Computing Aumann's Integral

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Robert Baier and FrankLempio

Abstrat. Quadratureformulaeforthenumerialapproximationof

Aumann'sintegralareinvestigated,whihareset-valuedanaloguesof

ordinaryquadratureformulaewithnonnegativeweights,likeertain

Newton-CotesformulaeorRombergintegration.

Essentially,theapproahonsistsinthenumerialapproximationof

thesupportfuntionalofAumann'sintegral byordinaryquadrature

formulae. Forset-valuedintegrandswhiharesmoothinanappropri-

ate sense,thisapproahyields higherordermethods,for set-valued

integrandswhiharenotsmoothenough,ityieldsfurtherinsightinto

well-knownorderredutionphenomena.

Theresultsareusedtodenehigherordermethodsfortheapproxi-

mationofreahablesetsofertainlassesoflinearontrolproblems.

Mathematis Subjet Classiation (1991): 34A60, 49M25,

65D30,65L05,93B03

Keywords: Aumann'sintegral,reahableset,

nitedierenemethods

1 Introdution

Themainobjetiveofthispaperistoinvestigatehigherordermethodsfor

thenumerialapproximationofAumann'sintegralandthereahablesetof

linear dierentialinlusions. Wehoosean approah basedessentially on

thenumerialapproximationofthesupportfuntionalofAumann'sintegral

byordinaryquadratureformulae,p.inthisonnetion[3℄. Butontraryto

[3℄,werestritouroutlineofbasierrorestimatesfromtheverybeginning

to quadrature formulae with nonnegative weights, moreover, we use the

weak regularity assumptions in the spirit of [17℄ to get higher order of

onvergene. Thus,wefollowadiretapproahtohigherorderquadrature

formulaeforset-valuedmappings,avoidingtheuseofembeddingtheorems

forspaesofonvexsets. Inthisrespet,ourpresentationdiersfrom the

indiretapproahindiatedin [10℄,whihisbasedon[16℄,[12℄,and[4℄.

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The underlyingideasareoutlined in Setion2resultingin thefundamen-

tal error estimate of Theorem 2.6. Moreover, the set-valued analogues

of losedNewton-Cotesformulaewithnonnegativeweightsresp.Romberg

method aregivenin Proposition 2.7 resp.2.8 together with allregularity

and smoothness assumptions required for higher order onvergene. Ap-

plying these results to smooth set-valued integrands, as in Example 4.1,

in priniplearbitrarilyhigh order of onvergeneanbeahieved, e.g.by

the set-valued analogueof Romberg's method. Inaddition, forset-valued

integrandswhiharenotsmoothenough,asinExample4.2,wegetfurther

insightintotheorderbarriernotied in[19℄.

In Setion 3 we will apply our results to the approximation of reahable

sets for linear ontrol systems and get higher order methods at least for

ertain problem lasses. In fat, smoothness in the sense of Setion 2of

thefundamentalsolutionmultipliedbytheset-valuedinhomogeneityisthe

ruialproperty,lakofitresultsinorderredutionphenomena. Thus,we

getatleastapartialanswertosomeopenquestionsdisussedin[9℄. These

eets are illustratedbyExample 4.3, whih isnotsmoothenoughin the

abovesense,henegivingadditionalinsightintotheorderbarrierdesribed

in [20℄, and by Example 4.4, whih is arbitrarily smooth, thus admitting

numerialapproximationsofthereahableset ofarbitrarilyhighorder.

Note that there isatheoretial approah desribedin [8℄ whih resultsin

order of onvergene greater than two, ifthe ontrol region is a ompat

onvexpolyhedron. But,onvertingtheseoneptualideasintoanumerial

algorithmisuntilnowonlypossiblefororderofonvergeneequaltothree.

Alltestexamplesaretreatedbytheabovemethodsbymeansofadualap-

proah,explainedmorepreiselyinSetion4. ThedevelopmentofeÆient

algorithms for higher dimensional problems, following this dual approah

or omputing diretly sumsof sets aording to the presented set-valued

quadratureformulae,isaninterestingandhallengingeldofresearh.

Inthefollowing,wedesribebrieytheonnetionbetweenAumann'sin-

tegralandlineardierentialinlusions.

Problem 1.1 (LinearInitialValueProblem) Let the n n-matrix

funtion A() beintegrable onI =[a;b℄and

G: I =)R n

beaset-valuedmapping.

