 Bochner, S. and Von Neumann, J. (1935). I.Ann.Math., 36, 255-290 Bohl, P. (1906). J.fur.Reine U.Angew.Math., 131, 268 Bohr, H. (1924). Acta Math., 46, 101 Bohr, H. (1925). "Sur les Fonctions Presque Periodiques", Comptes Rendus Acad.Sci., Paris Bohr, H. (1925). "Sur l'approximation des Fonctions Presque Periodiques par des Sommes Trigonometriques", Comptes Rendus Acad.Sci., Paris Bohr, H. (1952). "Collected Mathematical Works", Copenhagen Corduneanu, C. (1968). "Almost Periodic Functions", Wiley Interscience, New York Esclangon, M.E. (1902). "Sur les extension de la Notion de Periodicite", Comptes Rendus Acad.Sci., Paris Esclangon, M.E. (1913). "Sur les Fonctions Quas i-rpeir i.od i que s Moyennes Deduites d'une Fonctions Quasi-periodiques", Comptes Rendus Acad.Sci., Paris Favard, J. (1933). "Le¥ons sur les Fonctions Presque Periodiques", Gauthier Villars, Paris Fink, A.M. (1972). SIAM Review, 14, 572 Fink, A.M. (1974). "Almost Periodic Differential Equations", Springer Verlag lecture notes in Mathematics, No.377, New York Frechet, M. (1941). Rev.Scientifique, 79, 341-354 Jensen, B. (1949). Det.KGL~ Dan Ke Viden. Selskob.Mat-Fys.Meddel, 25, 1-12 Meisters, G.H. (1958). Amer.Math.Soc., 5, 683 Nakajima, F. (1972). Funkcial Ekvac., 15, 61-73 Yoshizawa, T. (1975). "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions", Appl.Maths. Sci., No.14, Springer Verlag, New York

Chapter 3

PROPERTIES OF ORDINARY DIFFERENTIAL EQUATIONS

3.1

Introduction Continuous dynamical systems with a finite dimensional state

vector are often represented by an ordinary differential equation. n 1 A real scalar t (called time) and an open set D in R + are ~n n volved, where R is the space of real n-dimensional column vecn tors. An element of D is written (x,t). Let f:D ~ R be continuous and let x denote dx/dt.

Then a general differential

equation of the type described is a relation of the form x(t)

f(t,x(t)),

(3. I)

or briefly x

==

f(t,x)

If x is a continuously differentiable function defined on (x,t) E D,

t E J

J c R,

and x satisfies (3. I), we say that x is a

solution of (3.1) on an interval J. Given

(x ,t ) E D,

containing t

o o

0

the problem of finding an interval J

and a solution x of (3.1) satisfying

is called an initial value problem for (3.1).

x(t ) o

= x0

If there exists a

solution to the initial value problem, we refer to this as the solution of (3.1) passing through (x ,t). itial value problem is written

o

0

Symbolically the in-

3. ORDINARY DIFFERENTIAL EQUATIONS f(t,x) ,

x

which

x(t )

x , o

o

t

E J

71

(3.2)

equivalent to

~s

t

x(t)

f

+

X

o

t

(3.3)

f(s,x(s»ds o

provided that f(t,x) is continuous 3.2

Existence and Uniqueness of Solution Our main concern in this section is to establish general the-

orems about the existence and uniqueness of solutions of the initial value problem (3.2).

The general questions of continuation

of solutions and continuous dependence of solutions on initial data are not considered here; the interested reader should consult Coddington & Levinson (1955) for the relevant discussion and technical details. The presentation in this section is adapted from Coddington & Hale (1969), Hille (1969), Rosenbrock & Storey

Levinson (op ci.ii.) s

(1970), Curtain & Pritchard (1977) and Birkhoff & Rota (1978). The first theorem we shall derive asserts the existence of at least one solution to the initial value problem (3.2) if f(t,x) is continuous in D. Theorem 3.1: Existence (Peano) Suppose that f is defined and continuous in D = {(x,t): Ilxn. x E: R , It-t I < a, t E: R}, also suppose that x 0 II < 13, 0 Ilf(x,t)11 <; m in D. Then there is at least one solution of (3. I) passing through (x ,t ) which is defined on the interval t +rJ o

r

o

where <

-1

min(a,l3m

0

)

[t-r 0

'

(3.4)

Proof: Hale (op oi.t., ) gives a proof which uses the Schauder fixed s

point theorem (see Chapter One).

The following proof is taken

from Hille (op.cit.), who demonstrates existence on the interval [to,to+rJ.

The extension to

[to-r,toJ

is immediate.

Divide

STABILITY OF LINEAR SYSTEMS

72

the interval t

[ t , t +r] o 0

to + j2

jk

-k

into 2

k

equal parts and set

r

(3.5)

Define a sequence of piecewise linear functions {~(t)}

by the

following formulae: x

~(t)

+ f(t ,x )(t-t )

000

0

t

~(t)

where

Jk O

~

t

t.

~

J +1,

0.6)

k 1,2,3, ... ,2 -I.

j

k

We now show that the functions equicontinuous in [t ,t +r]. o 0

are uniformly bounded and

~(t)

In the jth interval -k m2 r

(3.7)

which gives

~

(j+l)m2

-k

r

mr

~

< S

(3.8)

by the choice of r.

ded in [t ,t +r]. with

o

1~1-~21

0

~

2

-k

It follows that the Now let r.

~l

and

~2

~(t)

are uniformly boun-

be two points in [t ,t +rJ 0

0

Thus they are either in the same or in

adjacent subintervals.

In either case (3.9)

We can dispense with the restriction if

1~1-~21

> 2

-k

r,

the interval

subintervals of length < 2

-k

[~1'~2]

~

-k

2

r

because

can be broken up into

rand (3.9) can be used for each sub-

interval.

The resulting inequalities add to give a sum

(~1)-~(i;2)

lion the left, while on the right we still get

~

II~

73

3. ORDINARY DIFFERENTIAL EQUATIONS ml~1-~21.

Hence, for any choice of ~1,~2

any choice of k, (3.9) holds.

in [t ,t +r] and for o

0

Therefore the functions

~(t)

sa-

tisfy a fixed Lipschitz condition and form an equicontinuous family. Applying the Arzela-Ascoli theorem 1.2, we can find a subsequence

{~

Ki

(t)} which converges uniformly in [t ,t +r] to a con0

tinuous function x(t).

0

There remains the question of the differentiability of x(t). At the partition points t

the derivative does not exist, but jk we have left and right-hand derivatives (Rosenbrock & Storey, op.

cit.) and for large values of k these differ very little. ~

It

E

o

,t +r) 0

+

!!if ~(O for

For an

+

~§g

~

t

(3.10)

f (~,x(O)

x(O

o

Suppose now that <

> t

~

~

:S

is such that

o

t. J,k.

i.

Then we have

fg-~.

(0

(3. II)

f(t'_ 1 k ,xk (t'_ l k ) J 'i i J , i

~

!!lJ-x (0

!

k

if

.

fCt j , : .

''k. (tj,k.))

~

~

~ f t. k

J, i

if

(3.12)

t. J,k.

~

~

Now

II x k . (t j _ I ,k. ) -x (0 II ~

+

:S

II~. i.

~

II~.

(t j _ 1 ,k.

i.

)-~.

(0

11

~

II

(~)-x(O ~

-k' m2 ~r

+ II~.

(O-x(O

II

~

-+

0

as

i

-+

00.

(3.13)

74

STABILITY OF LINEAR SYSTEMS

This

that

implie~

f(t'_ 1 k ,~ J

'i

i

(t'_ l k» J 'i

since f 1S continuous in D.

~

(3.14)

f(~,x(~»

Thus ~( t ) exists for all

t E: [t 0'

and satisfies (3. I) .

to+rJ

I t 1S interesting to note that any limit function of the se-

quence {~(t)}

is a solution to the initial value problem.

Thus

if the sequence does not have a unique limit, the solution is not unique.

Clearly the conditions on f are not strong enough to gua-

rantee uniqueness and additional conditions are required.

A con-

venient condition on f is that it satisfies a local Lipschitz

condition, which is defined as follows: Definition 3.1: Local Lipschitz condition n+1

A function f(t,x) defined on a domain

Dc R

is said to

satisfy a local Lipschitz condition 1n x if for any compact set U c D,

there is a z such that

Ilf(t,x)-f(t,y)11 for

s

(3.15)

zllx-yll

(x,t),(y,t) cu. Note that if f(t,x) has continuous first partial derivatives

with respect to x in D, then fCt,x) is locally Lipschitzian in x. A basic existence and uniqueness theorem under the hypothesis that f(t,x) is locally Lipschitzian in x, is derived from the

Method of Successive Approximations.

This technique is usually

attributed to Picard, and leads to the following: Theorem 3.2: Existence and Uniqueness (Picard) n Suppose that f:D ~ R is defined and continuous 1n t): Ilx-xo II < B,

II f (t , x) II

s

n

x E: R ;

< a,

t

E: R}

and suppose that

m

Ilf(t,x)-f(t,y)11 there.

It-t o I

D = {(x,

s

zllx-yll

Then there is a unique solution of (3. I) passing through

(x ,t ) which is defined on the interval (t -r,t +r) where o

0

0

0

3. ORDINARY DIFFERENTIAL EQUATIONS <

r

-1

min(a.,Sm

75

)

Proof is adapted from Hille (op.cit.).

The initial value problem

(3.2) is replaced by the equivalent integral equation t

xt t

)

X

f f(z,x(s»ds

+

o

t

(3.16)

o

whose solution will be obtained as the limit of a sequence of } defined by

functions {~(t) X

o

x

(t; )

0

t

x

~(t)

+

0

f f(s'~_I(s»ds t

for

(3.17)

0

1,2,3, ...

k

Assume that

~_I(t)

Clearly, x

I(t) = x , but the induction hypothesis must also

k-

is well defined on the interval of interest. 0

0

include that ~_I(t) we see that

11~_I(t)-xoll

is continuous and

< S.

is well defined and continuous.

f(s'~_I(s»

Ilf(s'~_I(s»11

"

Thus Also

m

whence

f(s'~_1

II ( t

t

(s»dsll

" f t

o

<

by the choice of r. and satisfies

o

mr

<

b

This implies that

x, (t; ) k

Ilf(s'~_I(s»llds

0

=

x, 0

(3.18)

is also continuous,

~(t)

Ilxk (t)-x0 II

< S.

Thus the approxi-

mations are well defined for all k. Using the assumption that f(t,x) satisfies a local Lipschitz condition, we have

76

STABILITY OF LINEAR SYSTEMS t

11~(t)-~_I(t)11

f [f(S'~_I(S»-f(S,Xk_2(S»]dSII

1\

t

o t

S

z

f II ~-1 t

(s)-~_2

(3.19)

(s ) II ds

o

k = 1 we have the estimate

For

t

II xl (t)-x o (r )

II f f (s ,xo (s ) )ds II

II

t

o

for

I t-t o I

s

zmlt-t I o

(3.20)

< r

Proceeding by induction, we assume that

II~_I

II

(t)-~_2(t)

(3.21 ) for

It-t I < r o

and infer that

11~(t)-~_1

(t )

II

s

k-I k -z-mlt-t I k! 0

(3.22)

so the estimate holds for all k. It follows that the series 00

xo(t) + k:1

[~(t)-~_1

(3.23)

(t»)

whose k t h partial sum is ~(t), r

converges uniformly for It-tol < n. x:R ~ R From (3.22) we note

to a continuous limit function

that k-I

k

~ ~k' . m0 exp(z\t-t 0 \)!t-t I and observe that if It-tol is not large, ~(t)

(3.24) converges rapidly

to the limit x(t). The uniform convergence of

~(t)

to x(t) implies the uniform

3. ORDINARY DIFFERENTIAL EQUATIONS convergence of

to

f(t'~_I(t»

t

J f(s,x(s»ds

+

t

o

uniformly

and

t

J f(s'~_I(s»ds t

f(t,x(t»,

77

~n

t.

0

Hence x(t) satisfies the integral equation (3.3)

and consequently is a solution to the initial value problem (3.2). To show uniqueness, suppose that there are two solutions Xl (t), x 2(t) defined in the interval (t -r,t +r). o

0

Then we have

t

f [f(s,XI(S»-f(s,x2(s»]ds

II Xl (t)-x2 (t ) II

t

o

for

It-t I < r o

Hence t

Il x I ( t ) - X2 ( t ) 11

s

z

f

Il xI(S)-x 2(s)llds

to by the Lipschitz condition.

(3.25)

To complete the proof of uniqueness

requires the application of the following lemma: Lemma 3.1:

Gronwall-Bellman (Bellman, 1953)

Let g, f and X be continuous functions on [t ,tl] to R. f(t)

?

o

0

on [to,t l] and x(t) has the property that for

If

t s

[to,t l] t

x(t)

S

get) +

f f(s)x(s)ds t

(3.26)

o

then on the same interval

xC') , gC,)

+

f'gc,),c,)exp[f:cC)dT]d' t

o

C3.27)

s

Proof: This lemma is very important

~n

stability studies since

it yields an explicit inequality with respect to x from an implicit inequality.

The function

STABILITY OF LINEAR SYSTEMS

78 t

f f(s)x(s)ds

y (t.)

t 1S

(3.28)

o

continuous and continuously differentiable in

and

y(t) o

x(t)

=

S

t

E

[to,t 1 ]

Moreover we can rewrite (3.26) as

O.

get) + yet).

(3.29)

f (t)x(t)

(3.30)

Since y (t )

and f(t)x(t)

f(t)g(t) + f(t)y(t)

S

(3.31 )

it follows that yet)

S

f(t)g(t) + f(t)y(t)

(3.32 )

Consider now t

= y(t)exp[-

z (t )

f f(s)ds]

(3.33)

to

Then

t

y(t)exp[-

z(t)

f f(s)ds] t

o

f(t)g(t)exp[-

t

o

and

z(t ) o

= O.

t E [to,tl] S

f f(s)ds]

o

(3.34)

o

Integrating both sides of (3.34) between gives t

f f (s)g(s)exp [ f f(T)dT]ds t

f f(s)ds] t

t

z (t; )

- y(t)f(t)exp[-

t

t

Obviously

t

(3.35)

s

o

whence, by (3.33) t

yet)

S

f t

t

f(s)g(s)exp[ o

f s

f(T)dT]ds

(3.36)

3. ORDINARY DIFFERENTIAL EQUATIONS Therefore x( t which

t

get) +

t

f f(s)g(s)exp[ f f(T)dT]ds s to

)

«

~s

the required inequali t y ,

To return to the proof of theorem 3.2, we set x2(t)

II,

get) '" 0

II xl (t) - x2 (t.)

and

II

79

:s

f Ct ) '" z

x Ct ) '" IlxI (t.) -

in (3.26) and deduce that

0

(3.37)

so that

and uniqueness is proved. It must be emphasized that the above theorems establish only the local existence of solutions to the initial value problem. In general, the continuation of solutions for ficult and is not considered here.

It-t

o

I

>

r

is dif-

The continuous dependence of

solutions on initial data is contained in theorem 3.2 by virtue of the Lipschitz condition satisfied by f(t,x), although the question in general is complicated.

However, on infinite time inter-

vals we arrive at the concept known as stability in the sense of

Liapunov, a topic we will return to in Chapter Five.

Theimpor-

tant case when f(t,x) is periodic or more generally almost periodic can not be treated in this chapter since the existence conditions assume some form of stability property for the solution to (3.1) and is therefore left until Appendix One.

3.3

Linear Ordinary Differential Equations In the remaining sections of this chapter we are going to

examine in great detail the properties of linear ordinary differential equations.

These are of great importance because they are

very frequently used to represent the dynamical behaviour of many physical systems encountered in engineering practice.

Further-

more, we shall see that it is often possible to give explicit

80

STABILITY OF LINEAR SYSTEMS

solutions to equations of this type.

First of all we recast the

existence and uniqueness theorem (theorem 3.2) for linear equations. nary

The crucial point is that the solution of a linear ordidifferential equation is defined wherever the equation is

defined.

If we denote M as the set of nXn bounded and continun

ous real or complex valued matrix functions of the real variable t, the existence and uniqueness condition for the linear case is given by: Theorem 3.3: Existence and Uniqueness - Linear Case M is defined and continuous in J = n n {t: It-t I < a, t E R} and suppose that (x ,t ) E R x J. o 0 0 Then there is a unique solution of A:J

Suppose that

~(t)

-+

(3.38)

A(t)x(t)

passing through (x ,t ) which is defined on the whole interval jt-t

I o

o

< a s

0

00

Proof: As in the proof of theorem 3.2 the method of successive

approximations can be used.

The details are given in Brockett

(1970).

Note that on compact intervals

n 2N

la ..

if

~J

I

S N

II All

L: max i ,j t

=

the interval of interest.

~n

Ia .. (t ) I ~J

Hence the

Lipschitz condition IIA(t)x - A(t)Y11

s

(3.39)

zljx-yll

is clearly satisfied with

z

= n 2N and guarantees uniqueness.

The next step, having established existence and uniqueness for the linear case, is to consider the problem of constructing explicit solutions to the initial value problem.

The technique is

contained in the following: Theorem 3.4: Solution space Let S denote the set of all solutions of equation (3.38), of the equation x

A(t)x

with A defined and continuous as above.

In other words,

~.e.

3. ORDINARY DIFFERENTIAL EQUATIONS {x(t): ~(t)

S

= A(t)x(t),

81

(3.40)

t E J}

Then S is an n-dimensional vector space, and a basis

{Xl(t),

X2(t), ... ,x

of S may be obtained by letting xi(t) be the n(t)} unique element of S which satisfies the condition x , (t )

where

(3.41 )

e.

0

~

~

n. is the natural basis of R

{el,e2,e3" .. ,e} n

Proof: Clearly the function

x(t) = 0

for all

This function is called the trivial solution. ~,~

E

Rand

x(t),y(t)

S,

E

ddt U;;x (t ) + t;y (t ) ]

t

E

J

~s

~n

Furthermore, if

then

Z;;x (t; ) + t;y (t )

I;;A(t)x(t) +

~A(t)y(t)

A(t ) [I;;x (t ) + t;y (t ) ] so that

S.

[I;;x(t) + t;y(t)]

E

(3.42)

S., It

i

s a simple matter to verify

that S forms an additive group under addition and that scalar multiplication has the following properties: (i) (ii) (iii) (iv) where

I;;[x(t)+y(t)]

I;;x(t) + I;;y(t)

(I;;+t;)x(t)

I;;x(t) + t;x(t)

I;; [t;x(t)] 1x(t)

(3.43 )

[I;;t;]x(t) x(t)

I;;,t; E Rand

is the identity of R.

We now show that the x. (t) are a basis of S. ~

First of all

suppose that n I

i=1 Then

c. x. (r ) ~

~

0

Vt E J

n I

i=1

(3.44)

n c.x.(t) ~

~

0

I

i=!

c.e. i.

~

and so the scalars c. are all O. ~

linearly independent functions.

0

(3.45 ) This implies that the x. (t) are ~

Now, if x(t) is any element of S

STABILITY OF LINEAR SYSTEMS

82

then x(t ) can be constructed by taking linear combinations of the e

o

that is

i,

n

x(t )

e.e.

L

o

~

i=1 n

The function (3.46).

(3.46)

~

e.x. (t )

L

~

i=l

~

is

S and agrees with x(t) at t

~n

o

by

Hence it follows from the uniquess part of theorem 3.3

that

n L

x (t )

i=1

e.x. (t; ) ~

Vt

~

E:

(3.47)

J

and the assertion is proved. This theorem leads us to the important definition (Yakubovich and Starzhinskii, (1975»: Definition 3.2: Fundamental matrix Let J

be the nXn matrix whose jth column ~s

¢(t,t )

x , (t ) E:

S,

o

x.(t ) = e ..

with

J

That is to say, the columns of

J

0

the vector

¢(t,t ) are n-linearly independent solutions of (3.38) satisfying o

the initial condition

x.(t) J

0

=

e., j

=

J

1,2, ... ,n.

Then we call

¢(t,t ) the fund(ffTIental or transition matrix of the linear ordio

nary differential equation (3.38). Note that according to the above definition, where I is the identity matrix.

¢(t,t) = I, o

0

It is clear from the definition

of the fundamental matrix that it may be considered as the unique solution to the initial value problem ¢(t , t )

A(t)¢, Moreover, if

o

(x,t) o

0

0

E:

n R x J,

I,

(3.48)

E: J

t

it follows that the solution of

equation (3.38) passing through (x ,t ) is given by o

x

Ct)

¢(t,t )x o

0

(3.49)

0

Next we aim to show that the matrix ¢(t,t ) is nonsingular for all

t

E:

J.

o

Use will be made of the following lemma whose origin

is attributed to Abel, Jacobi and Liouville.

85

3. ORDINARY DIFFERENTIAL EQUATIONS

Lemma 3.2: Abel-Jacobi-Liouville Suppose that

) is a fundamental matrix for o is defined and continuous in J. Then

A:J + M n

x

~(t,t

A(t)x.

t

det

~(t,t

o

)

exp [

f t

for all

t,t

o

tr

(3.50)

A(S)dS]

o

E J.

Proof: Brockett (op.cit.) demonstrates this lemma by showing that det~(t.t •

w

=

0

scalar initial value problem

) is a solution to the

[tr A(t)]w,

W 1, to E J. This lemma is important in o stability studies for non-autonomous systems since it is clear

that the linear system (3.38)

s~nce

det~(t.t)

o

Theorem 3.5

unstable if

~s

lim Re{ltrA(T)dT} t-+oo

t

o

is unbounded.

Suppose that

~(t.t ) is a fundamental matrix for x o A:J + M ~s defined and continuous in J. Then n nonsingular for all t E J.

where

Proof: follows immediately from lemma 3.2, for all

det~(t,t

~.e.

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

o

o

)

) is

# 0

t t: J.

Having established that

) is a nonsingular mat~ix. it is o of interest to try to find its inverse. As a consequence of the ~(t,t

definition of

~(t,t ), which includes uniqueness, it is intuitiveo ly clear that fundamental matrices satisfy the following composi-

tion rule: ¢(t,t ) o

(3.51 ) for

Since

¢(t ,t ) o 0

I,

=

t,t1,t

o

E J.

it follows that (3.52)

For arbitrary same argument.

t

=

t1 E J

we have

~

-1

(t,t) o

~(t

o

,t)

by the

84

STABILITY OF LINEAR SYSTEMS

Lemma 3.3

Suppose that ¢(t,t ) is a fundamental matrix for where ¢

-1

o

A:J

-+ M

defined and continuous in J.

lS

n (t; , to) = ¢(t ,t) o

x = A(t)x,

Then the matrix

is a fundamental matrix of the adjoint dif-

ferential equation Y

-yA(t)

where

Y':J

-+

(3.53)

n, R

i.e. y is a row vector,

lS

continuous and dif-

ferentiable in J. ¢(t ,t)¢(t,t ) = I

Proof: Since

o

~(t ~(t

o

0

,t)¢(t,t ) + ¢(t ,t)¢(t,t )

0 0 0

o

o

0

,t)¢(t,t ) + ¢(t ,t)A(t)¢(t,t ) 0

0

0

det¢(t,t ) f 0

Hence, noting that

~(t

it fo llows that

o

-Nt ,t)A(t)

,t)

(3.54)

o

Recalling the definition of fundamental matrices, equation (3.54) is equivalent to saying that each row of the matrix ¢(t ,t) is a o

solution of (3.53). A simple transposition of equation (3.54) yields ~'(t

o

If we have

adjoint.

-A' (t)¢' (t;

,t)

A(t)

o

-A' (t)

,t)

(3.55 )

then equation (3.53) is called self

Such equations are found in the study of mechanics,

often in connection with oscillator problems (Venkatesh, 1977). The fundamental matrix of any self-adjoint equation is clearly

orthogonal.

This implies that every solution of (3.53) has a

constant norm as t varies. The final topic we include In this section concerns the addition of a forcing term to the initial value problem (3.38), that is to say we now have an inhomogeneous linear differential equation of the form x tt )

A(t)x(t) + h I t

)

(3.56)

3. ORDINARY DIFFERENTIAL EQUATIONS

85

is defined and continuous in J, x:J ~ Rn is n continuous and differentiable in J and h:J ~ R is continuous n in J, which satisfies the boundary condition (x ,t ) s R x J.

where

A:J ~ M

n

o

0

To construct a solution to (3.56) we appeal to the following: Theorem 3.6: Variation of constants Suppose that

~(t,t

o

) is a fundamental matrix for (3.38).

Then

every solution of (3.56) is given by the formula x rt

)

for any

t,t

t

~(t,t)x

o

0

S

J,

x

o

J ~(t,s)h(s)ds

+

o

(3.57)

to s Rn.

~-A(t)x

Proof: If we rewrite equation (3.56) as

h

and use

(3.54), then equation (3.56) is equivalent to d -1 -d [~ (s,t )x(s)) s 0

~

-1

(s,t )h(s)

(3.58)

o

Integrating both sides from t

to t we obtain

o

Jt~-l(S,to)h(S)dS

to A rearrangement of the terms 1n this expression yields equation (3.57) and the theorem is proved.

3.4

Constant Coefficient Differential Equations We shall now turn our attention to the special, but very im-

portant, case of linear ordinary differential equations with constant coefficients.

In other words, we shall examine initial

value problems of the form: Ax(t),

x ,

x(t ) o

o

t

o

s J

(3.59)

A s M , the set of nxn constant coefficient matrices, n n x:J ~ R is continuous and differentiable in J. To begin with

where

the fundamental matrix for (3.59) satisfies


A~(t,t

o

for all

t,t

o

S

J.

o

),

~(t

o

,t )

It follows that

(3.60)

I

0

~(t,t

o

) must satisfy the

86

STABILITY OF LINEAR SYSTEMS

integral equation t


f A
I +

t

(3.61 )

o

If we attempt to solve equation (3.61) by the method of successive approximations ¢(t,t) o

0

I +

I

k = 0,1,2, ... ,

then we observe that the expression for
t- t o )

+

----=--2! k

A

+

o)

is given by the

----:::-:--=--

3!

(t-t )

+ •••

k

o

k! which we recognise, as

k-+

oo

,

as the power series expansion of

the exponential matrix function.

By the ratio test (Rosenbrock &

Storey, op.cit.) this series converges for all this we deduce that 00


J = R

It-t o I

<

From

and

Ak (t-: t )k 0

L: k=O

k!

exp [A(t-t )], o

Vt,t

o

E:

R

(3.62)

so that all solutions to the initial value problem (3.59) can be expressed very succinctly as x

Ct)

exp [A( t- t )] x o

0

(3.63)

For convenience we recall some basic properties of matrix exponentials: (i) exp [At]exp [As]

exp [A(t+s)]

3. ORDINARY DIFFERENTIAL EQUATIONS (ii) (exp[At])

-1

exp[-At] exp[A+B]

(iii) exp [A]exp [B] d (iv) dt exp[At] (v) exp[P where

-1

A,B,P

87

AP]

iff AB

=

Aexp [At]

exp [At]A

-1 P exp[A]P

if

BA

(3.64)

detP f 0

M and t,s E R. n It is of interest to see the explicit form that the fundamenE

tal matrix ¢(t,t ) assumes in relation to the constant matrix A. o

From the theory of matrices (see for example Gantmacher, 1959) it is known that there exists a nonsingular constant matrix M

such that

n

P-1AP

P E

is the Jordan canonical form of A, i.e., (3.65 )

Then exp [At]

¢ (t , 0)

exp[PAJP

-1

Pexp[AJt]P

t] -1

(3.66)

and A has the form J diag [Al ,A z, ...

(3.67)

'~]

where each square block A.

diag[A.l,A. z"" ,A.

]

]]

]m

j

],

(3.68)

j=I,2, ... ,k

and each square elementary divisor block

A. ]

0

A. ]

A.]p

p=1 ,2, ... ,m. ]

A.

0

]

The latter has dimension r .. k

L

(3.69)

p]

m.

Using this notation for A J,

]

L

j=1 p=1

r

pj

n

(3.70)

88

STABILITY OF LINEAR SYSTEMS

We see that in general A will have k distinct characteristic J values or eigenvalues A., each being associated with m. elementJ

J

ary divisor blocks of dimension r ..

To within the ordering of

PJ

the elementary divisor blocks, this representation is unique. Theorem 3.7 Suppose that ¢(t,O) is a fundamental matrix for ~(t) n where x:R ~ R is continuous and differentiable in R. columns of

¢(t,O) = exp[AJtl q-I exp(A.t) 2:: J h=O

y(t)j,p,q for

q=I,2, ... ,r .,

e. h J,p.q-

Then the

are given by t

h

hT

(3.71)

ej,p,q_h

p=I,2, ... ,m.,

PJ

= AJx(t),

j=1,2, ... ,k.

J

The terms

are columns of the identity matrix.

Proof: Coddington & Levinson (op.cit.) and Gantmacher (op.cit,) give proofs for a general matrix

M n. To discover a little more about the structure of the funda-

mental matrix

exp[AJtl

J,p,q

€

we rewrite (3.71) as

exp (A. t ) t

y (t ) .

A

q-I

J

q-2

[(q_\)! ej,p,l

-t-

t h + 1- q

2::

h=O

(3.72)

x e.

J,p,q-h ]

and note that the term after the summation sign tends to zero as

t

~

J ,p,q

II

the brackets

~n

although the norm of the brackets re-

00

malns bounded away from zero. loglly(t).

h!

Therefore we have

x.t + (q-I)logt + logll['JII J

where [.J is the bracketed expression

~n

(3.72).

(3.73)

From (3.73) it

follows that x· J

lim {t-lloglly(t). II} J,p,q

and q-j

(3.74)

t~+oo

lim t~+OO

rOglly(t).J,p,q exp(-x.t) J log t

II}

(3.75)

89

3. ORDINARY DIFFERENTIAL EQUATIONS where

x·J

Re (A.)

(3.76)

J

It follows that the multiplicity of X·, which will be denoted by ~(x.),

J

is given by

J

m. J

(3.77)

L: r . PJ p=1

the matrix A possesses k distinct characteristic values.

s~nce

J

The relationship (3.76) between X. and A. leads to the following J

J

definition, originally due to Liapunov (1893) and generalised by Perron (1930):

Definition 3.3: Generalised Characteristic Exponents Suppose that

x(t)

n x:R ~ R

and continuous in R, in R.

