Stability of linear infinite dimensional systems revisited

Stability of linear infinite dimensional systems revisited

Citation for published version (APA):

Przyluski, K. M. (1982). Stability of linear infinite dimensional systems revisited. (Eindhoven University of Technology : Dept of Mathematics : memorandum; Vol. 8217). Technische Hogeschool Eindhoven.

Document status and date: Published: 01/01/1982

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•

EINDHOVEN UNIVERSITY OF TECHNOLOGY

Department of Mathematics and Computing Science

Memorandum 1982-17

November 1982

STABILITY OF LINEAR

INFINITE DIMENSIONAL SYSTEMS REVISITED by

K. Maciej Przytuski

University of Technology

Dept. of Mathematics & Computing Science P.O. Box 513, Eindhoven

, ,

STABILITY OF LINEAR

INFINITE DIMENSIONAL SYSTEMS REVISITED*)

by

.,

t

k' **)

K. Mac1eJ Przy us 1

Abstra~t. The paper is devoted to a study of stability questions for ... --...~-....,."..,

linear infinite-dimensional discrete-time and continuous-time systems.

The concepts of power stability and tP- stability for a linear

discrete-time sys tem ~+ 1 = A x

k (here xk € X, X is a Banach space, A is linear

and bounded) are introduced and studied~ Relationships between these

concepts and the inequality r(A) < 1 (r(A) denotes the spectral radius

of A) are also given. Next, the discrete-time results are used for a simple derivation of some well-known properties of exponentially stable

and LP-stable linear continuous-time systems described by !(t) - ~~(t)

(~ generates here a strongly continuous semi group of linear and bounded

operators on

X).

Some remarks on norms related to stable systems are

also included.

*>The final version of this paper was written while the author was visiting the Technische Hogeschool at Eindhoven.

**) _{Institute of Mathematics, Polish Academy of Sciences, Sniadeckich 8,}

, ,

2

-INTRODUCTION

The paper contains a concise study of stability theory for linear infinite dimensional discrete-time and continuous-time systems. Most of the results presented here are quite classical. The novelty lies mainly in the arrangement, and the proofs are new. In particular, it is shown that stability questions for continuous-time systems can be analysed by discrete-time methods. Thus, the intimate connections between the tra-ditionally separated theories of discrete-time and contiuous-time sys-tems are emphasized. Of course, such connections are known and they are more or less implicit in a number of works; see e.g. [10, Chap.II, Sec.3 and Chap. III, Sec.5]. It has been shown in [14] that discrete-time me-thods are useful in studies of asymptotic stability and LP-stability

of infinite dimensional continuous-time systems. We shall follow some ideas of this last paper.

The summary of the contents of the present paper reads as follows: In Section 1, we introduce some notations and basic definitions. In Sec-tion 2, we prove that tP-stabiLity and power stability of a discrete-time system ~+1 '" A~ (where A is a linear bounded operator on a Banach space X) are identical concepts. In fact, we prove that they are both equivalent to the condition r(A) < l;r(A) stands here for the spectral radius of A • The equivalence between the last condition and tP-stabi-lity was found by Zabczyk [16, Sec.S]. Among other things we prove that, for an Zp -stabZe discrete-time system~ two norms" 11·11 and

I-I

~ are

P

equivaLent. Here for alt x € X,

Ixl

:= (

p

co

I

UAkxUP)l/p k=O

" 1

.

,

3

-and II • II is the original norm on X. In Section 3, we consider continuous-time systems given by ~(t)

=

~~(t) (where

A

generates a strongly

-continuous semigroup {S(t)} + of linear and bounded operators on X).

t€lRQ

The concepts of exponential stability and LP-stability are defined. It is shown that exponential stability of a ~ontinuous-time system is equivalent with power stability of a (properly defined) dis~rete-time system. A similar conclusion holds for LP-stable systems. In this case a relation between two norms on X, ·11 • II and

I-I

,is given. Here

p p

II x U_p:=

(f

o

)

l/P

II S(t)xIlP dt

and

1·1

is defined as before, with A:= S(L) for some positive fixed~. p

The norms II • II and

I-I

~annot be (in general) equivalent; nevertheless

p p

it is possible to obtain (for every L) the following inequality:

Here a is a positive number depending on T.

~

The discrete-time results of Section 2 are used in Section 3 to obtain a simple derivation of a well-known (see e.g.[4],[6],[14]) reLa-tionship between exponential stability

and

LP-stability of a continuous-time system.

We note that in Section 1 a "frequency-domain" machinery to study discrete-time systems is developed and next used, in Section 2, to de-rive some characterizations of stable discrete-time systems. This, machi-nery can also be used to study other control-theoretic problems (see

·

,

4

-The importance of the paper is twofold: It provides a simple illu-stration of "frequency-domain" methods in infinite-dimensional context and, it proves that discrete-time methods simplify the theory of con-tinuous-time systems.

~c~p~e~m~t~ The author is indebted to Professor M.L.J. Hautus for many helpful suggestions.

