A stability result for parameter identification problems in nonlinear parabolic problems

A STABILITY RESULT FOR PARAMETER

IDENTIFICATION

PROBLEMS IN NONUNEAR PARABOLIC PROBLEMS

STANISLAW

MIGRSKI

Ja,gclloni,University ul. Naxwjki 11,30072 Cracow

(Received Hatch 15, 1993)

ABSTRACT. Asequencefidmttificatim

lrollems

of coecients in theparallic equationwitlt n<mlinearboundaryconditims is cmsidered. Theparameter(indexofanelement of thesequence) appears in thecost flnctionals aswell as},mndary data. It is pr,vcdthat tlte optimalsolutions exist anltha.t under Somecontinuouscvergmicefthe cost functima.lsand the convergenceof thedata, the setsofoptinml solutims cmverge insne smse to theset <f <ptinmlsolutions of the linit pr,bln,:

KEY WORDS AND PtIRASES.

Invm’se

1,r,l,lmt, system i(lelit-ific.ation, stability of sflutions,

1,arabolicdifferentialeqmtion,n,mlinom"l,mnda.rycondition. 1992AMS

SUBJECT

CLASSIFICATION.

35R3{),

35I(60.

1. INTRODUCTION.

In

therecent years tlmrehas

been.

anincreasing

interestin

the

imrameter

identification

(or

inverse)

problems

inw,lvig

differential equation constraints. Suchproblems arise inparticularin the coefficient estimationfor partiM differentiMequati,ms

(for

examplein

[2-3],

[14],

[17-18])s

wellasinthe theoryofstructural optimization.The identificationproblemsconsist indctennining

ofunknown parameters

(coefficients)

fromknownol,scrvationsofthemodelled processes.

In this paperweinvestigateaclass ofidentification

probhems

fi.r

thesecond ordernonlincar

parabolic system:

u’= -A(t).

in fl x

(0, T),

(1.1)

+

fl(u)

gg in

r

x

(O,T),

(1.2)

u(O)

o

in

,

(1.3)

where fl C R" withboundary

F,

0 <

T < +oo,

fl

is a maxi,nalmonotone gral)hinR x

R,

the

operator

A(t)

has the form

o

(,,

a)

and

o,,(,)

"(*’10,.

"

W (see

N()tati()n)

()f th(.: s()ltti(ns (.()(1.1) (1.3), we ;re itereste(l i, tl,e f()ll(,wig l)a.runeter

i(lentifictti()u

()

whereu(a,

,

)

denotestleweaks()luti()ui c()rresI)(),[ingto the(lath.n.,gan[

.

Here

a([

i whatfi)llows, weSll)p()setheft tlegral)h

fl

is fixed.

Our [dm is tol)r()ve the existence of()l)tinml s()lt.i()us

(i.e.n.n

clenet which realizes the

niuimnu) t()

()

udtosl()w tle stability()f_()t)tiuMs()lti()ns u(l(.rIWrtnrl)ati()usofthe g mt(l

:

welltmoftl,.,,’,,st ftutcti,,,tM_{ft. As} in(li,’n.te,l it,

[I],

[4-5]

,m,l

[I0-11]

the st.,d,ility

thiskind i)laysanimportant rolein

It sltould be noticed that a c()nxptu(.,ss ofa(hnissible subset of

lint&meters

is the cruciltl

tssunl>tion

in the identific[,tion

l)rol)lems (c()ml)tre

e.g.

[5]).

Sinee the cost fimction is not

c()nvex ingenerd) theuniq,enessof(.)i)tinml solutions isnot guarant(I.Therefot the stability

isunderstood in thesenseof continnity ofmttltivtd,edmtpl)ing.

Wenotethat the widelykn()wnq)pron.(’h tothe pn.rmneter identifictttioni)rol)lemsusedalto) i,, n,,m(,ric,,l methods

(see

[4-51,

[141)

is th,, outp,t le,tst sqnm’es formuh,tion

([10-11],

[13]).