= A(t)x(t),

where

A:R

~

M

n

~s

defined

is continuous and differentiable

For each nontrivial solution x(t) define a generalised

characteristic exponent X(x) 'of A(t) by x(x)

lim {t-

1

(3.78)

logllx(t)ll}

t~+co

Furthermore, if

A(t) = A s M ,

then the

n

real parts of the characteristic values

X. are precisely the J A., j=1,2, ... ,k of A. J

Generalised characteristic exponents play an important role

~n

stability theory, as we shall see in the next section and in Chapter Five.

A related concept is the type number of a matrix,

which was introduced by Marcus (1955):

Definition 3.4: Type Number Suppose that

x(t) = A(t)x(t),

and continuous in R, in R.

where

A:R

~

M

n

is defined

is continuous and differentiable

For each nontrivial solution x(t) define a type number

veX) df A(t) by lim {lOgllx(t)exp(-x t log t

vex)

t~+co

)

II}

(3.79)

Each X. may have many type, numbers associated with it and they J

are not necessarily distinct.

We use the notation V. to repreJW

90

STABILITY OF LINEAR SYSTEMS

) denote their multipliJW It is interesting to observe that in the case A(t) = A £

sent distinct type numbers and let city.

~(v.

J

Mn, the dimensions of the blocks A. , p=1 ,2, ... ,m. are deterJP J mined by the type numbers v. . In fact, the number of blocks A. JW JP of dimension 2 v. +1 is exactly ~(v. ). Thus the total strucJW JW ture of A is determined to within the imaginary parts of the J

characteristic values of A by the X., v , , J

3.5

JW

~(X.),

J

u Cv . ). JW

Periodic Coefficients and Floquet Theory Linear ordinary differential equations with periodic coeffici-

ents occur in many theoretical and practical problems concerned with rotational or vibrational motion (Stephens, 1966). lies their importance.

Herein

We consider the following initial value

problem

~(t)

A(t)x(t) ,

~ AP

A(t+w) n x:R ~ R

where

A~R

in R.

We note that if ¢(t,t )

and

n

o

then so is ¢(t+w,t).

n 1 (x ,t ) S R + (3.80) o 0

is continuous and differentiable ~s

a fundamental matrix of (3.80),

Thus we may write

o

¢(t+w,O)

A(t),

¢( t+w, w)¢ (w, 0)
(3.81 )

Since

(3.82)

This assertion has been demonstrated by Bellman (op.cit.) for example.

This leads to the following:

Theorem 3.8: Floquet Representation (Floquet 3 1883) The fundamental matrix of (3.80) can be represented in the form


P(t)exp[w

-1

t log
where pet) is a periodic nonsingular matrix and P(O)

(3.83) I.

91

3. ORDINARY DIFFERENTIAL EQUATIONS

Proof: We write pet)

-1

~(t,O)exp[-w

t

log~(w,o)]

Then pet) is nonsingular, and we have P (t+w)

-1

~(t+w,O)exp[-w

(t+w)log~(w,O)]

-1

~(t,O)~(w,O)exp[-log~(w,O)]exp[-w

t

log~(w,O)]

P (t )

For general ~(t,t

o

t

o

f 0

(3.84)

it follows that -1

)

P(t)exp[w

(t-t

o

)log~(w,O)]P

-1

(t) 0

(3.85)

Theorem 3.9

There exists a nonsingular periodic transformation of variables which transforms (3.80) into an equation with constant coefficients.

Proof: Suppose that

P'(t+w)

pet)

is defined by (3.83).

To

simplify notation, let -1

w

B

Let

x(t) y

Since B

=

(3.86)

log~(w,O)

P(t)y(t)

~n

P- 1 (t ) [A( t ) P ( t ) pet)

=

(3.80).

P(t) ] Y

W(t,O)exp[-Bt],

P-1 (t ) [A (t ) P (t.) -

The equation for y is (3.87)

it follows that

P(t ) ]

(3.88)

and this proves the result. Note that the characteristic values of W(w,O) are generally complex numbers and that their logarithms are also complex, although the real part is determined uniquely.

There are occasions

when A(t), pet) and B are real, even though the characteristic values of W(w,O) may be negative.

In such cases the matrix pet)

has period 2w and is of the form pet)

W(t,0)exp[-(2w)

-1

*

t(logW(w,O) + log W(w,O»]

where the asterisk denotes complex conjugate.

92

STABILITY OF LINEAR SYSTEMS

Definition 3.5: Characteristic Multiplier Suppose that ¢(t,O) is the fundamental matrix of (3.80).

Then

the characteristic values of ¢(w,O) are called characteristic mul-

tipliers of (3.80) and are denoted by P., j=1,2, ... ,n. Observe that Re(A) such that

J

p = exp[Aw]

is a generalised

characteristic exponent of the periodic matrix A(t).

We now have

the following: Theorem 3. 10 ,

1\.

If

w

p. = e J , j=J,2, ... ,n J

are the characteristic multipliers

of (3.80), then n

IT

j=l

p.

J

n L: A. j=J J

e xp [ (

"AC')d']

0

w

w

f trA(s)ds,

(mod 2;i)

0

Proof: The theorem follows immediately from the definitions of characteristic multipliers and exponents, and Lemma 3.2. It would appear that linear periodic equations share the same simplicity as linear equations with constant coefficients.

How-

ever, there is a very important difference - the characteristic exponents are defined explicitly only after the solutions of (3.80) are known, there being no obvious relation between the characteristic exponents and the matrix A(t). Clearly the problem of determining the characteristic multipliers or exponents of linear periodic differential equations extremely difficult one.

~s

an

In fact it has no known solution except

for some scalar second order systems and, more generally, Hamiltonian and canonical systems.

The problem is further examined

in subsequent chapters. 3.6

Notes Linear periodic differential systems have attracted consider-

3. ORDINARY DIFFERENTIAL EQUATIONS

93

able interest (Starzhinskii (1955), Cesari (1963), Yakubovich and Starzhinskii (1975), Venkatesh (1977)); an important special case ~s

the scalar second order differential equation (Hill's equation), x + f(t)x

0,

with

f(t+w) = f(t)

Liapunov (1949) has shown that if

f(t)

$

(3.89)

0, Vt,

then the char-

acteristic multipliers of Hill's equation are real and positive so that there are infinitely many unbounded solutions {x(t)} for t E R+.

However if

f(t) > 0, Vt

~

and

f f(T)dT w

$

4,

then

o

Hill's equation has purely complex characteristic multipliers with all solutions bounded for

t

E

R+.

Related to Hill's equaf(t) = 0 +

tion is the well known Mathieu equation, in which with

~cos2t

f(t) and

w = 2TI;

w

o+

= TI ~a(t),

and the Meissner equation in which for, aCt) = { 1 -I

o

$

t < TI

TI

$

t

$

2TI

for further references on these special periodic

equations see Cesari (1963) and McLachlean(1947). 3.7

References

Birkhoff, G. and Rota, G. (1978). "Ordinary differential equations" J. Wiley, New York Bellman, R. (1953). "Stability theory of differential equations", McGraw-Hill, New York Brockett, R.W. (1970). "Finite dimensional linear systems", J. Wiley, New York Cesari, L. (1963). "Asymptotic behaviour and stability problems in ordinary differential equations", 2nd Ed. Academic Press, New York Coddington, E.A. and Levinson, N. (1955). "Theory of ordinary differential equations", McGraw-Hill, New York Curtain, R.F. and Pritchard, A.J. (1977). "Functional analysis ~n modern applied mathematics", Academic Press, New York Floquet, G. (1893). Ann.Ecole Norm. Sup. 12, 47-79 Gantniacher, F.R. (1959). "The theory of matrices", Chelsea, New York Hale, J.K. (1969). "Ordinary differential equations", Wiley Interscience, New York Hille, E. (1969). "Lectures on ordinary differential equations", Addison-Wesley, New Jersey

94

STABILITY OF LINEAR SYSTEMS

Liapunov, A.M. (1893). Comm.Soc.Math.Kharkov (in Russian). (In translation: Ann.Math.Studies 17, Princeton, (1949)) Markus, L. (1955). Math.Zeits. 62, 310-319 McLachean, N.W. (1947). "Theory and applications of Mathieu functions", Clarendon Press, Oxford Perron, O. (1930). Math.Zeits 32, 703-728 Rosenbrock, H.H. and Storey, C. (1970). "Mathematics of dynamical systems", Nelson, London Starzhinskii, V.M. (1955). Amer.Math.Soc.Trans. I, 189 Stevens, K.K. (1966). SIAM J.Appl.Math. 14, 782 Venkatesh, Y.V. (1977). "Energy methods in time-varying system stability and instability analyses", Lecture notes in physics No.68, Springer Verlag, Berlin Yakubovich, V.A. and Starzhinskii, V.M. (1975). "Linear differential equations with periodic coefficients", J. Wiley, New York

Chapter 4

KINEMATIC SIMILARITY

4.1

Introduction

The concept of kinematic similarity of matrices as considered by Markus (1955) and others, defines an equivalence relation on the set of all n x n matrices' whose entries are continuous and bounded functions on the nonnegative reals R+.

Evidently this

idea has its origins in Liapunov's work (Liapunov, 1893) where it is referred to as reducibility, a term which still persists 1n more recent Soviet literature, e.g. Erugin, 1946.

Markus'

concept is too restrictive for studies concerned with almost periodic matrices (Langenhop, 1960).

Accordingly it is useful to

redefine the set of matrices of interest to account for functions which are bounded and continuous on the whole real line R.

The

resulting modification of Markus' concept Langenhop (op.cit.) is called complete kinematic similarity. The purpose of this chapter is threefold.

The first is to

give a full account of kinematic similarity, roughly following Markus' work.

The question of invariants under kinematic simi-

larity is raised and conditions for kinematic similarity stated. Proofs are only given in cases of special interest or significance; in all others the proofs may be found in the cited references. The second objective is to examine the consequences of kinematic similarity in the study of linear systems of ordinary differential

96

STABILITY OF LINEAR SYSTEMS

equations.

The result of great practical significance here is

that for linear systems uniform stability (see Chapter 5 for a detailed explanation of this concept) is invariant under kinematic similarity.

In the last section we give several examples of ki-

nematically similar matrices, each example arising from particular properties of different subsets of M . n

4.2

Liapunov Transformations and Kinematic Similarity

An important subset of M consists of nonsingular matrices n

whose derivative with respect to time on R.

lS

continuous and bounded

An analogous subset is defined on R+o

The matrices refer-

red to are called Generalised Liapunov Transformations or simply

Liapunov Transformations if the range of interest is R+. Definition 4.1: Generalised Liapunov Transformation

A matrix

pet) s M

n

is said to be a Generalised Liapunov

Transformation (GLT) if and only if: (i) pet) s M

n

for all

t

S

R 8 s R

(ii) There exists some constant

o

< 8 < !detP(t) I for all

t

S

such that

R.

In some applications, such as those concerned with stability theory, it is more convenient to confine interest to the halfline R+, in which case we have the following: Definition 4.2: Liapunov Transformation

A matrix

pet) s M n+ (LT) i f and only if:

lS

said to be a Liapunov Transformation

(i) P (t; ) E M for all t E R+ n+ (ii) There exists some constant 8 E R 0< 8 < IdetP(t)! for all

such that

t E R+o

Definition 4.3: Complete kinematic similarity (LangenhopJ Let

A(t) ,B(t) EM. n

Then

kinematically similar to B(t»

A(t)

~

B(t)

(read: A(t) completely

if and only if there exists a Gen-

eralised Liapunov Transformation pet) such that B(t)

-pet)

-1

•

[P(t)-A(t)P(t)]

for all

t E Ro

97

4. KINEMATIC SIMILARITY It is easy to see that we also have A(t) where

-Q(t) Q(t)

hypothesis.

=

-1

pet)

•

[Q(t)-B(t)Q(t)]

-1

.

Note that the indicated inverse exists by

The modifier "complete" is used to distinguish this

concept from that considered by Markus (op.cit.), namely:

Definition 4.4: Kinematic similarity (Markus) Let

A(t),B(t) EM.

Then

n+

A(t)

B(t)

~

if and only if there

exists a LT pet) such that B(t)

-pet)

-1

•

[P(t)-A(t)P(t)]

for all

t E R+.

If in the above definition the relationship between A(t) and B(t) ~s

only established to within an arbitrarily small degree there

results the condition known as approximate similarity, which was introduced by Lillo (1961) in the study of the stability of perturbed differential

Definition 4.5: Let

equation~.

Approximate similarity (Lillo)

A(t),B(t) EM. n

mately similar to B(t»

Then

A(t)

~

(read: A(t) approxi-

B(t)

if and only if, given any

exists a matrix P(c5,t) such that

P(c5,t), p(c5,t)

c5 > 0 -1

there

•

, P(c5,t) E M

n

and II-p(c5,t)-l[p(O,t)-A(t)P(c5,t)]-B(t)11 for all

t E R.

Here

11·11

<

c5

is the uniform norm.

This definition is included for completeness and will not be used in the sequel.

From definitions 4.3 and 4.4 it is easy to

verify that complete kinematic (c.k.) similarity and kinematic (k) similarity are equivalence relations on M and M n

tively.

n+

respec-

Theorem 4.1: (Martin> 1966; Markus> op.cit.) Complete kinematic (respectively kinematic) similarity

~s

an

equivalence relation on M (respectively M ). n

Proof: Consider only the case for M. n

It

n+ ~s

necessary and suf-

ficient to establish that c.k. similarity is symmetric, reflexive and transitive.

Let

A(t),B(t) E

Mu.

Under the identity matrix

98

STABILITY OF LINEAR SYSTEMS

A(t)

~

A(t)

and we have already noted that if

B(t)

~

A(t).

B(t)

-1

then

A(t) E: M

thus

•

[P(t)-A(t)P(t)J

it follows that for any GLT P(t),

B(t) E: M

satisfying the initial hypothesis. -PI (t )

and if

B(t)

~

From the equation -P(t)

B (t )

A(t)

C(t) E: M

n

-1

if

n

Finally, if

n

•

[PI (t)-A(t)P I (t)]

(by hypothesis)

is given by

C(t) then C(t) where

P 3(t) = P I(t)P 2(t).

Clearly P3(t) is a GLT, so

A(t)

~

C (t.) •

4.3

Invariants and Canonical Forms Our principal aim in this section is to clarify the relation-

ship between the characteristic exponents of a matrix and kinematic similarity and to show that real Jordan matrices with an assumed structure (permuted) are canonical forms for a subset of M , including constant matrices in M First of all we present n+ n, the following theorem concerning the invariants of matrices under kinematic similarity and then go on to remark that these form a complete set of invariants

for real Jordan matrices.

Theorem 4.2: Invariants of matrices under kinematic similarity (Markus~

1955)

The characteristic exponents and the multiplicities JW kinematic similarity. V.

~(x.)

J

X. of A(t) J

and

~(v.

JW

M , their types n+ ) are invariants of E:

Proof: Let -P(t)

-1

•

[P(t)-A(t)P(t)J

B(t),

define the kinematic similarity A(t) ~ B(t), and consider a n vector x:R+ ~ R satisfying x = A(t)x on the half-line R+

4. KINEMATIC SIMILARITY n y:R+ + R

and a vector shown that

= P(t)y(t).

x(t)

Then it ~s

easily

Consequently lim t-

x(P(t)y(t»

X(x)

y = B(t)y.

satisfying

99

1logllp(t)y(t)

II

t++oo

s

lim

t-

1logllp(t)

II

+ X(y)

(4. I)

t++oo

Since pet) is bounded, by hypothesis, it follows that On the other hand,

=

yet)

is also bounded, whence

pet)

X(y)

S

A similar argument shows that the numbers X., v. , J

JW

~(X.),

~(v.

J

-1

x(t)

S

and by hypothesis pet)

X(x)

and finally

v(X(x»

= v(X(y».

JW

X(x)

X(x)

=

X(y). -1

X(y).

In other words,

) are the same for A(t) and B(t).

The development of the idea that real Jordan matrices are canonical forms for a subset of M will proceed in stages. Theorem n+ 4.2 does not quite solve the problem although the link between Jordan matrices and kinematic similarity has been forged as a result of the invariants chosen.

First of all we consider the sim-

plest case and state the following well known result: Theorem 4.3: Static

- constant matrices

simi~arity

Each constant matrix

B E M

n

is statically, and thereby kine-

matically similar to a Jordan matrix

Proof: This is a well known result. (1959).

The assertion is that there exists a constant Liapunov

transformation

-P P It

~s

E M • J n For a proof see Gantmacher B

-1

-1

P E M n

such that

B E B J.

That is

[-BP]

BP

RADCLIFFe-

(4.2)

interesting to note that static similarity implies kine-

matic similarity, although the converse is not true.

The question

here is one of uniqueness, as may be verified by means of simple examples.

To clarify the situation consider the following:

Theorem 4.4: Kinematic 1955)

simi~arity

-

rea~

Jordan matrices

(Markus~

Let A and B be real, constant, Jordan matrices. Arrange the J J order of the blocks A down the principal diagonal of A (say, J j

STABILITY OF LINEAR SYSTEMS

100

IAI,

first by increasing

second by increasing Arg A and third by

dim A see Chapter 3). If A ~ B then A = B j p; J J J J. ppoof: It is sufficient to verify that the invariants X., v. , V(X.), V(v. ) determine the total structure of A J. J JW see section 3.4.

J

JW

For details

Thus we are led to a stricter version of Theorem 4.3, namely: Theorem 4.5: Kinematic similarity - constant matrices

Subject to the ordering assumed above, each constant matrix is kinematically similar to a real Jordan matrix B

B E M n

E

J

Mn .

Proof: By virtue of Theorem 4.3, the constant matrix B is statically and therefore kinematically similar to a possibly complex A

Jordan matrix

B E M Since Theorem 4.4 guarantees uniqueness, J n. subject to the ordering of submatrix blocks, it is sufficient to

prove that

B E B In fact, it is only necessary to consider J. J elementary divisor blocks of the respective matrices. The assertion is that A

b.

1

b.

0

b.

J

J

1

0

b.

J

J

o

A

1

I

b.

b.

0

J

J (4.3)

where

A

= b.

b.

J

J

+ i~ ..

Define the corresponding block of the as-

J

sumed Liapunov transformation by

o (IT(t»

.

exp(i~.t)

JP

J

o

Direct calculation shows that -(IT(t)

-1.

).

JP

[(IT(t».

JP

-

which completes the proof.

A

(B ) . (IT(t».] J JP JP

(B ) . J JP

(4.4)

4. KINEMATIC SIMILARITY

101

From this theorem we conclude that under kinematic similarity the unique canonical form B for B is obtained by taking a complex J Jordan matrix B ~ B and then deleting the imaginary parts of J the characteristic values of B . J Theorem 4.6: Complete set of invariants under kinematic similarity

(Markus, 1955)

For the subclass of matrices

~n

M

n+

which are kinematically

similar to constant matrices, the invariants consisting of characteristic exponents X., types v. , and their multiplicities and

~(v.

A(t)

~

) form a JW

B for

J

JW

complete set of invariants.

A(t) E M

n+

real constant Jordan matrix

~(Xj)

Furthermore, if

B E M , then there exists a unique n B which displays the invariants B J

and

of A(t).

Proof: follows from Theorems 4.2 and 4.5. 4.4

Necessary and Sufficien! Conditions for Kinematic Similarity To fully exploit the concept of kinematic similarity it is de-

sirable to have a set of necessary and sufficient conditions for kinematic similarity. quite difficult.

Unfortunately the general problem here seems

However, there are special situations for which

it is possible to give conditions sufficient to ensure that two given matrices are kinematically similar.

We treat the simplest

case first. Theorem 4.7: Erugin's Theorem (Erugin, 1946) Let A, B E M Then A ~ B if and only if A and B have J J n. the same distribution of I's on their superdiagonals, and for corresponding characteristic values a. of A and b. of B we have Re (a.) J

=

Re (b . ) , J

j

=

1,2, ... , n ,

J

Proof: follows from Theorems 4.4 and 4.5.

J

Note that the require-

ment on the characteristic values simply means that the two matrices have the same characteristic exponents. A more interesting case is that of matrices with time dependent elements kinematically similar to constant matrices.

It is clear

from the definition of kinematic similarity that the matrix

STABILITY OF LINEAR SYSTEMS

102

A(t)

M is kinematically similar to a constant matrix B s M n n+ if and only if there exists a Liapunov transformation pet) which S

satisfies

(4.5)

A(t)P(t) - P(t)B for all

t s R+.

It is easily shown that if pet) is a solution

of (4.5), then X(t)

P(t)exp[Bt]

is a fundamental matrix of

(4.6)

A(t)X(t) and if X(t) is a fundamental matrix of (4.6), then pet)

X(t)exp[-Bt]

is a solution of (4.5).

For X(t) and pet) related 1n this way,

it follows that det pet)

det X(t) det exp [-Bt]

(4.7)

and from Lemma 3.2 (Abel-Jacobi-Liouville) t

det pet)

detX(O)exp [

f

tr(A(s)-B)ds]

(4.8)

o If pet) is a solution of (4.5) and

X(t)

= P(t)exp[Bt]

then

p(t)-I exists if and only if the columns of X(t) are linearly 1ndependent over the field of complex numbers since this is true if and only if

detX(O) #

o.

Also we observe that if p(t)-I exists,

then the columns of pet) are certainly linearly independent over the complex numbers.

(4.5) such that pet)

pet) s M is a solution of -I n+ exists, then pet) s M if and only if n+

Moreover, if -I

(4.9)

is bounded for all det pet) det pet)

t s R+. -I

I"

Clearly

4. KINEMATIC SIMILARITY If pet)

-1

is bounded then det pet)

103

r s bounded away from zero so

that this, together with the boundedness of pet), implies through

(4.8) that (4.9) holds, i.e.

~s

bounded for all

det pet)

t

€

R+.

Conversely, if (4.9) holds, then

is bounded away from zero and this, together with the

boundedness of pet), implies that ding to Theorem

p(t)-l

€

4.5 we may write

M n+

Note that accor-

(4. 10) place of (4.9), where

B € M ~s a real Jordan matrix. By J n combining this argument and Theorem 4.6, Langenhop (op.cit.) ~n

proves the following: Theorem 4.8: Necessary and sufficient conditions for

(Langenhop, 1960)

A(t)

~

B

Then for A(t) ~ B it is necesA(t) € M and B € M n+ n sary and sufficient that there exist real numbers X. which are J characteristic exponents of A(t) and which have multiplicities Let

fl(Xj) with k L: fl(X.) j= 1 J

n,

such that k L: fl(X.)X.)dJ J J j=1

J

~s

bounded for all

t

€

(4. 11)

R+.

Proof: for details see Langenhop (ap.cit.). We call attention to the fact that Langenhop proves the above theorem for the whole real line.

We have set it in R+ to follow

the already established trend in this chapter, knowing that it is

STABILITY OF LINEAR SYSTEMS

104

generally possible to restate the results so far for R. Theorem 4.8 contains the most general result in this section. Specific results of wide interest and application will be considered in the next section.

The overall problem is essentially

one of estimating the characteristic exponents and their multiplicities.

4.5

Estimates for Characteristic Exponents We consider the simplest case first of all, that is let the

kinematic similarity be defined for matrices B

E

A(t) E M + 1

and

An obvious possibility is that

MI' t

i 0f A(s)ds where

B + n(t)

!n(t)!

as

0

-+

t

(4.12)

Then

-+

t

B

lim t->=

i f A(s)ds 0

In this case the Liapunov transformation

~s

a bounded solution of

(4.5) and is written as P (t )

exp

[(A(S)-l~ o

s

~

(A(OdfJ dS] 0

Clearly this result is applicable to those cases which can be treated as if

n

= I,

e.g. diagonal matrices, provided of course

that the limi t t

lim 1t->= t exists.

f A(s)ds o In the general case the relation

found by analogy with the scalar case. tion is defined by

A(t)

~

B

cannot be

Recalling that the rela-

4. KINEMATIC SIMILARITY

105

we obtain the linear differential equation pet)

A(t)P(t) - P(t)B

whose solution is a Liapunov transformation given by pet)

X(t)exp[-Bt]

where X(t) satisfies the linear homogeneous equation X(t)

A(t)X(t).

Assume for the sake of argument that the matrix

B

E

M

n

1S de-

fined by t

B

1 t

r A(s)ds

(4.13)

)

o

provided of course that the limit exists.

Then, reasoning as for

the scalar case, we are tempted to suggest that

(4.14)

pet)

This 1S generally incorrect for two reasons: (i) it 1S only possible to express X(t) as

X(t)

ex,

U~(')d']

1n certain cases, which will be specified later. (ii) Should it happen that

then pet)

X(t)exp[-Bt]

exp[I~(')d']

exp I

r-Bt.]

1S not always equal to (4.14), Slnce

STABILITY OF LINEAR SYSTEMS

106

exp[A]exp[Bl

# exp[A+B]

unless

AB

BA.

As may easily be verified, if t

A(t)

t

f A(s)ds

f A(s)ds A(t)

0

0

then neither (i) nor (ii) hold.

Furthermore, if

t

f A(s)ds

lim J..

B

t-+<x> t

(4.13)

o

exists, then P(t)

given by (4.14).

~s

In cases where the above limit does not exist, Vul'pe (1972) has succeeded in extending the above result by introducing a gene-

B

ralised limit

£

M

n

given by

lt ~ o f ([~ (A(~)d~ o

B

(4. 15)

J

••

which has the property that if B exists, then B exists and

B

B',

however the converse does not necessarily hold. We summarize the above discussion in the following: Theorem 4.9

t

A(t) £ M commute with f A(s)ds for each n+ 0 Assume that A(t) can be decomposed into A(t) = Ap (t)+A0' Let

A

o

£ M

n

fo Ap (s)ds t

Ao

and commutes with both A(t) and Then

E M

n+

A(t)

~

A

o

and

f

t

o

A(s)ds.

where

Suppose that

t

lim t-+<X>

ff

A(s)ds.

0

ppoof: To begin with certain commutation results must be estab-

lished.

First of all t

Ap(t)

f A (s)ds

o

P

107

4. KINEMATIC SIMILARITY

t

fo

A (s)ds A (t) p p

(4.16)

Similarly t

Ao

f

A (s)ds p

o

t

J AP (s)ds

A

(4.17)

0

o Let

It

~s

easily shown that

-pet) -1 CP(t)-A(t)P(t)) Since

f

=

A . o

t

o

A (s)ds p

E:

M , n+

both pet) and

pet)

-1

pet) possesses a continuous first derivative.

E:

M , n+

and clearly

Therefore

A(t)

~

A as required. o

To clarify the conditions under which a matrix commutes with its integral we introduce t

B. (r )

f A(s)ds

(4.18)

o where

A(t), B(t)

E:

M

n+

Assuming that

(i) there exists a nonsingular differentiable matrix pet) such that

108

STABILITY OF LINEAR SYSTEMS B(t)

where

(4. 19)

BJ(t)

M lS ln Jordan canonical form, n+ (ii) the distribution of l's on the superdiagonal of BJ(t) does E

not change for all

t E R+,

(iii) no difference between different characteristic values of BJ(t) vanishes in a subinterval of R+ unless it vanishes identically, (iv) if the difference between different characteristic values of BJ(t) vanishes identically then the superdiagonals in either elementary divisor block become O. Epstein has proved the following: Theorem 4.10: (Epstein, 1963) The matrix

B(t) E M n+

B(t)B(t)

which satisfies 0,

that is t

A(t)

t

J A(s)ds - f A(s)dsA(t)

o

(4.20)

o

o

and having a Jordan canonical form is constant for all

t E R+,

BJ(t) E M n+ is obtained

whose structure

(i) by finding all matrices X(t) satisfying

o

(4.21) (ii) determining the nonsingular solutions pet) of the matrix differential equation P (t )

X(t)P(t)

(4.22)

(iii) forming B(t)

(4.23)

The matrices X(t) form a linear space under addition which depends only on the structure of the elementary divisor blocks and the set of subscript pairs for which the difference between different characteristic values vanishes.

4. KINEMATIC SIMILARITY

109

Proof: Observe that, trivially (4.24) Differentiating (4.23) yields •

-1

pet) BJ(t)P(t) + pet)

B (t )

-).

BJ(t)P(t) + pet)

-).

BJ(t)P(t) (4.25)

. . B(t)B(t)-B(t)B(t)

whence

(4.26) If

B(t)B(t) - B(t)B(t)

0,

then

o

(4.27)

Taking into account (4.24), (4.26) becomes

which is precisely (4.21). An immediate consequence of Theorem 4.10 is the following: Theorem 4.11: (Epstein, 1963)

Let A(t)

A( t ) s M n+ t

f A(s)ds

0

commute with its integral, that is t

f A(s)ds A(t).

o

Then A(t)

[X(t)+BJ(t)+BJ(t)X(t)-X(t)BJ(t)]

where X(t), BJ(t) are defined as in Theorem 4.10.

(4.28) The Liapunov

transformation is of course pet).

Proof: follows directly from (4.25). For the particular class of matrices which are normaZ, that is statically similar, by a unitary transformation, to a complex

STABILITY OF LINEAR SYSTEMS

110

diagonal matrix, it is possible to derive a relation between the characteristic exponents of

A(t)

M and the averages of the n+ characteristic values of A(t) for each fixed value of t. E

Theorem 4.12: (Markus, 1955) Let

A(t) E Mn+ be normal for each fixed t on R. Let A(t) + be the maximum of the real parts of the characteristic values of A(t) and A(t) be the minimum of these real parts. t

-I

lim t t-+oo

Then,

t

f A(s)ds

x·J

o

~

limI-

t-+oo t

f A(s)ds

(4.29)

o

where X. are the characteristic exponents of A(t). J

Proof: It is known (Hamburger & Grimshaw, 1951) that A(t) and A(t) are real, continuous and bounded functions on R+. ~(t)

tion x(t) of r(t)2

= A(t)x(t)

For any solu-

define

x(t)*x(t)

(4.30)

Differentiating, we have

. *x(t) x(t) *.x(t) + x(t) * * * x(t) A(t)x(t) + x(t) A(t) x(t) whence

* "e * *-1 (x(t) A(t)x(t) + x(t) A(t) x(t))r(t)(2x(t) x(t))

~ (r )

Since A(t) is normal for each A(t) and

A(t )

(4.31 )

t

S

R+

~

* * * *-) (x(t) A(t)x(t)+x(t) A(t) x(t))(2x(t) x(t))

~

ACt)

~

r(t)r(t)

.