- : t

5

-1. DISCRETE-TIME AND CONTINUOUS-TIME SYSTEMS. BASIC RESULTS

The main purpose of this section is to describe the classes of systems which we will consider. We shall be concerned only with the dis-crete-time systems which can be defined by linear difference equations and with the continuous-time systems which can be defined by linear differential equations in a Banach space. Moreover, the systems consi-dered will be autonomous, i.e., the operators specifying the appropriate difr~rence or differential equations will be constant.

The section starts with the definition of a linear discrete-time system. The topic next dealt with is the formal discrete Laplace trans-form and its usage to describe linear discrete-time systems. The section contains also a discussion of some spaces of formal power series and germs of holomorphic functions, which will play an important role in the next section. Lastly, the definition of a linear continuous-time system is given. The section concludes with a simple but crucial result relating continuous-time systems with discrete-time systems.

Let

X

be a Banach space and L(X) denote the set of all linear and bounded operators X + X. The norm on X will be denoted by Ii '11. For the set of all nonnegative integers we shall write~~ i.e., ~~:= {O,I,2,. • .}.

Let A € L(X) be fixed.

~ It is said that a Linear di8crete-time 8Y8tem6(A) is given, if the linear difference equation

~+l = A~

+

•

•

6

-k + 0

Trivially, ~

=

A

Xo

for all k €~o; here A := It the identity

operator on X.

Now we introduce some new concepts needed in the sequel.

Let L(X)[[z]] denote the ring of formal power series in the inde-terminate z with coeffiaients in L(X). Similarly, X[[z]] will denote

the left L(X)[[zJJ-module of formal power series in the indeterminate

z with coeffiaients in X. For a discussion of formal power series we refer the reader to [3, Chap.III, §2, no.ll] •

!XI

Let W:= {Wk}k=O be a sequence of elements of L(X) (of X). The

for-mal power series in the indeterminate z with coefficients in L(X) (in X)

defined by

is called the formal discrete Laplace transform of

w.

A standard but useful result (whose proof is included here only for completeness) is given by the following

t~~osit~o~~ Let a linear discrete-time systemG(A) be given

and

X:= {xJ~_o be a sequence of elements of X. Denote by X(z) the formal

discrete Laplace transform of

x.

Then

...

(I - zA) X(z) ,. x

o

(as an equality of formal power series) if and only if the sequence X satisfies the linear difference equation defining the linear discrete-time syst~m ~(A).

..

•

•

7

-~. Note that in the ring L(X)[[z]],

Hence, the considered equation (I - zA) X(z)

= Xo

holds if and only if

It proves that (I - zA) X(z)

= Xo

if and

k . +

~

=

A X_O' L e. t ~+ 1 = A ~ t k €?LO'

+

only if t for every k €?L

O'

o

It is of special interest for the sequel to study these formal power series which satisfy some additional requirements. Therefore, we shall consider all formal power series W(z):=

I

zkw

k

i~

the indeterminate +

k.e?L

O

z with coefficients in L(X) (or X) such that the following standard growth condition holds:

"" "" k +

There e:r:ist M ~ 1 and p, 0 :S P < I, such that II W

k" :S Mp for aU k €?L

o •

A formal power series which satisfies the standard growth condition will be called a Hurwitz formaZ power series. Simple calculations show that the set of all Hurwitz formal power series with coefficients in L(X) is a subring of L(X)[[z]]; we shall call it the Hurwitz ring of L(X). Similarly, all Hurwitz power series from X[[zJJ form a left module over

•

•

8

-Let

D

1:= {s €

el

lsi $ I} be the

aZosed unit disa of

e

and let

H(D1;L(X»

denote the set of all functions

°

₁ +

L(X)

such that for every f(') €

H(D} ;L(X»

there exist an open neighbourhood N

f of i\and a holomorphic function gfCo):N

f +

L(X)

so that f(o) is the restriction of gf(·) to

D

I. Addition and multiplication of functions from

H(D};L(X»

can be defined

in the standard way and

H(D};L(X»

becomes a ring; it is called the

ring

of

ge~s

of L(X)-vaZued

hotomo~hia

funations on

D_t• Similarly, replacing

L(X)

by

X,

we can define

H(OI;X),

This set is a left module over

H(D1;L(X»,

if addition and (left) multiplication by elements of

H(D1;L(X»

are defined as usual.

H(D1;X)

is called the

H(Dt;L(X»-moduZe

of germs of X-vaZued

hoZomo~hia funations on

0t'

For a more general

treatment of the concept of germs the reader is referred to [2, Chap.I,

§ 4, no. t ].

Let W(z):=

L

z~k

be an element of the Hurwitz ring of

L(X).

The

kE2Z+

o

well-known Cauchy-Hadamard formula for the radius of convergence of a power series enables us to define a function

""

Dl 3 s)-.+W(s):=

i:

skw

k E

L(X) •

k=O

Actually, as it is easy to see, the function belongs to

H(OliL(X».