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Find an absolutelyontinuousfuntion y(): I !R n

with

y 0

(t)2A(t)y(t)+G(t) (1.1)

for almost allt2I and

y(a)=y

0 : (1.2)

Suh problems arise from awide range of appliations, e.g.from optimal

ontrolproblemsorfromperturbeddynamialsystemswithunknown,but

boundedperturbations.

The only propertyreally needed is that all solutionsof (1.1),(1.2)anbe

equivalentlyrepresentedintheform

y(t)=(t;a)y(a)+ t

Z

a

(t;)g()d

forallt2I withafundamentalsolution(t;)ofthehomogeneoussystem

d

dt

(t;)=A(t)(t;)

foralmostallt2I,satisfyingtheinitialondition

(;)=E

n

foraxed 2I,andwithanintegrableseletiong()ofG()onI.

Hene,thereahable setat timet2I

R(t;a;y

0 )=

z2R n

: thereexists asolutiony()on[a;t℄of

(1.1),(1.2)withz=y(t)

anberepresentedbymeansofAumann'sintegralas

R(t;a;y

0

)=(t;a)y

0 +

t

Z

a

(t;)G()d

forallt2I, wheretheintegralisdened aordingto

Denition 1.2(p. [2℄) Let

F :I =)R n

beaset-valuedmapping. Dene

R

I

F()d =

z2R n

: thereexistsanintegrableseletion

f()ofF()onI with z= R

I f()d

asAumann's integral ofF()overI.

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Consequently, our rst step towards higher order dierene methods for

linear dierential inlusions should onsist in the investigation of higher

order numerialmethods fortheomputation ofAumann's integral. This

is theentral subjetofSetion2.

2 Quadrature Formulae for Set-Valued

Mappings

Denition 2.1 A set-valued mapping F : I =) R n

with nonempty and

losed images is integrably bounded, if there exists a funtion k() 2

L

1

(I)with

sup

f(t)2F(t) kf(t)k

2 k(t)

for almost allt2I.

In fat,we intendto use thewell-knownmethod ofsalarization ofaset-

valuedsituation,just exploitingthefollowingfundamental fat.

Theorem 2.2 Let F : I =)R n

beameasurableset-valuedmappingwith

nonempty andlosedimages. Then

Z

I

F()d

isonvex.

If, moreover, F()isintegrably bounded, then

Z

I

F()d = Z

I

o(F())d

is nonempty, ompat, and onvex. Here, o() denotes the onvex hull

operation.

Fortheproofofonvexityandompatnesssee[1,Theorem8.6.3,pp.329{

330℄, the existene of a measurable seletion f() of F() is proven in [1,

Theorem 8.1.3, pp. 308℄. It follows from the integrably boundedness of

F()thatthisseletionisalsointegrableandhene R

I

f()d isanelement

of R

I

F()d. Compare[13,8.2,Theorem1,pp.334℄fortheequalityofboth

integrals.

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Denition 2.3 LetCR n

beanonempty setanddene

Æ

(l;C)=sup

2C

<l;>2R[f1g

for all l2R n

,where <;> denotes the usualinnerprodutinR n

.

Then Æ

(;C)isalledsupport funtionof theset C.

We list the following property of the support funtion, p. [1, Table 2.1,

pp.66℄whihwewillusebelow.

Lemma2.4 Let l2R n

andCR n

beanonempty set.

Then the following equalityholds:

Æ

(l;C)=Æ

(l;l(o(C)));

where l(o(C))denotes the losure ofthe onvexhull ofthe setC.

Obviously(seee.g. [1,Proposition8.6.2,pp.327℄),under theassumptions

of bothpartsofTheorem2.2wehave

R

I

F()d =

z2R n

: <l;z>Æ

(l;

R

I

F()d)= R

I Æ

(l;F())d

foralll2R n

:

This isthebasisofthesalarization,whihonsistsin approximating

Z

I Æ

(l;F())d

byaquadratureformulaJ(l;F)

Z

I Æ

(l;F())d =J(l;F)+R (l;F);

withremaindertermR (l;F)dependingonl2R n

andF().

