A WEAK INVARIANCE PRINCIPLE AND ASYMPTOTIC STABILITY FOR EVOLUTION

intcrnat. J. Math. & Math. Sci.

VOL. 18 NO. 2 (1995) 255-264

255

A WEAK INVARIANCE PRINCIPLE AND ASYMPTOTIC STABILITY FOR EVOLUTION

^EQUATIONS

WITH BOUNDED GENERATORS

By

E. N. CHUKWU

Department

of Mathematics Box 8205, North Carolina State University Raleigh N. C.27695-8205 U.S.A.

and

P.SMOCZYNSKI

Department

ofMathematicsand Statistics Simon

Fraser

University Burnaby B.C. V5A 1S6 Canada

(Received October 7, 1991 and in revised form April 6, 1993)

KEYWORDS Asymptoticstability,invarianceprinciple,

Lyapunov

functions AMS SUBJECT CLASSIFICATION CODES 34C11,34C20,36G20.

ABSTRACT.

If V is a

Lyapunov

function of anequation du/dt

u’

Zu in a Banach space then asymptoticstability ofanequilibriumpointmaybeeasily provedif itisknown that sup(V’) <0 on sufficientlysmallspherescenteredattheequilibriumpoint.

In

thispaper weak asymptotic^stabilityis proved for abounded infinitesimalgenerator

Z

under aweaker assumption V’

<

0 (which alone implies ordinary stability only)ifsome observability condition,involving

Z

andtheFrechetderivative of

V’,

is satisfied.Theproofisbasedonanextension of LaSalle’s invarianceprinciple,whichyields convergence ina weaktopologyand uses a stronglycontinuous

Lyapunov

funcdon. Thetheory is illustrated withan example ofan integro-differential equation ^ofinterest in the theory ofchemical processes.

In

this case strongasymptotic stabilityisdeducedfrom the weakoneandexplicitsufficient conditionsfor stability aregiven. In the case of a normal infinitesimal generator

Z

in a Hilbert space, strong asymptotic stability is proved under the following assumptions Z*+

Z

is weakly negativedefinite and Ker

Z

0}.Theproofisbased onspectraltheory.

256 E. N. CHUKWU ^{ANI) ).} SMOCZYNSKI

Section 1. Introduction

Awell-known theorem of A.M. Lyapunov see 4 or 11 forexample,assertsthat zero in R isanasymptoticallystable equilibrium point of an evolution equation

--(t)du Zu(t) t>0 (1.1)

dt

if V(x)>0 for x

:

0 V(0)=0 and

V’(x)=(Zxlc3V(x))

< 0 for x:0 (1.2)

where V R --> R is a suitable

Lyapunov

function and 0V is its Frechet derivative notation is further explained in the beginning of Section 2 asymptotic stability is recalled in Section3 ).R. Datko in 5 and 6 extended A. M.

Lyapunov’s

theory^toinfinite-dimensional Banach spacesunder a stringentassumption

W’(x) ZxI/)W(x) < -ct( Ilxll (1.3)

where ot 0,^+o,, ---> 0,^+,,o or(x) >0 for x>0 and ct(0) 0, see alsoremarksandpapers quotedin 18 ). If V(x) xlGx ), where G isasymmetric operator, then the above inequality means strictnegativedefiniteness ofthe operator Z*G +GZ,where Z* is the adjoint of Z.One is compelled^touseassumptionsas thisbythefact that boundedsetsin a Banach spaceareprecompactif andonlyifthespace ^isfinite-dimensional, see 7 However,if onemanages ^{tofind a} Lyapunov function that satisfies a weaker condition

V’(x)=(ZxlOV(x))

> 0 for x 0 (1.4)

then

Lyapunov’s

theoryyields only ordinary stability evenin thefinite-dimensionalcase 4].

Essential information on asymptotic behavior of solutions under (1.4) is given in LaSalle’s invarianceprinciple,see 13],accordingtowhichall thesolutionsconvergetothe maximalinvariant subset M of R x V’(x) 0 }.Todaythisprinciple^{is a}standard tool of investigation of asymptotic stabilityinBanachspaces,see 12], 11 I, 18] (thecontext of waveequation)^and 9 ], 2 ], 3 (thecontext of functional differentialequations).