More-over, simple calculations show that the resulting mapping from the Hur-witz ring of

L(X)

into the ring of germs of L(X)-valued holomorphic functions on Dl is a homomorphism of rings. The homomorphism is an iso-morphism of rings. Indeed, every element of

H(O);L(X»

is representable

(via the Taylor expansion at 0) by a (convergent on some neighbourhood of

°

_{1) power series which defines a Hurwitz formal power series; thus}

..

•

9

-the homomorphism considered is surjective. It is also an injective homo-morphism as follows from the principle of isolated zeros (see e.g.[7, Chap.IX, Sec.IJ). We shall denote the defined above isomorphism of the Hurwitz ring of L(X) onto the ring of ger.ms of L(X)-vaZued hoZomorphia funations on

D

1 by (j).

In a similar manner as above, replacing the Hurwitz ring of L(X) by the Hurwitz module of X, and H(D1;L(X» by H(D1;X), we Can define a bijective mapping ~ from the Hurwitz module of X onto H(D);X). As it is easy to see, the mapping is an isomorphism of the additive gpoup Of the Hurwitz moduZe of

X

onto the additive gpoup of the H(Dt;L(X»-moduZe of ger.ms of X-vaZued hoZomorphia funations on

D

_J•

Now, we Can observe that ~ is a bijective semi-ZineaP mapping (relative to the isomorphism (j» of the HuI'Witz moduZe of X onto the

H(D1;L(X»-moduZe H(D);X1; in other words, the ordered pair .(cp,l/I) of isomorphisms is a dimorphism of the Hurwitz moduZe of X onto

H(Dt;X).

Hence, the ordered pair (cp-l,lj/-l) of isomorphism is a dimorphism of the H(D1;L(X»-module H(D);X) onto the Hurwitz modu'A3 of X .. We shall not make the above statements more precise; the detailed definitions of semi-linear mappings and dimorphisms are to be found in standard text-books (see e.g.[3, Chap.II, §t, no.13J). Here we confine ourselves to

the simplest properties of the pair (cp,1/I) , which can be deduced directly from the definitions of cp and lj/.

Let us say that a for.maZ power series W(z) E: L(X)[[z]] (w(z) E: X[[z]])

defines a ger.m of holomorphia funtions on

D

_I, Le., an element of

H(D);L(X» (of H(D);X», i f the series is a Hurwitz formal power series. In this case the ger.m W(·):= cp(W(z» (w(·):= lj/(w(z») of hoZomorphia

"

.

•

- 10

-funations on 01 is said to be defined by the fonma~ power series W(z)

(w(z». Similarly, a genm of ho~omorphia funations on

n

_{l ,} W(·) E:

e

H(DliL(X»

(w(·) E

H(D

₁

;X»

defines a fonma~ power series by the

Taylor expansion at 0, Let denote the series by W(z) (w(z». It is a Hurwitz formal power series. We have W(z) -

~-l(W(.»

(w(z) m

~-l(w(.»

)

and the fonmal power series is said to be defined by the Taylor expan-sion (at 0) of the genm W(·) (w(·» of holomorphia funations on

D

_l,

Now, the basic properties of the dimorphism (cp,~) can be summarized as follows. If we have an equality (involving only well-defined sums and products) of formal power series which define some germs of holo-morphic functions on

D}

(i.e., this equality is an equality of Hurwitz

formal power series), the equality holds for the germs of holomorphic functions on

D)

defined by the formal power series. In other words, this equality holds for all s e

D

_l• Conversely, every equality (involving only well-defined sums and products) in

H(D);X)

or

H(D

₁

;L(X»

holds also for

the formal power series defined by the Taylor expansion (at 0) of the germs of holomorphic functions on O} occurring at both sides of the considered equality in

H(DI;X)

or

H(DI;L(X».