(4.32) -)

~

A(t)

(4.33)

Integrating, we obtain t

f A(s)ds

~

log ret) - log r t O)

o Now

x

~

t

JACs)ds

(4.34)

o lim t-+oo

t

logllx(t)

II

-.- 1 Li.m -

t-+oo t

log ret)

(4.35)

III

4. KINEMATIC SIMILARITY Therefore t

-.- I I im

t->=

t

I

-]

:;

.\(s)ds

Em- log r t t ) t t->=

0

S

(

-]

Emt t-->=

J

t A(s)ds

0

or t

-.- 1 hm t t->=

I

t :;

.\(s)ds

-1 lim t t->=

:;

X

0

JA(s) ds

0

which proves the theorem. Observe that if t

-]t f lim t->=

<

A(s)ds

0,

o

then every solution of

x

=

A(t)x

under the assumptions of the

theorem, approaches the origin as

t

~

00.

Theorems 4.9 and 4.12 are ,quite general and we take advantage of this fact to derive some interesting results for almost periodic matrices Theorem 4.13: If

A(t)

E:

and A(t),

f

If

0

t

J

(Markus,

AP n+

A(s)ds

(A(s)-Ao)ds A(t)

A(t)

E:

M

n+

,

A(t) ~s

almost periodic)

t

o then

1955)

(i. e.

A o almost periodic. ~

E:

and A all commute on R+ with o

A

o

M

n+

Moreover, the Liapunov transformation pet) is

Proof: follows from Theorem 4.9.

(For the definition and proper-

ties of the mean value operator M see Chapter 2.) t In the same spirit we have the following: Theorem 4.14: If

A(t)

E:

(Markus, 1955)

AP n+

and A(t),

f

t

A(s)ds

commute on R+ and A(t)

o

has an absolutely convergent Fourier series with exponents bounded away from zero, then

A(t)

~A

o

o.

Moreover, the Liapunov

112

STABILITY OF LINEAR SYSTEMS

transformation

almost periodic.

~s

Proof: By hypothesis jtA(s)ds is bounded and thus almost periodic.

o

Since A(s)ds

o

the result follows immediately. To establish bounds on the characteristic exponents of an almost periodic normal matrix, we use Theorem 4.12, which can be restated as: Theorem 4.15 : (Markus, 1955) be normal for each fixed t E R+ and let M n+ A(t) be almost periodic or have period T > O. Then 1\(t ) and Let

A(t)

E

A(t) are also almost periodic, or have period T respectively, and

f

t

• t1 I i.m t->oo

t

A(s)ds

o

x·J

lim t->oo

Proof: Let A(t) have period

J.-t J{ J\(s)ds o

T > O.

(4.36)

Then clearly both 1\(t) and

A(t) have period T. Suppose that A(t) period of A(t), Le.

~s

almost periodic and let T be an n-almost IIA(t+T)-A(t) II < n

for all

IIACt)

II

a s bounded, we can choose n so small that

1\(t) I

<

0 and

all

t E R.

IA(t+T)-A(t) I

<

0

t

E

R.

Since

!1\(t+T)

for a preassigned

0 > 0

Thus both 1\(t) and A(t) are almost periodic.

and

There-

fore the limits proposed in (4.36) do exist and the required result follows directly from Theorem 4.12. The problem of estimating the characteristic exponents of a periodic matrix has attracted considerable attention in the literature (Proctor, 1969; Sansone & Conti, \964; Bellman, 1953; Gantmacher, 1959).

Floquet's Theorem (Theorem 3.8) gives the most

obvious result, first stated by Liapunov. Theorem 4.16: (Liapunov, 1893) Let B with

A(t)

M be periodic with period n+ B E M given by n E

T

>

O.

Then

A(t)-

113

4. KINEMATIC SIMILARITY

(4.37)

B

where X(t) is a solution of

X(t) = A(t)X(t).

Moreover, the

Liapunov transformation pet) is periodic with period T.

Proof: follows from Theorem 3.8. It is unfortunate that the matrix differential equation

(4.37). using a

X(t)

must be solved before B can be constructed according to

A(t)X(t)

In some circumstances it may be possible to construct B approximation technique.

success~ve

the problem

A(t)o

~

The idea is to solve

where 0 is a small parameter and then see

B

if the solution is valid for

0 = 1.

According to Theorems 4.16

and 3.8, a periodic Liapunov transformation pet) is sought in the form

(4.38)

X(t)exp [-Bt]

P (t )

Assume that X( t ) can be exp r es s ed as a power

ser~es

~n

the para-

meter 0 00

L: ~(t)o k=O

X(t)

k

X

,

0

(4.39)

I

This converges for all finite values of 0 and

t S R.

At

t = T,

(4.40)

X(T) by definition. -1

B = T

Therefore

logX(T)

can be written as

(4.41 )

L: k=O

B

The convergence of this series corresponding majorant series. 00

pet) it

~s

L: ~(t)o k=O

~s

established by considering a From the relation

kook

exp(-

L: Bko t), k=O

clear that pet) can be written as

X o

I

(4.42)

STABILITY OF LINEAR SYSTEMS

114

pet)

(4.43)

which has the same radius of convergence as (4.41).

Substituting

the expressions for pet) and B int9 the differential equation A(t)P(t)O - P(t)B

P (t)

(4.44 )

and equating like powers of 6 yields k-l A(t)P

k_ 1

=

(t) -

m= 1

P (t)B - B k k -m m

(4.45 ) (4.46)

Since the matrices

A(t), PI (t) E M

n+

are real and periodic by

hypothesis, it follows from (4.46) that t

I

PI (t.)

[A(s)-B ]ds 1

0

and

T

B 1

I

T

I

A(s)ds

0

Therefore

t

T

t

J A(s)ds

PI (t )

T

0

I

(4.47)

A(s)ds

0

Similarly an expression for P

2(t)

can be obtained by the same

procedure and in general B k

+f

T

k-l [A(s)P

k_ 1

(s ) -

=

m=1

0

Pk_m(s)Bm]ds

(4.48)

and t

J

Pk(t)

0

Hence the

ser~es

k-l [A(s)P

k_ 1

(s ) -

L

m=l

P (s)B ]ds - Bkt m k -m

(4.49)

for pet) and B may be constructed and will con-

verge for sufficiently small values of O.

Note that the series

4. KINEMATIC SIMILARITY

115

will not converge for any values of 8 for which the matrix X(T) has characteristic values with negative real parts, since we have assumed throughout that the Liapunov transformation has period T. In the remainder of this chapter we shall consider linear timevarying systems with bounded and continuous coefficients, x

A(t)x,

A( t )

(4.50)

M

E

n

which are commutative, that is t

A(t)

t

J A(s)ds

JA(s)dsA(t)

to

to

for all t and establish conditions for the evaluation of the state transition matrix ¢(t,t ) and hence the stability of (4.50). o

The coefficient matrix

A(t)

E

M

can always be decomposed

n

into the non-unique representation (Wu, 1980) r

A( t )

L:

j=l

(4.51)

f. (t)F. J ~

r

where {f (t)}l are linearly independent scalar sets of functions i of t E R which are extracted from A(t) and F. EM. In this n

~

case the commutative property of A(t) yields

~

[ j=l

f. (t)F.J J J

[ft j=!~ t

f. (s)F .dS] J J

o

f.(t)F. J J

h.(t,t )F.] J

0

J

t

J A(s ) d sA( t ) t

o

h.(t,t )F. J 0 J

[~ j

=

1

f. (t)F.] J J

STABILITY OF LINEAR SYSTEMS

J 16

which can only hold iff the {Fj}~

are mutually commutative, that

is, F.F.

F.F. ~

J

J

for all

~

i, j

1,2, ... r,

where t

hj(t,t

f

o)

fj(s)ds.

to Also since the linear system (4.50)

commutative

~s

t

f

ex p[

(jl(t,t )

o

t

exp[

A(S)dS] o

~

h.(t,t )F.]

j=1

J

J

0

r

IT exp [h. (t , t ) F . ] J 0 J j=l

-

and from

{F.} E M

the state transition matrix of (4.50) can be

n

J

(4.52)

computed through (4.52) as though the system was time-invariant, We note in passing that if each F.

since h.(t,t ) are scalars. J

0

has n-distinct characteristic values

k

=

1, ... ,n;

J

Theorem 4.3 gives

for

P E M

n

a nonsingular Liapunov transformation (or similarity

transformation in this case). -1

P

r [I

j=1

f.(t)F.]P J

r

f. (t)p j=l J

I

Hence

J

-1

r

F.P J

j

I =

1

f.(t)Diag{Ak[F.]} (4.53) J

J

The transformation of A(t) has diagonalised both sides of (4.53) -1

and since A(t) and P

A(t)P must have the same characteristic

4. KINEMATIC SIMILARITY

117

values {A.(A(t»}, then (4.53) provides a simple relationship J

between the characteristic values of A(t) and the constant matF

r~ces

and the associated functions f (t), as, i

i

r

A. (Af t ~

2:

)

j=1

f.(t)A. [F.], J

We can summarise the above

1,2, ... n

i.

J

~

(4.54)

.

the following theorem:

~n

Theorem 4.16 Consider the linear time varying system M

n

x

= A(t)x,

if

A(t)

€

commutative then

~s

r

(i) A(t) can be expressed as

2:

j=1

f.(t)F. J

J

tually commutative, (ii) the system state transition matrix can be written as r


where

exp [h. (r , 1.'

IT

J

j=l

f

h.(t,t) J

0

t

0

)F.

J

]

t

f.(s)ds. o

The necessary and sufficient con-

J

ditions for stability can be determined directly from this expression for
o). (iii) If the characteristic values of A(t) are distinct then they

are given by r

A. (A(t»

2:

~

j=1

(iv) The system

x

for all (t,t ) and 0

lim h.(t,t)+ 0 t->= ~ 1,2,

,r;

,n. Example 4.1

and

f.(t)A.[F.], J

i.

= A(t)x, i

with

i

J

1,2, ... n .

is (a) stable i f

Ih. (t,t ) 0

~

I

<

1,2, ... ,r; (b) asymptotically stable if h.(t,t) > 0 ~

0

Re{\[F ] } < 0 j

for all (t,t ) 0

for all

j

and

1,2, ... ,r;

2,

Consider the linear periodic system

x

A(t)x

with

i

k

I,

STABILITY OF LINEAR SYSTEMS

118

6 sinwt]

o.c o suit;

A (r )

for

[

-osinwt

0,6 > 0,

(4.55)

acoswt

a,0,6 finite

w f O.

and

Clearly A(t) can be de-

composed into the form (4.51 ) as

At t )

acoswt

[: ~]

s i noit

+

:]

[ _0,

F 1,F 2 E M clearly commute and so by theorem 4.16 n the periodic system above is commutative. Also the system (4.55)

The matrices

~s

stable since

<

lacoswtl and

<

Isinwtl

for all

(w, t ) .

From equation (4.52) the state transition matrix of system

(4.55) is easily computed on noting that, for

t

o

o

t

f

acoswsds

aw

-I

sinwt,

0

t -I

f sinwsds

W

(I-coswt),

0

as cos(06w-1 (l-coswt» 1J(t,O)

=

-1

exp(aw

0-1 sin(06w-1 (I-coswt» ]

sinwt) [

-6

-I

-1

sin(oSw

-1

(l-coswt»cos(oSw

(I-coswt»

The characteristic values of F 1 are (1,1) and-the characteris1

tic values of F2 are ±j(OS)2, so from theorem 4.16 the characteristic values of A(t) are given by

4. KINEMATIC SIMILARITY

119

I

(a 2+o 2 S2)'exp(jwt) 1

(a 2 +o 2 S2 ) ' e xp (- j wt )

AZ(A(t»

which indicates that the characteristic values of A(t) are periodic and alternate between the left and right half s-plane; however x = A(t)x

the commutative periodic system w # 0,

long as

is always stable as

and so stabili ty is independent of the charac-

teristic values of {F. } whenever

{f.(t)} rL i.

~

MI'

When

w

=

a

the

periodic system (4.55) becomes time-invariant and unstable, which indicates that unstable linear time-invariant systems can be stabilised by sinusoidal modulation. The results of theorem 4.16 can be related to theorem 4.9, for the decomposition A (t)

p

= Ap(t) + Ao E Mn + ,

A(t)

as,

-

A E M o n

by expanding

m

A (t)

L

P

g.(t)G. J

j=l

J

The Liapunov transformation in this case becomes m IT exp [h. (t)G. J J J j= I

pet)

where

h. (t ) J

t

J g. (s)ds J

0

for

ition matrix for the system above and theorem 4.16 as

w(t,a)

x

j

=

=

(A (t)+A )x p 0

The state trans-

1,2, ... ,m.

follows from the

~

t { j=l eXP[h.(t)G.J}eXPA J J 0

(4.56)

which is readily computed once the decomposition of A (t) is fixed.

Finally from theorem 4.9 the system

•

P

x = A(t)x

matically similar to the time invariant system

y

=

is kine-

AoY'

A difficulty with theorem 4.16 is the non-uniqueness of the decomposition (4.51); however if the coefficient matrix

A(t) EM

n

120

STABILITY OF LINEAR SYSTEMS

A.r, (t), i = 1,2, ... ,n; Zadeh

has n-distinct characteristic values

and Desoer (1963) have shown that A(t) has a unique decomposition for all

t

R

E:

n

z

A(t)

A. (t)R. (t ) ~

i=1

(4.57)

~

R. (t )

where the (nxn) matrix

lim US-A. S-+A. (t.)

R. (t ) ~

E:

~

~

(t )

is given by

M n

(sI-A(t»

-1

}

~

T

e.(t)(e~(t»

(4.58)

J

~

-1

R. (t) is called the residue matrix of (sl-A(t»

for all t.

t.

associated with the characteristic value A. (t) and the set of i.

characteristic vectors {e

of A(t); {(ei (t»T}7 is the trans-

i(t)}7 n pose of the reciprocal basis of {e (t)}I' i that for all t E: R R. (t)R. (t.)

o.. R.(t)

R. (t)R. (t.)

J

~

J

From (4.58) we see

~

~J

(4.59)

~

and so the residue matrices are mutually commutative, and the state transition matrix of the time-varying system with

A(t)


o

E:

M

commutative with its integral, is given by

n

t

~

j=1

only of use if

f A.(S)dS]

[ex p R. ~

if A(t) is decomposed ~s

x = A(t)x

~n

t

o

its spectral form (4.57).

R. E M ~

(4.60)

J

n

for all

t E Rand

This result

i = 1,2, ... ,n.

Example 4.2: (Marcus-Yambe problem (1960)) Consider the linear periodic control system x

A(t)x + u

where

and - 1 -

A(t)

[ -]

- Cisintcost

-I

-+

.... Cisin 2t

(4.61 )

(4.62)

121

4. KINEMATIC SIMILARITY

{(a-2)±(a 2-4)}/2

The characteristic values of A(t) are independent of t.

If the system (4.61) were stationary we would

a < 2,

predict stability for

~n

fact it is known that (4.61)

unbounded solutions for

possesses

which are

a < 2,

and this apparent con-

tradiction is resolved as follows:Since A(t) is periodic it is kinematically similar to a constant matrix a suitable Liapunov transform is given by,

r

P (r )

cost

Sint]

l-sint

cost

so that the above transformation,

x(t)

P(t)y(t),

produces the

time invariant system By + E(t)u

y

(4.63)

where

I

cost

E(t)

B

L sint and the condition for stability is a

=

2

a < 1.

-Sint], cost Suppose however that

then the kinematically similar systems (4.61),

unstable; the problem is now to select a feedback

(4.63) are

u(t)

=

-C(t)x

such that x ~s

[A(t)-C(t) ]x

stable.

(4.64)

Using the same Liapunov transformation, pet), it will

be assumed that the constant matrix F is kinematically similar to [A(t)-C(t)] and given by

That is

F

~

A(t)-C(t)

therefore stable.

y(t.)

has characteristic values -2 and -J and

It follows that

Fy + P ( t )

-1

u (t )

(4.65)

122

STABILITY OF LINEAR SYSTEMS

with

= P(t)y(t).

x(t)

However, F is composed of two parts Band

some matrix D such that

B -

F

D

[

0.=2

o

d zz

so that

and

d 11

d1Z]

d 21

d zz

d 11

3,

for the selected F.

Thence [B-D]

~

A(t) - C(t)

or alternatively P(t)BP(t) But

B

~

-1'

+ P(t)P(t)

A(t),

C(t)

-1

- P(t)DP(t)

-1

A(t) - C(t).

therefore

P(t)DP(t)-l

31

-sintcost]

cosZt

L-sintcost

sinZt

which illustrates that the unstable periodic system (4.61) can be stabilised by a state feedback control law with time varying gain C(t). It A(t)

~s

possible to investigate the stability of the system

via another Liapunov transformation

pet)

=

exp(P1t),

x PI

M if A(t), A(t) sM. We know that A(t) ~ B if B = p(t)-l n n (A(t)P(t)-P(t», so using the above Liapunov transformation P-I(t)(A(t)P(t)-P(t»

exp(-P1t )(A(t )-P1)exp(Plt ) 0 0 0

which is independent of t for all t

o

and can be set equal to B.

The solution to the time invariant system y (t.)

exp{B(t-t )}y(t ) o

therefore x Ct )

P(t)y(t)

0

y=

By

is

S

4. KINEMATIC SIMILARITY

123

exp(Plt)exp(B(t-t ))exp(-P1t )x(t ) 0 0 0


(4.66)

Example 4.3 Consider again the Marcus-Yambe problem of example 4.2.

A

suitable Liapunov transformation matrix for the periodic matrix A(t) of (4.62)

l-I

exp 1

pet)

so that

for

t

o

.l

~s

B

~

O·,

A(t)

0

~s

given by

the same value for B as

~n

example 4.2.

Finally from (4.66), the state transition matrix is given by exp l Co--Lj t l co s t

exp (-t) s i n t

-exp[(a-I)t]sint

exp(-t)cost

]


which illustrates that the periodic system (4.61)

a <

I.

~s

stable for

124

STABILITY OF LINEAR SYSTEMS

References

Bellman, R. (1953). "Stability theory of differential equations", McGraw-Hill, New York Erugin, N.P. (1946). Trud.Mat.Inst.Steklov 13, 95 (in Russian) Gantmacher, F.R. (1959). "The theory of" matrices", Vol.I, II, Chelsea, New York Hamburger, H.L. and Grimshaw, M.E. (1951). "Linear transformations in n-dimensional vector space", Cambridge University Press Langenhop, C.E. (1960). Trans. Amer. Math. Soc. 97, 317-326 Liapunov, A.M. (1893). Comm.Soc.Math.Kharkov (in Russian).(In translation: Ann.Math.Studies 17, Princeton, (1949» Lillo, J.C. (1961). Proc.Amer.Math.Soc. 12, 400-407 Markus, L. (1955). Math.Zeitschr 62, 310-319 Markus, L. and Yambe, H. (1960). Osaka Math.J. 12, 305-312 Martin, J.F.P. (1966). Proc.Amer.Math.Soc. 17, 636-648 Proctor, T.G. (1969). Proc.Amer. Math. Soc. 22, 503-508 Sansone, G. and Conti, R. (1964). "Nonlinear differential equations", Pergamon Press, London Vulpe, I.M. (1972). Diff.Uravneniya 8, 2156 Wu, M.-Y. (1980). Int.J.Control 31, 11-20 Zadeh, L.A. and Desoer, C.A. (1963). "Linear systems theory", McGraw-Hill, New York

Chapter 5 STABILITY THEORY FOR NON-STATIONARY SYSTEMS

5.1

Local Equilibrium Stability Conditions

The stability of dynamical systems with respect to initial conditions and disturbances is the single most significant criterionin system design.

The equilibrium states of a system

(given by those points in the state space for which the time derivative of the state vector is zero) is stable if for small initial disturbances from this point the system remains within the vacinity of the equilibrium state.

Should the system res-

ponse eventually converge to this point then the stability property is also asymptotic within this domain of state space. Clearly these concepts are local and the domain

of attraction

or convergence about the equilibrium state is crucial in control system design.

Hopefully this domain of attraction includes the

whole state space, we then have asymptotic stability in the

large, whereby after any initial disturbance and initial state the system response will eventually converge to the equilibrium state. A,very large number of definitions of stability exist, (LaSalle and Lefeschetz, 1961; Hahn, 1963, 1967; Willems, 1970; Yoshizawa, 1975; and Venkatesh, 1977) but only those which are of practical use and are relevant to systems described by homogeneous nonstationary differential equations will be discussed

126

STABILITY OF LINEAR SYSTEMS

in this chapter.

The various definitions of stability can be

broadly classified as those which deal with the equilibrium of the null solution of free or unforced systems and those which consider the dynamic response of systems subject to various classes of forcing functions or inputs (these usually lead to input/ output stability criteria - see Desoer and Vidysagar, 1975). We will consider the general vector set of homogeneous differential equations, f (r , x) ,

x

x( t )

x

o

(5.1)

o

where x(t) is a n-dimensional continuously differentiable state n vector with x:R+ = [0,(0) -+ En, f:R+ x B=D-+E with f I t ,») = 0 for all t E R+ and f(t,x) satisfies a Lipschitz condition in x and B = {x ex

En,

E

Ilxll

ct > a}.

< ct,

In the following, the

equilibrium state x e of (5.1) can always be set equal to zero by a linear state transformation, so that the equilibrium state x e and the null solution to (5. I) are considered throughout as equivalent.

Definition 5.1: Stability of the equilibrium state (Willems, 1970) The zero solution x(t;x ,t ) = 0 for all t o

0

t

~

0

of the dif-

ferential equation ~ = f(t,x) is said to be Liapunov stable if and only if for each t such that <

6(~,t)

> 0 and each ~ > 0 there is a 6(~,t

o

0

) > 0

o

implies that Ilx(t;x ,t ) o

0

II

<

~

for all

t

~

t . o

The geometric interpretation of definition 5.1 is that by constraining the initial state to a sufficiently small sphere centred at the origin the resulting state trajectory remains for all t > t

o

in a prescribed sphere of radius

~.

If the describing differential equations (5.1) are linear: x

A(t)x;

for A(t) EM, n

x

o

= x(t ), 0

(5.2)

5. STABILITY OF NONSTATIONARY SYSTEMS

127

then the conditions of definition 5.1 can be simplified, since the state solution

x(t;x , t ) o

0

x

that is for any 6 S R, o' t ;:> t .

initiating from x

is linear in

0

x(t;6x,t) = 6x(t;x ,t) o

0

0

for all

0

o

Definition 5.2: Stability of linear systems

The zero solution

x(t'x,t) = 0 , 0 0

for all t

;:>

t 0 of the linear

system (5.2) is said to stable in the sense of Liapunov if and only if for each t

;:> 0 there is a finite constant N(t ) such

a

0

that for all

Ilx(t;x a ,t 0 ) II

(5.3)

t;:> t . o

Geometrically definition 5.2 means that the solution to (5.2) remains for all t ;:> t Theorem 5.1:

in the sphere defined by (5.3).

o

Stability of linear non-autonomous systems (Willems 1970)

~ = A(t)x,

The zero solution of

A(t) s M, n

is stable in

the sense of Liapunov if there is a constant N (which may depend upon t ) such that o

11
S;

o

for all

N(t) 0

t;:> t

(5.4)

o

where t ; hence a

a

=

inequality (5.4) follows.

a

1

Conversely if any of these solutions

is unbounded,

be found.

Proof: for t ;:> t

and the properties of the state transition

o matrix (section 3.4) we have,

'1Ix(t;t ,x ) II o

11
0

S;

S;

II
II· 11
11
0

0

Ilx 0 II

0

II

128

STABILITY OF LINEAR SYSTEMS

Hence definition 5.2 can be satisfied with

N(t) o

N II <1>(0, t )

=

0

II.

The remainder of the proof follows by contradiction: if inequality (5.4) is false, then

11<1>(t,o)

II

is unbounded on R+, hence at

least one ¢(t;o,I

is unbounded on R+. Consequently inequality k) (5.3) cannot be satisfied for any finite N(t ) and the zero soluo

tion must be unstable in the sense of Liapunov. For nonstationary systems it is important to distinguish between uniform and nonuniform stability properties.

The zero solu-

tion to (5.1) is uniformZy stable if the 0 in definition 5.1 is independent of t theorem 5.1

So for the linear system (5.2) the N of

o

independent of t , and the theorem's condition

~s

o

(5.4) becomes,

for all

11<1>(t,t o)11

t:2: t

(5.5)

o

and the stability of the linear system (5.2) is uniform.

ExampZe 5.1, (Massera 1949): Consider the linear scalar system x I t; ) o

(4t sin t - 2t)x,

x

x

o

with solution x

Ct ;« ,t ) o

x

0

o

exp{4 sin t - 4t cos t - t 2

- 4sint

+ 4t cost

o

0

0

+ t 2}. 0

We now show that whilst the equilibrium solution x

0 is stable

e

(in fact asympotically stable in the large) this does not imply uniform stability.

Now

x{(2a+l)n;x ,2na}

x

o

0

exp{(4a+l)n(4-n)}

is bounded for a given a, however no bound exists independently of a since

exp{(4a+l)n(4-n)}

can be made as large as we wish.

Consider now the nonlinear system (5. I) cient vector f(t,x) where

f:R+

x

B ~

~s

En

~n

which its coeffi-

periodic, that is f(t,x) and

f(t,o)

=

0

= f(t+w,x), w>o

for all t

E

R+.

In

this special case if the zero solution is stable it is also formly stable (Yoshizawa 1966), however if f(t,x) is almost

un~

129

5. STABILITY OF NONSTATIONARY SYSTEMS

then stability is not necessarily equivalent to uniform

~eriodic

stability.

This nonequivalence for

f(t,x) E AP(C)

was demon-

strated by Conley and Miller (1965), who constructed an almost periodic function

with the properties:

f(t) E AP(C)

T

f f(u)du

(L)

T

as

co

-+

-+

o

There exists real sequences {t }, {T } such that as n

(ii)

T

n

-+

"",

t

n

-+ ""

and

t

! n t

n

n

+T

f(u)du < -n

-+ "",

n = 1,2,

for

n

Example 5.2: Consider the scalar linear equation -f(t)x,

x

f(t)

the solutions are x(t;x ,t ) t

(i) above,

o

0

(5.6)

AP(C),

E =

x

exp(-!

t

0

t

f(u)du).

By property

0

exp(-J f(u)du) -+ 0 as t -+ "" and there exists a to t constant N such that exp(-! f(u)du) ~ N (where N may depend upon to)'

~o

Thus the zero solution of (5.6) is stable.

However

if we consider the solution of (5.6) through the point (t ,x ), n

x

o

> 0

then

x Ct

+ T ;x ,t )

t +T

n

non

x

exp(-

o

n

f

t

~

nf(u)du)

x

0

0

exp(n),

n

which shows that the zero solution of (5.6)

~s

not uniformly

stable. If the upper bound

N(t) o

II x II 0

~n

equality (5.3) is replaced

by a bound B(t ) which is independent of the initial state x , o

o

we then have bounded or Lagrange stability; if in addition this bound B is independent of t

o

the null solution

~s

uniformly boun-

ded, which is also equivalent to uniform stability for linear systems. 5.2 . Asymptotic Stability For the majority of practical system designs Lagrange or Liapunov stability is insufficiently precise and some kind of convergence of the state solution with increasing t to the

STABILITY OF LINEAR SYSTEMS

130

equilibrium state is necessary, hence we say that a system is asymptotically stable if it is both convergent and stable; to be more precise we have: Definition 5.3: Asymptotic stability The zero solution of

= f(t,x)

x

is asymptotically stable if

it is stable and if there exists a o(t)

oCto) >

°

such that

Ilxoll <

implies that

o

Ilx(t;x ,t ) II o

°

-+

0

as

t

-+

00.

For the linear system (5.2) the above definition holds for all x

and can be geometrically interpreted as requiring the trajec-

o

tory

{x(t;x ,t)} o

N(t ) Ilxoll o

to not only remain in the sphere of radius

0

(by stability in the sense of

Lj apunov )

but in addi-

tion for large t the trajectory is to come arbitrarily close to the origin of the state space.

II
mous system (5.2) since finite t o

as -+

t

°

-+

II

is a finite number for any

we can conclude that

for any fixed and finite t

00

as

o

and since from the properties of transition matrices

o

=

Moreover for the linear autono-

t

o

11
if and only if

II -+ ° II
The following theorem is now obvious:

-+

Theorem 5.2: Asymptotic stability of linear systems The zero solution of

~

A(t) E M t E R+ is asympn, totically stable if there exists a finite number a (which may de=

A(t)x,

pend upon t ) such that 0

II
~

II
-+

a

for all

°

as

t ::> t

0

and 0

t

-+

00

If in addition the bound a is independent of to and the above limit holds uniformly scalar

~

there is a

T(~)

~n

t , or equivalently for some positive o

which is independent of t

o

such that

11 t + T(O, then the linear system o 0 (5.2) is uniformly asymptotically stable.

Example 5.3: Consider the scalar system (Coppel, 1965)

x

=

a(t)x

5. STABILITY OF NON STATIONARY SYSTEMS x(t) = x

its solution is given by

131

exp(f.t a(u)du), hence from to definition 5.3 its zero solution is asymptotically stable if and only if fta(u)du-+-- oo as t-+-oo and similarly it is uniformly o

stable if and only if

j,ta(u)du

o

If the parameter a Ct) = sin log t + cos log t - S t 1 < S < 12; then a(u)du = t sin log t - St -+ -00, as t -+ o

t

to·

1 ~

f

the system is asymptotically stable. exp(2nn + 8 val

(8

Jt t

1

82 )

~

a(u)du

However selecting t

t = exp(2nn + 8

) ,

0: %0: 1

for so

00,

1

=

) where [8 ] is the inter1,8 2 2 sin8 + cos8 ~ y > S, then as n-+ 00

in which (y-S)(exp8

t ~

is bounded above for each

t1

-

2

exp8

1)exp(2nn)

-+

00,

1

and consequently the system is not uniformly stable.

It there-

fore follows that asymptotic stability does not imply uniform stability. An exception to the above' conclusion is for the general periodic system: x

x( t )

f(t,x) ,

for

f:R+ x B

w >

o.

x

0

D

-+ En ,

(5.7)

0

f(t,o)

and

0

f(t+w,x) = f(t,x)

In as much as if the zero solution of (5.7) is asympto-

tically stable then it is also uniformly stable (Yoshizawa, The restriction on

f(t,o)

196~.

