Heat Type

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Stability of Lipschitz Type ⁱⁿ

Determination of Initial Heat

Distribution*

SABUROU SAITOH

^aand

MASAHIRO YAMAMOTO

^b

Department

of Mathematics, Faculty of Engineering GunmaUniversity, Kiryu 376,

Japan

e-mail:ssaitoh@ ns518.eg.gunma-u.ac.jp

b

Department

of Mathematical Sciences, The University of Tokyo Komaba,

Meguro,

Tokyo153,

Japan

e-mail:

myama

@^tansei,cc.u-tokyo.ac.jp (Received12February1996)

Forthe solutionu(x,t) u(f)(x,t)of theequations

u’(x,t) zXu(x,t), x ft, >0

]

u(x,O) f^{(x), x ft}

u(x,t) O, xeOft, >O,

where ft C ]r, 2 < r < 3is abounded domain withCg-boundaryand foranappropriate subboundaryFof ft weproveaLipschitzestimate of IlfllL2(a) For/z (1,

45-)

^{and fora}

positiveconstantC

Thenorm IlB.(r(0,oo))is involved andstrong,but it isanaturalonein oursituation relating to atypicalandsimplenormforanalyticfunctions.Furthermore, it isacceptable^{in thesense} that B.(r(0,o

-<

ClIflIH(a^holds.

*Dedicated^toProfessorKyuyaMasudaonthe occasion of his 60thbirthday.

73

Keywords: Heat equation;observationproblem;theoryof control;stabilityofLipschitztype;

transformby Reznitskaya;real inversion formula for theLaplacetransform.

AMS1991 SubjectClassification: Primary: 35K05,Secondary:93B05

1

INTRODUCTION AND THE MAIN RESULT We

consider an initialvalue problemforthe heat equation:

u’(x, t)

zXu(x, t), u(x,

O) f ^(x),

u(x,t)

=0, xEO

t>O,

(1.1)

where _C ]r, 2 < r < 3 is a bounded domain with

C2-boundary

gt

,

^Ou

^A

^{theLaplacian, is a traceoperator(e.g.Adams}

^1]),

that is, if u E

CI(),

^then

Og(X) 1)i(X)-xi(X),

Ou ^{X 0’2,}

01) i=1

I)(X)

(Pl

(X) Pr(X))

beingtheoutward unit normal to 3fl atx.

For f L2(fl),

thereexistsaunique

(strong)

solution u

u(f) . ^C([0,

^0);

^L2())

^fq

^el((0,

^x:));

^L2())

such that Au 6

C((0,

oc);

L2())

and

u(.,t)lO

^{0, > 0 (e.g.}

Pazy

15

]).

In

this situation, we have thefollowingproblemswhen we consider the heat flux

v

^f)(x,

^t)

^{on a}subboundary

F

off2as measurementsfor > 0.

(I)

(Uniqueness)What kind of a set

F

C 0, does

Ou(f)

(x,

t),

xEF,

t>0 determine

f

^{(x), x e$2uniquely?}

(II)

(Construction) Wewish torepresentthe initial heat distribution

f ^(x)

on2 in terms of

v

^{f)(x, t),x e}

^F,

^{> 0.}

(III)

(Stability) Canwe estimate

IlfllL()

^by

v)(x,

^t),x

^F,

^>

^O?

Moreover

what normof ^Ou(f)_Ov

(x

t),x e

F

> 0should wechoose forthe estimation of

fll ^?

The determination of initial heat distribution is called an observation problem.The observationproblemisimportant alsoin thetheoryofcontrol, because it is a dualproblem to the controllability problem (e.g. Dolecki and Russell

[6]). For

theobservationproblemin the heatequation,we can refer to

Cannon [2]. For

similartypes of

problems,

the reader can consult Dolecki

[5],

Mizeland Seidman 13],andSakawa 17].

In

thispaper,

(I)

and

(III)

^aretreated,while

(II)

will be discussed in asucceedingpaper.