In accordance with the remarks above we note the following

~~~i!i0~J~~~ Let W(z)

e L(X)[[z]],

V(z) E

L(X)[[z]],

w(z) E X[(z]J

and v(z) e X[[z]] be given Hurwitz fonmal power series. Let W(·):= cp(W(z»

e H(D

₁;

L(X»,

V(·):x cp(V(z» E

H(D);L(X»,

w(·):= ~(w(z» E

H(D

₁

;X)

and v(o):_ ~(v(z» E

H(D1;X)

be the genms of holomorphia funations on 0l

defined by these fonmal power series. Then

.

, ,

•

- 11

-holds as an equality of formal power series if and only if

(**)

We

o) V(o) v(o)

=

w(o) ,

i.e., if and onZy if for all s €

iiI

(***) W(s) Ves) v(s) - w(s).

Conversely, let W(·) € fl(D

I ;L{X», V(o) € H(D, ;L(X», w(o) e H(D) ;X)

and v(o) € H{D};X) be given germs of holomorphic functions on

Dt"

Let W(z):= ,-I(W(e» € L(X)[[z]], V(z):= ,-I(V(·» e L(X)[[z]], w(z):=

w-1(w(o» e X[[z]J and

v(z):~

w-)(v(o» € X[[z]] be the formaZ power

series defined by the Tay lor expansion (at 0) of these germ'S of ho lo-morphic functions on

D

_t• Then equality (**) (i.e., equivalently, equality

(***» holds if and only if equality (*) takes place.

0

In the next section spectral properties of the operator A defining G(A) will be of some interest. Thus, we shall write cr(A) for the

spec-trum of A. Let rCA) denote the spectral radius of A, i.e., r(A):=

SUp{IAI IA E cr(A)}. We denote the following well-known (Beurling-Gelfand)

formula:

r (A) == lim (II Ak II) 11k •

k-+<X>

-1

Let RA(z):= (I - zA) € L(X)[[zJJ. An easy consequence of the

Beurling-Gelfand formula is the following

f.f.2.e2.U£.i£?llJ .}..:.

RA (z) defines a germ of holomorphic functions on

ii)

,

..

•

12

-Of course, D) denotes here the (open) unit disc of (.

Now we give some basic facts about continuous-time systems. Let ~

be a given linear operator: dom(~) + X with dom(~) c X. By dom(~) we

shall denote the domain of ~. Throughout the paper it will be assumed that ~ generates a strongLy continuous semigroup {S (t)} + of linear

tE1Ro

and bounded operators from

X

into

X.

In other words, ~ is the

infinite-+

simal generator of {Set)} + • As usual, :lRO denotes the set of all non-. tE:RO +

negative real numbers, ~.e. t :lRO := {t E :lR1 t ~ A}.

We shall study a linear differential equation !(t)

=

A~(t) for +

t € :lR

O' It is convenient to assume that the differential equation is satisfied in a weak sense. Therefore we shall consider a Cauchy problem with ~(O) not restricted to be in dom(~) but being from the whole space X. All solutions of the Cauchy problem are given by ~(t)

=

Set) !(O),

t €

01-:lRO' They are called the mil.d soLutions of !(t) '" ~!(t) , !(O) E X,

+

t e: _{:lRO' Details concerning the theory of linear differential equations} in Banach spaces are presented, for instance in [4J and [9J.

Now we are ready to make the following

~ It is said that a linear continuous-time system '*'(~) is given, if the mild solutions of the linear differential equation

:ie(t) = A x(t)

""

""-are considered for all t € :IR~ •

following (see also [)4J)

Using the semigroup property of {Set}} + we can easily get the tE:lRO

,

.

,

•

•

- 13

-""

~ Let T > 0 be a given number and X:== {xk}k_Obe a se-quence of elements of X. Assume A:= SeT). Then the sequence X satis-Neg the Unear difference equation x

k+ 1 == A xk for k €

ll~

i f and only

i f ~ .. ;!S(k'r) for aU k € ll~" where ;!S(t) .. S (t) ;!S(O) denotes the (mild)

solution of !(t) .. !;!S(t) , t € lR~ , with ;!S(O):= xO'

The above result describes a natural correspondence between con-tinuous-time and discrete-time systems. The correspondence can be used to study the problem of stability of continuous-time systems; it will be done in Section 3 •

•

14

-2. STABILITY OF DISCRETE-TIME SYSTEMS

This section provides an easily accessible and complete account of some of the most fundamental results in stability theory of linear discrete-time systems. We begin with the important definition of power stability. A theorem characterizing this notion is given. The presented proof of the mentioned theorem makes use of the spaces of germs of holo-morphic functions which have been defined in the previous section. Next, the concept of i.P-stability is introduced and studied. It is shown that

SeA)

is lP-stable if and only if the spectral radius of A is less than 1.

Whereas the result is (at least for p = 2) well known, the proof given in the text seems to be new. It is based on the fact that if rCA) is less than 1, one can find a new norm

1'1

on X which is equivalent to

the origi~al norm of X and such that IAI is less than 1. As a consequence

of the obtained results we note that power stability and i.P-stability are equivalent notions. Some remarks supply the main text.

The concept of power stabiltity (which will be defined below) is a discrete-time counterpart of the more common notion of exponential stabiltity of a continuous-time system. Thus, the following definition is quite natural.

R.~ij.Jlj.ti2..n....kJ~ A linear discrete-time system@(A) is said to be POWBF

stable

if for all

Xo

€

X

there exsist M ~ I and 0 s r < 1 such that

+ QO

for all k € ?l0' Here {~}k=O is defined by ~+1 • A~, k e: ?l0' Le.,

k

.

;

•

•

15

-The theorem given below links the introduced above concept of power-stability and spectral properties of the operator A E L(X) which defines ~A), More precisely, we have

~ Let a lin.ear' disarete-time system <!>(A) be given. The sys-tem is pOlJer stabZe i f and only i f o(A)" the speatrum of A" is aontain.ed in the unit disa D} of ¢" i.e., the in.equaZity rCA) < 1 takes pLaae •

Proof. LetQg(A) be power stable. Then» as easily can be checked, the "'-_""" -rw

linear operator (I - sA) is injective for all s E

D1'

We wish to show

that it is a surjective operator. For this, let X(z) be the formal

La-~ +