Using LaSalle’s invadanceprinciple J.P.Miller and A.N.Mitchell,see 14 provedin 1980 anextremely interesting^fact stabilityfor a linearsystemin afinitedimensional caseunder (1.4) isequivalentto observability of the pair (Z,

OV’

), observabilityis explained atthe beginning of Section3 ).This ideaprovedfruitful intheory of retarded differential equations, see 21, 3

Strong

compactness oftrajectories is characteristic for waveequations, functional differential equations and parabolic equations.Thisisrelatedwiththe fact that the generators are unbounded and in thisaspect bounded generators,withweak damping,arenoteasiertotreatthanunboundedones if compactness of the resolvent is lost thenonlyweakasymptotic stabilitymay be obtained. However, strong asymptotic stability may be easily deduced from the weak one if strong compactness of trajectoriesmaybeproved independently ofcompactnessoftheresolvent.

Evolutionequations (1.1) inBanach spaceswith a linearbounded infinitesimal generators

Z

areconsidered inthispaper.Weakasymptoticstabilityisprovedwiththehelpof a

Lyapunov

function withnon-strictly negativetime derivative. Section2 dealswith omega-limit ^sets.Itisshown that the LaSalle’s maximal invariantsetcoincides withthe unobservablespaceofthe pair Z,/)V’ and the weak invarianceprincipleisproved.Weakasymptoticstabilityisprovedin Section3 if,moreover, the pair Z,/)V’ isobservable.

By

means of asimple exampleit isshown that observabilityis not a necessarycondition ofweak asymptotic stability. The theory is applied to equations ^{with Volterra} integraloperatorsintheright-handsidesthatarise inthetheory of chemicalprocesses,^see ]. For this type of operator thekernel of /)V’ may beofcodimensiononeand observability is unexpected.

Simpleconditions onthe kernel of the integral operator that ensure strongstabilityare stated. Normal operators in Hilbert spaces are considered in Section 4 without the technique of invariant sets and compactness arguments It is also shown with the help of an example that under weak damping considered in thispapertrajectories tend toequilibrium

mucfa

more slowly thanin the case ofa strong

NEAK INVAR1ANCE PR[NCIPI,E AND ASY?,IPTOTIC STABILITY 257

damping correspondingto(1.3). R.Datkoprovedthatforstrictly negativedefinite generators in Hilbert spaces trajectories tend exponentiallyto zero. It isproved in [21] that exponentialweak asymptotic stabilityimplies exponentialstrongstability (the adjectiveexponential is omitted in[21]).

Section2.Weakinvarianceprincipleandnon-observability.

Weconsideralinearevolution equation

--(t)du Zu(t) > 0 (2.1)

dt

where Z B^---) B isaboundedlinearoperator in a Banach space B. See 16 for existence and uniquenessof solutions and for theproofthat u(t) eZ(t-t0)(u (to))foreverysolution u andall^t,to:,0.

We shalluse frequentlythe following notation. B is a Banach space over the field C of complexnumbers, B* its adjointspaceand (. I. Bx B* C the dualitymapping. If x B then its normisdenoted as Ilxll. We assume that (ZlX Izzy) ^=zlg2(xly) for all

z,z

₂ C and all x B, ye B*, where the bardenotes conjugation. A sequence z_n n 1,2 B converges weakly to z** B if limn._,**(z z** y 0 forall y B* and this will be written shortly as z** w-limn._,**z_n

Definition2.1.Let ^u 0,^+o,, B.Then x B belongstothe weak omega-limitset

of

^u,

shortly x wo)ls(u) if

liminft._,,,,,l(u(t) x y)1 0 forall y B* (2.3)

andbelongstothe strongomega-limit set

of

^{u,shortly x smls(u) if} liminft_,**llu(t) x 0.

Weneedtwolemmas in theproofof the invarianceprinciple.

Lemma

2.2.ONOMEGA-LIMIT SETS.Let u

C(

(0,+o,,),

B

bea boundedsoh,tion

of

^an

evolution equation (2.1) withabounded

infinitesimal

^generator

^Z

^{in a}

reflexive

^{Banach.space B.}

Then wo)ls(u) isa non-emptyweaklycompact invariantsubset

of

^B,so)Is(u)isa non-empty strongly compact invariantsubset

of

^Band so)Is(u) wo)ls(u).

The proof ofthis lemma is omitted since it is similar to the proofs of the corresponding statements in 11 and 18 ]. The second lemmaisquiteelementaryanditsproofis omitted,butit supports anessentialstep of theproofof the invarianceprinciple.

Lemma

2.3.

If

^f

cl((0,-o),

0,-t-oo f’ df/dt<0in O,+,,o and f" isunifirmly continuous in (0,+o,,) then limt_,**f’(t)=0.

