25
Internat. J. Math. & Math. Sci.
VOL. 18 NO. (1995) 25-32
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. A sequence f idmttificatim lrollems of coecients in the parallic equation witlt
n<mlinear boundary conditims is cmsidered. The parameter (index of an element of the sequence)
appears in the cost flnctionals as well as },mndary data. It is pr,vcd that tlte optimal solutions
exist anltha.t under Some continuous cvergmice f the cost functima.ls and the convergence of
the data, the sets of optinml solutims cmverge in sne smse to the set <f <ptinml solutions of
the linit pr,bln,:
KEY WORDS AND PtIRASES. Invm’se 1,r,l,lmt, system i(lelit-ific.ation, stability of sflutions,
1,arabolic differential eqmtion, n,mlinom" l,mnda.ry condition.
1992 AMS SUBJECT CLASSIFICATION. 35R3{), 35I(60.
INTRODUCTION.
In the recent years tlmre has been. an increasing interestin the imrameter identification (or
inverse) problems inw,lvig differential equation constraints. Such problems arise in particular in
1.
the coefficient estimation for partiM differentiM equati,ms (for example in [2-3], [14], [17-18])s
wellas in the theory of structural optimization. The identification problems consist in dctennining
of unknown parameters (coefficients) from known ol,scrvations of the modelled processes.
In this paper we investigate a class of identification probhems fi.r the second order nonlincar
parabolic system:
u’= -A(t).
in fl x
+ fl(u) g g
u(O)
o
in
where fl C R" with boundary F, 0 < T < +oo,
operator A(t) has the form
in
(0, T),
(1.1)
r x (O,T),
(1.2)
,
fl
o (,,
(1.3)
is a maxi,nal monotone gral)h in R x
R, the
a)
and
o,,(,)
"(*’10,. "
is the conormal derivative sociated to A(t). Above, v is the unit outward normal vector to P.
Given the set of admissible parameters
and the cost fltnctional
defined on the space
26
S.
W (see N()tati()n) ()f
th(.: s()ltti(ns
(.()(1.1)
MIG6RSKI
(1.3),
we
;re
itereste(l i, tl,e
f()ll(,wig l)a.runeter
i(lentifictti()u
()
,
.
Here a([
where u(a, ) denotes tle weak s()luti()u i
c()rresI)(),[ing to the (lath. n., g an[
i what fi)llows, we Sll)p()se theft tle gral)h fl is fixed.
Our [dm is to l)r()ve the existence of ()l)tinml s()lt.i()us (i.e.n.n clenet which realizes the
niuimnu) t() () ud to sl()w tle stability ()f ()t)tiuM s()lti()ns u(l(.r IWrtnrl)ati()us of the
well tm of tl,., ,’,,st ftutcti,,,tM ft. As in(li,’n.te,l it, [I], [4-5] ,m,l [I0-11] the st.,d,ility
g mt(l :
this kind i)lays an important role in
It sltould be noticed that a c()nxptu(.,ss of a(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 in generd) the uniq,eness of (.)i)tinml solutions is not guarant(I. Therefot the stability
is understood in the sense of continnity of mttltivtd,ed mtpl)ing.
We note that the widely kn()wn q)pron.(’h to the pn.rmneter identifictttion i)rol)lems used alto)
i,, n,,m(,ric,,l methods (see [4-51, [141) is th,, outp,t le,tst sqnm’es formuh,tion ([10-11], [13]). In
tiffs al)l)roa.ch the cost fimcti()md to I)e miniaturized h.s the
where C: W
Z is a.n observation ol)erttor defined on the sl)a.ee of sohtions, za is the desired
element (target) in the space of ol)servations Z. Such cases tre also included in the fi’e of
the paper. Next, it should I)e underlined that the identification of coefficients in partiM differentid equttions is, in genend, m mst,,.I,le l)r,.,1)lem ([1G-17], [13]). This is d,e to the theory of
hom()genization ([8], [18]) which slows thnt ,)perators with highly oscillatory coefficients c
"rel)lace(l" by very diflhrent ones a.n(l still giving the same response.