place transform of a sequence X:= {~}k;O where, for every k €~O' ~ is defined by the difference equation ~+l .. Ax

k • In accordance with Proposition 1.1, we have the following equality of formal power series:

(I - zA) X(z) .. Xo •

Having assumed~(A) to be power-stable we can note that X(z) defines an element of H

(D

l ;X). Trivially, the formal power series (I-zA) defines an ele-ment of

H(D

₁;L(X».Thus by Proposition 1.2, for all S E Dl we have

Of course, we have the same equality as above for all Xo E

X.

Hence, (I - sA) is surjective for all S E

D

l , Now the desired inclusion o(A) C DI is obvious,

To prove that rCA) < 1 implies that<;(A) is power stable note that,

•

- 16

-holomorphic functions on D

1. Thus, for all Xo E

X,

RA(z)xO is a Hurwitz formal power series, i.e., for every Xo E

X

there exist M ~ and r,

o

s: r < 1, such that

n

~ 11(= II A k Xo II) s: Mr k for all k E ZlO • +

~_~ It is possible to give a simple direct proof of the fact that the condition rCA) < 1 implies power stability of

SeA).

However, we feel that the reasoning given above is more natural in the context

of the rest of the proof of Theorem 2.1 •

In order to state an important lemma we need to define lP(X) for

s: p < ~. Thus, by lP(X) we shall denote the Banach space of all

se-~ +

quences {~}k=O such that for all k E~O' x

k € X and the sequence of

;pOI) ~ P

numbers {II~ Ir \ ... 0 is summable. The norm of X:= {~}k""O E l (X) is IIxU

lP

We have the following

k.~!P!l1~-b t~ The conditiona below are equivalent:

k ~ p

(i) For every p~ 1 s: p < OI)~ and for every x E X~ {A x}k=O € l (X),

o

1- "" p

(ii) There exists p~ 1 s: p < co~ such that for every x € X, {A "'x}k=O E l (X).

(iii) The spectral radius of A~ r(A)~ is less than 1.

(iv) There exist M

~

I and 0

s:

r < 1 such that II Ak II

s:

Mrk for every

~ The only nontrivial step is to show that (ii) implies (iii). As it is well known, for any given € > 0 we can find a new norm

I -I·

on X

•

17

-so that the inequality

IAI

s rCA) + E: takes place and the norms II· II and

I-I are ~quivalent. Since rCA) is not greater than IAI, we shall try to find a new norm

1·1

(equivalent with II· II) such that the inequality

IA[ < 1 will hold.

Assuming (ii), we have fixed a number p, I

s

P < ~, such that for

k ' " P

every x €

X,

{A x}k_O €

l (X).

Let

It is a well-defined norm on X. In fact, X equipped with the norm

I-I

p

is linearly isometric with the linear subspace LA of lP(X), where

We want to show that LA is closed in lP(X). To prove this assertation assume that X:= {~}~=O € £.P(X) is in the closure of LA' Then there

k,.,'

lim II A xJ - x. II = O. For k = 0 we

j-+o:> it

such that lim II xj - X II == O. Of course,

j-+o:>

+ In particular, for all k € 2Z0 '

have lim

"i

j -

Xo

II

=

O. This equality

j-+o:>

k,... k

implies lim "A xJ - A

Xo

II == 0 for every k

j-+«> + € 2Z0 • Now, k k k , . , ' k ' k k.... .• II A Xo -

~"

.. II A Xo - A xJ + A

x

J -

~

II

s

II A Xo - A -xJ II + I ' l' k +

t 1mp 1es that ~ == A

Xo

for all k €2Z 0' is a closed subspace of lP(X).

•

- 18

-To end the proof note that for every x € X we have the obvious

in-equaltiy II xII :s; Ixl • By the interior mapping principle (see e.g.(S,

p

Chap.II. Sec. 2]) there exists a positive number a such that, for every

X € X, Ixl s allxll. Evidently, it can be assumed that a is greater than 1.

p Now, we have

IAxI

-Ixl -lIxliS Ixl -a-1Ixl - (l-a-1)lxl •

p p p p p

The above inequality implies IAI < 1 and, in consequence, r(A) < 1.

P

o

Remark 2.2. There is another interesting way to prove Lemma 2.1. It

pro-~

ceeds as follows. Assuming (ii) ,we have that S : X + lP(X) ,

Sx:-is a well defined linear mapping. It can be checked that S Sx:-is a closed operator. Since the domain of S is the whole space X, S € L(X) by the closed graph theorem. Thereby, there exists a positive number a such that, nSxll S all x II. Since IISxll ... lxi, we get Ixl S allxll, i.e., the

lP lP p p

same inequality as in the final part of the proof of Lemma 2.1. It should be noted that the calculations which prove that S is a closed operator are exactly the same as these given in the proof of Lemma 2.1 to show that LA is a closed subspace of lP(X).

Use, in a similar context, of an operator which plays an analogous role in a continuous-time system theory as S for discrete-time systems has been suggested by A. Pazy (see(12, p.293]).

The concept of lP-stability (which will be defined below) is a dis-crete-time counterpart of the widely used notion of LP-stability of a continuous-time system. For p = 2 and X being a Hilbert space, the im-portance of the notion of iP-stability is well recognized; see e.g. (11J,

•

•

- 19

-~ Let p, I $ P < ~, be given. A linear discrete-time

sys-tem@(A) is said to be .e.P-stabZe if for all

Xo

€ X the sequence

belongs to .e.P(X). Here

{~}~=o

is defined by

~+l =~,

k

€~~

k

~ 101 A xO'

, i.e.,

As a trivial consequence of Lemma 2.1 and the definition of .e.P-stability we note the following

Jlte.2.t~kk The oonditions beZow are equivaZent: (i) For ever,y P, 1 $ P < ~, &(A) is tP-stable.

(ii) There exists p,} $ P < ~, such that

&

(A) is tP-stabZe.

(iii) The speotraZ radius of A, rea), is Zess than I.

(iv) There exist M

~

) and 0 $ r < I suoh that IIAkU $ Mrk for ever,y

+

k € ~O'

~~r~k3!. The equivalence between .e.P-stability of @;(A) and the inequa-lity rCA) < 1 has been established by J. Zabczyk (see [J6, Sec.S]). His proof of the result is based on a technique quite different from ours and makes use of the Banach-Steinhaus theorem. It seems that the method presented in the proof of Lemma 2.1 is not only simpler, but also a more natural one. However, it should be noted that Zabc~yk was able to prove a more general stability result.

By Theorems 2.1 and 2.2 we immediately get