In

tiffsal)l)roa.ch the costfimcti()md to I)eminiaturizedh.s the

whereC: W

Z

is a.nobservationol)erttordefined on thesl)a.eeof sohtions,

za

is the desired element

(target)

in the space ofol)servations Z. Such cases tre also included in the fi’e of the paper.

Next,

it should I)e underlined that theidentification of coefficients in partiM

differ-entid equttions is,in genend, mmst,,.I,le

l)r,.,1)lem

([1G-17],

[13]).

This is d,e to the theoryof hom()genization

([8],

[18])

which slows thnt,)perators with highly oscillatorycoefficientsc

"rel)lace(l"

by verydiflhrentones a.n(l stillgiving thesameresponse.

Finally,wei)ointoutthat tle

l)rol,lems

(:)fthetype

(1.1)

(1.3)

occurinmany mathematical models of

l)henomenu

studied in physics.

For exnml)le,

eq,ation

(1.1)

(lesebes the change of pressure during the flow ofviscous fluids in porous mediaor it governs the heat distribution

in a body occupying the w)lume ft.

It

is nt,nd to consider such problenm not only with the classical

(Dirichlet

and/or

Neumam)

I>oun(htry conditions, buttlsc)with themore

gener

ones.

The boundary condition

(1.2)

includes some particuhtr cses e.g. the Signorini condition, the

Stefan-Boltzmn

heat radiation law, the.Newt()n’slaw, the natural convection, the Midlis-Menten law. For these and otherimportmt examples of the condition

(1.2)

which appem-s in

mechanics, biologymdchemistry,werefer to

[12], [7], [9]

m(!thebibliographyinthem. We

recall that theevolution vtrittional iuequditiescanbeformulated inthe fi)rm

(1.1)

(1.3) (see

[6], [9],

[12],

[15]).

Forthegeneralidentification theoryi)resentedin an’abstractrammerwerefer

Theremainder ofthis note is divided in three parts.

In

Section 2 we give a sult on the

continuous dependence of solution to b(,un,htry vdue problem

(1.1)-

(1.3)

on the data. With this background, in Section 3, weshow that the problem

(P)

ha s()lution. The lt section is devoted tothe stabilityofoptimalsolutionsxvithspect to variations inthe givendata andthe

cost fimetional.

Notation. Let flbeaboundedopensubsetof

R"

withLipsehitzcontinuousboundm’y

F.

For

with rcp:ct t< t,is un<lert<,<,l ilthe dit, ri],tti<,lmleileand

dvJd

will]edel<te<lIy v

.

Tim

st.anl ht"tl,eim,er

l)rdtct

ii

L(F).

Givel,cl:Iets

A

alld

B

hla

Banach

space

X,

wedc6no

tl,edistance f,,,cti,,,,1,y ,I(.,:,A)=

+,,.1{11.,-

,,ll.x-

,,

A}

,,,,,

t.l,,

,,,,,,.,,ti,,,,

,,f ,,t st D O’

1,’(A,B)=

.s,,.+,{,(,,,

B)" a

A}.

(.5)

Thrttghtt thisl>l>era. s,aticnx’tmtitver

t,l,;te!

sdscrilt.s

is

a.dlt.etl.

2.

CONTINUOUS DEPENDENCE

ON

TIIE DATA.

The goal of this sectin is t study the qtestin ofcontitmous

dependence

f solutions t)

(1.1)

(1.3)

onthe data aij,9 m,l

:.

First wegive tire existence result n tire slutions to tltis

ltllem.

Tothisenl,weathqt thefllwi,g

DEFINITION

2.1.

(see

[7])

A

fimcti,,n u is avea.k s,,h,ti,,n t,o

(1.1)

(1.3)

if

if there existsafiutctin w

L()

such t.l,t

,,,(,, t)

/(,,.(-, t))

,,..,. ,,,,

C,

(2.

)

alld

<

.,,’(t),,,

>

+,,(t;.,,(t),,,)+

<

.,,(t),,

>v=<

/(t),.,

>r

w,

e

v,,,.c.

,.,,

(0,T),

-(())

:,

wherewehave set

a(t;z,v)

aij(x,t),lx,

Vz,,,

e

V,

a.e.

e

(0,

T).