E.g. for(omposite)Newton-Cotesformulaeof openorlosedtype,Gau

quadratureor Rombergintegration,J(l;F)hastherepresentation

J(l;F)= N

X

i=0

i Æ

(l;F(t

i )) (2.1)

withN 2N,

i

2R, andagrid

at t :::t b:

(2.2)

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ThisrepresentationandLemma2.4suggesttheinterpretationofthequad-

ratureformulaasasupportfuntion

J(l;F)=Æ

(l;

N

X

i=0

i

l(o(F(t

i )))

oftheset

N

X

i=0

i

l(o(F(t

i ))) (2.3)

whihispossibleifallweights

i

(i=0;:::;N)are nonnegative.

The followinglemma, relatingthe Hausdordistane haus(A;B) between

two sets A;B to the support funtions, an be found in [15, Satz 14.1,

pp.148℄.

Lemma2.5 Let A andB be nonempty, ompat, and onvexsets in R n

,

then

haus(A;B)= sup

klk

2

=1 jÆ

(l;A) Æ

(l;B)j:

Applyingittothesalarizedquadratureformulawithremaindertermyields

thefollowingerrorestimate.

Theorem 2.6 Let F : I =)R n

be a measurable and integrably bounded

set-valuedmappingwithnonemptyandompatimages,andletthequadra-

ture formula J(;) have nonnegative weights

i

, nodes t

i

2 [a;b℄ (i =

0;:::;N)andremainder term R (;).

Then the following error estimateholds

haus(

Z

I

F(t)dt;

N

X

i=0

i o(F(t

i

))) sup

klk

2

=1

jR (l;F)j:

Tobe moreonrete, we usethe omposite losed Newton-Cotes formula

of degreek 2 N overthe interval [a;b℄ whih is exat for polynomialsof

degreeatmostk. ChoosethenumberofsubintervalsN 2N ofthegrid

t

i

:=a+ih(i=0;:::;N); h= b a

; (2.4)

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suh that N

k

isan integer,and applythe losed Newton-Cotesformulaof

degreekoneahsubinterval[t

ik

;t

(i+1)k

℄, thenwearriveat

b

Z

a Æ

(l;F(t))dt = N

k 1

X

i=0 t

(i+1)k

Z

t

ik Æ

(l;F(t))dt=

= kh N

k 1

X

i=0 k

X

j=0 w

k j Æ

(l;F(t

ik +j ))+R

k

N

k (Æ

(l;F()))

Using the resultsof[17, Theorem 3.5, pp. 52℄(or theearlier resultsmen-

tionedin [7℄)inaslightlymodiedway,weouldestimatetheerrorby

jR k

N

k (Æ

(l;F()))j(1+ k

X

j=0 jw

k j j)

(

W

k +1

k +1 (Æ

(l;F());2 k

k +1 h)

1

; ifk isodd;

W

k +2

k +2 (Æ

(l;F());2 k

k +2 h)

1

; ifk iseven;

where

(f;Æ):=

(f;Æ)

1

is the averagedmoduliof smoothness oforder

dened in [17, Denition 1.5, pp. 7℄and W

denotes the -th Whitney

onstant.

Proposition2.7 LetF : I =)R n

beameasurableandintegrablybounded

set-valued mapping with nonempty, ompat, and onvex images. Let the

losed Newton-Cotes formula of degree k have oeÆients w

k j

0; j =

0;:::;k. Assume that the support funtion Æ

(l;F()) has an absolutely

ontinuous( 1)-st derivative andthat the -thderivative is of bounded

variation with respet tot uniformlyfor all l2R n

withklk

2

=1,where

2

f0;1;:::;kg; if k isodd ;

f0;1;:::;k+1g; if k iseven:

Integrating overthe grid

t

i

:=a+ih (i=0;:::;N); h= b a

N

;

andintroduingtheset-valuedmappingorrespondingto(2.3),thefollowing

errorestimates holds

haus(

b

Z

a

F(t)dt;kh N

k 1

X

i=0 k

X

j=0 w

k j F(t

ik +j ))

C(k;)(1+ k

X

w

k j ) sup

klk

2

=1 b

_

d

dt

Æ

(l;F())h +1

:

(8)

Here, b

W

a

()denotes the totalvariation and

C(k;)= 8

>

<

>

: 2

k +1 k

+1

1

Q

j=0 (k +1 j)

W

k +1

; ifk isodd ;

2 k +2

k +1

1

Q

j=0 (k +2 j)

W

k +2

; ifk iseven:

Proof. Apply the error estimates in [17℄ mentioned before and use the

estimates [17℄[(3),(4), pp. 8 and (7), pp. 10℄ for a bounded, measurable

funtion f:

k

(f;h) 2 k 1

1

(f;kh) (k2N) ;

k

(f;h) k

1

Q

j=0 (k j)

h

k (f

()

; k

k

h) (if f ()

exists andis

boundedandmeasurable, 2f0;:::;k 1g; k2N);

1

(f;h) h b

_

a

f (if f isofbounded variation)

Q.E.D.