~~~r~~ Let a Zinear disorete-time system ~(A) be given. It is power stable

if

and on7,y if it is .e.P-stable for some (for ever,y;' p,

o

•

20

-~ It is unnecessary to use the whole power of Theorem 2.1 to

get the above corollary. In fact, a simple direct calculation shows that power stability implies lP-stability. Hence, by Theorem 2.2. power

sta-bility implies the inequality rCA) < 1. Thus. to get Corollary 2.1 it

is sufficient to use only the part of Theorem 2.1 which says that the

inequality rCA) < 1 implies power stability. Note that the above

reason-ing provides a proof of the remainreason-ing (and more difficult) part od Theo-rem 2.1. However, the proof of TheoTheo-rem 2.1. presented in the paper has its own value. The "frequency-domain" method suggested by this proof can be used to study various control-theoreticquestioos. In particular, a solution of the so-called stabilizability problem can be obtained by

use of the mentioned method; see e.g. [IS].

The proof of Lemma 2.1 shows that the corollary below is true.

~ Let a Zinear discrete-time system~(A) , which is lP-stabl~,

be given. Then the system defines a new norm

1·1

on X which is given

p

(for

aZZ

x € X) by

Moreover, the norms, II· II and

1·1 ,

_p are equivaZent.

~~~1....?-,,1~ The statement of Corollary 2.2 is rather surprising in view

of a result by A. Pazy. He showed (see [12, Thm.2]) that, without

additonal assumptions. an analogous result for continuous-time ~ystems

is faZse. More precisely, it is shown in [12) that the following result

, ,

•

21

-holds: Let {S (t) } + be a stl'ongly oontinuous semigroup of Unea:l' and

tEE.

bounded .operators from X into X, I ~ p < 00 be fixed, and

00

((

)l/P

nxl~:;

\J

"S(t)xl~ dt

o

be finite for every x E X. Then II ~ II is a new norm on X. The norms" II· II

p

and II • lip" a:l'e equivalent i f and onl.y i f there exist

to

> 0 and

Co

> 0

such that

for aU x E X. To prove that the last inequality is necessary for II' II and II • II to be equivalent norms, some estimates of the norm II • II are

p p

+

made in [12J. They involve an analysis of small time intervals,[O,\] S ]RO' Clearly, we are not able to make something similar to it, if discrete-time systems are considered. This is a reason for which our Corollary 2.2 holds.

•

•

22

-3. STABILITY OF CONTINUOUS-TI~m SYSTEMS

In the previous section we presented stability theory for linear discrete-time systems. It turns out that this theory provides a very natural approach to the study of stability problems for continuous-time systems. These problems are to be studied in the present section.

We start with the well-known definition of exponential stability of a linear continuous-time system. Our first theorem of this section proves that exponential stability of a continuous-time system is equi-valent with power stability of a suitable discrete-time system. Next,

the important concept of LP-stability is defined. In order to study

LP-stability we derive some inequalities between some norms on

X;

the

norms are related to a discrete-time system. A theorem is given that expresses the fact that LP-stability of a continuous-time system can be explored by an analysis of tP-stability of a discrete-time system. Using Corollary 2.1 we get equivalence between exponential stability and LP-stability for linear continuous-time systems.

A number of papers have been devoted to stability theory for linear

continuous-time system; we mention only (1],(4],[S],[6],[J2] and [13].

We begin with the following

Definition ~-- -"""~~ 3.1. A _ _ _ w linear continuous-time system ~(A) is said to be exponentiaLLy stabLe if for all ~(O) € X there exist ~ ~ I and! < 0

such that

·

,

23

-+

for all t E ~O • Here ~(t) is the corresponding to ~(O) (mild) solution

of js( t) = ~(t), t E ~O' + i.e

a,