(2.2)

Weneed thefollowinghypotheses,,n tltcdata,fthe1-,rollem

(1.1)

(1.3):

(H,)

the coefficients

{aij},

i,j 1,...,n arefunctions fi’om

C(Q)

such that

a’ii

aij(:t,t)ij,

V

e

R"

(2.3)

a.e. in

Q,

forsomeconstant a

>

0,

(Hz)

isamaximM

(multivalued)

m,,notonegraph in

R

/ Rwhich satisfies

the condition 0

fl(0),

Itiswellknown

(see

e.g.

[61)

that

flis

asubdifferentialofaproper,convex,l.s.c,fimction

PROPOSITION 2.1.

In

addition to the hypotleses

(H)- (Hs)

we sume that j()

L(fl).

Then theproblem

(1.1)

(1.3)

hasaunique weaksolution u 6

W.

Morover,

the following

estimateholds:

II-IIw

+

I1,,’11<=)

(

+

Ilgll)

+

I1),

(2.4)

wherew6

LZ()

is aselectionof

fl

xvhich apl)earsin

(2.1)

andc

R

iindependentofg d

Theproofof this resultcanbe foundin

[7],

Proposition 1, where the overMlhypotheson

the coefficients aij were more restrictive.

A

careful look in that proofcan convince the reader thatit isstill truefor theceof coefficients satisfying

(H).

Let

{ai},

k6

N,

beasequence in

C()

satisfying

(2.3)

uniformlywithrespect tok and let

(1.3)

correspondingto

{aj

},

{(g,

)},

wehave

LEMMA

2.1. Under the ubove m,_ttttions,let us a.ssume that stisfies

(Hz)

dj()

L’().

It

(9,,

t,)

(9,

t2)

in

L2(E)

x

H,

as k +c, tlen

"t- ,t i

V

f3C’(0,

T;

H) a,lw,akly in

W,

,s _k

+c,

_{where L u(-i.,g,) is tiilu, s<lti (still it tl, sesc}fDefiit,im

2.1)

(1.1)-

(1.3) corrt,sp,,n,ling t,_ai.i, g a,l

.

PR()OE. N,te that iy h,wer senicott,iudty ,fj, it fi,llows tha.t j()

L().

Tlwrefore,

fr, m Pr,l,<,it.i<,u 2.1, wek,w tl,t t.l,weexists a mdqesolti,n to

(1.1)

(1.3)

c,,rresp,nding

to _aij,y alibi

.

Sulstr;t.ctiug tw,.qmt.ios whichItresatisfiedby

u

and u, weobtain

<

.,,()-

.,,’(.),.,,

>

+ ,(;.,,(.,,,)

+

<

,,,(.)-

,,,(),,

>r=

<

.()- (.),

>r

fi,r all v

V,

a.e. s (0, T), where’iv,,’k

L"(E)

aresm’h tlat

,,,(,}

e

z(.,,(,}),

.,,,{,)

f(u(,})

..

,,,-anda(.;

.,.), a(.;.,

.)

areofthe form

(2.2)

wit,h hecoecients ai,

a},

respectively.

Taking

u(s)

u(:)

theflmcti,n vaulintgrating lmth sidesweget:

lu(t)-,,(t)

+

,,(s;..(.s)-,,(),,,(s)-,,(s))ds +

+

< ,,()

,,,(),,,()

,()

>t. d.

gl

1

+

,,(;

,,(),,,()

,,())

k(;

,(), ,,/)

,).d]

+

(1

(.), ,)

)

r

ds.

Hence,

from

(H)

and

(t12)

we c)l.)tain:

fo’

’

o,,()

o.,,,,(s)

0.,()

+

(,’

,,)0,,

!1.0.i

0.

Id

+

Uing thecontinuity of the trace operator from

V

to

L(P)

madtiminequMity 2ab

N

a

+

b

wehave

lug(t)-,,(t)l

+,

II,,(s)-

u(s)llas

N

I

-1 +

/c.,

Ilull

I1"

t)

"ollc.(

+c:

I()

whereci, 1,2arepositiveconstructs indepen(lentof k.l’om this,

(2.4)

andfrom thehypotheses

wededuce that

u,u in

C(O,T;H),

as

k-l-c.

