Journal of Mathematical Sciences, Vol. 10I, No. 5, 2000

ON A UNIQUE PECOVERY OF LOWER-ORDER EQUATION FROM DYNAMICAL BOUNDARY

TERMS IN THE MEASUREMENTS

M. I. B e l i s h e v

WAVE

U D C 517.946

D e d i c a t e d to O. A . L a d y z h e n s k a y a on the occasion of her jubilee An approach to dynamical inverse problems, based upon their relations to boundary control theory (the so-called BC-method), is developed. This method is applied to the problem of recovering a vector field in a domain via tire response operator (the dynamical Dirichlet-to-Neumann mapping). A feature of the case under conside'mtion is that the opt, r, tor that gove~ns the evolution of the correspomting dynamical system, is non-self-adjoint. The paper gives a b'rief descT~;ption of the technique of the BC-method. Bibliography: 19 titles.

0. INTRODUCTION The possibility of reconstructing lower-order terms of the wave equation in the interior of a domain froln dylmmical nleasurelnents on its boundary is discussed. A f~atul'e of the case under consideration is that the evolnt ion of the dynamical system under s t u d y with a boundary control is governed by a non-self-ad,j o i n t operator. We a t t e m p t to extend our approach (the so-called BC-method, see [1, 5-9]) to r,his class of systems. Let fi C R" be a bounded domain, 0f~ = F. Consider the system

a . - A a - (b,V~,) = 0 in ~ x (O,T);

(0.1)

"1,--0 = ",l,:0 = 0;

(0.2)

'alr•

(0.3)

= f

with a smooth vector field b = { b l , . . . , b" }; let 'a = uf(.r, t) be a sohttion of it. The operator R'.v : .f ---, o,,~o,~Irx[0.Tl (V is a normal) plays the role of the data of the inverse problem. We show that the fiold b can be Ulfiqucly recovered fi'om the operator R ,T in a |)oundary layer tY = {x E Q I dist(:c,F) < ~, .hose thickness ~ = ~(T) < T is deternfined by the'geometry of f~. This layer is contained in the so-called Bardos zone described in this paper. The property that plays a crucial role in the recovery is the local boundary controllability of system (0.1)-(0.3) established by Bardos in 1994. As Nr as the author knows, the l ~ t known result in the reconstruction of lower-order terms is due to Nakamura [17]. We improve on this result tbr equations not containing the term ut: the time that we need for reconstruction is significantly smaller than the time used in [17] (see also [12, 13]). This paper is a step on the road to the extension of the one-dimensional version of the approach [1] to multidimensional non-selfadjoint inverse problems. It should be noted that the colnpletion of our results is still far fl'om the completion of their one-dimensional analogs. The results of the present paper were mmomlced in [7]. 1. GEOMETRY In this section, tile necessary material of a geometric nature is presented. ~Ve introduce subdomains (Bar(los zones) in which the inverse problem will be solved. Translated from Zapiski Nauchnykh SeminaTvv POMI, Vol. 249, 1997, pp. 55-76. Original article submitted October 6, 1997. 3.i08

1072-3374/00/1015-3408 $25.00 C) 2000 Kluwer Academic/Plenum Publishers

1 . 1 . T h e e i k o n a l a n d c u t l o c u s o f a d o m a i n . Let f~ C ]R'~ be a bounded dornain with s m o o t h (of class C ~~ b o u n d a r y F. T h e fimction r : Q -+ R + , r ( x ) := dist(x, F) is called an eikonal. Let g~ be a straight ray issuing out of a point ^/ E F and d i r e c t e d inward f] along the norlnal to the boundary; let G [ 0 , s] be a segment of the ray of length s > 0; let x(7, s) be the far end (fl'om F) of this segment. T h e n u m b e r s. = s . ( 7 ) > 0 is called the critical length, if r ( x ( 7 , s)) = s for s < s. and r ( x ( 7 , s)) < s for s > s.. In other words, the segment L/[0, s] minimizes the distance from x(7, s) to F if and only if s _< s.. The point x . ( 7 ) := x(7, s. (7)) is called the separation point of the ray G" The set w = U

x.(7)

3'EF is the separation set (cut locus) of the d o m a i n f~ with respect to its boundary F (see [10, 11]). T h i s is a closed set of m e a s u r e zero: c o = ~ ; m e s c o = 0 ; dist(co, r ) > 0 . (1.1) Consider an e x t e n d i n g family of s u b d o m a i n s (layers) in f~: ae :=

e

e},

T h e y cover the entire domain; the m a g n i t u d e T. := inf {( > OI f ~ = f~ } is called the filling time. The layers ft ~ are t)omlded by the level surfaces of the eikonal

(F ~ = F), which are equidistant to F. T h e sets F ~ \ co are s m o o t h (n - 1)-dimensional surfaces; typically, smoothness is lost on co. The eikonal m a y lose smoothness only o n w. 1.2. T h e s e m i g e o d e s i c c o o r d i n a t e s . For each point x E ~ \ w, we fix the (only) nearest point 7(x) E F on the boundary, so that dist(x,7(x)) = r ( x ) ; the pair (7(x), r ( x ) ) is called the semigeodesic (:oordinates (s.g.c.) of the point x (see [11]). Fix :r0 E ~ \ w; let 7(x0) = 7o, v(x0) = r0. In the vicinity of the point 7o E F, let us choose a local coordinate s y s t e m ~d* : F --+ R, # = 1 . . . . ,n - 1. The set of functions 71 o 7 ( ' ) , . . . ,7 ' ' - t o 7('), r(-) turns out to be a c o o r d i n a t e system in a neighborhood of x0. The infinitesimal Euclidean length and volume have a known f o r m in the senfigeodesic coordinates: ds '~ = g,,, d7 u dT" + (d'r)2;