= 0 can be removed by utilising the

following lemma for almost periodic systems with f(t,x) E AP(C):Lemma 5.1: (Yoshizawa 1975) For the almost periodic system if for any to E R+ and T(to'O ~ 0 for all

~

such that t

~

t

o

x = f(t,x), f(t,x) E AP(C) ,

> 0 there exists a OCto) > 0 and a

~

Ilxoll

+ T(t ,s), 0

< oCto)

implies

Ilx(t;xo,t

o) then the zero solution of the

II <

system is defined on R+. W~

note that the condition f(t,o)

=

0 is not assumed and that

for the existence of the zero solution it is sufficient to assume t

o

= o.

Now applying lemma 5.1 to the periodic system (5.7)

without the condition f(t,o)

= 0 we get:

132

STABILITY OF LINEAR SYSTEMS

Theorem 5.3: Uniform asymptotic stability for periodic systems

(Strauss 1969)

For the periodic system (5.7) (without

f(t,o)

=

0) subject to

the assumptions of lemma 5.1, the system's zero solution is uniformly asymptotically stable.

Proof: The proof is relatively simple, in that the existence of the zero solution to (5.7) on R+ follows directly from lemma 5.1. All that needs to be shown

~s

that the solution is unique to the

right, then the solution is both asymptotically and uniformly stable (Strauss, (1969». Returning to the linear homogeneous equation, x

A(t)x,

(5.8)

A(t) EM, n

its stability properties can be summarised completely

~n

terms

of its fundamental matrix X(t) as: Theorem 5.4: General stability conditions for linear systems Let

X(t)

be a fundamental matrix of the linear system (5.8),

then the zero solution to (5.8) is: (i) Stable if and only if there exists a constant N > 0 such that IIX(t) II

S;

for all

N

t 2 t

(5.9)

o

(ii) Uniformly stable if and only if there exists a constant N > 0 such that 1 II X(t)X- (s ) II

S;

N

for

t

0

S;

s

S;

t <

(5.10)

00,

(iii) Asymptotically stable if and only if II X( t ) II

-+

as

0

t

-+

(iv) Uniformly asymptotically

00

, s~able

(5. 11 )

if there exist positive con-

stants N, a such that IIX(t)x-1(s)

II

s;

Nexp(-a(t-s»

for (5.12)

Proof: Without loss of generality assume that X(t ) = I, so that X(t) =
5.2.

o

the state transition matrix of (5.8).

o (5.11) were established respectively in theorems

The solution to (5.8) which takes on the value

1:;

Conditions

5.1 and at time s

5. STABILITY OF NON STATIONARY SYSTEMS -1

x(t) = X(t)X

is

N

if

therefore since the solution

(s)~,

Nil ~II II X(t)X- l (s ) II

II x Ct) II 2:

II ~ II < 1,

II x (t ) II Setting

S

t

>

IIX(t)X

-1

= Nexp(-aT) ,

s+T.

-1

sN

>

II ~ II ,

which holds uniformly

Suppose now that (5.10) does hold then

and

II X(t)X for

(5.13)

and

N > s

stable

~s

s

<

~

and hence (5. 10) follows. for

133

(s)~11

~

IIX(t)x-l(s) II

N.

~

inequality (5.13) becomes -1

(s)~

II

~

Nexp(-a(t-s»)

s

~

So that the null solution to (5.8) is uniformly

asymptotically stable.

An equivalent condition (Willems, (1970»

to condition (iv) for uniform asymptotic stability of (5.8) is if there exist positive constants N a such that l, l for

II X(t) II

> t

2:

s

2:

t

o

.

The above inequality is a form of exponential stability and is clearly a sufficient condition for uniform asymptotic stability. We have seen in section 4.2 that the Liapunov transformation x(t) = P(t)z(t) where

-1

B =P

transforms the linear system (5.8) into

AP - P

-1'

P.

Thus if

X(t) = P(t)Z(t),

z = Bz

where X(t)

and Z(t) are the fundamental matrices of their respective system -1

Also

Z =P

-1

-1-1

=Z P and the boundedness of X implies boundedness of Z. If ~ = Bz is stable then it is also uniformly stable (that is IIZ(t)Z-l(T)11 ~ N) since B sM. equations, then

l IIX(t)X- (T) II

=

X,

0 s T s t,

n

1

IIp(t)Z(t)Z-l (T)P- (T) II s II P(t) II

x IIZ(t)Z-l(T)11 .llp-l(T)11 N' and

X

s

IIp(t)11 .N.llp-

1(T)11

~ N'

for some

and so (5.8) is uniformly stable if it is

stable and reducible (or equivalently a Liapunov transformation exists). For the special case of

A(t) = A EM,

rix of the linear system (5.8) is where

J

Diag{J.} 1

n

the fundamental mat-

X(t) = exp Ct.A) = S exp (tJ)S

-1

,

is the Jordan canonical form of the constant

matrix A and S is an invertible similarity transforming matrix

134

STABILITY OF LINEAR SYSTEMS

of A (Desoer, 1970).

From theorem 5.4 and the explicit represen-

tation of the exponential matrix in Jordan canonical form we obtain directly: Theorem 5.5: Stability conditions for linear time invariant systems The null solution to the linear time invariant system x = Ax,

(5.14)

is (i) stable if and only if every characteristic value of A has a real part not greater than zero, that is Re(A.) f 0 for all i, and those with Re(A.) r.

=0

t.

are distinct;

(ii) asymptotically stable if and only if every characteristic value of A has negative real part (that is Re(A.) < 0 for all i). ~

Clearly stability for linear time invariant systems implies uniform stability and asymptotic stability imples uniform asymptotic stability.

Theorem 5.5 is particularly significant

s~nce

the stability properties of linear time invariant systems depend only upon the characteristic values of the coefficient matrix A, unlike in the general linear nonstationary system (5.8) where knowledge of the fundamental matrix and thus a complete set of independent solutions is required.

The only exceptions to these

restrictions is when A(t) is periodic or has certain asymptotic properties associated with large t (Rapoport, 1954). We have shown in section 3.5 that Floquet theory allows the stability results of lii.ear time invariant systems with coefficient matrix A to be used for linear periodic systems with the characteristic value of A being replaced by the characteristic exponents of x

A(t)

A(t)x,

A(t+w), w > O.

Thus for the periodic system

A(t) = A(t+w) E M n,

(5.15)

we have the following theorems directly from theorem 5.5. Theorem 5.6: Uniform stability of linear periodic systems The null solution of (5.15) is uniformly stable if and only if A(t) has no characteristic exponent with positive real part, and

135

5. STABILITY OF NONSTATIONARY SYSTEMS

if the characteristic exponents with zero real parts correspond to Jordan blocks of order I in the Jordan canonical form of matrix B. The conditions of theorem 5.6 are equivalent to no characteristic multipliers of A(t) being greater than 1, and that all Jordan blocks in the Jordan canonical form of A which corresponds to characteristic multipliers of magnitude I, are of order 1. Theorem 5.7: Asymptotic stability for linear periodic systems The null solution of (5.15) is uniformly asymptotically stable if and only if A(t) has only characteristic exponents with negative real parts, or equivalently if A(t) has culy characteristic multipliers with magnitudes less than I. 5.3

Matrix Projections and Dichotomies of Linear Systems In this section we briefly digress to develop the theory of

matrix projections and dichotomy of solutions of linear systems; so that we can derive stability conditions which are characterised only by the elements of coefficient matrices

A(t) E M

n

near homogeneous systems: A(t)x,

x

x( t

o

x

)

o

for li-

(5. 16)

•

From a practical viewpoint unstable systems are as interesting as stable systems, particularly if the factors causing instability can be identified and compensated for in system design.

We

consider the situation whereby the system (5.16) exhibits two kinds of solutions in state space: one which is bounded by a decaying exponential as exponential as

t

~

+00.

t

~

+00

and the other bounded by a growing

Clearly the solutions continued backward

in time will follow opposite bounds.

So for any

t

E

R

the only

bounded solution to (5.16) is the trivial one which is at the opposite extreme from having all solutions bounded.

To clarify

the situation define E as the set of all points in the state oo space R which are the values for t = 0 of bounded solutions n

of (5.16); this set is clearly a non-void manifold since

0 E E

00

136

STABILITY OF LINEAR SYSTEMS t = 0

The value

for the definition of E

Following the solutions of (5.16)

and selected for convenience. t = 0

from

to E R,

to

quite arbitrary

~s

00

the set of values for

t = to

of the

bounded solutions of (5.16) is precisely

X(t)E , where X(t) o 00 By analogy with E define oo' as the set of all points in the state space which are values

is a fundamental matrix of (5.16). E

o

t = 0

for t

-+ +00;

o

E

of integral curves of (5.16) which tend to zero as

this set is a non-void linear submanifold of E

00

since

E .

In the following suppose that the state space R can be n and another subpartitioned and defined by the direct sum of E o

00

Here again the choice

space Ej, which is the complement of E of

00

o

t

Massera and Scheffer (1958) have shown

is arbitrary.

that if the state space is the direct sum of subspaces E and oo Ej, then every solution x(t) of (5.16) can be expressed uniquely as: x ( t ) = x (t ) + xl (t ) , o

wi t h x (t.) 0

E

X( t ) E

00

,x 1 ( t

)

E

and PI = I-P o -1 0 respectively (or X(t)P X (t) and

Alternatively, there exist projections P and E

ated with E

00

X(t)(I-P)X

-1

o

X( t ) E j . associ-

o

(t) when associated with X(t)E

00

and X(t)E j respec-

tively), such that

with

P

o

o

space R .

0

and

n (I-P ) = {a} o

Note that if

n

P 2

P

o

x(o) E E

00

o

for all x(o)

~n

the state

n E 1 we have

P x(o)

x(o)

o

P (I-P )x(o) o 0 so that

(5.17)

P x(o) + (I-P )x(o),

x(o)

E

00

n E

1

be supplementary.

=

{a};

=

(P -P 2)x(0) 0

0

0

in this case the projection ~s

said to

It is now obvious that every solution x(t) of

(5.16) can be written as x

Ct )

X(t) (I-P)X

-1

-1

(t )x(t ) + X(t)P X

000

(t;

)x(t )

000

(5.

18)

This representation of the solution of (5.16) as the sum of two

137

5. STABILITY OF NONSTATIONARY SYSTEMS sets of solutions starting from E ferred to as a

dichotomy.

and E

00

1

is unique and is re-

A linear transformation S leaves the

, E invariant if and only if the transformation como If the characmutes with each of the projections P , (I-P ). o 0 teristic values {A.} of S are contained in two disjoint subsets subspaces E

00

t.

G

and G

00

1,

then the state space

R

n

=

E

00

UE

E 1 ) are just the elements of G jection P

P

f

1 (2ni)

o

where

of R onto E n 00

o

(AI-S)

-1

00

(or G 1).

n

(or S

00

->-

The corresponding pro-

(5.19 )

dA

is a closed contour

~n

the complex characteristic value 00

r

(Riesz and Nagy,

Re(A.) < a,

such that

~

and each

are scalar numbers.

onto E

k > 0

E

r r

G 1 lie outside

S> a

->-

~s

plane that contains all elements of G

R

invariant un-

~s

S

der S, such that the characteristic values of

00

and E

1

, whilst all elements of

1955).

If each

Ai E G1 ,

A. ~

>

E

S

G

00

~s

where of

Then the projections

commute

such that

Ilexp(St)P II o Ilexp(St)(I-P

:s;

o

)11

k exp tc.t ) :s;

k exp Cfst )

0

for

t

for

t :s; 0

2:

An important lemma, based upon the above projection concepts, that is used to determine the stability of linear homogeneous differential equations with time-varying coefficients is: Lennna 5.2 Let

P

X( t ) E M be an invertible matrix for t 2: t , and let n o be a projection. If there exists a positive constant k such

o that

(

t

J

k

for

t

::> t

o

to then there exists a positive constant y such that

(5.20)

STABILITY OF LINEAR SYSTEMS

138

II X(t)P o II

yexp(-k

~

-I

t)

for

t

Proof: It follows that Coppel (1965). let

aIt

[f t

= !!X(t)P II-I

)

t

(5.21)

o

Suppose that

,,0,

and

II X(t)p o II

~

P

then from the identity

o

t

o

t

fto X(t)P X-I(s)X(s)P a(s)ds

a(S)ds]X(t)P 0

o

o

it follows that

[a( t ) j

-I

t

Setting

f a(s)ds,

bet)

to

aCt)

b(tl)exp(k

therefore,

IIX(t)p

o

-I

Then setting -I

t);

Y

0

~

fa(S)dS t

therefore

(t-t l »

II

for

~

[a(t)j-I

-I

-I

k[b(t)j-I

exp(-k

-I

exp(k t l) l) the lemma follows. ~

kb(t

k.

o

kb(t l)

yexp(-k

~

-I

(t-t l »

for

so large that

To return to the question of bounded solutions of the linear homogeneous system (5.16) we now make the following definition:

Definition 5.4: Exponential dichotomy The homogeneous linear differential equation

x = A(t)x

is

said to possess an exponential dichotomy if there exists a projection P

o

and positive constants

IIX(t)p X-let

)11

~

such that

k,~,a,6

kexp l-a(t-t)1

0 0 0

II ~ ~ exp !-6(t-t ) I 0 0 0

l IIX(t) (I-P )X- (t; ) for all

t

E

for

t

for

t

t

~

o

~

o

t

(5.22 )

R.

If in definition 5.4 the constants a=6=O, then the system 1S said to possess an ordinary dichotomy.

Note that for time inva-

riant systems the existence of an exponential dichotomy is equivalent to saying that the characteristic values of the coefficient matrix A lie off the imaginary axis.

It is then clear that two

time-invariant systems that are related by a similarity trans-

5. STABILITY OF NONSTATIONARY SYSTEMS

139

formation satisfy common exponential dichotomies with the same projections.

This is really a result about kinematic similarity

see also Chapter 4 and Appendix 2).

We have already noted in

theorems 5.6, 5.7 that all stability questions concerning linear homogeneous systems with periodic coefficients are given by results relating to time invariant systems with their characteristic values being replaced by the characteristic exponents of A(t). Also, like linear time invariant systems, linear periodic systems satisfy an exponential dichotomy with projection P

= I, so that

o

if the periodic system (5.15) has every characteristic exponent p. such that ~

Re(p.) < a

then there exists a constant S such

~

that

II X(t)X- 1 (s ) II or,if

:s

S exp (a(t-s»

for

t

?

for

t

:S S

s

Re(p.) > a, then i.

:s

Sexp(a\t-s»

where X(t) is the fundamental matrix of (5.15).

Also by theorem

5.4 asymptotically stable time invariant systems satisfy an exponential dichotomy with P

o

= I; the situation for other than

constant or periodic coefficients A(t) is quite different as suggested by the following example due to Fink (1974):

Example 5.4 Let A( t )

[

-] +

1L

-I -

2" s i n t cos t

3

COgLt

~

] -

•

-I +

cost •

~2. sin2. t

The characteristic values of A(t) are given by solution of Det(\(t)I - A(t» Yet the system

0 x

(-cost, sint)exp(!t)

for all t as the constants

A(t)x

~(-1

± ill).

possesses solutions of the form

whose norm

+

+ro

as

t

+

00

Fortunately there are several known conditions (Coppel, 1967 (2»

that give exponential dichotomy in the time-dependent case;

the following will be stated without proof:

STABILITY OF LINEAR SYSTEMS

140

Theorem 5.8: Exponential dichotomy conditions (Coppel 1967(2)) Let Re(A.)

S

r.

> O.

A(t)

M

S

n

possess m characteristic values A. such that ~

-a, a > 0

~

Then for

6(N,a+S,n),

and (n-m) characteristic values with ReCA.) t,

min(a,S) > n > 0

where

the system

x

=

II A( t

A(t)x

)

II

~

there is a constant

N, such that if

;>

6

6 then

satisfies an exponential dichotomy (5.22)

with

Po [:m:]

and k,£ depending only on N, a+S and n. This theorem indicates that A(t) was too large in example 5.4 and illustrates the dangers of determining system stability based only on the characteristic values of time dependent coefficients A(t).

However, rather conservative sufficient conditions for ex-

ponential dichotomy of (5.16) -based explicitly upon the diagonal dominance of the coefficient matrix A(t) can be developed from Gershgorin's theorem. values

A

1(A),

This theorem locates the characteristic

... An(A),

of the matrix A in the union of circles

in the complex plane centred along the diagonal elements {a .. (t)} ~~

of the coefficient matrix A(t) (Gantmacher 1959, and Chapter One). The radii r. of these circles are linear functions of the offi.

diagonal elements of A(t), and three types of diagonal dominance can be identified. Consider the linear homogeneous system,

(5.23)

A(t)x,

x

If the coefficient matrix A(t) ~s n l: la .. (t)! + 1;, J~ j=1

such that IRe(a .. (t»1 ~~

for all i and any I; > 0, then the matrix A is

j#i

said to be column dominant with Gershgorin circles la .. (t) J~

I.

;>

r.

i.

n l: j=1

j#i

If in addition we use the £1 norm for the state vector

141

5. STABILITY OF NONSTATIONARY SYSTEMS of (5.23) then the measure of the matrix operator is given by (see section 1.5), max {Re(a .. (t )

u 1 (A)

~~

j

a .. (t) < 0

if

for all i.

~~

n L

+

j=1 j#i

la .. (t)!}

<

J~

Utilising Coppel's inequality for

norm of x based upon ].l(A) (see section 1.5), we have II x f t

)

II

for

Ilx(t ) II exp(-t;(t-t » o

0

t 2 t

o

Hence the system (5.23) has an exponential dichotomy with projection

P

I

=

o

if A(t) is column-dominant with

a .. (t) < 0

for

i.i.

In addition the system (5.23) is exponentially stable by

all~.

the proof of theorem 5.4. If now the elements of coefficient matrix A(t) are such that n

IRe(a .. (t»12 ~~

for all i and

la .. (t)1 +,t;,

L

j=1

~J

~

> 0, then the

jii

matrix A(t) is said to be pow-dominant with circles

n L

r.

.i.

j=l jh

la .. (t)l· ~J

If we now select the £00 norm for x, the measure of the ]J00 (A)

matrix operator for this norm is

max{Re(a .. (t» ~~ i

+

n

L

j=1

la .. (t)IL ~J

Thus if A(t) is row-dominant with k(k s n) dia-

jii

gonal elements

a .. (r ) < 0

the remaining (n-k)

]Joo(A) s -t;,

for all t, then

~~

a .. (t ) > 0

for all t then

~~

and if

]Joo(A) s C

Hence the system (5.23) has an exponential dichotomy, since there are k independent solutions span(x

1

,

•••

with

~)

Ilx(t')llexp (~(t-t»

0

pendent solutions x) n

with

(-t;(t-t» o

Ilxll

Ilxll

xl' ...

~

such that

x

E

strictly increasing and

s Ilx(t)11

0

for

t 2 t,

0

and (ri-ik) inde-

such that x E span(x + l , k n) strictly decreasing and Ilx(t) II s Ilx(t ) II exp (x + ' k J

(Lazer, 1971).

•••

x

0

STABILITY OF LINEAR SYSTEMS

142

A third type of diagonal dominance of A(t) exists which is called mean dominance, if,

[Re Ca .. (t»!

2'

i i.

n

Ia J1.. (t ) I}

I:

j=l

+

E"

for all i and any

>

~

j#i

H

o.

n

I:

j=l j#i

la .. (t)1 + 1J

In this case the

n

Gershgorin circles are of radii

Then if we use the ~2 by

\1 (A) 2

= m~x 1

{

\

I:

r . 1

j=l j,ii

(Ia .. (t)[ + [a .. (t)I). 1J J1

norm for x then the measure of A is given

(A+A*)} :; 2

-~

a .. (t ) < 0 11

if

for all t and i.

Applying Coppel's inequality (I .32) to this measure gives the inequality

II x t t ) II

Ilx(t

o

)11

exp(-~(t-t

0

for

»

which is an exponential dichotomy with P

o for exponential stability for system (5.23).

t

2: t

o

,

I and a condition Indeed for any dia-

gonal dominant matrix A(t) with a .. (t) < 0 for all t and i, the 11 above inequality holds for any norm since all norms are equivalent.

In conclusion we see that

IRe(a .. (t»1 > r. is a suffi11 1 cient condition for exponential dichotomy of the linear homogeneous system (5.23). 5.4

Asymptotic Characteristic Value Stability Theory

A less conservative but more restrictive theory for the stability of linear nonstationary systems based upon the asymptotic properties of the characteristic values {A(t)} of A(t) can be developed via matrix projection theory.

Consider the linear

homogeneous system (5.23) but with the additional condition that lim A(t) = Aoo EM. n

t->=

It will be shown 1n the sequel that if Aoo

1S a distinct characteristic value of Aoo then A(t) has a unique characteristic value (exponent) A(t) in the neighbourhood of Aoo such that

lim A(t) = Aoo ' t->=

5. STABILITY OF NONSTATIONARY SYSTEMS

143

r be a closed contour in the complex characteristic value

Let

plane which includes Aoo' but does not include any other characteristic value of Aoo' then there exists a projection

(2~i)

Po

J r

(AI-Aoo)-l dA, For all points

which commutes with A00 (see also equation (5.19».

r

A on

II AI-Aoo II

~

II A(t)-AJI

S2

1 (2ni)

pet)

2'

E, > O.

t

for

J

2'

t

If

t

is chosen so large that

0

then

0

-1

(AI-A(t»

II AI-A(t) II

2'

§; 2

and (5.24)

dA

r is defined as a projection which commutes with A(t). (I-P(t»(I-P ) + p(t)p

S (r )

o

Setting

0

I + (P(t)-p )(2P -I), o

P(t)S(t)

so that

(5.25)

0

= p(t)p o = S(t)p 0

S-l(t) exists for all large t and

lim S (r) = I,

and pet)

follows that the characteristic vector

S(t)p S-l(t). ~

t

=

o

S(t)~,

to the characteristic value A(t) of A(t) inside -1

S

where

=

(t)A(t)S(t)~oo ~

00

then

t-+=

r.

It then

belonging

00

Thus (5.26)

A(t)~oo

is the characteristic vector of matrix A00 associated

with Aoo inside r. Then if A(t) is continuous and differentiable r times then by equations (5.24-26) so is pet), Set) and A(t). 00

Also if

f

IIA(s) lids <

t

pet)

~

(2ni)-1

f (AI-A(t»-l A(t)(AI-A(t»-l dA r

~

then differentiating (5.24)

00,

(2n)-1 1IA(t)11

f II (AI-A(t»-111 r

~

rr-1 IIA(t)

II

J II (AI':"'A

oo

r

)

-111

2dA

,

2dA

STABILITY OF LINEAR SYSTEMS

144

co

co

J IIp(s)llds

hence also

<

J IA(S)lds

and

co

t

<

for large t.

co

t

S(t.)

Similarly from (5.25),

Ils- 111 2 Iisil ... Iisil

pet) (2P -I)

II (S-l), II

then

0

for large t , and hence

J IIS(s)llds

<

5

00

t

We are now able to use the transformation matrix Set) to diagonalize A(t) into a Jordan type matrix.

We know that for Aco a

distinct characteristic value of Aco ' there exists a constant S1milarity type transformation matrix S such that

Aco Dco

Moreover

o D(t)

S-l A(t)S

=

are. linear combinations of the ele-

ments of A(t) with constant coefficients and are such that

f IID(s)\\ds t

<

we may then as sume that

co;

Aco and that Set) satisfies:

o Lemma 5.3: (Coppel 1965) Let

A(t) E M

n

such that

f II A(s) II ds

<

co

and l e t Aco be a

o

distinct characteristic value of

Am = lim A(t). t->=

Then there

exists for large t an invertible matrix Set) E M such that n

exists,

< 00,

and

5. STABILITY OF NONSTATIONARY SYSTEMS

-1

S

145

(t)A(t)S(t)

where A(t) is the characteristic root of A(t) such that lim A(t) = Aoo

and

t-+<x>

This lemma is now used to establish a theorem similar to theorem 5.6 for the general class of coefficient matrices A(t) E M

n

which are of bounded variation onthe interval [t ,(0). Theorem 5.9: Asymptotic uniform stability

x = A(t)x

The linear homogeneous equation

f

and

t

if:

I A(~h

lids <

o

(Conti~

1955)

with

A(t)

E

M

n

has uniformly stable solutions if and only

co

o

(i) the characteristic values of A(t) have non-negative real parts for t

~

t ; o

lim A(t), whose real parts

(ii) the characteristic roots of Aoo

t-+<x>

are zero, are distinct.

Proof: Let A. and A. (t) be respectively the charactpristic values ~

~

of Aoo and A(t) and are related by By lemma 5.3 a

Set)

E

M

n

lim A.(t), (i = 1,2, ... n).

Ai

t-->=

~

exists such that

o -1

S

(t)A(t)S(t) A

-m

where

A(t)

(t.)

Diag(A (t), ... ,Am(t». So by making this time a 1 varying transformation yet) = S-l(t)x(t) to the system equation A(t)x,

x

it

~s

A(t) EM,

(5.27)

n

uniformly stable at the same time as the system

y which

=

(S-l(t)A(t)S(t) - S-1(t)8(t»y;

~n

turn since

Jllso

1(s)8(S)

lids

<

is uniformly stable

STABILITY OF LINEAR SYSTEMS

146

at the same time as the system, y

(5.28)

S-l (t)A(t)S(t)y.

Let yet) be the principal fundamental matrix of (5.28) (i.e. yet ) = I) then o

z(r )

0

yet)

o where

l: (r )

Y

-m

Diag{exp(f t

So if

(t ) t

A1 (s ) ds •... , exp (

t

o

(i = 1.2 •..•• m)

Re{A.(t)} ~ 0 t.

f

t

A (s)ds)}. m

o

then by theorem 5.4

part (ii), the system (5.27) will be uniformly stable over [t,oo) if and only if for some N > 0, Ily

1

-m

~

(t)y- (s ) II -m

N.

for

00 > t 2 S 2 t

This inequality is satisfied since A

-m

o

.

(00) (and by similarity

Y (t)) has characteristic values with negative real parts, and -m 1(s) > t 2 S 2 t hence IIX(t)xII is bounded for and so o

(5.27) is uniformly stable by theorem 5.4.

A stronger result on uniform asymptotic stability of (5.27) can be derived if

A(t)

M

E:

and

n

IIA(t) II

is small (or equiva-

lently satisfies a Lipschitz condition in t): Theorem 5.10: Uniform asymptotic stabiZity (Lyascenko, 1954) Let T

A(t)

O.

2

t

k (r - t ) 2

1

2

E:

t

for

o

M

such that

n

N > 1

with t

1

2

Ilexp(TA(t))II

t ,

t

0

2

and 2

t

0

a > O. where

~

and

Nexp(-cn) IIA(t

for

1)-A(t) -1 2

k < a 2(NlogN)

.

II

~

If

X(t) is the fundamental matrix for x

(5.29)

A(t)x.

then IIX(t)Xwhere

1(s)

II

~

N2exp(-(3(t-s)) !

(3 = a - (kNlogN)2

asymptotically stable.

>

0,

for

and the system (5.29) is uniformly

5. STABILITY OF NONSTATIONARY SYSTEMS

Proof: For any

=

x

T

to'

~

147

(5.29) can be rewritten as

A(T)X(t) + (A(t) - A(T»X(t)

hence,

t

X(t)x(s) +

x( t )

s

Therefore Ilx(t)

f X(t)X-1(u) (A(u)-A(T»x(u)du.

II

t

"Nexp(-a(t-s»

Ilx(s)

II

+ N

f exp(-a(t-u»

x

s

1/ A(u)-A(T) 1/-1/ x(u) 1/ du Applying the Gronwell-Bellman lemma to the above inequality gives

t

Ilx(t)

II

"

f IIA(u)-A(T) II du}-llx(s) II

Nexp(-a(t-s»exp{N

s

for

t

~

s.

Setting

t

s+Y,

Lipschitz condition for y Ilx(s+y)

II "

So putting /lx(s+y)

where

a,

T = s+Y 2

k Nexp (-ay) exp ( N4y

/I

"

exp(-By) I/x(s) 1/

"

N2exp(-ay) Ilx(s)

"

IlX(t)X- (s ) II

"

~

Ilx(s)

(5.30) B and the condition

II

Finally substituting

in the above inequality for

~

N2 exp (-

in the above,

II

Clearly a

N2exp(-B(t-s»

follows from (5.30). 1

.

kNy 2 i exp (- - I 4 )

1

II

and utilising the

2} Ilx(s)/1

y

B = a-(kNlogN)2.

Ilx(t)

~

x(t)

X(t)X-I(s)~

=

an arbitrary vector B( t - s »

for

t

~

s

~

t

o

•

We note from theorem 5.4 part (iv) that this is the necessary and sufficient condition for the uniform asymptotic stability of (5.29).

There is an obvious connection between theorem 5.10

148

STABILITY OF LINEAR SYSTEMS

and theorem 5.8, since on setting p

= I

o

In theorem 5.8 all the

n

characteristic values of A(t) have negative real parts and given that

IA(t)

I

< 0

we satisfy the conditions of theorem 5.10.

Theorems 5.8-5.10 demonstrate that stability conditions for linear nonstationary homogeneous systems can be expressed in terms of the asymptotic properties of the characteristic values of the coefficient matrix A(t) and the fundamental matrix of the system equations; the question now arises can general nonlinear system equations of the form of (5.1) be treated in like manner? Suppose that the nonlinear vector f(t,x) can be decomposed by a process of linearisation into A(t) s M,

f:R xB +

n

a}.

f(t,x)

= D + En for

B

= A(t)x

= {x ;x E

+ g(t,x),

where

En, lixll «((,

(( >

The nonlinear equation A(t)x + g(t,x),

x

x

It )

x

o

(5.31 )

o

has a solution through x , which is not necessarily unique unless o

a condition such as

Ijg(t,x)

II

0 I- 0

~ oCt) jlxll where

integrable (Curtain and Pritchard, 1977) is imposed. rised element of (5.1) (or (5.31»

and The linea-

is the homogeneous linear

equation A(t)x,

x

(5.32)

A(t) EM, n

whose fundamental matrix

IS

X(t) with X(t ) = I.

We now estab-

o

lish results that relate the stability of the linear system

(5.32) to the stability of the null solution to (5.31). Theorem 5.11: (Caligo, 1940; Conti, 1955) If g(t,x) satisfies the inequality

II g(t,x) II

(5.33 )

where oCt) is non-negative and integrable over [t ,00) and if o

there exists aN> 0 such that ~

N,

for

co

>

t

2: S

2:

t

o

.