On

theother

hand,

owing to the estimate

(2.4),

which istmifortnwith respect to

k,

weobtain ut u weaklyin

3. EXISTEN(:E IIES[!I:I’ IN I1)I);NT1FI(IATIt)N.

Using tile rcstlt ,f I.(lna _2.1, w’. ,’,1 _{ln’,x.’’} t.l .xisl.,lc,,f sltt,in ,,f t.lm i,lettificati

lrl,l,n (D). Tlis isgiven i 11, till,wig

TItE()I/EM 3.1. I,’.t tl:

tsstttll.is

,t"

lrlsit.im

2.1 ltll. W, _Sttlltst’ t.lat the set f

almissillcltra,mt,’.rssatisti,b

L,t tle cst 5mctimal

l"

1 R1.,_{w,kly squ,nt.ially ltwerscmicmtimns}m

.

Then the

lrolhn

(’P)

has asllt,in.

PR()O. XXhe,l,l,lytlcdir,ct (’t,l,l ,fc.ad’ths,f’variat.i,,ts

(see

_e.g.

[1]).

Let 9, satisfy

(Ha) a,llet

{a}

1,e titiizig s,,le,,’t’r,

,

scltl,t

lint ,:/(u(,.,.,.q,9))

id’{,.’l(,,.(-.,9,)) -.

Ad}

),)..

Sin(.,c .<t.Iis c,)ml)act, tl(:’.r(.:exists a. std,s(qw.ceof tle seqteltCe

{a,},

rehd)eled again as

{a,},

<1a ,u 3d swh that (,. a(). It tl)ll()ws fl’<)m L(unnla.2.1 that inI)a.rticflar

R.EMARI, 3.1. Ingeneral, v,,itl,nt_coavxity a.ss,nqtins, wedo notexpect uniquenessof theol)timal soluti,,nin ide,tifica.ti,,

([1],

[16]).

4. STABILITY RESULT.

In this section we give the m,in rcsflt f this.lml)er on the dependence

(hence

also the

stability)of theoptimal clemntsfortheln’,,1,h:nn

(’P)

onthe dataa.swellasonthecost functional. We consider tlmsequence

(inlexed

by the

lm.rmneter

k

N)

oftheidentification problems:

find

a*={,,iS}

in sothat

()

where

u(a,g,k)

are the s,,luti,)ns in

to(1.1)

(1.3)

corresponding to the perturbed data

gk,k and

k

are theperturbedfunctionals. We show that theset of optima.1 solutions to

(Pk)

convergesinsonesense,to thesetof optimal solution to

(P).

We

need the following cmtimos convergence of5mctionals. Let

(A’,r)

be a topological

space and

k

:,l" R.

DEFINITION 4.1. We sa.y tha.t a. sepencc of functionals

{},

k

N,

is sequential continuously convergent (shortly,

C,-com,

erges) t,:, andwewrite

C,,(r-

X)

lim iffor everyx ,V andfor every

{xk}

C ,Vwhi,:h_r-c,,nvcrgest,,x,the sequence

k(x)

converges t,,

:(.,).

For

each k

N,

wedenoteby

S,

$ thesetsof optimal elementsto the

proMen

(P),

(P),

respectively,i.e.

li, l, $,,$ {},

(4.3)

wl.,rc h*(.,-)is1.,fi,,1i (1.5).

PR()()F. Wearg’

I,y

c,t.ri,lict,i. If(,t.3)is ,t,vcrifie,!t,l.u’eexistg> 0andasequeuce

I,:,, },

k

+

s,:l t.lt

<

h*(S.,.,S),

V

k.

N.

(’,lrly, thereexist

a.

S,.

sucl t.lt

<

d(,.L.,S

), V

:,.

c

N.

(4.4)

In view ,,fc,mpactuess,ff

M,

we !,.,1,’,tl.t tluccxist, _ulscqucnce of

{a.

},

that wewill

d:nt,e iuthe ameway,wh tlu.t

fi,rsomea*

M.

Let,ow

u.

u(a.,

_g.,

,.