In order to formulate precisely the invariance principle we recall the following notions.

Abounded linear operator A B B* in a Banach space

B

is called synmetric if x

Ay

y Ax

)-

forall x,y B,wherethe bar denotes conjugation, hence x

Ax

isreal.

A symmetric bounded linear operator A is called weakly negative (positive)

definite

^if y

,y

< >_ 0 for all y B,andiscalled strictly positive if y

Ay

^_>

II

y ² for all ye B

Let G B

--

^{B* be asymmetric bounded linear operator. Let u}

^C((

^0,+**),

^B

^be

asolutionof(2.1) andlet V(x) xlGx Then thetimederivative of V(u(t)) maybeexpressed asfollows

dV(u)

V’(u(t)) (t) (u(t) (Z*G + GZ)u(t) (2.5)

dt

where Z* B*--->B* istheadjoint of Z. V(x)^--)R forall x in

B

bysynmaetry of G,similarly V’(x) R hence V B ---> R mayserveas a

Lyapunov

function.

The major advantage of the result below over Theorem 2.3 in 18 is that the

Lyapunov

functionmaybeonly stronglycontinuousand needsnotbeweaklycontinuous.

Theorem 2.4.WE,d( INVARIANCE PRINCIPLE.Let u

CI((0,+,,,,), B

be a solution

of

^(2.1)

with a bounded

infinitesimal

^generator

^Z

^{in a}

reflexive

Banach space

B Let

G:B B*

bea symmetricboundedstrictlypositivelinearoperatorsuchfhat the operator Z*G + GZ

B --

^{B* is}

258 E.N. CHUKWb ^ANI) P. S>IOCZYNSK[

weaily negative

definite.

^{Let M be the maaimal invartant set}

of

^{(2.1) contatned tn}

Ker(Z*G+GZ).Then u(t) approaches lVl in thefollowingsense

inf{ liminft_,,,,l/u(t)-xly)l x M }=0

for

^all y B* (2.6)

Prmf Let us te V(x) x Gx as a Lyapunov function. Then

u(t)I1 < V(u(t) <V(u(0) <_{+ since} V’(u(t) < 0 by (2.5) and weaknegativedefiniteness f Z*G+GZ,so u(t) isbounded. All theotherassumptionsof Lemma2.2 are also atisfied hence wtols(u) isa non-empty andwe’,ffdycompactnvariant set.If we01s(u) Ker(Z*G + GZ) then (2.6) followsfrom (2.3) since wools(u) M bythe definition of M and invariance of wools(u) Therefore theproof^willbecomplete^{if we show}that wools(u) Ker(Z*G + GZ).

So,let x w-limn._,,,u(tn) wools(u) Using (2.5) we obtain thefollowing identity

V’(u(tn)-X)=(

u(tn) I(Z*G+GZ)u(tn) ^{+ x}I(Z*G+GZ)x

-( u(tn)l(Z*G+GZ)x

(xI(Z*G+GZ)u(tn)

=V’(u(tn)) ⁺ V’(x)

2Re(u(tn)

I(Z*G+GZ)x)

The left-hand side isnon-positive bythe weaknegativedefinitenessof Z*G +GZ The firsttermin theright-hand^sideconvergestozeroby Lemma (2.3),for V’(u(t) is anon-positive and uniformly continuous function of t, thanks to boundedness of G and Z. The third term converges to

2Re(x (Z*G+ GZ)x =-2V’(x), where Re denotes thereal part.Thustaking limsup of bothsides weobtain that 0 >_ V’(x) 2V’(x) -V’(x) hence V’(x) > 0. This,togetherwith V’(x)<0, which followsfrom (2.5) and theweaknegativedefiniteness of Z*G +GZ implies V’(x) 0.

Letusconsiderthe following functionof arealvariable g(s)=V’(x+ys)=(x+ysl(Z*G+GZ)(x+ys))

where x,y B arefixed.Since V’ takes only non-positive values by theweak negativedefiniteness of Z*G + GZ then g<_ 0.Itwasshown above that g(0) V’(x) 0, thus g attainsitsmaximum value for 0, hence 2Re( y (Z*G+GZ)x g’(0)=0. Replacing y with

/(-l)y

weobtain in a similarway 2Im( y Z*G + GZ)x 0,therefore y Z*G +GZ)x 0, whichimpliesthat (Z*G+GZ)x 0, for y may be any elementof B.Thuswehave showed that wools(u) is asubset of Ker(Z*G+GZ).

Q.E.D.