These resultsonlyapplyiftheoeÆientsw

k j

(j=0;:::;k)arenonnega-

tivewhih istheasefortrapezoidalrule(k=1), Simpson'srule(k =2),

and fork=3;:::;7;9,sothat themaximalorderof onvergeneis10 for

thelosedset-valuedNewton-Cotesformulae(p.[11,Table6.2.1,pp.268℄).

Similarresultsanbeahieved,ifweonsiderompositeNewton-Cotesfor-

mulaeofopentypewithdegreek,wheree.g.theoeÆientsarenonnegative

forthemidpoint-rule(k=0)andfork=1;3(p.thetablein[14℄).

Let us briey mention Romberg's method of extrapolation whih is de-

sribedinmoredetails in[5℄,[11℄, [18℄. Computetheintegral

b

Z

a Æ

(l;F(t))dt

bytheompositetrapezoidalruleforasequenzeofstepsizes,say

h

0

=b a; h

1

= 1

h

0

; :::; h

r

= 1

r h

0

;

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orrespondingtothesequeneofgrids

a=t

i;0

<t

i;1

<:::<t

i;2

i=b; t

i;j

:=a+jh

i

(j=0;:::;2 i

);

and startwiththerstRombergolumn

T

i0 (l)=

h

i

2 2

i

1

X

j=0 (Æ

(l;F(t

i;j ))+Æ

(l;F(t

i;j+1

))) (i=0;:::;r):

Using thereursiveformulaforj=1;:::;i; js

T

ij (l)=T

i;j 1 (l)+

T

i;j 1 (l) T

i 1;j 1 (l)

4 j

1

(i=1;:::;r);

weareabletodenethesets

T

ij

(F)=fy2R n

j <l;y>T

ij

(l)foralll2R n

with klk

2

=1g (2.5)

for j=0;:::;i; j s; i=0;:::;r. It isknownthat eah T

ij

(l)ouldbe

writtenintheform(2.1)withnonnegativeweights(p.[11,Theorem8.3.1,

pp.381℄)andN=2 i

,heneweanapplyallobtainedresultsandget

Proposition2.8 LetF : I =)R n

beameasurableandintegrablybounded

set-valued mapping withnonempty, ompat, and onvex images. Assume

that the support funtion Æ

(l;F()) has an absolutely ontinuous (2s)-th

derivative and that the (2s+1)-st derivative isof bounded variation with

respettot uniformlyfor alll2R n

withklk

2

=1.

Then the estimate holdsfor the set-valuedmapping introduedin(2.5)

haus(

b

Z

a

F(t)dt;T

ij (F))

j

Y

=1 1+(

1

2 )

2

1 ( 1

2 )

2

ij

j

Y

=0 h

2

i

for all j =0;:::;i; j s; i=0;:::;r, where

ij

an be hosen indepen-

dently ofall stepsizesh

i j

;:::;h

i .

Proof. Everything follows from a generalized Euler-Malaurin summa-

tion formulawhih gives(underthe statedweakerassumptions)thesame

asymptotiexpansionoftheompositetrapezoidalrulefortheintegral

b

Z

a Æ

(l;F(t))dt

asdesribedin[5℄,[18℄. Q.E.D.

Hene,eahofthes+1olumnsofRomberg'stableaudenessuessively

integration methods of order 2;4;6;:::;2s+2 for the approximation of

Aumann'sintegral,ifthesupportfuntionÆ

(l;F())issuÆientlysmooth.

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3 Approximation of Reahable Sets for

Linear Control Systems

WereturntotheLinearInitialValueProblem1.1,assumingthatitisgiven

byalinearontrolproblem ofthefollowingstandardtype.

Problem 3.1 Let the nn-matrix funtion A() and the nm-matrix

funtion B() beintegrable onI,andthe ontrolregion

UR m

benonempty,ompat,andonvex.