)andu* ..(.*,g,),h.,n,,tethes,:,hti,,ns t,,

(1.1)- (1.3)

which c,,rresl,oul tothe triph,s

(..,.,

9,.,

.,.

)a,l("*,9,

),

r"sl’cctively. Fr,,m (4.1) and

(4.8)

2.1 gives

u.

u* w,a.ldy iu

W,

as

k

+.

Since the fimctiouals

d

C’,:,-’nv’rg,s

t

J,

wc

J(.,,’)

i,,:h,.(,,L

).

(4.6)

Le

us fix auarbitrary

.

M.

Let ’u,.

u(.,9,,,.)

and u

u(a,9

)

be the solutions

(1.1)

(1.3)

with theinlicateddata. Frcm theccmt,_{iuts lepeudeuce}ou thedata,weconclude that

u.

’u wcaklyiu

W,

as

kv +.

Hence,

asabove,wehave

Since

a.,.

arein

S,,,,

wehave

J(,t)

liu

Jr,,.

(u,,.).

(4.7)

In

the limit,as k,,

+oo,

onegetsfrom

(4.6), (4.7)

s(,,(,,’,,j,))

_<

s(,,(,,,,)), v

,,

e

.

Fromthe arbitrarietyofa

,

welu.w,* S and thisimpliesthat

,(,,Z.,,s)

II,L

--*11.

(4.8)

But nowfrom

(4.5),

we,:,btain tha.t

(4.8)

c,.,tra,licts

(4.4).

Thisproves

(4.3)

,! _completes the

’-FI; c,verg’n’e (4.2) lulls, f’,v itt’’, f’w tl,’ _{’tu’c f"}f’uct, iu;,lff t.h’ fiwn

(1.).

t"

l’rtuled

listrilnttetl ols.l’vatis il "H cv.rgilg tza, xve havthat(4.2),is satisfiedwith ,1 tiffseml,ed,lingis c,nl,;,ct. (c,,l,a,v,’

[15]).

in..L

k

+,

tlmn usig tlu cnxqatct,t,ss,ftlt tx’ace ’nl,llitgfi’m W int

L2(E),

weeasily gt

REhIARK 4.1. ()te c g’u’ra,lize t.l" ’,slts _lres,nt,.l al,w to t,lecasewhen

A(t)

isa litt:reti;l_ol)erat{w tf timt)

0 0

()tlrt.lu:vy,withsmeniuwchngcs,cntltntlle ittx;’rs,

lnlh’tns

inwlving theidetifica.tion of

REIARK

4.2.

A

futher gcter;,liza.t.i f nr r,slts can lm oltained considering the ilentificatim prolh-.s fr

(1.1)

(1.a) witl the ixl lnmtlary cmditicns

instea(l ,,f

(1.2),

wheregi

Pi

(0,

T)

ani

Pi

arethe (lisj,int parts ofF. Theexactfi)nmdatim

ftheresultswith olviCnts mliticatims is h,f’t t, the: ilfl.:rested reader.

ACKN()WLEDGEIvlENT. TI,= l,al)(’r ws wtitt,n wlfih: tim mth)t" was visiting the Scmla Normale Sul)eriore di Pisa and was S,tl)l))rtel I)y tlu: Istitut Naziniale (li Alta Matematica FrancescoSeveri,

Rome,

Italy.Tiffs w()rkwas1)a,rtiallytim(ledbyaNational Research Projectin Mathematics"Eqmzioni Differenziali eCa.lc()h) dell(: \’aria.zioni" fthe Ministero(lell’Universit,4 e(ldlaRicerca ScientificaeTecn)hgia.,40% c(mtra.cts.

REFERENCES

AHMED,N.U.

Qptinfizat,iota,t____[IIdett, ifica.t.i,n ,fSystemsGv,,ened

__

Ew’l,,tionEq,,ations

onBana(:lt Space, Pitman R,s(’,.rch N()U:s i,t M,.t.hema.tics 184,

Longman,

Boston

London- Melbourne, 1988.