Remark.If B isseparablethenformulas (2.6) and (2.3) maybegivena metricformas in 18,Corollary2.1 ].

The maximal invariant setfrom LaSalle’s invarianceprinciple maybe describedpreciselyusing the notion ofnon-observabilityfromcontroltheory. Let W "B B* and

Z B -- ^B

^betwobounded

linear operators.Then theclosedsubspaceof

B

Nobs(Z, W

I’nO

^{Ker WZ (2.7)}

is the non-observable space of the pair (Z, W). (Compare this definition with the sufficient condition of observability (3.1) in Section 3 ). It seems that the following simple fact remained unrecordedtill this time.

Lemma

2.5. ON MAXIMAL INVARIANT ^SET

If ^Z

^{and W are bounded then}

Nobs(Z, W coincides with the maximal invariantsubset

of

^(2.1)thatis contained in Ker W.

Proof.

In

thefirstpartweprove^that Nobs(Z, W is an invariantsubset of Ker W.

By

its definition Nobs( Z W is a subspace of W, hence it remains to show invariance that is,

eZtx

Nobs(Z,W) for all e R if x e Nobs(Z,W).Thisfollows from the factthatfor abounded

/EAK INVARJANCE PRINCIPLE AND ASYHPTOTIC STABILITY 259

operator Z,ezt lm_{K_} (Zt) k! inthesenseof the uniform operatortopology,see 17 ],hence

=0

K

by (2.7)

WZneZtx=limK__,=y(zt)

^{/k! =0}

k=t.

Second part"maxmallty. Let M denote themaximalinvariant subset of Ker W Maxmalty of M implies that Nobs(Z,W)

_

^{M Itremains to showthe conver,,,e inclusion. Let uscm,,tleran}

arbitraryelement x of M Then

WeZtx

0 for M is invarant and M c:: KerW. The function

WeZtx

is infinitely manytimesdifferentiable since Z and W are bounded, hence all its derivatives areequalzerofor >_0,thatts

WZneZtx

0 for all n > 0 and > 0.Taking 0 weconcludethat x e Nobs(Z,W),thus Mc:Nobs( Z, W), for x wasarbitrary.Therefre M Nobs( Z, W).

Corollary 2.6.ON LASALLE’SMAXIMALINVARIANT SET.Under the assumptions

of

^Theorem

2.4 the maximalinvariantset M coincides with Nobs( Z, Z*G+GZ).

Proof. Thisfollows from Theorem2.5 with W Z*G+GZ.

Section3. Weak asymptotic stabilityandobservability.

Weakasymptotic stability forthe evolutionequation (2.1) ^is provedin this secuon withthe helpofobservability.Thepair Z, W iscalled observable if foreverysolution u of (2.1) the condition Wu(t)= 0 for all >0 implies that u(t) 0 for all >0. A necessary and sufficient condition forobservability for bounded operatorsisthat see 19,firstpart of Theorem 5.1.1

Nobs(Z, W

[’n>0

Ker WZ 0 (3.1)

Inordertomake communicationprecise,let us maintainthe following well-known terminology.Zeroin B the equilibrium point of (2.1),is

a)

c

stable iffor every e> 0 thereexists a > 0 such that foreverysolution u we

havethat Ilu(t)ll< _{for all} > 0, whenever Ilu(0)ll< 8 This iscalled also ordinary stability.

asymptotically stable if limt_.,**llu(t)ll 0 forevery solution u This iscalledalsostrongasymptotic stability.

weakly asymptotically stable if w-limt._,**u(t) ^{0 for}everysolution u Theprincipalresultofthispaperis asfollows

Theorem3.1.ONWEAKASYMPTOTIC STABILITY.Let u

C(

(0,+,*,),

B

be a solution

of

(2.1) witha bounded

infinitesimal

^{generator Z in a}

reflexive

^{Banachspace B.Let G:B}

--

^B*

beasymmetric bounded strictly positivelinear operatorsuchthattheoperator Z*G+ GZ

B --+

B* is weaklynegative

definite

^{andthepair Z,Z*G+ GZ is} observable. Then zeroin

B

isa weakly asymptotically stable equilibriumpoint

of

^(2.1).

Proof. All theassumptionsof Theorem 2.1 aresatisfied, hence u(t) convergesinthe sense of formula (2.6) tothemaximal invariantsubset M of Ker(Z*G + GZ).Corollary2.6 implies^that M Nobs( Z,Z*G+ GZ hence M 0 bythe observability assumption. Therefore, by (2.6), wehave that foreverysolution u of (2.1) liminft__l(u(t) y )1 0 for all y B*.