Find an absolutelyontinuousfuntion y(): I !R n

with

y 0

(t) 2 A(t)y(t)+B(t)U for almostall t2I;

y(a) = y

0 :

With thenotationsfromtheintrodution,thereahablesetat timebis

R(b;a;y

0

)=(b;a)y

0 +

b

Z

a

(b;)B()Ud:

Nowapply a quadratureformula on anequidistantgrid (2.4)of the type

(2.3)witherrorestimate

haus(

b

Z

a

(b;)B()Ud;

N

X

i=0

i (b;t

i )B(t

i )U)

sup

klk

2

=1

jR (l;(b;)B()U)jonst

1

N

p

;

i.e. aquadrature formulaoforder pwith respet to thedisretization pa-

rameterN 2N. Suhformulaeexist,ife.g.

Æ

(l;(b;)B()U)

has an absolutely ontinuous (p 2)-nd derivative and if the (p 1)-st

derivativeisofboundedvariationuniformlywithrespettoalll2R n

with

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klk

2

=1(p.Propositions2.7and 2.8).

Choose inaddition adierenemethodfor theomputationof thefunda-

mentalsystem(b;)onthesamegrid(2.4)whihomputesapproximations

~

(b;t

i

)(i=0;:::;N)alsooforder p

sup

0iN k

~

(b;t

i

) (b;t

i )k

1

onst

1

N

p

:

This ispossiblee.g.for theAdams-Bashforthmethodof degreep 1(p.

[17, Theorem 6.3, pp.126℄), ifA() has anabsolutely ontinuous(p 2)-

ndderivativeandifthe(p 1)-stderivativehasboundedvariation. Under

slightlydierentassumptionsonehassimilarresultsforRunge-Kuttameth-

ods aswell(p.[6℄,[7℄).

Using the samenotation kMk

1

for thelub-norm ofamatrix M with re-

spettothesupremumnormkk

1 inR

n

andforthenormofasetSR n

kSk

1

=sup

s2S ksk

1

;

thefollowinginequalityholds

haus(

N

X

i=0

i (b;t

i )B(t

i )U;

N

X

i=0

i

~

(b;t

i )B(t

i )U)

N

X

i=0

i

haus((b;t

i )B(t

i )U;

~

(b;t

i )B(t

i )U)

N

X

i=0

i k(b;t

i )

~

(b;t

i )k

1 kB(t

i )Uk

1

;

iftheweights

i

areallnonnegative. Moreover, N

P

i=0

i kB(t

i )Uk

1

isbounded

uniformly forallN 2N, ife.g.B()isboundedon I. Hene,wearriveat

thefollowingresult.

Theorem 3.2 Considerthe Linear ControlProblem 3.1,andassumethat

A()andÆ

(l;(b;)B()U)haveanabsolutelyontinuous(p 2)-ndderiva-

tiveandthatthe(p 1)-stderivativeisofboundedvariationuniformlywith

respettoalll2R n

withklk

2

=1.

Assumemoreover,that N

P

i=0

i kB(t

i )Uk

1

isuniformlyboundedfor N 2N.

Then, ombining a quadrature formula with nonnegative weights of order

p withadierenemethod oforderpin the sensedesribedabove yields a

methodof orderpfor the approximation ofthe reahable setattimeb.

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4 Test Examples

In the following, we present a series of model problems, illustrating the

resultsofSetion2and3. AllnumerialtestsweremadeonanHPApollo

workstation, Series 400. For every supporting hyperplane, the expliit

knowledgeof at leastone boundary point of theset-valuedintegrand be-

longingtothathyperplaneisexploited fortheomputationoftheplots.

Inalltables,weuseLemma 2.5togetherwithuniformlydistributedpoints

l

i

(i=1;:::;)ontheboundaryoftheunitballtoapproximatetheHaus-

dor distane of twononempty, ompat, and onvexsets C ;D R n

in

thefollowingway:

max

i=1;:::;

jÆ

(l

i

;C) Æ

(l

i

;D)jhaus(C ;D):

Inthe abovesense,weuseadual approah forthealulationof thepre-

sentedset-valuedquadratureformulaeandfortheveriationoftheorre-

spondingerrorestimates.

Example 4.1 ComputeAumann'sintegral

2

Z

0 A(t)B

1 (0)dt=

2

Z

0

e t

0

0 t 2

+1

B

1 (0)dt;

where B

1

(0)denotes the losedunit ballinR 2

.

Thenthesupportfuntion

Æ

(l;A(t)B

1

(0))=Æ

(A(t)

l;B

1

(0))=kA(t)

lk

2

isarbitrarilyoftenontinuouslydierentiablewithrespettot withboun-

ded derivativesuniformlyforalll2R n

withklk

2

=1.