For

theuniqueness,the answer is known under acomfortable assumption:

PROPOSITION 1

u cr. _If

Let F

C 8 satisfy

F

8f2 N

U 5 ^O for

^{an open set}

Ou(f)

(x,t) =0, x E

F,

> 0, then

f ^{(x) O,}

^x

This is proved by the eigenfunction expansion of the solution and the unique continuationtheoremfor theelliptic operator (e.g.Mizohata [14]), and we can further refer to

Georg

Schmidtand Weck[8]for theproof.

Now

weproceedtothestability.

It

iseasilyexpectedthat thestabilityis verydelicate, because themap

f

^{advancestheregularitybythe}

"smoothing" property^oftheparabolic equation

(e.g.

Friedman

[7]). We

have totakeastrongnorm

II,

for

v

^{f)(x, t),x}

^F,

> 0 in ordertogetan upperestimateof

fll. ^More

^precisely,there are twoways.

(a) We

search fora norm

II,

of functions on

F

x (0,

x)

sothat

Ou(f)

Ov

(b)

Takinga usual norm for

O(vf,)

^{such as}

Ov

we search for a stability modulus co

C[0, o)

which is monotone increasingand

co(0)

0 so that

L(F^x(O,

cx)))

In (a)

we insist onthe stabilityofLipschitz type,while we have to admit the choice of astrong norm

I1,. In (b),

we insist on a usual norm for measurements^Ou(f) atthecostof a worsestabilitymodulus^09.Thelatterway

(b)

seems morepursuedinexisting

papers

(e.g.Exercise11.4(pp.

144-145)

in

Cannon [2]),

and the estimate of thetype

fllL=() ^O ([log IIDatall-1]),

for some constant a >0,istypical (e.g.seep. 147in

[2]).

Thepurposeof thispaperis topursuetheway

(a).

Ourchoiceof the norm

I1" ^II,

is, of course,strongerthan the

L2(F (0,

c))-norm, butis not extreme in the sensethat

We

shall usethefollowing analyticextensionformula

PROPOSiTiON

2([16])

Fortherighthalfplane

R

⁺

{Z; Z

p+iq, p >

0}

andIx ^>

1/2,

^{we have}the identity,

for

^theBergman-Selbergspace

H ^{(R +)}

comprising all analytic

functions f ^(Z)

^on

^R

^+with

finite

^norm

IlfllH,R+

F(2/z- 1)7r

+

If(Z)12(2p)2Z-2dpdq

< cxz,

ilfll2

¹

c n

t(R+)

E nF(n + ^2/x + ¹⁾ IOP(Pf’(P))I2p2n+2g-ldp"

n’--O

Conversely, any C

function f

^(p)onthe real positive line withaconvergent sum inthe right hand side canbe extendedanalytically onto

R+

and the analyticextension

f ^(Z)

^satisfying

^limp__. f

^{(p) 0 belongsto}

Hu(R+)

andtheidentity holds.

We

shall define

B,(r’x(0,))

p-

g x,

pp

_n,(,+)dS, therighthand sidebeing convergent.Then we obtain

THEOREM Foranarbitrarily

fixed xo

^]R

r,

we set

F {x

^Of2;

(x -xo). v(x)

>

0} (1.2)

and take

We

assume

r

(=

thespatial dimension) < 3 and

f

^1t2

^("2)

^CI

I--I( ^().

Then, thereexists a constantC C (f2,

F, tx)

> 0 such that

(1.4)

(1.5)

C-1 _Ilfll=(a)

^< _Ol)

CIIfllH(a>. ^(1.6)

B.(F^x(0,cx))

In

Theorem,ifr 1,then 1-’ is taken as oneboundary pointof the interval

. ^Here

^forsimplicity,weassume

(1.4),

thatis, thespatialdimensionisless than or equalto3. This is not essential andthe condition

(1.5)

should be replaced by

f

^e

^79(Ac)

^whereot

[] +

^1,

^{(Au)(x) --Au(x)}

^with

79(A) {u He();

_Ul0a

0}, [fl]

denotesthegreatestinteger among ones notexceeding

2 PROOF OF THEOREM

The

proof

willbedivided into threesteps.

2.1 First

Step

We

shall discuss the regularity of solutions to the wave equation ^corre- spondingto

(1.1).