Finally, we i)oint out that tle l)rol,lems (:)f the type (1.1) (1.3) occur in many mathematical
models of l)henomenu studied in physics. For exnml)le, eq,ation (1.1) (lesebes the change of
pressure during the flow of viscous fluids in porous media or 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, but tlsc) with the more 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’s law, the natural convection, the MidlisMenten law. For these and other importmt examples of the condition (1.2) which appem-s in
mechanics, biology md chemistry, we refer to [12], [7], [9] m(! the bibliography in them. We
recall that the evolution vtrittional iuequdities can be formulated in the fi)rm (1.1) (1.3) (see
[6], [9], [12], [15]). For the general identification theory i)resented in an’abstract rammer we refer
The remainder of this 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, we show that the problem (P) h a s()lution. The lt section is
devoted to the stability of optimal solutions xvith spect to variations in the given data and the
cost fimetional.
Notation. Let fl be a bounded open subset of R" with Lipsehitz continuous boundm’y F. For
H
La(fl),
by [[.[[, [.[, the norms in V and H, respectively. By V’ we denote the
dnal to V. Following e.g. [15], we define the Banach spaces: Y La(0, T; V), H La(0,T; H),
Y’ La(O, T; V’) (the spaces of the square s,mmmble fi,nctions defined on (0, T) with the vMues,
we denote
27
NONLINEAR PARABOLIC PROBLEMS
with rcp:ct t< t, is un<lert<,<,l il the dit, ri],tti<,lml
st
D O’
.
will ]e del<te<l Iy v Tim
,,,,,
ii
1,’(A, B)= .s,,.+,{,(,,, B)"
Thrttghtt this l>l>er
2.
dvJd
L(F). Givel, cl:I ets A alld B hl a Banach space X, we dc6no
t.l,, ,,,,,,.,,ti,,,, ,,f ,,t
1,y ,I(.,:,A)= +,,.1{11.,- ,,ll.x- ,, A}
st.anl ht" tl,e im,er l)rdtct
tl,e distance f,,,cti,,,,
and
eile
a. s,ati cnx’tmtit ver
a
(.5)
A}.
t,l,;te! sdscrilt.s
is
a.dlt.etl.
CONTINUOUS DEPENDENCE ON TIIE DATA.
The goal of this sectin is t study the qtestin of contitmous dependence f solutions t)
(1.1) (1.3) on the data aij, 9 m,l :. First we give tire existence result n tire slutions to tltis
ltllem. To this enl, we athqt the fllwi,g
is a vea.k s,,h,ti,,n t,o
DEFINITION 2.1. (see [7]) A fimcti,,n u
if there exists a fiutctin w L() such t.l,t
,,,(,, t) /(,,.(-, t))
alld
< .,,’(t),,, >
,,..,. ,,,,
(1.1) (1.3) if
(2. )
C,
,.,,
+,,(t;.,,(t),,,)+ < .,,(t),, >v=< /(t),., >r w, e v,,,.c. (0,T),
-(())
:,
where we have set
a(t;z,v)
aij(x,t),lx,
We need the following hypotheses
(H,)
(Hz)
Vz,,, e V,
a.e.
e (0, T).
(2.2)
(1.1) (1.3):
the coefficients {aij}, i,j 1,...,n are functions fi’om C(Q) such that
a’ii aij(:t,t)ij, V e R"
a.e. in Q, for some constant a > 0,
is a maximM (multivalued) m,,notone graph in R / R which satisfies
the condition 0 fl(0),
,,n
tltc data ,f the 1-,rollem
(2.3)
It is well known (see e.g. [61) that flis a subdifferential of a proper, convex, l.s.c, fimction
PROPOSITION 2.1. In addition to the hypotleses (H)- (Hs) we sume that j()
Then the problem (1.1) (1.3) has a unique weak solution u 6 W. Morover, the following
L(fl).
estimate holds:
II-IIw + I1,,’11<=)
( + Ilgll) + I1),
(2.4)
LZ() is a selection of fl xvhich apl)ears in (2.1) and c R i independent of g d
The proof of this result can be found in [7], Proposition 1, where the overMl hypothes on
where w 6
the coefficients aij were more restrictive. A careful look in that proof can convince the reader
that it is still true for the ce of coefficients satisfying (H).
Let
{ai}, k 6 N, be a sequence in C() satisfying (2.3) uniformly with respect to k and let
(1.3) corresponding to {aj }, {(g, )}, we have
LEMMA 2.1.
L’(). It
Under the ubove
ct
m, ttttions,
aii
in
let us a.ssume that
C(),
i,j,
stisfies
(Hz) d j()
S.