BALAKRISHNAN,

A.V. hlentificati()t,fdistrilmtedparameter systmns: non-computational aspects, Proceedings

IFIP

\;VorkigC)nference, 1R(me, 1976, A.R.uberti,Ed., Lecture NotesinControl andhfl)rnm.tion_{Sc.ienccs,\&)l.1, Springer-Verlag, Berlin, 1978,}1-10.

BAMBERGER, A., CHAVENT,

G. and

LAILLY,

P. About the stability of the inverse problemin1-Dwaveequati(ms apl)licati(ms totheinterpretatim ofseismicprofiles, Appl. Ma.tl. O!)t.i. 5 (1979), 1-47.

BANKS,

H.T. On a va.riati()lal a.1)l)r()ach t()somepa.rmneterestimationprol)lems, ICASE

Rep.

No. 85-32, NASA Langley Res.

Ctr.,

Ha.nl)t(m,

VA, June,

1985.

BANKS,

H.T.

and

ILES,

D.W. OnCOml)actness ofa(lmissil)leparametersets, Proceedings

BARBU,

V. B,,ulary c<,xtt.rl _lw,i,l’t. wi/l ,tli’tr st.;t: _{.lti<t, SIAM} ,l. Ct.rl

()ptil. 2f.._)) (1982), 125-1-t3

9.

BREZIS,

It. Pr(l,h.l(:s_{lil;,t(’r,lx,} .l_. F,I,II,.Pir(.s

A!!)I.

5_.j.l (1972), 1-168.

10.

11.

1"2.

13.

14.

15.

16.

17.

18.

C()L()NIUS,

F. (l

KUNISCII,

_{K. ()ttl)t h:ast squares stal)ility fi)r estimationof the} (liihsi()n (’()(Ili(:i(,nt _{in (,llil)tic C(luti()}

Pr()(-(.(li IFIP W()rki ()Itferece,

Gainsvillc, 1986, l.Lnsi(..(-k, R.Triggia.6 E(Is., L(.wtu’c N()tesin C(mtr()la(l Inf()nm-ti(m_{S(’i(ttc(s, V(,I.97, Si)rig(.r-V(wlg, B(’rli, 1987, 185-195.}

C()L()NIUS,

F. ii(l KUNISC,

II,

K. Stld.)ility f()r Pm’a.n(.:t(r Estimation in Tw() P()it

B()n,lary\rl,, l"r()l,h.,s, 3__{R__,,i),(:A__.’(.w.} hl,tl. 37()

(1986),

_1-29.

Paris, 1972.

KRAVARIS,

C,.m!

SEINFELD,

J.II. hl(:ntifica.i()t()fl)a.rameters in(listril)ute.(!I)armnct(:r

systcuis

I,y

r(:g,da.rizati,,n, SIAMJ. C()tt.r()l ()!)t,io23

(1985),

217-241.

KUNISCII,

K. I,h.:(.ificiti() l(lEstila.tii)n()fl-’ara.neters in Abstract Ca.u(,hyProblemu,

Matlmnuai(’I C,>t.r()l Tl,.,)ry. ll;,la,q C(’t,’.r Pl)li(’ati,,n, V,)1.14.,

PWN, Warsaw,

1985, 277-298.

LIONS,

J.L. K’.t.l,o(h,s (I(,_2 rSs()lti(, (h,s l)r()l)l(,n(:s ____x linfit.es ()n lin(:.aires

Dm()(l-Gttli(:r Villrs, P,ris, 1969.

LIONS,

J.L. S,)ne Aspc.cts (,t’ Mo(h:lli,g Pr,)l,lems in Distri]mted Parameter Systems,

Procee(ligs

IFIP

W(,rking(-_’.,)fi’(,.’e,

Rome,

1976, A.R.uberti, E(l., Lecture N)tcs

in Contr()landhff(>rnn.ti(,n S:i(,(:(:s, V()I.I,Sl)ringer-Verhtg, Berlin, 1978, 11-41.

MUR.AT,

F. Contre-cx,:tq,h:sp(,tr(liversl)rolA(,m(:s(nile(’(mtroh:intervient(hmsles

co(,f-ficients

A,. M;,t. I>un. A)121I.11__2

(1977),

,19-68.