(5.34)

Then there exists a solution to (5.31) which satisfies IIx(t)

II

5

Sllx(s)

II

for all

(5.35)

149

5. STABILITY OF NONSTATIONARY SYSTEMS

B> o

where the constant addition

lim X(t) t-+=

-1

II x (s ) II < B a. lim x(t) = O.

0 is such that

then

If

in

t-+=

Corollary I: If the linear equation (5.32) is uniformly (and asymptotically) stable, and if condition (5.33) holds, then the null solution to (5.31)

~s

uniformly (and asymptotically) stable. g(t,x) = B(t)x,

Corollary II: If g(t,x) is such that B(t)

S

M n

and integrable over [to'oo), then the linear system (A(t) + B(t»x,

x

where

(5.36)

for

is uniformly (and asymptotically) stable if and only if the linear system (5.32) is uniformly (and asymptotically) stable.

Proof: Follows similarly to that of theorem 5.10.

The solution

of (5.31) is t

x(t)

X(t)X-

1(s)x(s)

+,f X(t)X-

1(u)g(u,x(u»du.

(5.37)

s

Taking norms and utilising the properties (5.34), (5.35) we get on applying the Gronwell-Bellman lemma to (5.37), t

II x (t ) II

S;

N II x (s )

II

+ N

fs o(u)

Ilx(u) Iidu

t

S;

Nllx(s)llexp{N

f o(u)du} s

for

t

~

s,

t

where so that

B

Nexp{N

J o(u)du}. s

X(t) = I

Xes)

o

Finally setting s = to

~n

(5.37)

and taking norms yields t

Ifx (t ) II

s;

Ilx(t)11

Ilxoll

+

II

J X(t) t

o

X-

1(u)o(u)llx(u)lldull

150

STABILITY OF LINEAR SYSTEMS t

II X(t) II

Ilx o II + Nsllx 0 II

f o(u)du. t

So if

lim X(t) = 0,

(5.38)

o

it follows directly from inequality (5.38)

t-+«>

that

lim x(t)

=

0,

since o(s) is integrable by definition.

t-+«>

Corollaries I, II follow immediately from condition (5.34) and inequality (5.38) on application of theorem 5.4. By further requiring that the left hand side of inequality

(5.34) be integrable over [t ,t], then X(t) is bounded and in particular

lim X(t)

=

t-+«>

°

o

(see lemma 5.2); also if oCt) is inde-

pendent of t a simple condition for asymptotic stability of

(5.31) follows:Theorem 5.12: Asymptotic stability of systems with small non-

linearities

If the fundamental matrix X(t) of the linear homogeneous systern (5.32) is such that t

J IIX(t)x-1(s)!lds t

::;

N,

for

t ?: t

o

and

N > 0, (5.39)

o

and g(t,x) satisfies the inequality Ilg(t,x)11 ::; ollxll for 0 < 1, Nthen the null solution to (5.31) is asymptotically stable.

Proof: follows directly from application of lemma 5.2 to inequality (5.39).

Also if the 0 >

° is sufficiently small for A(t)

a periodic or time invariant coefficient matrix, the asymptotic stability of the linear system (5.32) implies asymptotic stability of the null solution to the nonlinear system (5.31) since inequality (5.39) is automatically satisfied if (5.32) is asymptotically stable for A(t) periodic or time invariant.

In addi-

tion the asymptotic stability of systems (5.31) and (5.32) for A(t) periodic or time invariant is uniform.

A result for uniform

asymptotic stability of the nonlinear system (5.31) for a more general class of coefficient matrices A(t)

E

M can be established n

5. STABILITY OF NONSTATIONARY SYSTEMS

151

by condition (iv) of theorem 5.4:Theorem 5.13 If the fundamental matrix of (5.32) satisfies thE inequality

Ilx(t)x- 1 (s ) II

for

Nexp(-S(t-s»,

s;

for positive constants N, S, and if N-1S.

00

> t

Ilg(t,x)11

s

~

s;

t

~

ol[xll

o

(5.40)

for

0<

Then every solution of (5.31) is defined for all t ~ t

o

and satisfies

II x (t ) II and

s;

N exp(-y(t-s»

Ilx II < N-1a, o

where

flx(s)

II,

for t

a is

~

t ,

(5.41 )

o

y = S - oN > O.

Corollary I: If the linear system (4.32) cally stable and if

s

~

~s

uniformly asymptoti-

sufficiently small, then the zero solu-

tion of (5.31) is also uniformly asymptotically stable. Corollary II: If the linear system (5.32) is uniformly asymptotically stable and

g(t,x) = B(t)x

0, then the system

for

B(t)

~ = (A(t)+B(t»x

S

M

n

and

lim B(t) =

t-Ko

is also uniformly asympto-

tically stable.

Proof: By theorem 5.4 (iv), inequality (5.40) is a necessary and sufficient condition for uniform asymptotic stability of the linear system (5.32).

By taking norms of the solution (5.37) and

substituting inequality (5.40) into the result, the GronwellBellman lemma yields inequality (5.41) directly.

And since ine-

quality (5.41) is the condition for exponential asymptotic stability for x(t) (Yoshizawa, 1966), it is sufficient to imply uniform asymptotic stability for x(t) and corollaries I, II therefore follow directly.

For a collection of examples of

corollary I I see Cesari (1940), in which B(t) is not necessarily stable but is integrable over R+.

Example 5.5: This example, due to Perron (1930) demonstrates that although the linear system

~

=

A(t)x

is asymptotically

stable, it is not necessarily uniformly stable and a linear system

x = (A(t)+B(t»x

with

B(t) S M n

and integrable over R+

STABILITY OF LINEAR SYSTEMS

152 with

o

lim B(t) t->=

can be unstable.

Let A(t) s i.n f log t ) +

with

1 < 2a < 1 + exp(-n).

°tO' (log t )

[

X

]

z (0) exp(t sin (log

t )

2at

-

which tends exponentially to zero as t

+00.

But if we take

:1

B(t)

x = (A(t)+B(t»x

then the solutions to

Ct)

are

(o)exp(-at)

X

x

J

Whence the solution to (5.32) is

x 1 (0 ) exp (- a t )

Ct)

x

-

]

1

t

[

exp(tsin(10gt)-2at) [xZ(0)+x 1 (0) fexP(-uSin(lOgU»dU] to Select a S such that 0 < S < 2z and then cosS > (2a-l)expn. So if

t

r

exp(2r-Dn,

t expn

fr

then for

r=I,2, ... ,

t

r

<;

S

<;

t expS

exp Cr-u sin(log u»du

>

o

fr t

exp Cu c o s Bj du

r

>

since

s i n Clog u )

<; -

it then follows as r Ix Z (t r expn)

I

co s S, +

00

Also s i nc.e

sin(log(trexpn»

that

Ix1(0)lt r (expS-l)exp(yt r )

+

t expS, r

5. STABILITY OF NONSTATIONARY SYSTEMS if

xl (0)

0,

~

y

since

153

(1-2a)expn + cosS > o.

=

Example 5.6: consider the scalar almost periodic differential equation x

-(a(t)-b(t»x,

-1 L: k-2' s i.n Cn r k ) k=l

a(t)

0

b (t)

1

ot

Clearly

Ib(t)

I

-1

4

~ 8

and

x(t)

E:

AP(C)

and

t

E:

R+, (5.42 )

for

0

~

for

t

:2

and

t

~

and

o.

lim b(t) t-+=

t

is

a(t)

3

00

where

for

xoexp{f (b(S)-a(S»ds}.

8

:2

0

The solution to (5.42)

For the given a(t), positive

o

constants at

1 2

a,S

exist such that for t

:2

t :2

f

1

a(s)ds

:2

St 2 •

-a(t)x

~s

asymptotically stable, but since

0

so that

x

=

t

f

b(s)ds

1

3

4-1) "3 (4t

the system (5.42 ) is unstable for every

0

<5 >

5.5

o. Stability in the Large

The stability properties of the previous sections were all local, that is there exists a closed domain in state space that includes the equilibrium state such that all solutions initiating in that region are stable or asymptotically stable.

In the case

of asymptotic stability, where there is convergence to the equilibrium state, the region of validity of convergence is called

154

STABILITY OF LINEAR SYSTEMS

the domain of attraction.

Should the domain of attraction include

the whole state space we then have global or stability in the

large. Consider the general nonlinear system f(t,O) = 0

f(t,x),

x

f:R xB = 0

and

+

-+

En,

(5.43 )

Definition 5.5: Asymptotic stability in the large The zero solution of (5.43) is asymptotically stable in the

large or globally stable if it is stable and every solution of (5.43) tends to zero as t

-+

00

Definition 5.6: Exponential asymptotic stability in the large The zero solution of (5.43) is exponentially asymptotically a > 0

stable in the large if there exists a there exists a

Ilx(t;x ,t ) II o

such that if

N(B) > 0 5.

0

Ilxoll

N(B) exp(-a(t-t » Ilx 0

0

B > 0

and for any 5.

II

B then

for all

t

2

t . o

Definition 5.7: Weakly uniformly asymptotic stability in the

large

The general solution yet) of (5.43) defined on R+ is said to be weakly uniformly asymptotically stable in the large if it uniformly stable and if for every

to E R+

and every x

o

~s

defined

on R+ we have lim !Ix(t;x ,t ) - yet)

t-+oo

If

0

f(t,x)

0

= f(t+w,x),

ly asymptotic stability

~n

II

0 W >

0

~s

periodic then weakly uniform-

the large is equivalent to uniform

asymptotic stability in the large (Yoshizawa, 1975).

However as

we shall see in the following example (Seifert, 1968) this equivalence is not the case for almost periodic functions

f(t,x) E

AP(C).

Example 5.7: Consider the scalar system, x

x,

for

-1 + (I-2f(t»(x-I),

for

-f(t)x,

for

1

0 5. x 5.

< x 5. 2

2 < x

(5.44)

5. STABILITY OF NONSTATIONARY SYSTEMS where

= -f(t,-x)

f(t,x)

and

f(t)

E

AP(C)

155

is the almost peri-

odic function constructed by Conley and Miller (1965) and discussed in example 5.2.

The zero solution to (5.44) is uniformly

asymptotically stable.

Assume that

-f(t)x.

If(t)1 < 1

Comparing the solution of

x

= -f(t)x

f(t,x) ~

then

(see example 5.2)

with that of (5.44) we see that every solution of (5.44) tends to zero as t

+

00, and thus the zero solution to (5.44) is weakly

uniformly stable in the large.

Considering the solution of (5.6)

through (t ,x ) we showed that the solution to (5.6) is not n

0

un~-

formly bounded and hence solutions to (5.44) are not uniformly bounded and the zero solution of (5.44) is not uniformly asymptotically stable. Finally if we now consider the linear nonstantionary system

~ = A(t)x,

(5.45 )

A(t) EM, n

a variety of stability conditions are equivalent and are given without proof (Yoshizawa, 1975):Theorem 5. 14 If the zero solution of the linear system (5.45) is asymptotically stable it is asymptotically stable in the large.

Moreover

if the zero solution of (5.45) is uniformly asymptotically stable it is exponentially asymptotically stable in the large and the N(S) of definition 5.7 is independent of S. Theorem 5.15 For the linear system (5.45) (i) Asymptotic stability and ultimate boundedness are equivalent. (ii) Uniform asymptotic stability

~n

the large and uniform ulti-

mate boundedness are equivalent. (iii) If A(t) is periodic in t, asymptotic stability implies uniform asymptotic stability in the large.

5.6

Total Stability and Stability under Disturbances Consider the general nonlinear system

~n

the large

STABILITY OF LINEAR SYSTEMS

156

x

f (t , x) ,

with

f r Rxf, -+ En

(5.46 )

where

L < B

=

{x:x E En, Ilxll < ex, ex > A}.

Definition 5.8: Total stability Let yet) be a solution to (5.46) such that all

t

for

(3

<;

Then the solution yet) is said to be totally stable

O.

~

Ily(t) II

if for any

t

and any

o(~)

> 0

with

o

~

0

there exists a

such that if get) is any continuous function on [t ,00) Ilg(t)11 < 0(0

for all

"y(t o ) - z 0 11 <

o(~)

t

~

t

~

~

0

and if

o

z

€

0

L

o

satisfies

then any solution z(t) through (t ,z ) of o

the system

0

f(t,z) + get)

z

(5.47)

satisfies Ily(t) - z I t ) II < ~

for all

t

~

t

o

.

If we restrict the function f(t,x) such that it satisfies a Lipschitz condition in x with f(t,o) = 0, then if the null solution to (5.46) is uniformly asYmptotically stable it is also totally stable.

Clearly total stability implies uniform stabi-

lity but the converse is not in general true.

Moreover total

stability does not necessarily imply asymptotic stability; an exception to this if

f(t,x)

A(t)x

with

A(t)

€

In which case we have:

M n

on R+.

Theorem 5.16: (Mas sera, 1958)

If the null solution to (5.45) is totally stable then it is uniformly asymptotically stable and exponentially stable

the

~n

large. Proof: From definition 5.8, if the null solution

~s

stable then there exists a

II z o II

solution

z(t;z,t) o

z

0 > 0

such that i f

of

0

totally < 0

the

A(t)z + oz

satisfies

Ilz(t;z ,t) o

II

< I.

But the solution of the above dif-

ferential equation and (5.45) are related by z(t;z ,t ) o

0

x(t;z ,t )exp(o(t-t o

0

0

»

for

t

2: t

o

,

5. STABILITY OF NONSTATIONARY SYSTEMS Ilx(t;z ,t )11 < exp(-o(t-t

then

o

0

0

».

157

Consequently by theorem 5.4

(iv) and theorem 5.14 the null solution of (5.45)

~s

both uni-

formly asymptotically stable and exponentially asymptotically stable in the large. We shall now relate the concept of total stability to I-stability and stability under disturbances for almost periodic systems. f(t,x) s AP(C)

Consider the system (5.46) but with x s L

odic in t uniformly for compact set such that g s R(f)

For

Q

and for all

L c B,

C

t

almost peri-

O.

~

Let Q be a

yet) s Q for all

and

(the hull of f - see section 2.2) and

t

~

0.

h s R(f)

let r(g,h:Q)

(5.48)

sup {II g(t,x) - h(t,x) II} s R+ x s Q t

which we now use in the following definition for the stability of solutions of (5.46) under

f(t,x) s

from the hull of

di~turbances

AP(C). Definition 5.9: Stability under disturbances from the hull (Sell, 1967 )

If for any

~

>

°

there exists a

Ily(t+T) - x(t;x ,g,O) II ~ ~ o ~ 0(0

IIY(T) - x II o

where

x(t;x ,g,T)

g,O) = x

o

o

through

t ~

>

o(~)

°

T

a solution of

(T,X) 0

and

such that g s R(f),

for some

~ = g(t,x)

x(t;x ,g,T)

Then the solution yet) of (5.46) for

°

whenever

ref ,g;Q) ~ 0(0

and

~s

for

0

with

Q

S

f s AP(C)

T ~ 0, x(O;x ,

for all

o

t

~

T.

is said to be

stable under disturbances from H(f) with respect to Q. This definition of stability for almost periodic systems

~s

formally equivalent to the I-stability introduced by Seifert (1966).

An obvious conclusion from this definition is:

Theorem 5.17: Given that yet) is a solution of (5.46) for such that

II yet) II ~ S < (XI

totally stable for

t

~

0

from R(f) with respect to

for all

t

~

0.

f s AP(C)

and

Then if yet) is

it is also stable under disturbances Q = {x: Ilxll

<:

y,

and

~s

158

STABILITY OF LINEAR SYSTEMS

the solution yet)

~s

asymptotically almost periodic in t.

A parallel set of results hold for f(t,x) periodic by which an exact equivalence exists between stability under disturbances and uniform stability.

w > 0,

Since in the case of

f(t,x)

=

f(t+w,x),

total stability of (5.46) implies uniform stability and

theorem 5.]] holds equally for f(t,x) periodic in t.

This equi-

valence is not in general true for almost periodic f(t,x), although some exceptions do exist (see Kato, 1970; Yoshizawa, 1975). 5.7

Sufficient Conditions for Stability The majority of necessary and sufficient conditions for stabi-

lity of linear non-stationary homogeneous systems

x

= A(t)x

involve the fundamental matrix X(t), which in turn implies full knowledge or computation of the solution of the systems equations. Only when the coefficient matrix A(t) is periodic, diagonal dominant or time invariant can stability conditions be directly vestigated from the elements of A(t).

~n

However it is possible to

generate a set of inequalities (called Wazewski's inequalities, 1958) for the sufficient conditions for stability of the linear homogeneous system x

A(t)

A(t)x,

Theorem 5.18:

S

M , n

x(t ) = x o

(5.49)

0

Sufficient conditions for stability (Wazewski, 1958)

A (t ) and A. (t ) are the largest and smallest characmax rm.n teristic values of the sYmmetrical matrix H(t) = A(t) + A* (t), If

then any solution of (5.49) satisfies, t

r

. (S)dS} " Ilx(t;xo,to)ll" ) Arm.n t

Ilxollexp{~JAmax(S)dS} t

o

Proof: The derivative of the inner products along the solution of (5.49)

lS

aCt)

t

o

(5.50)

x * (t)x(t)

159

5. STABILITY OF NONSTATIONARY SYSTEMS

*0

*

x x

x H(t)x.

A. (t )

Then from the definitions of

m~n

and

(c.f. Rayleigh quotients)

A . (t)x *x m~n

~

x *Hx

A (t.) max

of

H(t)

1,

Amax (t)x x,

~

that is

A • (t.)

a a

m~n

-1

A (t.) , max

which on integrating gives inequality (5.50).

The following suf-

ficients conditions for stability of (5.49) are as a result of theorem 5.18. Corollary The null solution to the linear system (5.49) is (i) stable if for all

t

E

o

R,

t

limf A (s) ds t-+oo max t

<

N (t;

o

J

o

and uniformly stable if N

independent of t .

~s

o

(ii) Unstable if t

lim t-+oo

f t

A . (s ) ds

+ 00

,

m~n

o

(iii) asymptotically stable if for all

t

o

E R

t

f

lim A (s ) ds max t-+oo t

-co

o

and uniformly asymptotically stable if the above holds uniformly with respect to t . o These sufficient conditions are highly conservative and are dependent upon the particular state space representation used. The Abel-Jacobi-Liouville lemma (3.2) can be used to establish similar sufficient conditions for stability (and instability), but in this case the trace of the coefficient matrix A(t) is

160

STABILITY OF LINEAR SYSTEMS

utilised, rather than the characteristic values of A* (t).

H(t) = A(t) +

Since the trace of a matrix is the sum of its characteris-

tic values the following instability condition is obvious from the above corollary: Theorem 5.19 The null solution of (5.49) is unstable if t

lim

f trace

(A (s)) ds

+ 00

t-7
o

and not asymptotically stable if for some t such that

o

there is a bound

S

t

lim

t-7
5.8

f trace t

(A(s))ds

2

-So

o

Notes and Input-Output Stability Throughout this chapter the majority of stability results have

been for linear nonautonomous systems; in almost all cases there has been a requirement to compute the system fundamental matrix, the only exceptions being in the cases of Coppel's theorem, Gershgorin's type results and for periodic systems where time invariant type results based upon system characteristic values are appropriate.

Other approaches to the stability of differen-

tial equations exist, these are essentially energy methods of which Liapunov's direct approach is most notable (Venkatesh, 1977; Willems, 1970; Yoshizawa, 1966; LaSalle and Lefschetz, 1961).

Of particular relevance to the study of periodic and al-

most periodic systems is the text of Yoshizawa (1975) which is based entirely upon Liapunov's direct method. An alternative approach (Desoer and Vidyasagar, 1975) to Liapunov stability is based upon the input-output properties of a system rather than its internal structure as specified by differential equations.

The input-output approach to stability is

5. STABILITY OF NONSTATIONARY SYSTEMS

161

motivated by engineers' desire to consider systems in the frequency domain via transfer functions rather than in the time doma1n.

The major disadvantage of the input-output technique is

that it yields only the conditions for global asymptotic stability and estimation of the domain of attraction in local stability questions is not viable. Input-output stability conditions for linear time-varying sysP tems are readily derived from the properties of L Norms. Consider the forced linear time-varying system

where =

(5.51)

A(t)x + B(t)u,

x

O.

A(t) EM, n

) u E Lp [0,00,

B ( t ) EM, m

m

x

E

En

x(t )

and

0

The solution to (5.51) can be written 1n terms of the state

transition matrix ¢(t,t ) of the homogeneous equation o

x

=

A(t)x

as t

f ¢(t,t

x(t)

t

o

)¢(s,t )-l B( s ) u ( s ) d s 0

o t

f G(t,s)u(s)ds

(5.52)

-00

where

Q ¢(t,t )¢(s,t) 0 0

G(t,s)

bounded matrix for all

-1

B(s)

t E [t ,00), o

1S a (nxm) nonanticipative (that is

G(t,s)

= 0 for

The system (5.52) is said to be LP[O,oo) bounded inputP n output stable when an input u E L [0,00) produces an output s > t).

= Gu

m

P

or equivalently there exists a constant n P such that Ilxll p s IIGul1 p S Yllull p whenever u E Lm

x

E L

o

<

for

Y< I

<;

Taking norms of (5.52) and utilizing Holder's inequality for 1+1 __ 1

p

q

• g1ves (

Ilxll

t

J to

IIG(t,s)11

Ilu(s)llds

p

STABILITY OF LINEAR SYSTEMS

162

I

t

I

J IIG(t,s)IIPIIG(t,s)llqllu(s)llds

s;

t

o

{f

s;

t

t

1

IIG(t,s)11

Ilu(s)II

P

dS}P{J IIG(t,S)lI d s t

o

t

Setting

> a

00

:2

JIIG(t,s) lids t

for

t

E

Ilx ll p

a

q

t

p

a

a

x E: LP n

Clearly

all

t E: R+

0

t

P

then inequality

0

t

J lIu(s) liP ds J IIG( T,s)11

(1 0;

o

Ilu(s) liP dS} dT

IIG(T,s)11

t

t

(5.53)

vector norm,

p

t

J{J

q

}
0

(5.53) becomes, on taking the 9.

S;

1

t

t

o + q E)

0

for

lI ull p'

dr

1

1

P

q

(- + -)

(5.54)

and the system (5.51 ) is LP-input/output stable. n (

The boundedness condition on

J t

t

for all

IIG(t,s) lids

t

E:

R

+

is

o

a necessary and sufficient condition for L. n

Unlike time invari-

ant systems, linear time-varying systems can be Ln-stable but not 1-stable.

L For linear time invariant systems L 1 stability is a n o o n necessary and sufficient condition for L stability, in addition n

G(t,s)

=

G(t-s) and the system (5.51) is asymptotically stable

in the large if and only if it is LP-input/output stable (for any I

0;

P

0;

(0).

n

For recent results on input/output stability for

nonlinear multivariable systems see Harris and Owens (1979) and Valenca and Harris (1979).

5. STABILITY OF NONSTATIONARY SYSTEMS

163

References 0

Caligo, D. (1940). Atti 2 Congresso Un.Mat.Ital., 177-185 Cesari, L. (1940). Ann.Scuola Norm.Sup.Pisa. (2) 9, 163-186 Conley, C.C. and Miller, R.K. (1965). J.Differential Eqns. I, 333-336 Conti, R. (1955). Riv.Mat.Unv.Parma. 6, 3-55 Coppel, W.A. (1965). "Stability and Asymptotic Behaviour of Differential Equations", Heath, Boston Coppel, W.A. (1967). Ann.Mat.Pura Appl. 76, 27-50 Desoer, C.A. (1970). "Notes for a Second Course on Linear Systems", Van Nostrand Reinhold,New York Desoer, C.A. and Vidysgar, M. (1975). "Feedback Systems: InputOutput Properties", Academic Press, New York Fink, A.M. (1974). "Almost Periodic Differential Equations", Lecture Notes in Mathematics No.377, Springer Verlag, New York Gantmacher, F. R. (1959). "The Theory of Matrices", Vols. I, II, Chelsea, New York Hahn, W. (1963). "Theory and Application of Liapunov's Direct Method", Prentice Hall, New Jersey Harris, C.J. and Owens, D.H. (1979). "Multivariable Control Systems", IEE Control and Science Record, June 1979 Kato, J. (1970). Tohoku Math.J., 22, 254-269 LaSalle, J.P. and Lefeschetz, S. (1961). "Stability by Liapunov's Direct Method with Applications", Academic Press, New York Lyascenko, N.Ya. (1954). Dokl.Akad.Nank.SSSR, 96, 237-239 Massera, J.L. (1949). Ann. Maths. 50, 705-721 Massera, J.L. (1958). Ann. Mathe. 64, 182-206 Massera, J.L. and Schaffer, J.J. (1958). Ann. Maths. 67, 517-572 Perron, O. (1930). Math. Zeits. 32, 465-473 Rapoport, I.M. (1954). "On some asymptotic methods in the theory of differential equations". Kiev. Izdat.Akad. Nauk.Ukrain SSR Ri e s z , F. and Nagy, B.Sz. (1955). "Functional Analysis", Ungar, New York Seifert, G. (1966). J.Differential Eqns., 2, 305-319 Seifert, G. (1968). J.Math.Anal.Appl., 21,136-149 Strauss, A. (1969). J.Differential Eqns. 6, 452-483 Valenca, J.M.E. and Harris, C.J. (1979). Proc.IEE, 126, 623-627 Venkatesh, Y.V. (1977). "Energy Methods in Time-varying System Stability and Instability Analyses", LNI Physics No.68, Springer Verlag, Berlin Wazewski, T. (1958). Studia Mathematica 10, 48-59 Willems, J.L. (1970). "Stability Theory of Dynamical Systems", Nelson, London Yoshizawa, T. (1966). "Stability Theory by Liapunov's Second Method", The Math.Soc.Japan, Tokyo Yoshizawa, T. (1975). "Stability Theory and the Existence of Periodic Solutions and Almost Periodic Solutions", Appl.Maths. Sci. No.14, Springer Verlag, New York

Chapter 6

ASYMPTOTIC FLOQUET THEORY

6.1

Introduction In this chapter we continue the investigation of kinematic

similarity by examining the concept in the context of almost periodic matrices with particular application to the study of linear differential equations with almost periodic coefficients.

It has

been suggested that a generalisation of classical Floquet theory to include the almost periodic case should be of considerable use in obtaining new theorems on kinematic similarity and the stability of differential equations. sation is known to exist.

Unfortunately no such generali-

The situation is illustrated quite

simply by means of the following example: The proposal say, with

1S

that for the scalar equation

a(t):R + AP1,

~(t)

= a(t)x(t),

an analogy with the purely periodic

case is sought whereby the solutions of the equation exist and assume the form x(t)

p(t)exp(bt)

(6. 1)

In this representation (c.f. equations (3.83) and (4.38»

pet)

possesses the characteristics of a Liapunov transformation and may even be almost periodic.

Also

b S Ml'

We observe that pet)

formally satisfies the differential equation [a(t)-b]p(t)

(6.2)

6. ASYMPTOTIC FLOQUET THEORY with the implication that

aCt)

~

b

on R.

165

Moreover, one solution

of (6.2) is t

ex p[

pet)

f

(6.3)

(a(S)-b)dS]

o

where the lower limit of integration has been taken somewhat arbitrarily as

a

for convenience.

For pet) to be a generalised

Liapunov transformation it must be bounded on R. b)ds

need not be bounded, even though

and

ftf(s)ds

aIt

are almost periodic, then

o

)

However,

t

J

o

(a(s)-

For if f(t)

s API'

a(f,a) = 0

is obviously

necessary, otherwise the integral would contain a term a(f,O)t It I

which becomes unbounded as

f

t

o

+

00.

The Fourier series of

~s

f(s)ds

a(f,O)t + L

a(f A) iA

exp(~At)

.

and an immediate question is' the sufficiency of

f

o

t

f(s)ds to be almost periodic.

a(f ,0) = 0

In fact it is not.

for

To see this

consider the series f(t)

(i t)

1 exp -L --

00

~

k=l k 2

(6.4)

k2

This series converges uniformly and so its sum f(t) is almost periodic.

However,

t

J f(s)ds

~

~ ~

k=1 ~

o

exp (it) k2

which is not almost periodic

(6.5)

s~nce

the coefficients violate

Parseval's equation. We shall see later that if almo~t

periodic.

IAI ~ m > 0,

then

fo t f(s)ds

This is the only known simple condition on the

Fourier series which yields the almost periodicity of except for the obvious condition

Lla(ft"')I

<

is

00.

f

o

t

f(s)ds,

166

STABILITY OF LINEAR SYSTEMS In the remainder of this chapter we consider partial analogues

of Floquet's theorem by imposing particular conditions on the almost periodic matrix F(t).

The problem here is to be distinguished

from the almost constant coefficient case in which F(t) is decomposed into two parts as follows: F(t)

B + A(t)

It has been shown (Bellman, (1953» characteristic values A and

that if

jIA(t) II

-+

0

as

B E M t

-+

n 00,

has simple then corres-

~(t)

ponding to each characteristic value there is a solution of [B+A(t)]x(t)

satisfying

lim t-1logllx(t) t-+oo

=

II

Re A

(6.6)

that is to say, we are able to evaluate the generalised characteristic exponents of F(t). For the case in which

A(t) t AP, n

the requirement on A(t)

can be relaxed to the extent that IIA(t) II need only be small (Berkey, 1976) or that some kind of nonresonance condition be satisfied (Coppel, 1967).

Alternatively, small parameter tech-

niques have been employed (Shtokalo, 1946, 1960; Kohn, 1976) to construct Liapunov transformations as formal power series with quasi-periodic coefficients.

Other results for second order sys-

tems and systems whose structure is canonical can be found in Lyascenko (1956), Gel'man (1957, 1959, 1965), Adrianova (1962) who generalises Gel'man's earlier work and Merkis (1968) who deals with the case in which the matrix of

coefficie~ts

commutes with

its integral (see similar results in Chapter 4, Theorems 4.13, 4.14) . An essential part of the framework required to prove Coppel's theorem (op.cit.) is contained in the next section.

We begin

with the observation that new almost periodic functions can be generated from a given one by convoluting it with other functions. This result is used to prove the Coppel-Bohr lemma which is the first step in showing that the integral of an almost periodic

167

6. ASYMPTOTIC FLOQUET THEORY function whose exponents are bounded away from zero is almost periodic.

The Cappel-Bohr Lemma and Linear Differential Equations with Almost Periodic Coefficients

6.2

The generation of almost periodic functions by convolution is especially interesting if Fourier transforms are used.