28
(9,, t,)
as
MIG6RSKI
(9, t2) in
L2(E) x H,
+c, tlen
k
"t-
,t
i
V f3 C’(0, T; H)
w,akly in W,
a,l
.
k
+c, where L u(-i.,g,) is tiilu, s<lti (still it
(1.1)- (1.3) corrt,sp,,n,ling t, ai.i, g a,l
,s
tl, sesc f Defiit, im
2.1)
N,te that iy h,wer senicott,iudty ,fj, it fi,llows tha.t j() L(). Tlwrefore,
Pr,l,<,it.i<,u 2.1, we k,w tl,t t.l,we exists a mdqe solti,n to (1.1) (1.3) c,,rresp,nding
to aij, y alibi
Sulstr;t.ctiug tw, .qmt.ios which Itre satisfied by u and u, we obtain
PR()OE.
fr, m
.
< .,,()- .,,’(.),.,, > + ,(;.,,(.,,,)
+ < ,,,(.)- ,,,(),, >r= < .()- (.), >r
fi,r all v
V,
(0, T), where
a.e. s
’iv,,’k
L"(E)
are sm’h tlat
,,,(,} e z(.,,(,}), .,,,{,) f(u(,})
..
,,,-
and a(.; .,.), a(.;., .) are of the form (2.2) wit, h he coecients ai, a}, respectively.
the flmcti,n v aul intgrating lmth sides we get:
Taking u(s) u(:)
,,(s;..(.s)-,,(),,,(s)-,,(s))ds +
lu(t)-,,(t) +
+
< ,,() ,,,(),,,()
,() >t. d.
k(; ,(), ,,/)
,).d] +
gl 1 +
(1
,,(; ,,(),,,() ,,())
(.), ,) ) r ds.
Hence, from (H) and (t12) we c)l.)tain:
+
fo’
’
o,,() o.,,,,(s)
(,’ ,,)0,,
!1.0.i
Uing the continuity of the trace operator from V to
0.,()
0. Id +
L(P) mad tim inequMity 2ab N
a+
b we have
lug(t)-,,(t)l +,
/c.,
II,,(s)- u(s)llas N
I1"t) "ollc.(
Ilull
+c:
I -1 +
I()
where ci,
1,2 are positive constructs indepen(lent of k. l’om this, (2.4) and from the hypotheses
we deduce that
u,u in C(O,T;H), as k-l-c.
On the other hand, owing to the estimate (2.4), which is tmifortn with respect to k,
ut
u
we obtain
weakly in
The uniqueness of solutions of problem (1.1)- (1.3) implies that the whole sequence u converges
to u in the sense of (2.5). This completes the proof of the lemma.
29
NONLINEAR PARABOLIC PROBLEMS
EXISTEN(:E IIES[!I:I’
3.
IN
I1)I);NT1FI(IATIt)N.
Using tile rcstlt ,f I.(lna 2.1, w’. ,’,1
11, till,wig
ln’,x.’’
t.l .xisl.,lc, ,f sltt, in ,,f t.lm i,lettificati
lrl,l,n (D). Tlis is given i
I,’.t tl:
TItE()I/EM 3.1.
almissillc ltra,mt,’.rs
satisti,b
L,t tle cst 5mctimal
l" 1
tsstttll.is
R
1.,
,t"
W, Sttlltst’ t.lat the set f
lrlsit.im 2.1 ltll.
w,kly squ,nt.ially ltwer scmicmtimns m
.
Then the
lrolhn (’P) has a sllt, in.
PR()O.
a,l let
(Ha)
XXhe ,l,l,ly tlc dir,ct (’t,l,l ,f c.ad’ths ,f’ variat.i,,ts (see e.g.
1,e t itiizig s,,le,,’ t’r,
scl tl,t
,
{a}
lint
Sin(.,c
<1
.<t.I is c,)ml)act,
a ,u
,:/(u(,.,., .q, 9))
tl(:’.r(.: exists a.
3d swh that
(,.
a().
It
id’{,.’l(,,.(-.,9,)) -.
std,s(qw.ce of tle
seqteltCe
Ad}
satisfy
),)..
{a,},
tl)ll()ws fl’<)m L(unnla. 2.1 that in
[1]). Let 9,
rehd)eled again as
{a,},
I)a.rticflar
R.EMARI, 3.1. In general, v,,itl,nt coavxity a.ss,nqtins, we do not expect uniqueness of
the ol)timal soluti,,n in ide,tifica.ti,, ([1], [16]).
4.
STABILITY RESULT.
In this section we give the
rcsflt f this. lml)er on the dependence (hence also the
stability) of the optimal clemnts for the ln’,,1,h:nn (’P) on the data a.s well as on the cost functional.