Firstby

(1.5)

^{wesee that}

u(f) CI([0, cx)

x

)

and

Au(f) C([0, cx)

x

)

(e.g. Theorems 4.3.5 and 4.3.6 in

Pazy [15]

and the Sobolevembedding theorem(e.g.Adams 1])).

We

shall consider a correspondingwaveequation:

{w"(x,t)=Aw(x,t),

^w(x,

^O)

^O,

^{w’(x, O) f (x),}

^/2,x

_ ^t>O } ^(2.1)

w(x,

t) O,

x e Of2, > O.

Again applying the regularity assumption

(1.5),

by the eigenfunction expansion of the solution w andthe Sobolev embedding theorem, we see that there exists auniquesolution

w

w(f) e C([0, oo); H3(Q)

^f)

H( (f2))

^f)

cl([0,

CX));

H2(f2)

^f’)

C2([0,

oo);

H ^{(f2)), (2.2)}

and

IIw(f)(’,t)llH(a), IIw(f)’(’,t)lln(a) IlfllH(a),

^{> 0}

(e.g.Theorem 1.1.1 in Komornik

[10]).

Theinequalitiesin

(2.3)

follow from conservationofenergy.

We

set

2

W(t) Ijr ^{( Ow(f)}

^(x,

^t)

^dS

^{=- Ow(f)(,}

^Ov

t)ll

²

By (2.3)

and the tracetheorem(e.g.Adams[1]),wehave

t>0.

W

E

cl[0, O), IWZ(t)], W(t)

^<

Clllf Iln2(),2

^>

^{O, (2.4)}

where

C1 C1 ()

> 0is aconstantindependentof > 0.

In

fact,

W

E

C[0, cx)

and

W(t)

^<

C111fll2n(

^is straightforwardfrom

(2.3)

and the trace theorem.

Next,

by

(2.3),

Ow(f))’

^Ov

^cO([o,

^o);

^L2(F))

and

(x t)

dS ^<

C ^f Iln2(,

²

^t>O,

Ov sothat

Wt(t)

2

fr Ow(f)(x’t)(

^{O Ov (x,t)dS, t>O.}

Hence W

6

C[0, )

andbySchwarz’sinequality

Iw(t)l- <2110w(f)

_< 2111fll

(.,t)

L2(r) Ov

H,/C IIflIH=,

^>

^O,

L2(I")

whichproves

(2.4).

It

follows from

(2.4)

and

w(f)(., 0)

0that

W(0)

0.Furthermore,we have, using

(2.4)

^andthe mean value theorem

W(t) W(t)- W(O)

^< sup

IW’(s)l

0<s<t

<

Cltllfll2(a),

^{> 0.}

^(2.5)

By (2.5)

and

(1.3)

^wehave

W(t)t3-4dt W(t)t3-4Zdt + W(t)t3-4Zdt

<

Clllfl[2H2(f) ^{t4-4Zdt + t3-4Zdt} c ^f,

_{(a 5}

4/z +

4/x

⁴

IIH().

That is,

( )2

fofr Ow(f)(x’t)Ov ^t3-41XdSdt

^<

^C211fllm

^().

^(2.6)

2.2

Second Step

By

the transform formulaby Reznitskaya,weget

u(f)(x, t)

²

_"-

^l

fo

^r/exp

⁽ --- ^{rl2 )} w(f)(x,o)do,

x f2, > 0

(2.7) (e.g.

Section5in

Chapter VII

in Lavrentiev,

Romanov

and Shishat.skii

11]).

By (2.2), (2.3)

andtheSobolevembedding theorem,wehave

Ow(f)

_(x,

_t)

Oxi _< C1 IlfllH(a),

^x6,t>O,

that is,

Ow(f) (x,t)

Ov

C IlfllH(a), ^xeF,

^t>O.

Thereforein

(2.7),

we canexchange

f0 ^...dr/and ,

^sothat

2nt ^{30u(f) (x, t)}

^{r/exp (x,}

xF,t>0.

Setting andchanging independentvariablesbys

r/2,

^wehave

x, exp(-sp)

2

p

^8v

Ow(f)

(x,

/-)ds,

Ov

x 6

F,

p > 0.