Fink (1974)

illustrates the idea as follows: to multiply the Fourier coeffif(t) € API by a sequence {b is equivalent to convok} f (t ) luting with a function whose Fourier transform is b at the k frequencies ;\.

cients of

Lemma 6. I: (Fink, 1974) Let ~

~

f(t)

Ll(R)

€

API

€

and

~

be a complex valued function such that

(the ~ "hat" notation means Fourier transform) and the

transform exists.

~nverse

Define

00

f f(s+t)~(t)dt

h(s)

(6.7)

_00

Then

h(t)

h(t)

€

API Z

~

and

(6.8)

a(h,A)~(A)exp(iAt)

In addition T(n,h(t)) where

T(~,

\I-Ill

n

f It ) )

(6.9)

is the Ll norm.

Proof {see Fink, (op.eit.)) The Coppel-Rohr lemma is really a result about trigonometric polynomials.

It is used as the starting point for the Approxi-

mation Theorem (Theorem 2.13) which extends the result. Lemma 6.2: (Coppel, 1967) Let f(t) be a trigonometric polynomial N

f (t)

L:

k=l

with

IAkl

2

(6.10)

akexp(iAkt)

m for all k.

Let

STABILITY OF LINEAR SYSTEMS

168

ak

N

.

L iA

get)

(6. II)

exp(~\t).

k=l"k

Then there exists a constant d independent of f, g and m such that -)

II gil

dm

(6.12)

[l f l},

Proof is taken from Fink (op.cit.). Noting that get) is the unique integral of f(t) with mean value zero, the idea of Lemma 6.1 is employed. m = I

the case when

and define

it ¢ (t )

for

1

(-it)

s f 0

For

Consider first of all

-)

t

t

t

It I

for

:<:;

(6.13) ~

we can estimate the Fourier transform of ¢(t) as fol-

lows: 00

J

A

21T¢(S)

(6.14 )

¢(t)exp(-ist)dt

Integrating by parts once yields

= ils

21T¢(S)

f ~(t)exp(-ist)dt

(6.15)

-00

and once again

f f exp(-ist)dt )

i ex p(-i 2st)

exp(-ist) dt t 3

(s t )

-)

(6.16 )

2i

2i [exp(-is)-exp(is)] s2

s2 If

It I

then

~

21T¢(S)

:<:;

2s

-2

t3

~

+ 2s

JIt I ~

!(1+t 2 ) . -2

II

I

where c is some constant, and

I :>

(6.17)

Hence dt

t

exp(-ist) dt t3

I

!( l+t 2)

cs

-2

(6.18)

6. ASYMPTOTIC FLOQUET THEORY if lsi ~

For

and

s

If ~

2 +

# 0

Is I

?

169 (6. 19)

J.

we have exp(-ist) dt t

itexp(-ist)dt + iJ

It I ? J sin st dt t

(6.20)

1

2 + 'IT

1

(6.21)

Since ¢(s)

invertible, we have

~s

N

l: k=1

get)

ak¢(Ak)exp(iAkt) 00

~ ~

k=1

f ¢(S)e~p(i\S)dS

exp(i\t)

(6.22)

f

(6.23)

f(s+t)¢(s)ds

whence (6.24 )

Note that 11¢11 constant.

does not depend on f, g or A but ~s

1

For the general case, suppose

Then

and consider (6.25 )

h (t )

w (t.)

IAkl? m

an absolute

t

mg(-) m

(6.26)

170

STABILITY OF LINEAR SYSTEMS

A

N

h(t)

k exp(i --;;;- r )

L ~ k=1

-.

and N

w(t)

exp(i -

L k=1

m

Clearly the exponents

~k

(6.27)

(6.28)

t )

m

satisfy

I~kl

2: I.

Using the

above result we then have Ilwll

I ~ Illllhll

s:

(6.29)

so that -1

Ilgll

m

Replacing 11~lll

Ilwll

s:

-1

m

~

(6.30)

11¢lllllfll

by d gives the required result.

Lemma 6.3: (Levitan, 1953) Let

f I t ) E API

f(t)

L

~

1\1

with

2:

~

such that

exp(iAkt)

m > 0

for all k.

Then

t

f

f(s)ds

E

API·

0

If

t g( t )

f f(s)ds

with

a(g,O)

0,

0

then Ilgll

s: dm-lll f II

(6.31 )

where d is an absolute constant.

Proof: The lemma has already been proved for trigonometric polynomials.

The Approximation Theorem extends the result as requi-

red (see Fink (op.cit.) and Coppel (op.cit.). We are now in a position to say something about solutions of the inhomogeneous linear equation

6. ASYMPTOTIC FLOQUET THEORY

171

(6.32)

Bx(t) + f(t)

;; (t.)

where

BE: H, f(t) = col(f l(t),f 2(t), ... ,f (t», fk:R -+ API' n n n x:R -+E . Scalar equations will be considered initially and then the vector equations will be built from short sequences of the scalar equations.

The aim is to show that the vector equa-

tion (6.32) can have almost periodic solutions, even if the corresponding homogeneous equation has almost periodic solutions, provided that the Fourier exponents of these solutions are not arbitrarily close to the Fourier exponents of any of the f

k.

This

is clearly a nonresonance condition. Lennna 6.4: (Cappel, 1967) Suppose

b

is

=

for all

A E: A f. k lution x(t) to ~(

such that Ilxll

Ax S;

~s

=

Af"

-1

dm

IS-Akl 2 m > 0

Then there exists a unique almost periodic so-

(6.33)

bx(t) + f(t)

t )

where d

for some real S such that

Moreover,

[l f II

the numerical constant of Lemma 6.3.

Proof: The change of variables y (r )

(6.34)

exp(-iSt)x(t)

transforms equation (6.33) into exp(-iSt)f(t)

get)

(6.35)

At this point we observe that Ilgll

II f II

ll y ll

Ilxll

(6.36)

with A

y

(6.37)

A - S x

and similarly A g

A - S f

If the nonresonance condition

(6.38) ~s

satisfied then the exponents of

STABILITY OF LINEAR SYSTEMS

172

get) are bounded away from zero by virtue of (6.38).

Lemma 6.3

asserts that yet) is the unique integral of get) with y:R

-+

API

t

and -1

ll y ll

dm

= 0,

M (y)

ll s ll-

Thus the sets of exponents of yet) and get) are the same, so by reversing the change of variables (6.34) we obtain the desired result. The following lemma deals with the case

which b is complex:

~n

Lemma 6.5 Suppose that

Re(b)

~

0

and

f:R

-+

API'

Then there exists a

unique almost-periodic solution x(t) to (6.33) such that

Ax

A f.

Moreover,

II x II

<::

IRe (b)

1-1 II f II

(6.35)

Proof: He seek bounded solutions to (6.33) for possibilities arlse.

The first is

Re(b) > 0,

Re(b)

~

O.

Two

in which case

00

x

- f exp(b(t-s»f(s)ds

Ct)

(6.36 )

t ~s

the required solution with Ilxll

<::

Re(b)-Illfll

(6.37)

The other possibility is that

Re(b) < 0

and then

t

f

xf t )

(6.38)

exp(b(t-s»f(s)ds

-00

with Ilxll

s

-Re(b)-Illfll

The change of variables

(6.39) u = s-t

~n

(6.36) gives

00

x It )

- f o

whence

exp(-bu)f(u+t)dt

(6.40)

6. ASYMPTOTIC FLOQUET THEORY

IX(t+T)-X(t)I Clearly x I t

Re(b)

<::

-1

173 (6.41 )

Ilf(t+T)-f(t)/1

is almost periodic with

)

T(n,x(t»

T(nRe(b),f(t»

::>

A = A f x

To show that

(6.42)

we consider the Fourier series

00

x( t )

R;

~(t)

a

0

L:

R;

k=1

+

L:

k=1

i\~

a

k

exp (Uk t

(6.43)

)

(6.44)

exp(i\t)

The differential equation then gives

ba

a(f,O)

o

for

k "2

for

k

(6.45)

0

- b f 0 for all k, it follows that a necessary and k sufficient condition for ~ f 0 is that a(f,A f 0, hence k) Ax = A£" The above can be extended to equations where x and f have van lues in R or En and b B € M A suitable change of variables Since

iA

=

n

can be imposed to triangularise B and then the scalar result applied successively.

This was first proved by Bohr and Neugebauer

(1926) .

Theorem. 6. I

Suppose that S of

!S-iAkl"2 m > 0

for all characteristic values

B and exponents

A € A£" Then there exists a unique k almost periodic solution x(t) to (6.32) with Ax = A f. Furthermore, there exists a polynomial r of degree less than or equal to n with no constant term, depending only on the matrix B and an absolute constant d, so that (6.46)

Proof: Following Fink (op.cit.), we first demonstrate existence. Noting that for any matrix B € ~ there exists a P € such

Mu

174

STABILITY OF LINEAR SYSTEMS

that

PBP-

abIes

y

1

=T

is a lower triangular matrix, the change of var~-

in equation (6.32) gives:

Px

Ty(t) + g(t) where

g

Pf.

(6.47)

It is clear that

A c A by the change of variy x Hence A A. Similarly y x

abIes and the reverse is also true. 1

Ilxllllp- 11-

1

:S

ll y l]

:S

The first equation in (6.47)

Ilpllllxll ~s

(6.48) which is of the form of the scalar equation discussed in Lemma 6.4.

Because of the form of T, tIl is in fact a characteristic

value of B.

Subject to the hypothesis of the theorem we apply

Lemma 6.4 and deduce that there exists a unique almost periodic solution of (6.48) with

and

(6.49) The second scalar equation

~n

(6.32) is

(6.50)

t22YZ(t) + hz(t) where hZ(t)

c A . With the above hypothesis f h2 we get a unique almost periodic yz(t) whose exponents A yz ~z c Af which satisfies the estimate

is almost periodic with

Ilyzll

-1

dm

-1

dm

-1

dm

-1

dm

-1

dm

A

IIh zll ( II g z II + It z i I • "y 111) -1

(lIgzll+ltzlldm ( II g II + I t

-1

Z 1 I dm

II g II )

-1

Ilgll(I+ltzlldm

IIg 1 11)

)

(6.51 )

6. ASYMPTOTIC FLOQUET THEORY

175

Similar estimates can be found for the remaining elements Y3.. .• y

n

of y and finally we obtain a solution yet) with

A c A y

f

which satisfies the estimate

(6.52) where reo)

~s

a polynomial of degree n with no constant term.

Evidently the coefficients of r depend on the matrix T and therefore the matrix B.

Using the relationship between the norms of

x and y, the estimate (6.52) can be transferred back to the original equation. The uniqueness of yet) (and hence of x(t»

rema~ns

to be shown.

If there are two solutions xa(t) and xS(t) of (6.32) with

then

xa(t) - xS(t)

is also an almost periodic solution whose ex-

ponents are contained in A f. xs(t) are the following

The Fourier series of xa(t) and

co

x (t) a

~

xS(t)

~

L:

k=1

~exp(L\t)

(6.53)

co

L:

k=1

bkexp(iAkt),

A k

E:

A f

whence x (t) - xS(t) a

~

~ (t) - ~S (t ) a

~

L:

k=l

(ak-bk)exp(i~t)

(6.54)

co

Since

.

.

x (t) - xS(t) ex.

L:

k=1

iAk(~-bk)exp(i~t)

B(xa(t)-xS(t»

it follows that

According to the hypothesis of the theorem iAk is not a charac-

176

STABILITY OF LINEAR SYSTEMS = 0 for all A E A . By the k k f xa(t) = xS(t). Linearity follows from unique-

teristic value of B so uniqueness theorem

~

- b

ness. The final step before discussing Coppel's theorm (ap.cit.)

~s

to consider the matrix differential equation BU(t) - U(t)B + C(t) with

U(t) EM, n

C(t) E AP

n

(6.55)

and

B E M

The equation can be

n

rewritten so that the right hand side of the homogeneous part is a linear mapping with characteristic values Obviously 0 belongs to the set {Sj-S£},

(Sj-S£),

j,£=I, ... ,n.

This implies that in

order to avoid resonance, the exponents of C(t) must be restricted in some way and in fact they are required to be bounded away from zero by a distance m.

Theorem 6.1 can now be applied to give

a unique almost periodic solution of (6.55) with

A

u

A

c

which

satisfies the estimate

(6.56) where 11-:1 is the appropriate matrix norm.

The case in which B

is a diagonal matrix whose characteristic values have distinct real parts is also very interesting.

Here we can apply Lemma 6.5

directly and obtain the estimate

uu Ii

min l:Sj,£:Sn

II C II

(6.57)

j#

6.3

Coppel's Theorem For the almost periodic case, one of the most significant par-

tial analogues to Floquet's theorm is Coppel's theorem (ap.cit.). Theorem 6.2: (Cappel, 1967) Consider the linear homogeneous differential equation ;;(t)

[B+A(t) l x f t )

(6.58)

6. ASYMPTOTIC FLOQUET THEORY with

x:R

En,

7

B

M

£

and

n

A:R

7

AP. n

177

Let the exponents of

A(t) be denoted A and let SeA) denote the set of all numbers k which are linear combinations of A with non-negative integral k coefficients, at least one of which is positive. Suppose that dist[Bj-B£, is(A)l for all

j,£

>

I, ... ,n,

istic values of B.

(6.59)

0 where

B£(£=I, ... ,n)

are the character-

Then a fundamental matrix X(t) of (6.58) is

of the form

where

(6.60)

P(t)exp(Bt)

X(t)

pet) = I+Q(t),

Q:R

7

AP n

and

A Q

C

SeA).

Proof: The linear transformation (6.60) applied to equation (6.58; yields the following equation for pet): BP(t) - P(t)B + A(t)P(t) We have already examined this· kind of equation. C:R

AP

7

(6.61 ) In fact, for

it is known that

n

U(t )

BU(t) - U(t)B + C(t)

(6.62)

admits an estimate II U II

II C II r (dm- 1)

c;

This will be used repeatedly. also that AA are all one sign.

(6.63) Assume that

A C SeA). Assume C Starting from Uo(t) = I we de-

fine a sequence {Uk(t)} inductively by taking Uk(t) to be the unique almost periodic solution of (6.62) with A(t)U

C (t )

k_ 1

(t )

(6.64)

so that AU

AAU

k

(6.65)

k_ 1

with /\

AU

C

k_ 1

/\A +

1\k-]

(6.66)

178

STABILITY OF LINEAR SYSTEMS

Therefore

Au

c

S(A),

this being true for all k.

The hypothesis

k

B.-B. =

that

AE

if

J

0

J

S(A)

is a positive distance from is(A) implies that

IAI

then

~ m.

satisfies the inequality ween {B.-B~}

and iA

J

uk

Thus any exponent of Uk(t), ~ say, ~ km.

Clearly the distance mk betincreases with k so that for sufficiently I~I

large k we obtain the estimate f(d~

-1

(6.67)

) IIAII'IIUk_111

Since f(') is a polynomial with no constant term and

where

IIAII p

IIUkl1

=

<

grows

k > N

like k, an N may be chosen so large that for

f(d~-l)

~

.e.k

(6.68)

d 2I1AII-. The inequality m p s k IIU k_ III

implies that

LllUkl1

pk is majorised by L ! . k

P(t)

I

Therefore the series

+ Q(t)

(6.69)

converges uniformally on R and is almost periodic with Note that Q(t) is almost periodic and

Ap

C

S(A).

C S(A). It is easily Q seen that P(t) is a solution of equation (6.61). Thus X(t) =

P(t)exp[Bt]

A

is a solution of equation (6.58).

Similarly we can find an almost periodic solution

to P(t)

BP(t) - P(t)B - P(t)A(t)

with

A C S(A). Q1 the equation

It follows that

BU(t) - U(t)B Therefore P1 (t)P(t)-I

(6.70) P1(t)P(t)

is a solution of

(6.71)

179

6. ASYMPTOTIC FLOQUET THEORY

is a solution of the same equation and its exponents are contained in SeA).

Hence it must be the zero solution.

Thus

P1(t)P(t) = I

and the matrix I+Q(t) has an inverse of the same form.

Therefore

pet) is a fundamental matrix solution of (6.61). This theorem can be interpreted immediately as a result on kinematic similarity.

Also the analogy with classical Floquet

theory is clearly demonstrated. Theorem 6.3 Suppose that

A:R

AP and B £ M Let the exponents of n n A(t) be denoted A and let SeA) denote the set of all numbers k which are linear combinations of A with non-negative integral k coefficients. Then for [B+A(t)] ~ B it is sufficient that ~

dist[Sj- S1 ' is(A)]

>

0

j,l = 1,2, ... ,n,

for all

where

Sl(l=I, ... ,n)

are characteris-

tic values of B.

Proof: follows immediately from Theorem 6.2. Berkey (1976) considered the same equation as Coppel but with different assumptions.

He proved that each independent scalar

solution of the equation approaches the form

as

t

~

with qk almost periodic and

00

the mean value of a

~

certain almost periodic function. Theorem 6.4: (Berkey, 1976) Suppose Sl) i 0

for

A:R ~ AP J

i 1.

n

and

B = diag(Sl'" .,S ) n

EMn

with

Re(S.-

Then for IIA II sufficiently small, equation

(6.58) possesses n independent solutions

~(t)

J

of the form

t

~(t)

(Pk(t)+ek)exP(Sk t +

J Vk(S)dS]

(6.72)

o

where

e

= col(Olk,02k, ... ,Onk)' k odic function and

vk(t)

~s

a scalar almost peri-

co I (Pk (t ) , ... , P k- (r ) , 0 , Pk k ( t ) , ... , P (r ) k, 1 01 , +1 k ,n (6.73)

STABILITY OF LINEAR SYSTEMS

180

where

Pk

Pk it) and

Proof: (B~rkey>

it) are almost periodic for

op.cit.)

It is clear that the characteristic exponents of [B+A(t)] admit the estimate lim t

-1

(6.74)

logll~(t)\1

t-+=

where t

lim t-

jr

1

t-+=

v(s)ds.

o

The requirement that

IIA(t) II

-7-

coefficient case) would force 6.4

~

0

as =

t

0,

-7-

k

(the almost constant

00

1, •.. ,n.

=

Almost Periodic Matrices Containing a Parameter We consider the almost periodic matrix A(t,a) which is analy-

tic in a: A(t,a)

A

(6.75)

o

with

A S M ~:R -7- AP n, n o real or complex parameter.

for

k = 1,2, ...

and a is a small

It is assumed that A(t,a)

periodic in t uniformly with respect to a in a domain lytic in a

~n

verge for

101

E* :0:

The power series

L:11~llok

and in particular for

P

~s

almost

E*

and ana-

is assumed to con-

101

>

o.

Theorem 6.5 Suppose that the matrix A(t,o) is almost periodic in t uniformly with respect to 0 in a domain E and analytic in 0 in E, and can be represented by the power series A(t,a)

+

A

~ ~(t)Ok

k=l

o

which converges for

101

< p.

Assuming that the matrix

A S M o

is diagonal and possesses characteristic values with distinct real parts and A(t,cS)

"\:R B(t,O)

-7-

AP

n

for all

k

=

1,2, ... ,

n

then (6.76)

181

6. ASYMPTOTIC FLOQUET THEORY where

B(t,O)

lEI

for

B

<

matrices.

UA

k

Bk

C

P~

I: Bk(t)O k=1

+

o

k

(6.77)

B E M o n

p,

A o Furthermore

and

(6.78)

S(kUAA_) -1<

The proof of the theorem will be preceded by two lemmas. Lemma 6.6: (Geltman, 1959; Blinov, 1965) Suppose that the power series (6.79) is convergent, where

~

E R+

and

E R.

~

If the equation (6.80)

possesses at least one root, then there exists a unique positive root

which is the least in absolute value of all the roots.

~

Lemma 6.7: (Geltman, 1959; Blinov, 1965) The function of the real variable a(O

p(O where

-1

[1-a(O-[1-2a(OJ

!2J

~

(6.81)

is a power series of the form (6.79), is single valued

a(~)

P~

for all

I~I

(6.79).

Furthermore, if the equation

<

p.

Here p is the radius of convergence of

(6.82)

2a(~)

has at least one root, then p this equation otherwise

p

=

~s

the smallest positive root of

p.

Proof of Theorem 6.5 It is necessary and sufficient to show that there exists a Liapunov transformation which assures the required similarity. To complete the analogy with .classical Floquet theory we suppose that P(t,o) is the required transformation which admits a formal

STABILITY OF LINEAR SYSTEMS

182

power series expansion as follows: 00

P(t,o) with

Pk:R

(6.83)

possessing diagonal entries equal to zero. The n aim is to show that the series (6.83) is convergent and to esti~

AP

mate its radius of convergence. The equation B(t,o)

-P(t,o)

-]

.

[P(t,o)-A(t,o)P(t,o)]

defines the supposed similarity, and may be written as A(t,O)P(t,o) - P(t,o)B(t,o)

(6.84)

We solve this equation; the solution will be almost periodic in t uniformly in 0, analytic in 0 in some domain and have an inverse. Substituting (6.75), (6.77) and (6.83) into (6.84) yields the identity 00

L:

k=1

• k Pk(t)o

00

[ Ao +

[ I +

L:

k=l

k~l

00

,\(tl,k] [ I +

L:

k=l

P (tl,k ] k

00

Pk(tl,k] [ Bo

+

L:

k=l

Bk(t),k

]c•.

a5)

and by equating like powers of 0 we obtain A

o

B

o

and a sequence of differential equations:

(6.86)

6. ASYMPTOTIC FLOQUET THEORY

AoPk(t) - Pk(t)Ao + ~(t) + Al (t)P

k -1

2

(t)P1(t) + ...

(t)

- Bk(t) - Pl(t)B - P k-

+ ~-l

183

k_ 1

(t) - P2(t)B

k_ 2(t)

-

(t)B 2(t) - P (t)B (t ) 1 k -1

(6.87)

The choice diag[Al(t)]

diag[~(t)

+ ~-l

(t)P1(t) + ... + Al(t)P

k_ 1

(t)] (6.88 )

ensures that the right hand sides of equations (6.87) have diagonal entries equal to zero.

This implies that each of the equa-

tions (6.87) is essentially decoupled.

Therefore the initial

hypothesis concerning A guarantees, through Lemma 6.5, the exiso

tence of unique almost periodic matrix solutions to (6.87). fact,

s

II PIli

hllA111

with ApI

AAI

s

1IP21!

h[I!A211 + 211A 111·IIP 1111

with A P2

c

AA U[AA +A ] 2 1 PI

In

184

STABILITY OF LINEAR SYSTEMS

with

A

AA U[A +A

c

P3

3

Aj

P2

JU[A +A

A2 Pj

1 ... U[AA +A i

Pj

+A p 1 j

and since the operation + is distributive through unions

II Pk II

h [ 111\ II +2 ( II Ai II II Pk-j II +

~

0

+111\_ j I1oIIP j ll )

+II P211-IIPjll)+ (6.89)

with (6.90)

The real constant h h

=

~s

given by

IRea£-Rea I j

m~n l<:j,£~n

j#£

where

a.,j = 1,2, ... ,n J

Setting

a

k

=

111\11

are characteristic values of A . and

be rewritten as

b

k

=

2a , k

0

the estimates (6.89) can

IIP211

<:

h[aj+bjIIPjIIJ

II P 3 II

<:

h [a 3+(b j II P 2 II +b 2 II P 1 \I ) +a j ( II P 1 II

0

II P 1 II ) 1

185

6. ASYMPTOTIC FLOQUET THEORY

+a

k_

3 ( II PIli II P2 II 0

+ •.. +ak-2(IIPllloIIPlll)] Similarly set

IIPkll

=

Pk;

(6.92)

then (6.92) becomes (taking the equa-

lity) PI

ha 1

(6.93) Now we define

a(o)

=

2h

~

k=1

00

II~IIok.

The series

I k=!

IIPkllolokl

which consists of norms of the terms of the series (6.83) 00

majorised by the series

(6.93) and

E;

=

101,

p(E;)

=

k

I PkE; k=!

in which the P satisfy k

whose generating function is (6.81).

According to Lemma 6.7 this function is single valued for

p

where

P is

any, otherwise

the unique positive root of

-p =

~s

p.

2a(E;) =

lE;I <

i f it has

This defines the radius of convergence

of p(E;) which in turn implies the convergence of (6.83).

We con-

clude that there exists an almost periodic solution to equation

186

STABILITY OF LINEAR SYSTEMS

(6.84) for

lo[ < p ~ p

and it remains to be shown that this so-

lution is fundamental and has an almost periodic inverse. Recall that a solution is fundamental if and only if 00

L Pk(t)ok)

det[I +

# 0

(6.94)

for all t

k=1

Since 00

II L

Pk(t)ok ll

k=1

if

It;;l < p,

fulfilled.

~

pet;;)

and

pet;;) < I

then the condition that

det(P(t,o))

# 0 is clearly

That the matrix P(t,O)-l is almost periodic has been

proved by Fink (1971) who observes that it is sufficient to show that

det[p(t,e))-l is almost periodic.

Each component of adjoint

P(t,e)

~s

a polynomial in the components of P(t,e) and for that

reason

~s

almost periodic.

almost periodic.

For the same reason det[P(t,e)) is

The required result follows from application of

the Abel-Jacobi-Liouville lemma.

This completes the proof of

Theorem 6.5 To complete this section we derive further results from Theorem

6.5 by restricting the kind of exponents allowed in S(UA A). k k

But

before doing so we extend a previous result which will be of further use. Theorem 6.6 Let the exponents Ajk) of the diagonal matrices satisfy the inequality

Fk:R

7

AP n

(6.95) and assume that the series 00

F(t,e)

L Fk(t)e

k

(6.96)

k=1

converges uniformly.

Then the integral

f F(s,o)ds t

G(t,c)

o

(6.97)

6. ASYMPTOTIC FLOQUET THEORY is almost periodic for all

/01

P

<

where

p

lim(dM(k)-1 )k-

187

(6.98)

1

k-where d is the absolute constant of Lemma 6.2 and p is the radius of convergence of (6.96).

Proof: Since the matrix F(t,o) is diagonal, we may apply the scalar result given as Lemma 6.3.

In fact we have

t

s~p I J f k i i (s)ds I

d

( k ) Ilfk .. m.

o

11

(6.99)

~~

~

where f .. (t) are the diagonal elements of Fk(t). Therefore k ~~ t t k ~ sup L: sup f.. (s,o)ds I J f k .. (s)dso ~~ I I t k=1 t ~~

J

0

0

,00

~

L:

k=!

d

( k ) Ilf k .. 11 Ich m• i i.

(6. 100)

~

Taking the bound min l~i~n

m.

(k)

i.

and applying the Cauchy-Hadamard theorem completes the proof. Returning to the extension of Theorem 6.5, we use Theorem 6.6 to prove the following: Theorem 6.7 Suppose that the almost periodic matrix A(t,a) satisfies the conditions of Theorem 6.5. nents contained in each circle

(k)

A.

J

s A Bk

S(UA~)

are bounded away from zero, so that

k (k) is greater than M in modulus.

lim(dM(k)-1 )k- 1 k-A(t,a)

In addition suppose that all expo-

Then in the

(6.101 )

STABILITY OF LINEAR SYSTEMS

188

Proof: By hypothesis we observe that A(t,O)

A

~n

the circle

101

< p ~

p

o

with

UA

k

Bk

C

S(kU AA)' -1<.

Since all exponents of Bk(t) are bounded away from zero for all k also by hypothesis, then according to Theorem 6.6 this implies that t

f k~1

Bk(t)ok

0

is almost periodic for p

lim(dM(k)-I) k

-1

k->oo

It is obvious that the matrices

A ,

and

o

all commute for

t s R+

s~nce

[A

o

+

they are all diagonal.

Application

of Theorem 4.13 completes the proof. Note that if the conditions of the theorem are fulfilled and if it is possible to put

0 = I,

then the result is similar to

The fundamental importance of Theorem 6.7 is

Coppel's theorem.

seen from the following:

Theorem 6.8: (Blinov 3 1965) Let the ordinary differential equation

~(t,O)

A(t,o)x(t,O)

(6.102)

possess a matrix of coefficients which satisfies the conditions of Theorem 6.6.

If we suppose that the real parts of the charac-

teristic values of

A

o

s M

n

are all negative, and that a. is the J

6. ASYMPTOTIC FLOQUET THEORY

189

characteristic value of least modulus, then the solutions of (6. 102) are asymptotically stable for those values of 0 for which the inequalities 00

la·1 J 101

>

L:

2

k=l

<

lolk

l ~l

(6.103)

p

hold.

101

Proof: By hypothesis we observe that in the circle A(t,O)

A

~

< p ~ p,

o

Since the matrices A and Bk(t) are diagonal, the simple applio cation of the Gronwall-Bellman inequality to solutions of the linear system kinematically similar to (6.102) shows that they are asymptotically stable for those values of

la.1 J

101

<

P

p

~

satisfying (6.104)

>

(see for example Cesari (1971), pp.35-36).

However it is easily

shown that

II

k B (t)o II k

L:

k=1

(6. lOS)

Moreover,

L: IIPk

k=1 p(t;) < I

II

for

101

00

II

lolk < p

~

~

L:

is majorised by

p.

L:

p (t;)

k=l

P E;k k

and

Therefore

00

L:

k=1

Bk(t)ok ll

k=1

"",\"

and the assertion is proved.

1

0 1k [I +p (E;) ]

~

2

L:

k=1

II~II

lolk (6. 106)

Theorem 6.9 Let the ordinary differential equation (6.102) possess a matrix

190

STABILITY OF LINEAR SYSTEMS

of coefficients A(t,8) which satisfies the conditions of Theorem 6.7.

If we suppose that the real parts of the characteristic va-

lues of

A

€

o

<

M

are all negative, then in the circle

n

p

lim(dM(k)-I)k k-+co

the solutions of (6.102) are uniformly asymptotically stable.

Proof: By hypothesis we observe that fined.

A(t,8) ~ A

for 181 as de-

o

The assertion follows immediately from the assumptions

regarding A .

Example 1

o

Consider the homogeneous system of linear equations

(6.107)

with

8 < 1

a real parameter.

Equations of this form frequently

occur in the study of parametric amplifiers and related devices (Venkatesh, 1977).

If a and S are noncommensurable, then the

matrix of coefficients of (6. 107) is clearly almost periodic.