Suppose

that for some y B*

limsupt_,,,.l(u(t) y )1 > 0 (3.2)

Thenthereexists asequence tn, n 1,2 as in Definition(2.1) suchthat limn._,_(u(tn) yl exists and is equal to r. Now boundedness of u, reflexivity of

B

and the Eberlain-Shmulyan theoremimplythat there exists asubsequence

tnj,

^{1,2 such that u**}

w-limj._,,,.u(tnj)

^exists

andthen I( u**l y )1

:

0 by (3.2) hence _u**

:

0. But u** isanelement of wtols(u) which isa subsetof M. Thus a non-zero u.,, belongs to M,which,by observability, consists of zero alone

a contradiction.Therefore w-limt._,**u(t ^0.

Q.E.D.

Remark 3.2.J.P.Miller and A.N.Mitchell proved-in 14] that,under (1.4),observability is anecessaryand sufficient conditionof asymptotic stability.Theorem3.1 assertsonlysufficiency of

260 E. N. ^CHUK,’U AND P. SHOCZYNSKI

this condition for weak a,ymptotic stability Itfollows from the Example below thatoh.ervahthty is not a nece.ary condttton

of

^wealt a.ymptotic ,tabtlity Theorem 3.3 sht,ws h,wever that strong asymptoticstability implies observabilty.

Example Let B*=B=L 10,11, C andlet Zx(,,)

"4(-I

).s.x(s),

G(x,y)

Jlo.l

x(o)y(ojdo.Letusconsideranequation

0tu(t,s)

Zu(s)

"/(-1

).s.x(s). Then W(x) G(x,x) x x -IIx

J[0,zi ^12do

mayserve as a

Lyapunov

function. Next Z*G+GZ 0,hence V’(u(t) 0 by (2.5).Therefore all the solutions are bounded. The solution ofthe above equation isgiven by the following formula"

u(t,s) u(0,s)e

q(-),

therefore

limt-,**,[IO.l

u(t,s).v(s)ds

limt..,...[[0,1

^u(0,s)e(

^4(-)st

^{)-v(s)ds 0 forevery ve B,}

hence w-limt_.,,u(^t,.)=0 thus zero is an asymptotically stable equilibrium point However the observabilitycondition (3.1) is not satisfiedforinthiscase

Nobs(Z,Z*+Z)=(’nzoKer(Z*+Z)Z n=Ker(Z*+Z)=H{0}

Theorem 3.3. ON OBSERVABILITY FROM ASYMPTOTIC STABILITY Let G" B-B*

be a symmetricbounded strictly positivelinear operator.

If

^{Z is abounded}operator on a Banach

,wace

^B such that the operator Z*G+GZ" B B* isweaklynegative

definite

^{andzero in H is}

astrongly asymptotically stable equilibriumpoint

of

(2.1) then the pair Z, GZ + Z*G ^is observable.

Proof. Let 0:x Nobs(Z,GZ+Z*G).Then

u(t)=eZtx

Nobs(Z,GZ+Z*G) ^for Nobs(Z,GZ+Z*G) ^is an invariant subspace of (2.1) by Theorem2.5, hence d(Gulu)

(ul (GZ + Z*G)u) 0 since Nobs(Z, GZ + Z*G c: Ker(GZ+Z*G) hence dt

(Gu(t)lu(t)) (Gu(0)lu(0)). Thus u(t) does ^nottendstronglytozero.Thiscontradicts theassumption of strongasymptotic stability.

Q.E.D.

Section 4.

Strong

stability forintegralgenerators.

Ifwetake a Hilbert space

H

as the Banach space

B

inTheorems 2.4 and 3.1 then B*

H

by the Riesz representation theorem and the duality mapping coincides with the inner product. Wetake G tobethe identity operatorsothat V(x) x x x

’

Theorem3.1 isnow applicableif theoperator W Z*G+ GZ Z*+

Z

isweakly negativedefinite and thepair Z,W satisfiestheobservabilitycondition(3.1). We applythistheorytosimple operators.A subspace S of a Hilbert space is an invariant subspace ofa bounded linear operator

K" H H

if KS S.

A

boundedlinearoperator K H H iscalled simple if

K

and K* have nocommon invariant subspaceon which theycoincide. K is simpleifandonly ifthe following controllability condition holds

closure

tn

^Ran

^{Kn( K-}

^{K* H,see 8,chapter 1,section}7, point 1, property 3 This,by duality (see 19],theproofof Theorem5.1.1),isequivalenttoanobservabilitycondition

(’nz,

OKer K-K* )K^*n 0 (4.3)

The following special case of Theorem 3.1 is of some interest in the theory of integro- differential equations.

