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


-

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



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>+!

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,12

~

Ilx11

2

(1.35)

,

which is known as Bessels inequaZity. to show that for the space (X,1

II xii.

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


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 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.




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


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
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


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->=


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,

(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


+ 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

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)!}


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
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

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
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

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


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


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

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