In Figure 4.1 we show theboundary of the set reated bythe omposite

trapezoidal rulewith stepsizes h=2:0 (the biggest set), h=1:0;0:5 (the

two smaller sets), and the referene set (the smallest set) omputed by

Romberg'smethodwith10rowsandolumns. Figure4.1illustratesorder

2oftheomposite trapezoidal rulewhihis onrmedbyTable4.1where

weshowtheapproximatedHausdordistane betweenthesets alulated

bydierentnumerialintegrationmethodsandtherefereneset.

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−10 −8 −6 −4 −2 0 2 4 6 8 10

−8

−6

−4

−2 0 2 4 6 8

Fig.4.1: Compositetrapezoidalrule withh=2:0;1:0;0:5

omparedwiththe refereneset

InTable4.1,oneanlearlyobserveonvergeneorder2fortheomposite

trapezoidalruleandorder4forompositeSimpson'srule.

Example 4.2 Thisexample waspresentedin [19 ℄ asanegativeresultfor

the approximation ofAumann'sintegral

2

Z

0

A(t)[ 1;1℄dt= 2

Z

0

sin(t)

os(t)

[ 1;1℄dt:

Inthisexample,thesupportfuntion

Æ

(l;A(t)[ 1;1℄) = Æ

(A(t)

l;[ 1;1℄)=

= jA(t)

lj=jl

1

sin(t)+l

2 os(t)j

isonlyabsolutelyontinuous,anditsderivativehasboundedvariationuni-

formly foralll2R n

withklk =1.

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−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

−5

−4

−3

−2

−1 0 1 2 3 4 5

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

−5

−4

−3

−2

−1 0 1 2 3 4 5

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

−5

−4

−3

−2

−1 0 1 2 3 4 5

−6 −5 −4 −3 −2 −1 0 1 2 3 4 5 6

−5

−4

−3

−2

−1 0 1 2 3 4 5

Fig.4.2: CompositeSimpson's rulewithh=

2

;

4

;

8

omparedwiththereferene set

Table4.2illustratesonvergeneorder2fortheompositetrapezoidalrule

and also only onvergene order 2 for the omposite Simpson's rule, be-

ause lakingsmoothnessof thesupportfuntion preventshigherorderof

onvergene. Wereferto Figure 4.2where theresultsofompositeSimp-

son's rule are plotted for stepsizes h = 0:5;0:25;0:125 together with

theresultofompositetrapezoidalrulewithh=0:02(therefereneset).

Simpson's rulereatespolytopeswithinreasingnumberofedges. Thisis

thegeometriexplanationgivenin [19℄fortheobservedorderredution.

(15)

Example 4.3 ThisexampleisalsoduetoVeliovandwaspresentedin[20 ℄.

Consider the linearontrol system

y 0

(t) 2

0 1

0 0

y(t)+

0

1

[ 1;1℄ for almost allt2[0;1℄;

y(0) =

0

:

Thenthefundamental solutionis

(t;)=

1 t

0 1

;

and thereahableset attimeb=1is

1

Z

0

1

[ 1;1℄d:

Inthisase,thesupportfuntion

Æ

(l;(1;)

0

1

[ 1;1℄)=j(1 ;1)lj

is only absolutely ontinuous, and its derivative is of bounded variation

uniformly foralll2R n

withklk

2

=1.

Hene,orderofonvergeneatmostequalto2anbeexpeted.

ThenumerialresultsinTable4.3wereomputedwiththeexpliitlyknown

fundamental solution,sothatnoerrorsourbytheapproximationofthe

fundamental solution. Nevertheless, we observe theexpeted onvergene

order for the rst method and a breakdown of the onvergene order of

ompositeSimpson'srule,whihisillustratedgraphiallyinFigure4.3.

(16)

−0.75 −0.50 −0.25 0.00 0.25 0.50 0.75

−1.5

−1.0

−0.5 0.0 0.5 1.0 1.5

−0.75 −0.50 −0.25 0.00 0.25 0.50 0.75

−1.5

−1.0

−0.5 0.0 0.5 1.0 1.5

−0.75 −0.50 −0.25 0.00 0.25 0.50 0.75

−1.5

−1.0

−0.5 0.0 0.5 1.0 1.5

Fig.4.3: Composite Simpson'srule withh=0:5;0:25

omparedwiththereferene set

(17)

Example 4.4 Considerthe linear ontrol system

y 0

(t) 2

0 1

2 3

y(t)+B

1

(0) for almostallt2[0;2℄;

y(0) =

0

with the losedunit ballB

1 (0)R

2

.