~(t) ... Set) ~(O).

In order to prove Theorem 3.1 (to be given below) we need some

no-tations. By T we shall denote a given positive number.

Now, let

a. := sup{IIS(0) II

Ie

E [O,T[}.

T

It is well known that a. is finite. (The result follows from the

prin-T

ciple of uniform boundedness [8, Chap.II, Sec. IJ and strong continuity

of {S(t)} + .) We shall also use the notation [ t \ for the entire

tE~o +

part of t/t. Note that t E ~~ implies [tJ

r € llO' Let 0r € [O,r[ be

given by

o

:= t - T[t] •

T • T

We will prove the following

~ Let a Linear continuous-time system~(A) be given and T be a fized positive number. Let A:= S(T). Then the continuous-time sys-tem~(~) is exponentially stabLe if and onLy i f the discrete-time system ~(A) is power stable.

f.!.<?2.t ...

Using Proposition 1.4 we have the equality ~ ... !(k(T» for all

+

k Ell

O; here !(t)

=

Set) ~(o) is the (mild) solution of !(t)

=

~(t),

t €

~~,

and

~+l

""

~,

k E

ll~

, with :5(0) == xO'

Assume ~(~ to be exponentially stable. Then, the inequality 1I~(t) II

S

t!

exp (r t) is fulfilled for all t E ~~ and some

t!

~ I, ! < 0.· Since the

24

-holds for all k e~~, the condition of Definition 2.1 is satisfied with M:- ~ and r:= exp(r t) • The evident inequality

o.

~ exp(t t} < 1 proves that (C(A) is power stable.

Conversely, let~(A) be power stable. Note that

lIas(t) II ... "as(t[t], + 0,) II =- _IIS(0,)xCt] II S a-rllx[t] II •

T ,

Now the power stability o£~(A) enables us to write the inequality

+

which is valid for all t e: lRO and some M ~ 1, 0 S r < I. We may assume +

that r :{: O. Then, for all t e: lRO '

II as(t) II S

M

exp(t t)

where ~:= (a,/t)M and r:= (11,) In(r). Clearly, the inequalities ~ ~ 1 and I < 0 are satisfied, and~(A) _{,... ,...} is exponentially stable.

~e~~k~~l~ The proof presented above suggests a simple way to prove the following well-known result: ~(k) is exponentiaLLy stabLe i f and

+ onLy i f the1"e exist ~ ~ 1 and ! < 0 suah that, fop aU t e: lRO '

II Set) II S M exp(r t). To show the nontrivial part of this result it is ,.. ,..

sufficient to note that, by Theorem 3.1 and Corollary 2.1, we may use condition (iv) of Theorem 2.2 for A:= Set). Then

o

[tJ [tJ

IIS(t)II=IIS(T(t] +0)1I=IIS(0)A 'II Sa ... Mr 'tsMe:xp(!t)

t T t •

.-with ~ and! expressed by the same formulas as in the final part of the proof of Theorem 3.1. Hence, the desired estimate of II Set) II is obtained.

25

-The important co~cept of LP-stability is defined as follows.

~ Let p, 1 S P < "" be given. A linear continuous-time system~(~ is said to be LP-stable if for all ~(o) €

X,

(

J

1I~(t)U P dt is finite.

o

Here *-(t) is the corresponding to ~(o) (mild) solution of ~(t)

=

+

Q(t), t € lRa' Le., 1f,(t) .., Set) ~(O).

The proposition and theorem given below are a slight generalization of [14, Prop.9J. The notations are exactly the same as introduced be-fore Theorem 3.1.

~ Let a linear continuous-time system ~(A) be given and~

for. a fixed pO$itive number T~ A:= S(T). For a given x € X, tet {xk}:=O

denote the corresponding to xO= x solution of the differenae equation . whiah desaribes the disarete-time system <S(A)~ and Zet /S(t)J1 t € lR~ ~

be the aorresponding to ~(O)

=

x (miLd) solution of the differential equation whiah defines

~(A). Then~

the numbers

f~~(t)

uP

dt and

I

U

~

uP

o

k-O

are simuLtaneousZy finite or infinite. Moreover~ the foUowinfj inequaUty

holds. Here p~ 1 s P < ""~ is a fixed number.

An immediate consequence of Proposition 3.1 is the following inte-resting

.

'

26

-~ Let a l.inear continuous-time sysi:em <£(~) be given and • be a fix~d positit'e number. Let A:= S ('t). Then the continuoU8-time system

~(~)

is LP-stabZe i f and only i f the discrete-time

system~(A)

is

lP-stabLe. Here p, I ::;; P <

al,

is a fixed number.