Theorem 4.1 ON ASYMPTOTIC STABILITY FOR SIMPLE GENERATORS

if

^the

infinitesimal

generator

Z of

^theevolution equation (2.1) is aboundedweakly negative

definite

^{operatorin a}

Hilbert space such that

K 4(-1)Z*

is a simple operator, then zero in

H

is a weakly asymptotically stable equilibrium point

of

^(2.1)

WEAK INVARIANCE PRINCIPLE AND ASYMPTOTIC STABILITY 261

Proof. K ",,(-I)Z* is smple hence (4.3) mplesthe following observability ctntlititn

(’n)

Ker(Z+Z*)Z

n=

0 ,thus the pair (Z,Z+Z*) is )bservable.All the assumptms f Theorem3.1 are satisfied if one takes the identity operatoras the operator G the result fi)llows.

Q.E.O.

Intheproofof thenexttheorem we shall need a theoremonapproximations bycompactsets.If X, Y aretwoboundedstronglyclosed subsets of a Banachspace B then their

Hausdorff

^{distance 0(X,Y) sup}

x

^{inf y x-y +sup yinf e}x ^x-y

iswell defined and it induces a metrictopologyinthe family of boundedclosedsubsets^of B.

Lemma 4.2 ON COMr’ACT SETS. The family

of

^{compact sets is a} closed subset

of

^the

topological space

of

^{closedsetsequippedwith}

Hausdorff

^metrics

Theideaofproof. Thedefinition "asubset ofametric spaceis stronglycompact if firevery positive thereexistssuch afinitesetof balls of radius

:

that covers it" means that the ^center.,,of the balls approximate the setinHausdorffmetric,thereforethe family of compactsetsisthe closure under Hausdorff metric of thefamilyof all the finitesets,hence a closedset.

Theorem 4.3 ON ASYMPTOTIC STABILITY FOR INTEGRAL GENERATORS Let z(x,s) be ameasurable

function defined for

^{all s, x [0,1] suchthat z(x,s) 0 for >x and z(x,x)}

:

0

for

all x [0,11 and let theintegraloperator in

L2(

[0,1l,C) givenby Zu(x)

JlO.xlz(x,s)u(s)ds

^be

weaklynegative

definite.

Then zero in

L2(

[0,1l, C) isa weaklyasymptoticallystable equilibrium point

of

)tu(t,s) Jt0,xlZ(X,S)U(s)ds.

^(4.4)

If,moreover,z(x,s) is continuouswith respect to

s,/)xZ(X,S)

^isarneasurable

function

^{(for <x)and}

the operator

u ,v)--.>

f[O,xlZ(X,S)U(s)ds ftO,xltgxz(x,s)u(s)ds

⁺

z(x,x)J’10,xlV(S)ds

^(4.5)

isalsoweaklynegativede,hire,then zeroisstrongly asymptotically stable.

Proof.SinceZis weaklynegative definite, then weak stability follows from Theorem 4.1 if weprovethatK

/(-1)Z*

isasimple operator.FromTheorem7.1 ,secdon7 of 8 itfollowsthat Kissimple

if Jto.xlZ(X,S)U(s)ds

⁺

Jtx.wlZ(S,X)U(s)ds

⁰

for

^{all x,w} [0,11 implies u(s) ⁰

for

^all

[0,1].Taking w x we obtainthat the firsttermhastobezero,hence the secondhastobe also zero.The continuity assumption and the requirement z(x,x)#0 implythat u(s) 0 forall s [0,1], hence

/(-1)Z*

issimpleand weakasymptotic stabilityisproved.

Nowletusconsider(4.4)inanotherHilbertspace

H

^{u e}

L2(

[0,1],C :/)u e

L2(

[0,1], C),u(0)=0 }.

The equation (4.4) in this space is equivalent to a system of two equations in

L2(

[0,11, C @

L2(

[0,1], C consisting of(4.4)and anequationfor the derivative

)tOxU(t,s) j’tO,xl/)xz(x,s)u(t,t)do

+

z(x,x)j’tO.xl/9,tu(t, oMo.

Thereforezero in

L2(

[0,1],C @

L2(

[0,1],C is aweaklyasymptotically stableequilibriumpoint of the systembythe alreadyprovedpart on weak stability, hence alsoin

H.