Thenthefundamental solutionis

(t;)=

2e (t )

e 2(t )

e (t )

e 2(t )

2e (t )

+2e 2(t )

e (t )

+2e 2(t )

;

and thereahableset attimeb=2is

2

Z

0

(2;)B

1 (0)d:

Inthisase,thesupportfuntion

Æ

(l;(2;)B

1

(0))=k(2;)

lk

2

is arbitrarilyoftendierentiablewith boundedderivativeswith respetto

uniformlyontheset fl2R 2

: klk

2

=1g.

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

−1.5

−1.0

−0.5 0.0 0.5 1.0 1.5

Fig.4.4: CompositeSimpson's ruleombinedwith Runge-Kutta(4)

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Fourth order of onvergene of omposite Simpson's rule is learly indi-

atedbyTable4.4 andillustratedbyFigure4.4, whereaveryroughstep-

size h = 0:5 gives a remarkably good approximation (whih nearly does

notdierfromthereferenesetwithinplotting preision). Comparingthe

resultsusingtheexpliitlyknownfundamentalsolutionwiththeombined

methods usinganumerialapproximationofthefundamental solution,we

observethesameorderofonvergeneinTable4.4,buthigherstartinger-

rors. Notiethatwehavehosenappropriatemethodsfortheomputation

of the fundamental solutionwhih have thesameorder of onvergene as

theintegrationmethod.

Table4.1: ResultsforExample4.1

approximated

numerialmethod stepsize Hausdordistane

omposite 2:0 1:9999999999999991

trapezoidalrule 1:0 0:5237537789937194

0:5 0:1325540105506304

0:25 0:0332417225019865

1:0 0:5237537789937194

0:1 0:0053233262580985

0:01 0:0000532420454213

0:001 0:0000005324213319

0:0001 0:0000000053242024

omposite 1:0 0:0316717053249587

Simpson'srule 0:5 0:0021577697845663

0:25 0:0001401261042604

1:0 0:0316717053249587

0:1 0:0000036242154220

0:01 0:0000000003630625

0:001 0:0000000000000480

(19)

approximated

omposite 1:0 3:9999999986840358

0:25 0:2077622028099686

0:2 0:1324688024984382

0:02 0:0013160325315322

0:002 0:0000131581652458

omposite 0:5 1:9056048962908503

Simpson'srule 0:25 0:4246439169047305

0:125 0:0876439543002339

0:2 0:1324688024984382

0:02 0:0026324128852804

0:002 0:0000263176810722

approximated

omposite 1:0 0:2254227652525390

0:25 0:0155000388904730

0:125 0:0038983702734540

0:1 0:0023997874829183

0:01 0:0000246670782419

0:001 0:0000002481024457

0:0001 0:0000000023598524

omposite 0:5 0:0686732300277554

Simpson'srule 0:25 0:0180416645285239

0:125 0:0049492014035796

0:0625 0:0012651981868981

0:1 0:0026219672162807

0:01 0:0000263963285972

0:001 0:0000002264152427

0:0001 0:0000000016145246

(20)

appoximated

omposite 1:0 0:9433330463362816

0:01 0:0000243687539239

0:001 0:0000002436654987

0:0001 0:0000000024125214

omposite 1:0 2:5354954374884917

ombinedwith 0:01 0:0000496413103789

themethodof 0:001 0:0000004919838970

Euler-Cauhy 0:0001 0:0000000048984064

omposite 1:0 0:1335888228107664

Simpson'srule 0:5 0:0224859067427672

0:25 0:0016216053482911

0:125 0:0000845026154785

1:0 0:1335888228107664

0:1 0:0000332142695469

0:01 0:0000000030953542

omposite 1:0 0:5738839013456635

Simpson'srule 0:5 0:0130316902023255

ombinedwith 0:25 0:0008327343054384

Runge-Kutta(4) 0:125 0:0000457276981323

1:0 0:5738839013456635

0:1 0:0000180766372746

0:01 0:0000000018105748

(21)

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RobertBaierandFrankLempio

LehrstuhlfurAngewandteMathematik

UniversitatBayreuth

Postfah101251

D-8580Bayreuth

Germany