(2.8)

We

definethe

Laplace

transform of g

Lo

^c(0,

⁾

^by

(12g)(p)"

(g)(p)

e^-spg(s)ds, p >

O,

the integral existingfor p > 0.Thereforewecan rewrite

(2.8)

as

q/-ff

10u(f)(1)

2

p

^Ov

^X,pp =fi’(x,p)=(/2N)(x,p),

x6F, p>0,

(2.9)

where

)(x, s) Ow(f)

_(x,

_V),

_x

_I’,

_{s > O.}

Ov

Then, byusing an isometrical identityfortheLaplacetransform in

Byun-

Saitoh[3],weobtain

t))2tl-2tZdt II’(x .) IIH.(R+),

^{2 X 1-’,}

provided thateitherof the bothhandsides isconvergent.

Since

(2.10)

((x, t))2tl-2dt

2

-Or

^(x,

^s)

itfollows from

(2.6)

and the Fubini theoremthat

s3-41Zds

t))2tl-2dt

< o

foralmost all x 1-’.Consequently applicationof

(2.10)

yields

\

^{Ov (+)’ aoeo}

^xF,

andhence

frfo+ ( ^{Ow(f)(x t) t3-4UdtdS -ff p-}

^x,

8 _B.(r"x(0,))

2

dS

(2.11)

2.3

Third

Step

In

thisstep,weapplythestabilityestimateforthe waveequation

(2.1):

PROPOSITION 3 (Observability inequality) and let

Let r

C Of2 be

defined

^by

^(1.2)

T

> 2 sup

Ix ^{x01 (2.12)}

xE2

where

xo

^{]i{ is} a point which ^is arbitrarily chosen

for

^specifying

the observation subboundary

F.

Then there exists a constant

C3 C3 ^{(fl, T, F)}

^{> 0such that}

Ou(f)

Ilfllr(a C3 ^(2.13)

0P _L2(Fx(0,T))

The estimate

(2.13)

is

proved

in

Ho [9]

and Lions

[12]. See

also Komomik

10].

Now,

by combining

(2.6)

with

(2.11),

wehave the secondinequality in

(1.6). Next,

^wefix

T

> 0 satisfying

(2.12).

Then,by Proposition 3,since

fFfo

^r

( Ow(f)(x’t)Ov

weobtain

t3-4ZdtdS

by

(2.11),

which isthe firstinequalityin

(1.6).

Thus wecompletetheproof of Theorem.

3

CONCLUDING REMARKS

(1) For

the inversion oftheLaplace

transform/2g

h,thecomplexone iswell-known(e.g. Chapter31 in Doetsch

[4]).

Thecomplexform is, however,notadequatein ourproblem,because the observation datahis real-valued and we have to extend the dataanalytically,which makes the stabilityunclear.

For

ananalyticalreal inversion formula for theLaplace transform,seeByun-Saitoh [3].

(2) One

of ourkeysisthetransform formulabetween aheatequationand a wave equation, through which we reduce the stability in the heat

problem

tothe one in the wave problem.

A

similartechniqueis used also in

Yamamoto

19].Thus we do not use theeigenfunction expansion of the solution to the heatequation, whichisused in Exercise 11.4 in

Cannon [2],

Dolecki[5],MizelandSeidman

[13]

and Sakawa[17].

The norm

[10u(f)1[

for observations is taken over the whole

(3)

_av

B.(rx(O,oo))

time interval

(0, oo). So

far, we do not know whether or notwe can reduce the observation time interval to a finite one withkeeping the estimate oftype

(1.6).

(4) In Vu

Kim

Tuan

and

Yamamoto

18],for a similar observationproblem, thetransform formula is considered in terms of a Mellin convolution transform and anotherstabilityofLipschitz typeisobtained.

Acknowledgements

This

paper

is anoutputofthe second author’s stayatFacultyofEngineering of Gunma University in October 1995.

He

thanks the first author for his

kind invitation. Thesecond author ispartiallysupported bythe Grant-in-Aid forCooperative^Researches

(No.06305005)

^{from the}

Japanese

Ministry of Education, Science,

Sports

and Culture, andSanwa

Systems

Development

Co.,

Ltd.(Tokyo, Japan).

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