Trajectoriesareboundedin

H

^{hence by Sobolev imbedding} theorem these are precompact in the strong topology of

L2([0,1],

C), so they tend to their strong omega-limit sets in the sense of strong topology of

L2(

[0,1], C).Butthe strong omega-limitsetiscontainedintheweak omega-limitsetthat consist of zeroalone. Thus_strongasymptotic stabilityisprovedprovided the trajectorystartsfrom

H.

^Nowlet

x

L2(

[0,1], C). Since

H

^isdensein

L2(

[0,1], C), then there exists asequence xj

H

^for

1,2 whichstrongly convergesin

L2(

[0,1], C to x and therefore

eZtxj

converges stronglyto

eZtx

for

eZt

isbounded independently of t. Thus thesequence of trajectoriesconverges under Hausdorff metricinthetopological spaceof closed subsets

f L2(

[0,1], C tothetrajectory starting

262 E. N. CHUKWU AND P. SMOCZYNSKI

from x. Each trajectory starting from an xj is strongly precompactand hencealso the trajectory starringfrom x isstronglyprecompactbyLemma4.2.Q.E.D.

Proposition 4.4.ONNEGATIVENESS.The operator

from

(4.4)isweaklynegative

definite iffor allfinite

non-decreasingsequences

x,

^{1,2,3 N+N det(M+M*)_>0,where}

^{Mtj z(x,}

^,xj

for

< < <N and Mj ⁰

for

^< < < N.Similarly, the operator

from

^{(4.5) isweaklynegative}

definite if

det(L+L*) > 0 where

L,,j Ml,j for

^{_<i,j < N,}

Ltj OxZ(X

,xj

for

^{N< N+j<_ <N+N}

and

,j=Ofor

N<i<N+j<N+N

l,LaJ=0 for

^{<i<N and} N<j<_N+N

l,La,J=z(x,,x,) fir

N<j<i<N+N and

Lij=0 for

N<i<N+j<N+N

Theconditions for negativeness ofthe operator from (4.5)are satisfied if the operator from (4.4) isnegative-definiteand the kernel z(x,s) is independent of thefirstargument.

Section5.

Strong

asymptotic stability fornormalgenerators.

Equivalence of asymptotic stability and observability for equations (2.1) with weakly dissipative normal infinitesimal generators Theorem 5.3 is proved in this section The proof is decomposed into three parts two of these are interesting in themselves and are stated as separate theorems.The firstoneofthese isindependent of normalness.

Abounded linearoperator

Z

is normal ifit commutes with its adjoint Z*Z ZZ*. The second step of theproofof Theorem5.3 onequivalenceofasymptotic stabilityand observability for normal operatorsisasimplificationof observability condition

Theorem 5.1.

If ^Z

^isanormal boundedlinearoperator then the observability condition (3.l) with W=(Z*+Z) isequivalentto Ker(Z*+Z)={0}.

Proof.If

Z

is anormallinearbounded operator then

Ker Z*+

Z

)Zⁿ Ker

Zn(

Z* +

Z :

^{Ker Z* +}

^Z

^{forall n}

^>

^{0. (5.2)}

Toshow the reverse inclusion weargueasfollows.Let x Ker Z(Z*+Z).^Thenbynormalness of

Z

we obtain II(Z*+Z)xll² (xlZ(Z*+Z)x)+ (Z(Z*+Z)xlx) 0. Thus x

Ker

Z*

+ Z Hence

Ker (Z*+Z):2Ker Z(Z*+Z) Ker(Z*+Z)Zⁿ Thisand (5.2) imply Ker(Z*+Z)Zⁿ Ker(Z*+Z) for all integers ^n_>0.Q.E.D.

Thethirdstep of theproofof Theorem 5.3 isasymptotic stabilityforselfadjointoperators. The proofisbasedonthetheoryof spectralmeasures, see 15].

Lemma

5.2. ON ASYMPTOTIC STABILITY FOR SELFADJOINT OPERATORS.

If ^A

^{isa weakly}

negative

definite

bounded selfadjointlinear operator in a Hilbert space

H

such that

Ker A

0 then limt..,**ll

eAtx

0

for

^{all x H}

Sketchof theproof.Thespectralmeasure

I.t

^ofthe operator A isprojector-valuedmeasureon thesetof all Borel subsets of

-IIAII,

0 (since

A

isweaklynegativeselfadjoint operator) such^that

eAtx [-IIAII,

0

e;t [t(d)x

^(5.6)

Regularity property ofspectralmeasuresimplies ^thatforevery e >0 there existsanatural number N(x,e) such that

VJ’(-Un .o )(l.t(dC,)x,

^x)=

.[(-,/n.

o

)l.t(dC.)x

^x-

]’[-,,A,,

.-Un

l.t(dC)x

^{11-< whenever} n>N(x,e).