It

can be rewritten as 00

0

0 A(t,8)

k

L: cos at8

k

k=1

(6.108)

+ 00

-2

k k -2 L: cos St8 k=1

-3

0

or simply as the series 00

A(t,8)

A

o

+

L:

k=l

which is convergent for €

AP2

for

k

~(t)8

k = 1,2, ... ,

181

€

[0,1)

with

A o

€

M2

and

By means of a similarity transformation the matrix A

o

~(t)

can be

6. ASYMPTOTIC FLOQUET THEORY

191

diagonalised; using the same notation we rewrite A(t,a) as

o

-I

-2

= (cos

k

= (cos

k

00

at

+

k k cos Bt)a

k=1

A(t,a)

+

o

00

-2

k=1

k k at + 2cos St)a

00

-2

= (2cos k at

k=1

k

+ cos St)a

k (6. 109)

k

k

2 = (cos at + cos Bt)a k=1 00

k

Clearly the system of equations (6.107) possess asymptotically stable solutions for

a = 0; for other values of a stability is

difficult to assess.

It is also obvious that the conditions of

Theorem 6.5 are satisfied so that the matrix ACt,a) is kinematically similar to a diagonal matrix B(t,a), almost periodic in t lal <

for

p<

We determine p as follows.

I.

We recall from (6.82) that p is the smallest positive root of the equation 2a(E.:) l.n which 00

2h

a(o)

=

k=1

For this example

1I~(t) II~

(t )

Ilak,

II

=

10

for all

k

1,2, ...

and h

I.

Therefore 00

4

z

k=1

IIA(t)

II

lal

k

so 00

40

z lelk

k=1

40';

~

(6. I 10)

192

STABILITY OF LINEAR SYSTEMS

giving

p

= 0.024

(approximately).

The exact form of the matrix B(t,o) is difficult to obtain analytically although the first few terms in the power series B(t,o)

B

(6. 1 I 1)

o

can be found immediately.

[-II

B

0

In fact

_°2]

(6. 112)

and

l'(CO'"' :

Bl (t )

0

cosBt)

2(cosat +

The evaluation of the remaining terms only feasible by numerical methods.

MSJ

(6.113)

the series (6.111) is

~n

Nevertheless it is possible

to determine a little more information about the stability behaviour of system (6.107) by applying Theorem 6.8. that the system (6.107) is asymptotically

-

< P

of

This tells us

stable for those values

for which the inequality

la·1 J

(6. 1 14)

is satisfied. of A

o

Here a. denotes the negative characteristic value J

with least modulus (all characteristic values of A must

have negative real parts according to the Theorem).

o

Therefore

we require that

but this inequality is satisfied for all

101

p

20

= 0.024,

L: k=1

1o Ik

=

1

01

< p,

since for

we have 0.49

<

I.

We conclude that the system (6.107) is asymptotically stable

6. ASYMPTOTIC FLOQUET THEORY

101

for all

<

193

P.

The same result can be obtained Vla Theorem 6.9 and Theorem 6.7. k

We observe that for a and

= 1,2, ...

Sf

0, all exponents of Bk(t),

are bounded away from zero in modulus by some posi-

tive constant.

Therefore the integral of

is almost periodic (i.e. bounded) for

lsi

p <

p

lim(const)k

(6. I 15)

k-+= It

follows that for

101

<

P,

the matrix A(t,o) is kinematically

similar to A and that system (6.107) is asymptotically stable. o References Adrianova, L.-Y. (1962). Vestnik Leningrad Univ., 17, 14-24 Bellman, R. (1953). "Stability theory of differential equations", McGraw-Hill, New York Blinov, I.N. (1965). Differentsial'nye Uraveniya, 1, 1042-1053 Bohr, H. and Neugebauer, O. (1926). Nachr.Ges.Wiss.Gottingen,

Math.-Physics Klasse, 8-22

Cesari, L. (1971). "Asymptotic behaviour and stability problems in ordinary differential equations", Springer Verlag, Heidelberg Coppel, W.A. (1967). Ann.Mat.Pura Applic., 76, 27-50 Fink, A.M. (1971). Proc. Am. Math. Soc. ,27, 527-533 Fink, A.M. (1974). "Almost periodic differential equations", Lecture Notes in Mathematics, 377, Springer Verlag, New York. Gel'man, A.E. (1957). Dokl.Akad.Nauk.SSSR, 116, 535-537 Gel'man, A.E. (1959). IZV LETI im Ul'yanova (LeninaJ, 39, 285-293 Gel'man, A.E. (1965). Differentsial'nye Uraveniya, I, 283-294 Khan, S. (1976). SIAM Appl.Math., 30, 749-767 Levitan, B.M. (1953). "Al mo s t periodic functions", GIZTTL, Moscow Lyascenko, N.Ya. (1956). Dokl.Akad. Nauk. SSSR, III, 295-298 He r k i.s , V.M. (1968). Litovsk.Mat.Sh., 8,101-112 Shtokalo, I.Z. (1946). Mat.Sh., 19,236-249 Sh t oka l o , I.Z. (1960). "Linear differential equations with va r i able coefficients". Izd.Akad.Nauk.Ukr.SSSR, Kiev (in English, Gordon Breach, (1961» Venkatesh, Y.V. (1977). "Energy methods in time-varying system stability and instability analyses", Lecture notes in physics No.68, Springer Verlag, Berlin r-

Chapter 7

LINEAR SYSTEMS WITH VARIABLE COEFFICIENTS

A surprisingly large number .of problems encountered in engineering practice can be formulated in terms of linear differential equations with nonconstant coefficients.

In many cases the coef-

ficients depend on time, in others the dependence is spatial. This chapter is intended as a stimulus for applications and further fundamental research based on the contents of the previous six chapters, and to satisfy the need to solve practical problems of an increasingly general nature.

We cite several examples of

such problems in the introduction and subsequently examine one in some detail.

The following survey is certainly not exhaustive,

many other examples will be found in the associated literature. 7.1

Introduction and Survey of Applications The most commonly encountered equations with nonconstant coef-

ficients are those due to Hill and Mathieu.

These two equations

conveniently describe the behaviour of a great many physical systems and yet they cannot be solved in closed form.

Solutions are

usually computed numerically using a wide variety of techniques (see for example, Friedmann & Hammond, 1977).

(aJ Pendulum with moving support The equations of motion of a pendulum with moving support are easily derived from the Lagrange equations

7. VARIABLE COEFFICIENT SYSTEMS

aT

d (aT)


ae

d

(aT)

av

--+-

ae

ae

195

o u


where 0 and U are generalised forces,

V

mg[£(l-cose) + u]

and the meaning of other symbols can be deduced from figure 7.1

mg Figure 7.1: A pendulum with moving support Applying the principle of "virtual work", we conclude that 0=0

and

U = F.

Substituting the expressions for T and V into

196

STABILITY OF LINEAR SYSTEMS

the Lagrange equations and differentiating, we obtain the following scalar nonlinear differential equations:

0

m£2e + m£usin8 + m£gsin8

For 8 sufficiently small, these equations reduce to 8+£-1(g+u)e u

0

Fm- 1

+ g

Assume that the displacement of the pivot point acoswt.

~s

given by

u

Then the second equation gives

which represents the force necessary to produce the motion. Furthermore, subject to this assumption the other equation becomes

For

a

=

0

this is the well known equation of a simple harmonic

oscillator.

We observe that the equation with

equilibrium

e

=

0

which

~s

a

I 0

admits the

stable for the simple pendulum (a -1

0). However, it is known that for certain values of g£ and 2£-1 aw the same equilibrium is rendered unstable by moving the pivot support. Setting g£ -1 = 0 and -aw 2 £ -1 = 2~, let w = 2 and

e

=

x.

Then the motion of the pendulum with moving support

satisfies the Mathieu equation

x+

(0 + 2~cos2t)x

0

which is studied in detail ~n

McLachlan (1947).

(b) Parametria Amplifiers Parametric amplification in a simple LC parallel circuit (inductive and capacitive) was one of the first demonstrations of a useful system which can be modelled by a linear differential equation with variable coefficients.

Parametric energy transfer is

usually achieved by varying periodically the capacitance in the

7. VARIABLE COEFFICIENT SYSTEMS

197

circuit and by choosing the correct frequency the circuit will sustain oscillations.

These devices, in a modified form, are

used extensively in modern communications systems (Keenan, 1968; MacDonald & Edmundson,

1961; Gardner, 1969; Bell & Wade, 1960).

(c) Clamped-clamped column under periodic axial load The equation of motion for the clamped-clamped column under periodic axial load shown in Figure 7.2 is (Iwatsubo et al., 1973)

and the boundary conditions are v(x,t)

dV(X,t) dX

°

where EI is the bending rigidity.

at

x

=

O,L

Hamilton's principle can be

used to replace the partial differential equation above by a system of coupled Mathieu

equat~ons,

where the partial properties of

• V Figure 7.2: Loaded clamped-clamped column

198

STABILITY OF LINEAR SYSTEMS

the column are discretised using the finite element methods (Friedmann & Hammond, op.cit.).

(dJ Electrons in a periodic potential The quantum mechanical problem of an electron in an electric field that varies periodically with position occurs

~n

a crystal

lattice, in which the periodic field is due to the uniformly spaced ions of the lattice.

A qualitative view of the problem is obtained

if we consider only one dimension. the wave function

The Schrodinger equation for

of the electron is then

~(z)

d2~ 2-2 --- + 8rr mh [E-V(z)J~ 2 dz

0

in which E is the total energy, m is the mass of the electron, V(z) ~s

its potential energy as a result of the electric field and h

is Planck's constant.

Let a be the spatial period of the electric

field (a parameter of the crystal lattice) and write V - V

V(z) Setting

o

<5

1

2

-2

= 8a mh

cos (2rrza- 1 ) [E-V 0]'

~

2

-2

= 4a mh Vl '

t = rrza

-1

and

1jJ

=x

the Schrodinger equation is reduced to the Mathieu equation.

(eJ Spacecraft attitude control in the presence of gravity gradient and aerodynamic torques An inertially referenced cylindrical spacecraft with rectangular solar panels is considered.

The vehicle is oriented with its

axis of symmetry perpendicular to the orbital plane.

Mendel (1968)

has shown that the linearised equations of motion of such a spacecraft in circular earth orbit and subject to gravity gradient and aerodynamic torques are 2a 21jJcos2w t - a 2sin2w t z 0 z 0 .0

e

where the inertia tensor I. 0' ~J

i,j

x,y,z

is diagonal, and

7. VARIABLE COEFFICIENT SYSTEMS a2 x

3w 2(1 0

3w 2(1 o

3w 2(1 o

~s

yy

-I

XX

XX

-I

-I

zz Zz

yy

)(21 )(21 )(21

XX

yy zz

199

)-1 )-1

)

-1

the i'th axis component of the aerodynamic torque act-

ing on the cylindrical portion of the spacecraft, I .. A~ ~~

~

i = x,y,z

is the i'th axis component of the aerodynamic torque acting on the solar panels,

i

wo

~s

~,e,¢

is an Euler angle sequence.

= x,y,z

the orbital angular velocity They define the spacecraft

roll, pitch and yaw angles, respectively. 7.2

Beam Stabilisation in an 'Alternating Gradient Proton Synchrotron The problem discussed in this section is that of damping trans-

verse oscillations of a proton beam in an alternating gradient (AG) synchrotron.

A feedback system is proposed which operates

by deflecting the beam at point 2 (refer to Figure 7.3) by an amount proportional to a detected position error at point 1 (Steining & Wilson, 1974). The need for some kind of beam "damper" in proton synchrotrons was first recognised when it became possible to inject and accelerate beams whose intensity was above the threshold of "resistive

wall" instability (Laslett et aL, 1965).

This effect is essen-

tially an interaction between the proton beam and the metallic walls of the synchrotron vacuum chamber.

Although a model ror the

"resistive wall" instability will not be included in the "damper" model; the response of the "damper" to initial condition type inputs is often used to establish the feedback gain required to suppress the instability once its growth rate is known.

Therefore

we will concentrate on the problem of designing a "damper" to

200

STABILITY OF LINEAR SYSTEMS

achieve a prescribed response to given initial conditions

Kicker

Figure 7.3: Beam deflection in an alternating gradient proton synchrotron

It is not our purpose here to derive in detail the single particle equations of motion in an AG synchrotron, many excellent texts already exist on that subject (Courant & Snyder, Bruck,

1966).

1958;

We shall say that the required equations of motion

in a circular accelerator are derivable from the relativistic Hamiltonian (Goldstein,

1950) I

H

e¢ + c[(p-eA)2 + m 2c2]' o

(7.1)

where e is the proton charge, ¢ is a scalar potential of the electromagnetic field, A is a vector potential of the electromagnetic field, p is the canonical momentum of the particle, c is the velocity of light in vacuo and m is the proton rest mass. In the a absence of field imperfections the transverse motion of the proton

7. VARIABLE COEFFICIENT SYSTEMS

201

satisfies the Hill equation

o

dx(s) _ A(s)x(s) ds

(7.2)

where A(s)

A(s+2rrR)

(7.3)

in which R is the machine radius and e dB dX

k(s)

(7.4)

P

defines the magnetic focussing strength of the guide field (transverse component).

Courant and Snyder show that a formal funda-

mental matrix for equation (7.2) has the form

I

'¥(s,s )

cos~(s)+a(s)sin~(s)

8(s)sin~(s)

L-y(s)sin~(s)

o

, cos~(s)-a(s)sin~(s)

]

(7.5)

which is usually referred to as the Twiss matrix in accelerator literature.

If the differential equation (7.2) possess linearly

independent solutions, which will be assumed, then the Wronskian W is equal to the determinant of '1 (Hale, 1969) and is a constant of the motion; its value

W= I

by normalisation.

The matrix

'¥ can be rewritten as
Since

Jsin~(s)

(7.6)

the unit matrix and

~s

a(s) J

+

Icos~(s)

o

8(s) ]

[ -y(s) det
I,

(7.7)

-a(s) we observe that

8(s)y(s) - a(s)2

(7.8)

-I

(7.9)

and

202

STABILITY OF LINEAR SYSTEMS

Assumption 7.1 The parameters a, Band y of the Twiss matrix (7.5) are real, continuous and bounded on R. The mfth power of ¢, which

the fundamental matrix for m

~s

turns around the machine, is thus

[Icos~(s)+Jsin~(s)]m

¢m

(7.10)

Icosm~(s)+Jsinm~(s)

It follows from (7.10) that if ~(s)

is real, the elements of ¢m

remain bounded with increasing m, in fact they oscillate. other hand, if bounded as

m

is not real,

~(s) ~

cosm~(s)

and

become un-

sinm~(s)

Therefore the motion is stable if

00.

On the

~(s)

is real,

or Itrace

2

:S

<1>1

(7. 1 1)

which is the same thing.

Note. that by virtue of (7.8) and Assump-

tion 7.1, B(s) can never vanish, that is, it is bounded away from zero.

Assumption 7.2

B of

The parameter

the Twiss matrix (7.5) possesses a real,

continuous and bounded first derivative on R. We now add the feedback structure to the Hill equation as follows: dx(s)

A(s)x(s) + bu(s)o(s-2nR)

(7.12)

q(s)

cx(R8)

(7.13)

u(s)

q(s)

(7.14)

ciS

in which

b: (0

g)',

c : (1

0),

g is the control gain and the

azimuth location of the control actuator or kicker has been set arbitrarily at equation

~n

s : 2nR.

Equation (7.12) is clearly a forced Hill

which the forcing term is impulsive.

The measurement

equation (7.13), simply states that the beam position (transverse coordinate) is measured at the azimuth location R8 and equation (7.14) defines the feedback law.

Combining equations (7.12),

7. VARIABLE COEFFICIENT SYSTEMS

203

(7.13) and (7.14) yields the following equation for the closed

loop beam dynamics dx(s) ds in which

A(s)x(s) + Ex(R8)o(s-2TIR) E

~

bc.

(7.15 )

It is possible to reduce equation (7.15) to a

more convenient form using the Liapunov transformation

pes)

(7.16)

A diagonal matrix B(s) is obtained which is kinematically simiTo show this we use the fact (Courant & Snyder, op

lar to A(s).

cit.) that 8 satisfies the equation

(S(s)

= ~~

etc.)

8(s)8(s) _ S(s)2 + k(s)8(s)2 2

(7.17)

4

It follows that the closed loop beam dynamics can now be expressed as yes)

P(s)x(s) + P(s)x(s) P(s)A(s)x(s) + P(s)Ex(R8)6(s-2nR) + P(s)x(s) P(s)A(s)P(s)

-1

yes) + P(s)EP(R8)

-1

y(R8)6(s-2TIR) +

P(s)p(s)-l y(s) [P(s)A(s)P(s)

-1

•

+P(s)P(s)

-1

]y(s) +

P(s)EP(R8)-l y(R8)o(s-2nR) B(s)y(s) + P(s)EP(R8) where B(s)

B(s)

~

A(s)

~s

-1

y(R8)6(s-2TIR)

(7.18)

defined by

P(s)A(s)P(S)-1 + P(s)p(s)-l 8(S)-1[0 -I

IJ

0

(7.19)

204

STABILITY OF LINEAR SYSTEMS

The solution of equation (7.18) can be derived immediately using the variation of constants formula as follows:

For

°

Y(S,T)P(T)EP(R8)

Y(s,O)y(O)

yes)

-1

y(R8)o(T-2nR)dT (7.20)

o ~

s < 2nR we have (7.21)

Y(s,O)y(O)

yes)

s = 2nR we have

whereas for

2nR

Y(2nR,0)y(0) +

y(2nR)

J Y(2nR,T)p(T)EP(R8)-l y(R8)o(T-2nR)dT o

(7.22)

Y(2nR,0)y(0) + p(2nR)EP(R8)-l y(R8) Y(2nR,O)y(0) + P(2nR)EP(R8) [Y(2nR,0)+p(2nR)EP(R8)

-1

-1

Y(R8,0)y(O)

Y(R8,0)]y(0)

[Y(2nR,0)+DY(R8,O)]y(0)

(7.23)

The fundamental matrix Y is given by

CO'[ 'B~~)] Y(s,O)

and

D

-'iO(( S~~)]

= P(2nR)EP(R8)

'iO[ 'B~~)] co,

[I' B~~)]

-1

(7.24)

(7.25)

Using the definition 1

Q = 2n

2nR do

J 13(0)

(7.26)

o

it follows that Y(2nR,0)

cos2nQ [-sin2nQ

sin2n Q] cos2nQ

(7.27)

7. VARIABLE COEFFICIENT SYSTEMS

205

Clearly the discrete closed loop system equation (7.23) defines the beam dynamics over one turn of the machine.

The asymp-

totic behaviour of the beam over many turns is determined by the characteristic values of the matrix Y(2TIR,O) + DY(R8,O)

S

(7.28)

A well known necessary and sufficient condition for asymptotic stability is that all characteristic values of S lie within the unit circle in the complex plane (Willems, 1970), i.e.

Iz·1 1

i

< I,

z=

1,2,

characteristic values of S

(7.29)

However, we note that for S defined by equation (7.28), the same necessary and sufficient condition is expressed by (Kalman & Bertram, p , 397, 1960)

I det S I

(7.30)

<

Hence, for asymptotic stability we require that cos2TIQ det

[[-sin2TIQ

where

R8 do S(o)

f

Sin2TIQ]

[

0 g/SlS2

cos2TIQ +

and

0]

o

[c~s~

-sln~

S(2TIR).

Sin~]]

<

cosW

(7.31 )

o

By evacuating the determinant it follows that the requirement for asymptotic stability becomes

I [1-g/SlS2

sin(2TIQ-W)]

I

<

(7.32)

Evidently asymptotic stability can be guaranteed by selecting the control gain g, the beta functions at the kicker and pick-up locations and the phase shift W between the kicker and pick-up to place the characteristic values of the system within the unit circle.

206

STABILITY OF LINEAR SYSTEMS

26.6 55

(value for the C.E.R.N. SPS)

0

104m

g

-5

10

rad/cm

80m We find that

det S = 0.906.

Thus the e-folding time (oscilla-

m

tions decrease as [det SJ2) is approximately 20 revolutions. References Bell, C.V. and Wade, G. (1960). IEEE Trans. Circuit Theory, CT-7, 4-11 Bruck, H. (1966). "Accelerateurs circulaires de particles", Presses Universitaires de France ,Paris Courant, E.D. and Snyder, H.S. (1958). Ann.Phys. 3, 1-48 Friedmann, P. and Hammond, C.E. (1977). Int.J.Numerical Methods in Engineering II, 1117-1136 Gardner, W.A. (1969). IEEE Tran~.Circuit Theory, CT-16, 295-302 Goldstein, H. (1950). "Classical Mechanics". Addison-Wesley, New York Hale, J .K. (1969). "Ordinary Differential Equations". WileyInterscience, New York Iwatsubo, T., Saigo, M. and Sugiyama, Y. (1973). J.Sound and Vibration, 30, 65-77 Kalman, R.E. and Bertram, J.E. (1960). Trans.ASME J.Basic Eng. Keenan, R.K. (1968). Proc.IEEE, 56, 1395 Laslett, L.J., Neil, V.K. and Sessler, A.M. (1965). Rev. Sci. Inst. , 36, 437 MacDonald, J.R. and Edmundson, D.E. (1961). Proc.IRE, 49, 453-466 MacLachlan, N.W. (1947). "Theory and Application of Mathieu Functions". Clarendon Press, Oxford Mendel, J.M. (1968). IEEE Trans.Automatic Control, AC-13, 362368 Steining, R. and Wilson, E.J.N., (1974). "Transverse collective instability in the NAL 500 GeV accelerator", Nuclear Instrum. Methods, 121, 206-228 Willems, J.L. (1970). "Stability Theory of Dynamical Systems", Nelson, London

Appendix I EXISTENCE OF SOLUTIONS TO PERIODIC AND ALMOST PERIODIC DIFFERENTIAL SYSTEMS

Existence Conditions for Periodic Systems

AI

Consider the general homogeneous periodic system

where B

f:D

+

{x:llx-x

=

f(t+w,x) = f(t,x),

f(t,x),

x

n R

II o

for

s,

<

x

°

w>

(AI. I)

D = JxB, J = {t: It-t I < a, t E: R} and o n Assume that if any solutions to E: R },

(AI. I) exist then they are unique.

By utilising Schauders fixed

point theorem (for generalisations appropriate to periodic systems see Browder (1959», Massera (1950) has shown that if the system (AI.I) is scalar, a solution which exists and remains bounded in the future implies the existence of a periodic soZu-

tion of period w. f(t,x) with

A:J

period w.

Additionally if the system is linear such that

+

M

n

and

both continuous and periodic of

Then Massera's result extends to this n-dimension li-

near periodic system. 0,

(AI.2)

A(t)x + h(t),

then given some initial condition (x ,t ) a solution

x(w;x , t ) o.

0

h(t) 1

The proof follows by assuming that o

0

through (x ,t ) is by (3.57) as 0

0

I

x

w

w

x

w

X(w)X-1(t

o)

+ X(w)

X-1(s)h(s)ds,

(AI.3)

o where X(t) is the fundamental matrix of the linear homogeneous

208

STABILITY OF LINEAR SYSTEMS x

mation of x equations

= A(t)x.

=

= Gx + b Tx where G and b Woo are defined by the right hand side of (AI.3) and r is a transfor-

equation

Setting

x

now suppose that the system of linear algebraic o; (G-I)x+b = 0 have no solution then (G-I) must be sin-

gular and there exists a fixed vector y such that y' (G-I) 0 k and y'b f O. Since y'G = y' then also y'G y' for k I , 2, ... , x,

I<

and by applying the transformation r repeatedly we have = Gkx + (Gk-] +Gk-z +... +I)b and hence y'~ = y'x + 0 0 o But since y'h f 0 then as k + 00 y'~ + 00 which implies

= rkx

ky'b.

that the solution to (AI.3) is unbounded.

x. = x(kw;x ,t )

But

I<

0

0

is bounded by definition so by contradiction the linear equation must have a periodic solution. Consider now the more general case of (AI. I) for n an arbitrary positive integer.

By applying Browders (1959) fixed point theorem

to the periodic system (AI. I), the following is a generalisation of Cartwright's (1950) result for second order systems:

Theorem AI. I: Existence of genepal pepiodic solution If the solutions of (AI. I) are bounded for some bound N, then there exists a periodic solution of period w such that N

for

Ilx(t) II s

t E: R.

So far an assumption on the uniqueness of solution has been necessary; however by imposing a stability condition upon the periodic solution, the uniqueness requirement can be dropped:

Theorem AI.2: (Yoshizawa, 1975) Given that f(t,x) is continuous on D and that the periodic system (AI. I) has a solution y f t ) such that for all

t E: R+.

of period rw(r I!y(t )-x o

0

II

2

< N

Ily(t)

II

s B* < B

Then there exists a periodic solution of (Al. I) 1,2, ... ) if there exists aN> 0 implies that

Ily(t)-x(t;x ,t ) 0

0

II

+

such that 0

as

t

+

00.

We note that the existence of bounded uniformly asymptotically stable solution to (AI. I) does not necessarily imply the existence of a periodic solution of the same period w.

Continuing the sta-

bility approach, Deysach and Sell (1965) have shown that if y(t)

AI. EXISTENCE OF SOLUTION

209

is uniformly stable then there exists an aUnost periodic solution to (AI. I); that is we cannot necessarily obtain a periodic solution to a dynamical system with periodic coefficients without posing additional constraints.

~m

Theorem 5.17 has demonstrated

that if yet) is uniformly stable, then yet) is stable under disturbances from the hull R(f) and so the following theorem follows: Theorem A1.3: (Halanau, 1962) If the solution yet) to (AI. I) is uniformly stable then yet) is asymptotically almost periodic and the system (AI.I) has an almost periodic solution which is also uniformly stable. Finally, Sell (1966) has similarly shown that a periodic solution of period rw (r

~

I) to (AI. I) exists if there is a bounded

solution yet) which is weakly uniformly asymptotically stable (equivalent to uniform asymptotic stability in the case of periodic systems).

This result is a special case of Theorem AI.3

when uniform convergence is used. A2

Existence Conditions for Almost Periodic Systems Consider the almost periodic system f(t,x)

x

n, f:RxD ~ R

where

formly for R+.

(AI. 4)

x

€

AP(RXD» is almost periodic in t unin, B = {x: x € R Ilx-x II < S*} and for all t € (f €

-{<

*

0

Let F be a compact subset of B , and let yet) be a solution

of (AI.4) such that

!IY(t)

II

<

S*,

For some positive sequence {a uniformly on RxF, then

hsR(f).

and k}

let

yet)

€

F

for all

t

€

lim f(t+ak,x) = h(t,x) k->=

Moreover assume that

lim y(t+ k->=

a

k)

= z(t) x·=

uniformly on R+ where z(t) is a solution to

h(t,x),

h s R(f)

(AI.5)

the dual differential system (AI.4), (AI.5) leads us to the definition of inherited properties.

210

STABILITY OF LINEAR SYSTEMS

Definition A.I: Inherited properties of almost periodic systems

(Fink, 1972)

If yet) has a particular property with respect to system (AI.4), and z(t) has the same property with respect to the dual system (AI.s), then this property is said to be inherited. For almost periodic systems total stability and stability under disturbances are inherited properties; in addition for periodic systems, uniform stability and uniform asymptotic stability are also inherited properties.

However it should be noted that

for almost periodic systems, uniform stability and uniform asymptotic stability are inherited properties only if the uniqueness of solution is assumed.

An example that demonstrates this res-

triction is given by Kato (1970), it also shows that uniform asymptotic stability does not necessarily imply total stability in almost periodic systems whilst it does for periodic systems. For periodic systems the boundedness of solution implies the existence of a periodic solution, whilst for almost periodic systems this is not the case.

Indeed, several examples of almost

periodic systems have been constructed by Opial (1961) and Fink and Frederickson (1971) such that the almost periodic system (AI.4) has no almost periodic solutions yet its solutions are uniformly ultimately bounded.

Thus in discussing the existence

of almost periodic solutions, stability properties of some kind must be implied.

Based upon the assumption of uniqueness of solu-

tion to (AI.4), Miller (1965), required that the bounded solution is totally stable for the existence of solution, Seifert (1966) required the E-stability of the bounded solution, whilst Sell (1967) required stability under disturbances from the hull (see also section 5.6).

All of these results can be achieved without

the condition of uniqueness of solution by utilising the property of asymptotically almost periodic functions (see section 2.6). Typical of these results are the following two theorems which are stated without proof:

211

AI. EXISTENCE OF SOLUTION Theorem AI.4:

Existence of an almost periodic solution (Coppel, 1967)

If the almost periodic system

x

=

f(t,x),

f € AP(RXD),

has

a bounded solution on R+ which is asymptotically almost periodic, then it has an almost periodic solution. Theorem AI.S:

(Yoshizawa, 1975)

If the bounded solution yet) of the almost periodic system (Al.4) is asymptotically almost periodic then for any there exists a positive sequence {~}

such that

z(t)

h

=

€

R(f)

lim y(t+ k~

a ) is an almost periodic solution of the system (AI.S) uniformly k on R+. These theorems tell us that if the almost periodic system (AI.4) has a bounded asymptotically almost periodic solution then the dual system (AI.4-S) also have almost periodic solutions.

Also

since stability under disturbances from the hull is by Theorem

5.17 a sufficient condition for asymptotic almost periodicity then if the system (Al.4) has a bounded solution which is stable under disturbances from R(f), then it also has an almost periodic solution yet) which is also stable under disturbances from R(f).

A

corollary to this result is that if the above solution yet) is totally stable then yet) is asymptotically almost periodic and the system (AI.4) has an almost periodic solution which is totally stable. In the previous section we have seen that if a periodic system has a bounded and stable solution then there exists an almost periodic solution. (Yoshizawa,

Examples of periodic systems can be generated

1975) whereby the system has a quasi-periodic solution

and thus the module of the almost periodic solution is tained within the module of the system.

not con-

This demonstrates that

uniform stability and stability under disturbances from the hull do not give module containment for almost periodic systems.

In

the following we develop the ,conditions, based upon uniform stability of solutions, for the existence of almost periodic solutions

212

STABILITY OF LINEAR SYSTEMS

to the almost periodic linear inhomogeneous system: x

where t on R.

x

A(t)x + f(t), A:R

-+

M

n

(AI.6)

and

are almost periodic uniformly in

Corresponding to (AI.6) is the homogeneous linear system A(t)x,

(AI.7)

and the equation on the hull x

G(t)x,

(AI.8)

G E R(A).

Clearly if the almost periodic system (AI.6) has a bounded solution on R+ which is uniformly stable then the null solution of (AI.7) and (AI.8) are also uniformly stable.