We consider tlm sequence (inlexed by the lm.rmneter k N) of the identification problems:
()
m,in
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
and
the
are
We show that the set of optima.1 solutions to (Pk)
functionals.
perturbed
gk, k
k
converges in sone sense, to the set of optimal solution to (P).
We need the following cmtimos convergence of 5mctionals. Let (A’,r) be a topological
R.
space and k :,l"
DEFINITION 4.1.
We sa.y tha.t a. sepencc of functionals {}, k
N, is sequential
t,:,
and we write
continuously convergent (shortly, C,-com,erges)
C,,(r- X) lim
if for every x ,V and for every {xk} C ,V whi,:h r-c,,nvcrges t,, x, the sequence k(x) converges
t,,
:(.,).
For each k N, we denote by S, $ the sets of optimal elements to the proMen (P), (P),
respectively, i.e.
ith the above notation we have
MIG6RSKI
S.
30
li, l,
wl.,rc
h*(.,-)is
arg’
I,y
If (,t.3) is ,t, vcrifie,! t,l.u’e exist g > 0 and a sequeuce
c,t.ri,lict, i.
+ s,:l t.lt
I,:,, }, k
< h*(S.,.,S), V k.
(’,lrly, there exist
a.
S,.
,,f c,mpactuess ,ff
N.
sucl t.lt
< d(,.L.,S ), V
In view
(4.3)
{},
(1.5).
1.,fi,,1 i
We
PR()()F.
$,, $
M,
we !,.,1,’,
:,. c N.
tl.t tluc cxist,
(4.4)
ulscqucnce of
{a. }, that we will
d:nt,e iu the ame way, wh tlu.t
fi,r some
a*
M.
,.
u.
Let ,ow
u(a., g., )and u* ..(.*, g, ),h.,n,,te the s,:,hti,,ns t,, (1.1)- (1.3) which
)a,l ("*,9, ), r"sl’cctively. Fr,,m (4.1) and (4.8)
c,,rresl,oul to the triph,s (..,., 9,.,
.,.
2.1 gives
u* w,a.ldy iu W,
u.
d C’,:,-’nv’rg,s
Since the fimctiouals
J,
t
J(.,,’)
.
as
+.
k
wc
i,,:h,.(,,L ).
u(.,9,,,.) and
Le us fix au arbitrary
M. Let ’u,.
(1.1) (1.3) with the inlicated data. Frcm the
ccmt, iuts
(4.6)
u
u(a,9 ) be the
lepeudeuce
ou the
data,
solutions
we conclude
that
u.
Hence,
as
above,
’u
wcaklyiu W, as
we have
J(,t)
Since
a.,.
are in
In the limit,
S,,,, we
+oo, one gets from (4.6), (4.7)
s(,,(,,’,,j,)) _< s(,,(,,,,)), v ,, e
as k,,
,
we lu.w ,*
,(,,Z.,,s)
now from
(4.5),
liu Jr,,. (u,,.).
we ,:,btain tha.t
proof of the theorem.
(4.7)
have
From the arbitrariety of a
But
kv +.
(4.8)
.
S and this implies that
II,L --*11.
c,.,tra,licts
(4.4). This proves (4.3)
(4.8)
,! completes the
31
NONLINEAR PARABOLIC PROBLEMS
’-FI; c,verg’n’e
f’,v
(4.2) lulls,
itt’’, f’w tl,’ ’tu’c f" f’uct, iu;,l ff t.h’ fiwn (1.).
t" l’rtuled listrilnttetl ols.l’vatis il "H cv.rgilg t za,
,1 tiffs
xve
hav that(4.2), is satisfied with
eml,ed,ling is c,nl,;,ct. (c,,l,a,v,’ [15]).
in..L
k
+, tlmn usig tlu cnxqatct,t,ss
,f tlt tx’ace
’nl,llitg fi’m W int
L2(E),
we
easily gt
REhIARK 4.1. ()te c g’u’ra,lize t.l" ’,slts lres,nt,.l al,w to t,le case when A(t) is
ol)erat{w tf tim t)
0
0
a
litt:reti;l
()tlr
t.lu:vy, with
sme niuw
chngcs,
cnt
ltntlle ittx;’rs,
lnlh’tns inwlving the idetifica.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
(1.2), where gi Pi (0, T) ani Pi are the (lisj,int parts of F. The exact fi)nmdatim
f the results with olviCnts mliticatims is h,f’t t, the: ilfl.:rested reader.
instea(l ,,f
_
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
Francesco Severi, Rome, Italy. Tiffs w()rk was 1)a,rtially tim(led by a National Research Project in
ACKN()WLEDGEIvlENT.