Proof of Pro os'tion 3.1. From Proposition 1.4 we have the equality

+

E llO' Now, note that :set) == S(8.) S([tJ.) :s(0) ...

o

~ == :s(kc) for all k

[d

t S(8 ) A x. Hence, T[d for all t

E lR~, we have the inequality (I/a

T)P llx(t) liP

0> ::;; IIA

·x

liP. Assuming that

L

II x

k liP is finite and noting that

k=O

we get immediately the desired inequality

...

O/at)p

f

"as(t) liP dt

~.

I

II~IIP

•

o

k=O

Clearly,the number

r'n~(t)

liP dt is finite.

o

To prove the rcmai.ning part of Propos i don 3. 1 note that for every

t E:

:R~

we can write the following equalitY:T([t\ +

Noting that (. - aT) E [O,T[ we get for all t E lR~

co

1) == (T - 8 ) + t.

•

•

Now, assuming that

J.II~(t)

liP dt is finite we have the inequality

o

a:

J

1I!(t) liP dt.

27

-co

Evidently, the number

l

lI"kUP is finite. k=O

~ Let ~(~) be LP-stable and A:= S(T). Let.for every

x

€

X,

IIxll := P

( J

o

) l/P IIS(t)

xu

P dt •

It is a new norm on X. Using Proposition 3.1 and Theorem 3.2 we have the following inequality

between the norms II • II, 11·11 and

I-I .

The last norm depends on T. The

p P

inequality above suggests the following question:

o

Are the norms II'" and

I-I

equivaLent norms on X? To answer this

ques-P p

don note that by Corollary 2.2, the norms 11'11 and II

-n ,

are equivalent.

p

On the other hand, a result by Pazy (see our Remark 2.5) says that the norms 11'11 and II -lip are equivalent i f and only if there exist to > 0 and

Co

> 0 such that IIS(tO)xll ~ call xii for all x € X. Thus the following

re-sult holds: For a given LP-stabLe aontinuous-time system ~(A), the norms

II • lip and

I-I

P are equival.ent i f and onLy i f there exist to > 0 and Co > 0

suah that

for all x € X.

As a consequence of Corollary 2.1, Theorem 3.1 and Theorem 3.2 we note the following well-known (see [61[12J and [14])

28

-~ Let a linear continuous-time system ~(~) be given. It is

exponentially stable

if

and only if it is LP-stable foY' some (foY' every)

29

-!

References

~

[1] A. Bensoussan, M.C. Delfour and S.K. Mitter. Representation and Control of Infinite Dimensional Systems. Forthcoming monograph, M.l.T. Press.

[2] N. Bourbaki, Theories Spectrales, Chapter 1-2. Hermann: Paris, 1967.

[3] N. Bourbaki, Algebra I, Chapters 1-3. Reading: Addison-Wesley, 1974.

(4] R.F. Curtain and A.J. Pritchard, Infinite Dimensional Linear Systems

Theory. Springer-Verlag: Berlin 1978.

[5] R. Datko, "An extension of a theorem of A.M. Lyapunov to semi-groups of

operators", J. Math. Anal. Appl., vol.24, pp. 290-295, 1968.

[6] T. Datko, "Extending a theorem of A.M. Lyapunov to Hilbert space", J.

Math.Anal. Appl., vol. 32, pp. 610-616, 1970.

[7] J. Dieudonne, Foundations of Modern Analysis. Academic Press: New York,

1960.

[8] N. Dunford and J.T. Schwarz, Linear Operators, Part I. Wiley-Interscience:

New York, 1966.

[9] G.E. Ladas and V. Lakamikantham. Differential Equations in Abstract

Spaces. Academic Press: New York, 1972.

[10J P. Lax and R. Phillips, Scattering Theory. Academic Press: New York,

1967.

[11] K.Y. Lee, S.-N. Chow and R.O. Barr, liOn the control of discrete-time distributed parameter systems", SIAM J.Contr., vo1.lO, pp.361-376, 1972.

30

-[12] A. Pazy, "On the applicability of Lyapunov' s theorem in Hilbert space", SIAM J. Math. Anal., vol.3, no.2, pp. 291~294t 1972.

[13] A.J. Pritchard and J. Zabczyk. "Stability and stabilizability of infinite dimensional systems", SIAM Review, vol. 30, no.1, pp. 25-52, 1981.

[14] K.M. Przy.!uski, "The Lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space", Appl. Math. Optim., vol. 6, no.2, pp. 97-112, 1980.

[15] K.M. Przyluski. to appear.

[16J J. ZabC!zyk, "Remarks on the control of discrete-time distributed para-meter systems", SIAM J. Contra vo1.12, no.4, pp. 721-735, 1974.