Under these condi-

tions the following theorem due to Favard (1933) shows that (AI.6) has an almost periodic solution: Theorem AI .6

If every nontrivial solution x(t) of (AI.8) on the homogeneous hull of the linear almost periodic system (AI.6) is bounded on R and satisfies solution on

Inf Ilx(t) tER t ER+,

II

> 0,

then if (AI.6) has a bounded

there exists an almost periodic solution

yet) of (AI.6) such that

mod(y) c mod(A,f),

where mod(y) is the

module of y and mod(A,f) is the set module of A on f.

Proof: (Outline) It is straightforward to show that a bounded solution y (t) to (AI.6) exists and each equation in the hull of o

(AI.6) has a unique solution with minimum norm. simple matter to show that

It is then a n), yet), y (t) E AP(RXR and finally o

by module containment (see definition 2.4) that

mod(y) c mod(A,f).

By similar reasoning Bochner (1962) derived as a corollary to Farvard's theorem (AI.6): Theorem AI. 7 If every equation in the homogeneous hull of the linear almost periodic system (AI.6) is almost periodic, then every bounded solution of (AI.6) is almost periodic. Bochner's result also holds for a variety of important special

AI. EXISTENCE OF SOLUTION

213

cases; (i) the Bohr-Neugebauer theorem (1926) for linear constant n coefficient almost periodic systems (A s M , f s AP(R (ii) when

f(t) :: 0

periodic, that

and 1S

A(t) s AP(M ), A(t+w)

A(t)

n

»,

n

and (iii) when A(t) is purely

for some

w > O.

References Bochner, S. (1962). Proc.Nat.Acad.Sci. (USA) 48, 2039-2043 Bohr, H. and Neugebauer, O. (1926). "Uber lineare differentialgleichungen mit konstanten koeffizienten und fastperiodischer rechter seite", Nachr.Ges.Wiss.GiJttingen, Math.-Phys.Klass., 8-22 Browder, F.E. (1959). Duke Math.J. 26, 291-303 Cartwright, M.L. (1950). "Forced oscillations in nonlinear systems", contri. to "The theory of nonlinear oscillations", Ed. S. Lefschetz 1, Princeton University Press Cappel, W.A. (1967). Ann.Mat.Pura Applic. 76, 27-50 Deysach, L.G. and Sell, G.R. (1965). Michigan Math.J. 12, 87-95 Favard, J. (1933). "Lecsons sur les Fonctions Presque Periodiques", Gauthier Villars, Paris Fink, A.M. (1972). SIAM Review 14, 572-581 Fink, A.M. and Frederickson,'P.O. (1971). J.Diff.Eqns. 9, 280-284 Halanay, A. (1962). Uspeh Mat.Nauk. 17, 231-233 Kato, J. (1970). Tohoku Math.J. 22, 254-269 Massera, J.L. (1950). Duke Math.J. 17, 457-475 Miller, R.K. (1965). J.Diff.Eqns. 1, 293-305 Opial, Z. (1961). Bull.Acad.Polon. Sci. Ser. Sci.Math. Astron. Phys. 9, 673-676 Seifert, G. (1966). J.Diff.Eqns. 2, 305-319 Sell, G.R. (1966). J.Diff.Eqns. 2, 143-157 Sell, G.R. (1967). Trans. Math. Soc. 127, 241-283 Yoshizawa, T. (1975). "Stability theory and the existence of periodic solutions and almost periodic solutions". Appl.Maths. Sci. No.14, Springer Verlag, New York

Appendix 2 DICHOTOMIES AND KINEMATIC SIMILARITY

The concepts of exponential dichotomies and kinematic similarity were respectively introduced in Chapters 5 and 4.

We now demon-

strate that kinematically similar systems satisfy common dichotoUnfortunately the problem of establishing that the exis-

m~es.

tence of a dichotomy implies kinematic similarity is not completely resolved, although some results are available. Definition A2. I: Kinematically Sim'ilar Systems The linear autonomous differential systems B(t)y,

with

lar systems if

B(t).

~

= A(t)x,

Y

are said to be kinematically simi-

A(t), B(t) E M A(t)

x

n

Our first formal observation is that two kinematically similar systems satisfy common exponential dichotomies. Theorem A2. I If a homogeneous linear equation has an exponential dichotomy then any equation kinematically similar to it likewise has an exponential dichotomy with the same projection Po and the same constants

a,S.

Proof: If

yet) = B(t)y(t)

the change of variables transformation, then

IY(t.) Po Y(s) -1 I

~

~s

x(t)

obtained from =

yet) = pet)

-1

X(t)

cons t IX(t ) P X(s ) -1 0

~(t)

= A(t)x(t)

by

where pet) is a Liapuno v

P(t)y(t),

and

I

A2. DICHOTOMIES AND KINEMATIC SIMILARITY !Y(t)(I-P )Y(s)-I!

const!X(t)(I-P )X(s)

S

o

-1

o

215

[

The assertion then follows trivially. The interesting problem is to show that a system satisfying an exponential dichotomy is kinematically similar to some other system.

One result for this is due to Coppel (1967) and is concerned

with conditionally stable systems (the same result for non-conditionally stable systems is trivial).

First of all we require the

following: Lennna A2. I

Let P

be a projection and let X(t) be a continuous nonsingular -1 matrix such that X(t)P X(t) is bounded for all t. Then there 0

0

exists a continuous nonsingular matrix P (t.) such that P(t)P pet)

-1

X(t)P X(t)

-1

(A2. I)

o

o

which is bounded, together

its

~ith

~nverse

for all t.

Proof: Suppose, without loss of generality, that P gonal projection, that is

P

P *.

o

0

o

~s

an ortho-

Since any positive Hermitian

matrix has a unique positive square root, there exists, for each t, a unique

= R(t)*

R(t) P

> 0

* X(t) X(t)P

such that ok

(A2.2)

+ (I-P)X(t) X(t)(I-P )

0 0 0

0

Moreover, since R(t)2 commutes with P , so does R(t). Let o Imatrixl 2 = tr(matrix* matrix). It follows at once from the definition of R(t) that

and hence

s

[X(t)R(t) -11

[X(t)P R(t)-11 + !X(t) (I-P )R(t)-11 o

S

0

(2n)

1 2

(A2.3) On .the other hand, if s~,

gives

where

~

=

J

IX(t)p X(t)-I[

II + n'.

0

Thus

S

ll,then IX(t)(I-P )X(t)-11 0

Z]6

STABILITY OF LINEAR SYSTEMS

(AZ.4) Finally R(t) is continuous. we take

pet)

= X(t)R(t)

Notice that if

=

P

-1

The lemma now follows at once if

.

0 or 1,

then

o The result of importance is: Theorem

= A(t)x(t)

~(t)

suppose that there exists R projection P

P(t )

1.

o

with

such that

A(t) s Mn ,

X(t)p X(t)-1 0

Then the given system is kinematically similar to the sys-

n

tern

and

AZ.Z

Given the homogeneous system

sM.

x( t )

R(t)

= B(t)y(t)

yet)

with

B(t) s M

n

such that

P B(t) 0

B(t)P

o

for all t.

Proof; Subject to the conditions of the theorem, the matrix functions R(t) and pet) of the previous lemma are continuously differentiable.

The change of variables

x(t)

=

P(t)y(t)

defines

the kinematic similarity where -pet)

B(t)

-1

•

[P(t)-A(t)P(t)].

Since the new system has R(t)as a f.m., mutes with P . o Because A(t)

lS

com-

B(t)

bounded, there exists a positive constant 8

such that for every t -81 If

P

o

A(t) + A* (t )

:0: =

P ,~

:0:

(A2.S)

81

then from the definition of R(t) we have

0

R(t)R(t) + R(t)R(t)

P X( t o

)

*

,,;'r>

(A(t) + A(t) )X(t)P

0

(A2.6) It follows that -8R(t)2

:0:

R(t)R(t) + R(t)R(t)

:0:

8R(t)2

(A2.7)

and hence

(A2.8) Therefore

217

A2. DICHOTOMIES AND KINEMATIC SIMILARITY •

{R(t)R(t)

-1

+ R(t)

-1'

R(t)}

2

e2 I

5

(A2.9)

and 5

We are going to deduce that

en

(A2.10)

B(t) ~ R(t)R(t)-l

is bounded.

Suppose that G and H are Hermitian matrices and

G 2 O.

Since

(GH + HG)2 and the trace of a product is unaltered by cyclic permutation of the factors, then tr(GH + HG)2 But

G ~ S2

for some Hermitian matrix S and hence

tr(GH)2

tr(SHS)2

ISHSI2

0

2

(A2. II)

Therefore IHGI 2

~IGH

+ HGI 2

(A2.12)

Thus (A2.10) implies that IB(t)1

5

I

enn]2

and the assertion is proven. We observe that the form of the matrix B(t) in the above theorem is

o (A2.13)

B (t )

o

B2 (t.)

where B1(t) and B2(t) are matrices of lower order than B(t). These correspond to two sets of solutions of the equation yet) B(t)y(t) or I

with different exponential growth behaviour.

we have the trivial case

A(t)

=

B(t)

with

B(t)

or B2(t). Theorems A2.1 and A2.2 can be combined as follows:

For =

P

o Bl (t.)

~

0

218

STABILITY OF LINEAR SYSTEMS

Theorem A2.3 If a homogeneous system possesses an exponential dichotomy with projection P

o

then it is kinematically similar to a system which

also possesses an exponential dichotomy and whose coefficient matrix commutes with P . o

We note that theorems A2.1, A2.2 and A2.3 are equally applicable to systems possessing an ordinary dichotomy.

Other results on the

relationship between dichotomies and kinematic similarity are contained in the papers of Sacker and Sell (1974), Sell (1974) and In particular Coppel (op.cit.) considers systems

Coppel (1968).

with almost periodic coefficients; further results on kinematic similarity in this context can be found in Fink (1974), and in Chapter 6. We have already noted in section 5.3 the relationship between stability and dichotomies; this suggests that the existence of a Liapunov function implies an exponential dichotomy.

To see this

equivalence consider the linear differential equation x

A(t)x,

A(t)

(A2.14)

M n

E:

If X(t) is the fundamental solution of (A2.14) with

X(o)

=

I,

define the Hermitian matrix function (X(t)p Xes)

Q (t )

-1

o

*

) (X(t)P Xes)

-1

0

)ds

t

- 2 «I-P )X(t)

-1

o

*

) (I-P )X(t)

-1

0

- 2 It (X(t) (I-P )X(s)-l)*(X(t)(I-P )X(s))ds o

0

(A2.15)

o

Assume that (A2.14) has an exponential dichotomy (see (5.22)) then

II Q (t ) II

2,Q,k(a+6+1) (a+6)

-1

which provides an upper bound to Q(t). Now since

X

= A(t)X then from (A2.15),

(A2.16)

A2. DICHOTOMIES AND KINEMATIC SIMILARITY

Q + QA(t) + A(t) * Q where

R

= X(t)P o X(t)

-2R * R - 2(I-R) * (I-R)

-1

projection operator P

219

.

But in

(A2.17)

of the properties of the

v~ew

(see section 5.3) the right hand side of

o

(A2.17) bounds the identity matrix, therefore

Q+

s

QA(t) + A(t)*Q

Clearly if

IIACt) II <

00

-I then

for

t

11611 <

00

~

(A2.18)

O.

since

IIQII <

00.

Finally we note that the quadratic function (a Liapunov function) V

x *Qx

has a negative definite derivative

.

V

x * [Q + QA(t) + A(t) * QJx

s

-llxll 2

(A2. 19)

We conclude that an exponential dichotomy for (A2.14) is sufficient for the existence ?f a quadratic Liapunov function with a negative definite derivative.

The converse is not true unless

the solution to (A2.14) is bounded.

The connection between

Liapunov functions and exponential dichotomies was first noted by Maizel (1954) and later by Massera and Schaffer (1966). References Coppel, W.A. (1967). J.Differential Eqns., 3, 500-521 Coppel, W.A. (1968). J.Differential Eqns., 4, 386-398 Fink, A.M. (1974). "Almost periodic differential equations", Lecture Notes in Mathematics No.377, Springer Verlag, New York Maizel, A.D. (1954). Ural.Politehn.Inst.Trudy, 51, 20-50 Massera, J.L. and Schaffer, J.J. (1966). "Linear differential equations and function spaces", Academic Press, New York. Sacker, R.J. and Sell, C.R. (1974). J.Differential Eqns., 15, 429-458 Sell, C.R. (1974). "The Floquet problem for almost periodic linear differential equations", Lecture Notes in Mathematics, No.415, Springer Verlag, New York

Appendix 3 BIBLIOGRAPHY

The papers here are collected in three sections; on Differential Equations with Almost Periodic Coefficients, on Reducibility and Kinematic Similarity, and on Stability. (i) References on Differential Equations with Almost Periodic Coefficients Abel, J. (1970). On the almost periodic Mathieu equation, Quapt. J.Appl.Math. 28, 205-217 Amerio, L. (1967). Almost periodic solutions of the equation of the Schrodinger type. Atti Acad.Nay Lincei Rend.Cl.Sci.Fiz.Mat. Natur. 8, 147-153 Amerio, L. and Prouse, G. (1971). Almost periodic functions and functional equations. Van Nostrand, New York Artjusenko, L.M. (1968). The application of Fourier series for finding almost periodic solutions of equations with mean values. Izv. Vyss.Ucebn.Zaved Matematika 5, 21-27 Barbalat, I. (1961). Solutions presque-periodiques des equations differentielles non-linearies. Com.Acad.R.S.Romainia 11, ]55159 Berezanskii, Y.M. (1953). On generalised almost periodic functions and sequences, related with the difference-differential equations. Mat.Sb. 32, 157-194 Birjuk, G.I. (]954). On a theorem concerning the existence of almost periodic solutions for certain nonlinear differential systems with a small parameter. Doklady.Akad.Nauk.SSSR 96,5-7 Blinov, I.N. (1965). Analytic representation of the solution of a system of linear differential equations with almost periodic coefficients which depend upon a parameter. Diff.Uravnenija 1, 1042-1053 Blinov, I.N. (1965). An analytical solution of a linear system of differential equations with periodic coefficients depending upon

A3. BIBLIOGRAPHY

221

a parameter. Diff.Equations 1, 679-691, 812-821 Blinov, I.N. (1971). The regularity of a class of linear systems with almost periodic coefficients. Diff.Equations 3, 764-768 Bochner, S. (1933). Homogeneous systems of differential equations with almost periodic coefficients. J.London Math.Soc. 8, 283288 Bogdanowicz, W.M. (1963). On the existence of almost periodic solutions for systems of ordinary differential equations in Banach spaces. Arch.Rational Mech.Anal. 13, 364-370 Bohr, H. (1952). Collected Mathematical Works, Vols.I, II, III, Dansk.Mat.Forening, Kobenhavn Boruhov, L.E. (1947). A linear integral equation with almost periodic kernel and free term. Doklady. Akad. Nauk.SSSR. 57, 647-649 Boruhov, L.E. (1954). On almost periodicity of solutions of some linear differential systems with almost periodic coefficients.

Nauk.Ez.Saratovsk.Univ. 659-660

Burd, V.S. (1965). Dependence on a parameter of almost periodic solutions of differential equations with a deviating argument. Akad.Nauk.Azerbaidzen.SSR.Dokl. 21, 3-7 Burton, T.A. (1966). Linear differential equations with periodic coefficients. Proc. Amer.Math. Soc. 17, 327-329 Bylov, B.F. (1965). The structure of the solutions of a system of linear differential equqtions with almost periodic coefficients. Mat.Sb. 66, 215-229 Bystrenin, V.V. (1941). On almost periodic solutions of certain ordinary differential equations. Doklady.Akad.Nauk.SSSR 33, 387-389 Cameron, R.H. (1938). Linear differential equations with almost periodic coefficients. Acta Math. 6, 21-56 Cartwright, M.L. (1971). Almost periodic solutions of differential equations and flows. Global differential Dynamics, Proc.Conf. Case Western Univ., U.S.A .. Springer Verlag Lecture Notes, Math. 235, 35-43 Chang, K.W. (1968). Almost periodic solutions of singularly perturbed systems of differential equations. J.Diff.Equations 4, 300-307 Coppel, W.A. (1967). Almost periodic properties of ordinary differential equations. Ann.di Mat.Pura ed Appl. 76, 27-50 Conti, R. and Sansone, G. (1964). Nonlinear differential equations. Pergamon, New York Corduneanu, C. (1968). Almost periodic functions. Interscience Publishers, New York. Doss, R. (1965). On the almost periodic solutions of a class of integro-differential-difference equations. Ann.Math. 81, 117.123 Emzarov, K. and Tulegenov, M. (1966). An existence theorem for almost periodic solutions of a differential equation with a small parameter in a Banach space. Vestnik Akad.Nauk Kazah. SSR 22, 42-44 Erugin, N.P. (1966). Linear systems of ordinary differential

222

STABILITY OF LINEAR SYSTEMS

equations with periodic and quasi-periodic coefficients. Academic Press, New York. Ezeilo, J.O.C. (1966). On the existence of almost periodic solutions of some dissipative second order differential equations. Ann.Mat.Pura Appl. 74, 399 Favard, J. (1933). Le~ons sur les fonction Presque-periodiques. Gauthier-Villars, Paris Favard, J. (1963). Sur certains systemes differentiels scalaires lineaires et homogenes coefficients presque-periodiques. Ann.Mat.Pura Appl. 4, 61 Fink, A.M. (1972). Almost periodic functions invented for specific purposes. SIAM Review 14, 572-581 Fink, A.M. and Frederickson, P. (1971). Ultimate boundedness does not imply almost periodicity. J.Diff.Equations 9, 280-284 Fink, A.M. and Seifert, G. (1971). Nonresonance conditions for the existence of almost periodic solutions of almost periodic systems. SIAM J.Appl.Math. 21, 362-366 Frechet, M. (1941). Les fonctions asymptotiquement presque periodiques, Rev.Scientifique 79, 341-354 Goldberg, R.R. (1957). Convolutions transforms of almost periodic functions. Riv.Mat.Univ.Parma. 8, 307-312 Golomb, M. (1958). Expansions and boundedness theorems for solutions of linear differential systems with periodic or almost periodic coefficients. Arch.Rat.Mech.Analysis 2, 284-308 Gunzler, H. and Zaidman, S. (1969). Abstract almost periodic differential equations. Abstract Spaces and Approximation. Birkhauser, Basel, 387-392 Gurjanov, A. (1970). On sufficient conditions for the regularity of second order systems of ordinary differential equations with uniform asympotically almost periodic coefficients. Vestnick Leningrad Univ. 25, 23-27 Halanay, A. (1960). Almost periodic solutions of systems of differential equations with a lagging argument and a small parameter. Rev.Math.pures et Appl. 5, 75-79 Halanay, A. (1963). Almost periodic solutions of systems with a small parameter in a certain critical case. Rev.Math.pures et Appl. 8, 397-403 Hale, J.K. (1964). Periodic and almost periodic solutions of functional differential equations. Arch. Rat.Mech.Anal. 15, 289304 Hale, J.K. (1969). Ordinary differential equations. Wiley-Interscience, New York. Harasahal, V.H. (1960). Almost periodic solutions of nonlinear systems of differential equations. Prikl.Mat.Meh. 24, 565-567 Hermes, H. (1973). A survey of recent results in differential equations. SIAM Review 15, 453-468 Ivanov, V.N. (1965). On linear differential operators in the space of almost periodic functions. Trudy Saratovsk.Inst.Meh.Selsk. 38, 141-149 Jakubovic, V.A. (1966). Periodic and almost periodic limit regimes

a

A3. BIBLIOGRAPHY

223

of automatic control systems with some discontinuous nonlinearities. DokZady Akad.Nauk.SSSR 171, 533-537 Kapisev, K.K. (1966). Quasiperiodic solutions of nonlinear systems of differential equations containing a small parameter. V~stnik Akad.Nauk Kazah.SSR 22, 42-47 Kempner, G.A. (1968). Almost periodic functional differential equations. SIAM J.AppZ.Math. 16, 155-161 Kovanko, A.S. (1965). Almost periodic solutions of certain differential equations with almost periodic right sides. Visnik Lwow.Univ.Ser.Math.fasc. 2, 3-8 Krasnoselskii, M.A. and Perov, A.I. (1958). A principle concerning the existence of bounded periodic and almost periodic solutions for systems of ordinary differential equations. DokZady Akad. Nauk.SSSR 123, 235-238 Langenhop, C.E. and Seifert, G. (1959). Almost periodic solutions of second order nonlinear differential equations with almost periodic forcing. Proc. Am. Math. Soc. 10, 425-432 Levitan, B.M. (1937). On linear differential equations with almost periodic coefficients. DokZady Akad.Nauk.SSSR 17, 285-286 Lillo, J.C. (1959). On almost periodic solutions of differential equations. Ann. Math. 69, 467-485 Lisevic, L.N. (1960). Extension of Favard's theorems to the case of a linear system of differential equations with analytic almost periodic coefficients. Dovopidi Akad. Nauk.. Ukra-in, SSR, 148-149 Ljubarskii, M. (1972). The extension of Favard's theory to the case of a system of linear differential equations whose coefficients are unbounded and almost periodic in the sense of Levitan. Soviet Math. 13, 1316-1319 Lyascenko, N.Ya. (1956). An analogue of the theorem of Floquet for a special case of linear homogeneous systems of differential equations with quasi-periodic coefficients. Doklady Akad.Nauk. SSSR 111, 295-298 Malkin, I.G. (1954). On almost periodic oscillations of nonlinear non-autonomous systems. PrikZ. Mat. Mech. 18, 681-704 Marcus, L. and Moore, R.A. (1956). Oscillations and disconjugacy for linear differential equations with almost periodic coefficients. Acta. Math. 96, 99-123 Massera, J.L. and Schaffer, J.J. (1958). Linear differential equations and functional analysis. I.Ann.Math. 67, 517-572 Miller, R.K. (1965). Almost periodic differential equations as dynamic systems with applications to the existence of almost periodic solutions. J.DifferentiaZ Equations 1, 337-345 Millionscikov, V.M. (1965). Recurrent and almost periodic limit ~rajectories of nonautonomous systems of differential equations. DokZady Akad.Nauk.SSSR 161, 43-44 Mitropolskii, Y.A. and Samoilenko, A.M. (1965). On constructing solutions of linear differential equations with quasi-periodic coefficients by the method of improved convergence. Ukrain.Mat. Z. 17, 42-59

224

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O'Brien, G.C. (1972). Almost periodic and quasi-periodic solutions of differential equations. Bull.Aust.Math.Soc. 7, 453-454 Oleinik, S.G. (1969). The investigation of linear systems of differential equations with almost periodic coefficients. Math.Phys. Nauk.Dumka~ Kiev.~ 6, 139-149 Opial, Z. (1961). Sur une equation differentieble presque-periodique sans solution presque-periodique. Bull.Acad.Polon.Sci. Ser. Sci. Math. Astr. Phys. 9, 673-676 Phillips, R.S. (1940). On linear transformations. Trans. Am. Math. Soc. 48, 516-541 Ragimov, M.B. and Zadoroznii, V.G. (1970). Almost periodic solutions of multidimensional differential equations. Akad.Nauk. Azerbaidzen.SSR.Dokl. 26, 8-11 Rjabov, J.A. (1963). On a method of finding a bound for the region of existence of periodic and almost periodic solutions of quasilinear differential equations with a small parameter. Izv. Vyss. Ucebn.Zaved.Matematika 33, 101-107 Sanchez, D.A. (1969). A note on periodic solutions of Riccati-type equations. SIAM J.Appl.Math. 17, 957 Sansone, G. and Conti, R. (1964). Nonlinear differential equations, Pergamon Press, London Seifert, G. (1966). Almost periodic solutions for almost periodic systems of ordinary differential equations. J.Diff.Eqns. 2, 305-319 Seifert, G. (1972). Almost periodic solutions for limit periodic systems. SIAM J.Appl.Math. 22, 38-44 Shtokalo, I.Z. (1960). Linear differential equations with variable coefficients. Izd.Akad.Nauk.Ukr.SSSR~ Kiev. (also Gordon-Breach, 1961) Svinbekov, K.D. (1964). On the analytic form of solutions of linear systems of differential equations with quasi-periodic coefficients. Izv.Akad.Nauk.Kazah.SSR Ser.Fiz.Mat.Nauk. 2, 69-71 Talpalaru, P. (1969). Solutions periodiques et presque-periodiques des systems differentiels. An.sti.Univ.Al.I.Cunza Iasi n Ser Sect.Ia 15, 375-385 Turcu, A. (1965). Almost periodic solutions of the equations of Duffing in the case of resonance. Studia Univ.Babes Bolyaik. Ser.Math.-Phys. 10, 83-94 Umbetzanov, D.U. (1970). The almost periodic solution of certain classes of partial differential equations with small parameters. Differencial nye Uravenija 6, 913--916 Urabe, M. (1972). Existence theorems of quasi-periodic solutions to nonlinear differential systems. Funk. Ekv. 15, 75-100 Vaghi, C. (1968). Soluzioni limitate, 0 quasi-periodiche, di un' equazione di tipo parabolico nonlineare. Boll.U.M.I. 4-5, 559580 Valeev, K. (1969). Linear differential equations with quasi-periodic coefficients and constant retardation of the argument.

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STABILITY OF LINEAR SYSTEMS

Erugin, N.P. (1941). A remark on Shifner's article. Izv.Acad.Nauk

SSSR Ser.Mat. 5,79-86

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SUBJECT INDEX

Abel-Jacobi-Liouville lemma, 73,102,159 Accumulation point,S Almost constant coefficients, 166 Almost periodic, 35 depending upon a parameter, 58,180 differential equations, 2a7 functions, 38 Alternating gradient proton synchrotron, 199 Approximate similarity, 97 Arithmetric means, 32 Arzela-Ascoli theorem, 13,73 Asymptotic almost periodicity, 64,209 Asymptotic characteristic values, 142 Asymptotic Floquet theory, 164 Asymptotic stability, 117,129, 150,205 for linear systems, 130 for liner periodic systems, 135 in the large, 154 uniform, 145

Banach space, 9 Beam stabilisation, 199 Berkeys theorem, 179 Bessels inequality, 29,48 Bochner almost periodicity, 62 approximation theorem, 52

Bohr almost periodicity, 39 transforms, 44 Bounded input-output stability, 161

Canonical forms, 98 Cauchy sequence, 6 Cesaro sums, 32,53 Characteristic exponents, 104 bounds, 110, I 12 estimates, 104,112 of almost periodic matrices, 109 of periodic matrices, 112 Characteristic determinant, 20 Characteristic multiplier, 92 Characteristic values, 88 Columns under periodic loading, 197 Commuting integrals, 109,1 I I matrices, 106 Compact spaces, 12,60 Complete Kinematic Similarity, 96 Complete set of invariants, 101 Constant coefficient differential equations, 85 Convergence almost everywhere, II in the mean, II of sequences, 11 pointwise, I I Contraction mappings, 13 Coppel-Bohr lemma, 167

234

SUBJECT INDEX

Coppels inequality, 25,142, theorem, 176

Decomposition of coefficients, lIS

Dinis condition, 32 Dirichlet kernel, 32 Dichotomies of linear systems, 135,214 Diophantine approximation, 51 Domain of attraction, 125 Dominance, 141

E-neighbourhood, 3 Eigenvalues, 88 Electrons in periodic potentials, 198 Equicontinuous sequences, 13 Equivalent norms, 12 Equivalence relation, 97 Erugins theorem, 101 Euclidean space, I Existence and uniqueness, 71, 74,207 Exponential dichotomy, 138,140, 214 Exponential asymptotic stability, 154 stability, 133

Fejer summability, 52 kernels, 33,53 Fixed point theorem, 14 Floquet theory, 90,164 representation theorem, 90 Fourier series, 29,36 coefficients, 28,49 of almost periodic functions, 37,43 Functional, 18 Fundamental matrix, 82,85

Generalised characteristic exponents, 89 Generalised Liapunov transformation, 96 Gershgorins theorem, 22 circles, 140 Gronwell-Bellman lemma, 77,147 149

Hamiltonian, 200 Hermitian matrices, 22 Hilbert spaces, 26 Hills equation, 93,194,201 Hull of a function, 63

Inclusion interval, 38 Induced matric, 17 norm, 17 Inherited properties, 209 Initial value problem, 70 Inner product, 26 Input-output stability, 160 Invariants, 98,101

Jordans condition, 32 canonical forms, 87 matrix form, 22 for Kinematic Similarity, 101

Kinematic Similarity, 95,97,203, 214,216 for periodic coefficients, II~ static matrices, 100

P

L -spaces, 10 Lagenhops condition, 194 Lagrange equation, 103 Liapunov function, 219 stability, 79,126 transformations, 96,105,119, 133,203

SUBJECT INDEX Linear differential equations, 79 independence, 8 operators, 16 transformations, 19 Lipschitz conditions, 73

Marcus-Yambe problem, 120 Mathieu equation, 93,194,196 Matrix invariants, 98 measure, 23,142 projection, 135,143 Mean values, 43,46 characteristic values, 110 Meissner equation, 93 Metric space, I Modal matrix, 21 Module containment, 4~,61,212 of quasi-periodic functions, 66

235

Reducibility, 95,214 Relative dense sets, 38,41 Residue matrix, 120 Right hand derivatives, 73

Schrodinger equation, 98 Set of exponents, 49 closure,S Similar matrices, 20 Solution spaces, 80 Spacecraft attitude control, 198 Stability of linear systems, 127 in the large, 117,153 nonlinear systems, 148 under disturbances, 155,157, 210 Static similarity, 101 Successive approximations, 74 Sufficient conditions for stab i. l i ty, 117,158

Non-resonance, 166,171 Normal spaces, 9 Normed metric spaces, 7

Ordinary dichotomy, 138,218

Parametric amplifiers, 196 Parsevals equation, 29,36,50, 165 Peano existence condition, 71 Pendulum with moving support, 194 Periodic differential equations, 90,207 solutions, 208 Picard existence condition, 74 Property L,P, 65

Quasi-periodic functions, 65

Topological space, 3 Total stability, 155 Translates, 62 Translation number, 38 operator, 62 Transition matrix, 116 Trigonometric polynomials, 37, 52 Twiss matrix, 201 Type number, 89

Uniform asymptotic stability, 132,146,151 bounded, 72 continuity, 4,60 convergence, 60 metric, 7 stability, 96,128,130 Unitary matrices, 22

236

SUBJECT INDEX

Variation of constants theorem, 85,204 Vector space, 7 Wazewski inequalities, 158

Weakly uniform asymptotic stability, 154,209 Weierstrass approximation, 33 Wronskian, 201