TI,=
l,al)(’r ws wtitt,n
Mathematics "Eqmzioni Differenziali e Ca.lc()h) dell(: \’aria.zioni" f the Ministero (lell’Universit,4
e (ldla Ricerca Scientifica e Tecn)hgia., 40% c(mtra.cts.
REFERENCES
AHMED,N.U. Qptinfizat, iot a,t____[I Idett, ifica.t.i,n ,f Systems Gv,,ened
Space, Pitman
London- Melbourne, 1988.
on Bana(:lt
R,s(’,.rch N()U:s i,t M,.t.hema.tics
Ew’l,,tion Eq,,ations
184, Longman, Boston
BALAKRISHNAN, A.V.
hlentificati()t ,f distrilmted parameter systmns: non-computational
aspects, Proceedings IFIP \;Vorkig C)nference, 1R(me, 1976, A.R.uberti, Ed., Lecture
Notes in Control and hfl)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
problem in 1-D wave equati(ms apl)licati(ms to the interpretatim of seismic profiles,
Appl. Ma.tl. O!)t.i. 5 (1979), 1-47.
BANKS, H.T. On a va.riati()lal a.1)l)r()ach t() some pa.rmneter estimation prol)lems, ICASE
Rep. No. 85-32, NASA Langley Res. Ctr., Ha.nl)t(m, VA, June, 1985.
BANKS, H.T. and ILES, D.W. On COml)actness of a(lmissil)le parameter sets, Proceedings
IFIP W()rking Conf(.rell(’e, Gainsville, 1986, I.Lasiecka, R.Triggiani Eds., Lecture
Notes in Control and
142.
Info,ma.tim
Sci:n(’es, Vol.97, Springer-Vcrlag, Berlin, 1987, 130-
32
S.
BARBU, V.
B,,ulary c<,xtt.rl lw,i,l’t.
(1982), 125-1-t3
()ptil. 2f.._))
9.
10.
BREZIS, It.
MIGRSKI
Pr(l,h.l(:s
wi/l ,tli’tr st.;t: .lti<t,
lil;,t(’r,lx, .l_. F,I,II,.
Pir(.s
SIAM
,l.
Ct.rl
A!!)I. 5_.j.l (1972), 1-168.
C()L()NIUS, F.
(l KUNISCII, K.
()ttl)t h:ast squares stal)ility fi)r estimation of 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()tes in C(mtr()l a(l Inf()nmS(’i(ttc(s, V(,I.97, Si)rig(.r-V(wlg, B(’rli, 1987, 185-195.
ti(m
11.
C()L()NIUS, F.
ii(l KUNISC, II,
B()n,lary \rl,, l"r()l,h.,s,
K.
3_
Stld.)ility f()r Pm’a.n(.:t(r Estimation in Tw() P()it
R__,,i),(: A__.’(.w. hl,tl. 37() (1986), 1-29.
1"2.
Paris, 1972.
13.
KRAVARIS, C,. m! SEINFELD, J.II. hl(:ntifica.i()t ()f l)a.rameters in (listril)ute.(! I)armnct(:r
systcuis I,y r(:g, da.rizati,,n, SIAM J. C()tt.r()l ()!)t,io 23 (1985), 217-241.
14.
KUNISCII, K.
I,h.:(.ificiti() l(l Estila.tii)n ()f l-’ara.neters in Abstract Ca.u(,hy Problemu,
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.
15.
LIONS, J.L.
Dm()(l-Gttli(:r
K’.t.l,o(h,s (I(,_2 rSs()lti(, (h,s l)r()l)l(,n(:s ____x linfit.es
Villrs, P,ris, 1969.
16.
LIONS, J.L.
17.
MUR.AT, F. Contre-cx,:tq,h:s p(,tr (livers l)rolA(,m(:s (nile (’(mtroh: intervient
ficients A,. M;,t. I>un. A )121I. 11__2 (1977), ,19-68.
18.
MURAT, F. Control in c,),:ffi(’i(,nts, S,st(:s ,i(.___[l
Pergamon Press, 1989, 8()8-812.
()n
lin(:.aires
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()l and hff(>rnn.ti(,n S:i(,(:(:s, V()I.I, Sl)ringer-Verhtg, Berlin, 1978, 11-41.
C,()tr()I
(hms les co(,f-
Encvc!()l)e.lia M.G.Singh E(I.,