Usingthisand (5.6) weapproximate u(t) with

Un(t) eat.[[-,,A,,.-l/n It(d;)

^x

.It-,,A,,.-l/n ^{e;t .(d;)}

^x

Therefore limt_. Un(t) 0. Theprooffollows easily fromthis.

Q.E.D.

The following theorem generalizes the interesting result ofR. K.Miller and A.

N.

Mitchell on evolutionequations in Hilbert spaceswithboundednormal infinitesimal generators.

Theorem 5.3 ONEQUIvALENCE OFOBSERVABILITYANDASYMPTOTICSTABILITYFOR NORMAL OPERATORS.

Let Z

be anormal bounded linearoperatbrina Hilben space

H

such that Z*+Z

WEAK INVARIANCE PRINCIPLE AND ASYMPTOTIC STABILITY 263

isweakly negative

definite.

^Then the pair (Z,Z*+Z) is observable

if

^{and only}

if

^{zero in}

^H

^is

astronglyasymptotically stableequilibriumpoint

of

^(2.1).

Proof.

Strong

asymptotic stability implies observability by Theorem3.3 andobservability is equivalentto Ker Z*+Z 0 by Theorem5.1. Itremains toshowthatthis condition implies asymptoticstability. So let x B be arbitrary. Bynormalityof Z, e^Zt also commutes with its adjoint e^z*t see 17], hence

eZtx

e(^(z-z*)t/2 (z+z*)t/2)X (e^{(Z-Z*)t/2 e(z+z*)t/2})x efz+Z*)t/2x II,

for e(Z-Z*)^t/2 _isaunitaryoperatorsince (Z-Z*)t/2 isskew-symmetric.

But

(Z+Z*)/2 isselfadjoint and itskerneliszero,henceby Lemma5.2 limt_,**ll

eZtx

limt_,**ll e(Z+Z*)t/2x ^{0, whichmeans} asymptotic stability,for x wasarbitrary.

Q.E.D.

Remark 5.4. R. Datko provedin [5 and [6] thatin Hilbert space strict negative def’miteness of Z* + Z impliesanexponentialdecayofsolutions to zero. Undertheassumptions^of Theorem5.3 whereonlyweak negative definiteness isrequired,solutionsmay decaytozero much slower. Thisisillustratedbelow.

Example. Let ^us consider an equation of the form (2.1) in

H=L:([0,1I,C)

with Zu(s) K +

13s

^{)u(s) where K is a}nonnegativereal numberand a iscomplexwith

Re(13)

_> 0. The solutionmay be easily found u(t,s) u(0,s)e-(K^+Is)t.Itfollows that Z*u(s) (K +

[3*s)u(s)

^where

13"

^is the number conjugate to

13

^Theoperator Z* +Z is strictly negativedefinite if

K

>0 for

Z* + Z )u(s) -2( K+

sRe(13)

)u(s) In this caseDatko’s theory is applicable hencenorms of all solutionstendexponentially^tozero.Indeed

u(t,.

e-Ktx/{ Ji0,1le-2Re(13)stl

^{u(0,O)12do < u(0,.}

^II

^e-Kt

Let K 0. Then Z*+Z isweakly negativeif

Re(13)

>0,but itis notstrictlynegativedefinite,hence Datko’s theoryis notapplicable,sincezerobelongstothe spectrum of Z* +

Z. However

Z* and

Z

commute. Therefore Theorem 5.3 implies strong asymptotic stability in this case. Letus takefor example u(0,s) ^--a where 0< o<1/2, asinitialdata. Thenthetollowingis correct

limt_,. ^TM u(t,.)112

limt..

^t-2a

Jto.]

^{e2Re(l)sts’2ads}

(Se())

^2a-1limt_,,,

ft0,ge(l;)tl ^e-Zs

^{s-2a ds}

(Re(l]))

^2a-

to,--I ^{e-2s s-Za}

^ds

thus u(t,. tendstozero like ta-l/2

Convergence

maybearbitrarilyslow if

x

^iscloseto 1/2.

If Re()=0 then the above estimates break down andoneobtainsweakasymptotic stability ^{as was} shownthe last exampleinthesection2, despite thatobservabilityislost.

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