dx = fl dF dr,

(1.2)

where dF is the Euclidean measure on F;

fl = 3(% r) :=

-detG

r) }

x/z

det {.q,,, (% 0) }

is tile .Jacobian of tile passage fi'om tile Cartesian coordinates to tile semigcodesic coordinates. 1.3. T h e p a t t e r n o f a d o m a i n . W i t h the semigeodesic c o o r d i n a t e s we ~ussociate the m a p p i n g i : f~ \ co --~ F x [0. T.], i(:v) := (7(x), w(:c)). It is a bijection onto its image. Tile set (-) := i(f~ \ co) is called tile pattern of the domain ft (see [6]). This is part of the cylinder r x {7 _> 0}, bounded from above by the coast 0 := U (7, s.('y)), which is a graph of critical length s . ( - ) . "yEF 3409

1.4. I m a g e s . Denote _'2":= F x [0,T,]; let X("/,T) be a point in f~ \ co with given s.g.c. (%'r). Let y(. ) be a function on ft. The time[ion ;~ defined on E by the relations

.Y(%T)

(%

S

/

0

e;

,

is called the image of the function y. Consider (real) Hilbert spaces 7-t := L2(f~) and ~ := L2(Z) and the subspace )co := {.f E ~= I s u p p f C (9} that contains fimctions localized on tile pattern. The operator I : 7-g --+ 5-, Iy := ~, which is well defined by (1.1), is called the irrmge operator Let us list its properties (see [9]): (i) the operator I is isometric; the relations (Iy, Iv)a= = ( y , v ) ~ ;

RanI =bee;

I * I = 1I~;

II* = X o

hold; here, X o : 5- --+ ~-e is the projector that cuts the functions off to the pattern; (ii) the adjoint operator I* : 5" --+ 7-t is given by the formula

(I* f)(x) = ~-1/2(7(x), r(x)) f(7(x), r(x)),

.~" E f t \ ~o;

(I.3)

it annihilates the functions f E ~" e Yo localized outside the pattern. 1.5. B a r d o s z o n e s . Fix T > 0 and consider a z o n e /3 T in ft t h a t is adjacent to the b o u n d a r y and is defined as follows. Let x E f*, and let g,,~ be the ray issuing out of x and directed along a E Sn-1; let r.~.,~ be the time that a point moving along tile ray at a unit velocity takes to reach the bomldary; let r be the smallest of tile two times r,,a and %,-a- The set

BT:= {XE['' sup~c''~ 0. The zones B r may be defined for an art)itrary Riemannian mmfifold with boundary; however, in tile general case, they may be emi)ty or leave part of a manifold uncovered. This is (:aused t)y closed geo(lesics that separate the boundary from the remaining part of the manifold. 2. DYNAMICS

Dynmnical system (0.1)-(0.3) is considered. The property of controllability of the system under study, which t)lays a key role in solving the inverse problem, is described. 2.1. I n i t i a l b o u n d a r y - v a l u e consider problem:

p r o b l e m . Let b = {b L. . . . ,b"}, bk(-) E C ~ ( ~ ) , 'tttt -- A'tt - - ( b , VTI, ) = 0

"'1,=o =

in QT;

be a vector field in ft: (2.1)

u,l,=o = 0;

(2.2)

U[ET = .f ,

(2.3)

where QT := f t x (0, T), ET := r x [0, T], .f = f(7, t) is a boundary control, and u = uf(x, t) is a solution (wave). Let us list known properties of the solutiou (they c a n be extracted fi'om [14-16]): (i) let .A4r : =

{

.fEC~176r) I

0-t

'f It=~

j--0.1 ....

}

be th(' lineal of smooth controls: fbr .f E 3-'tT problenl (2.1)-(2.3) has a unique classical solution a f E (ii) the mapping f --+ "uf is continuous when regarded as a mapping from L2(E T) to C([0, T]; L.2(ft)); this enables us to define a generalized solution fbr f E L2(zT); (iii) the lnapping .f -+ -571E T (u is the outward normal to F) defined on Ad T is continuous when regarded ~us a mapping from {.f E H 1(E T) [./Jt=0 = 0} to L2(E T) ( H ~ ( . . . ) are Soholev classes).

O,.J"

3-110

2.2. Dynamical

s y s t e m s . Consider problem (2.1)-(2.3) as a dynanfical system (below it is denoted by c~T), providing it with standard attributes of control theory: spaces and operators. (i) T h e space of controls (inputs) .T T := L2(E T) is called the outer .space of the system a T. Select in it a family of expanding subspaces: .~-T,~:.~__.{fe.~TlsuppfCP

x [ T _ 4 , T]},

0_<~
formed of delayed controls (~ is the time that a control belonging to . f T,r takes to act). The projectors X r'r : . ~ T ~ . ~ T , ( cut of[ the functions: ( X T'~ f ) ( . , t )

0
=O(t-(T-~))(.,t),

(0(...) is the Heaviside function). The outer space contains the smooth lineal M T. (ii) T h e space 7-/ = L2(fi) is called the inner space; the waves (states) uS( 9, t) are elements of it (see Sec- 2.1, (ii)). The family of subspaces 7-/~:={yET"/lsuppyEf~},

0<~
is selected in 7-/. The projectors G~ : 7-[ --* 7-/~ cut off the functions to fi~: ( a ~ y ) ( x ) = 0(4 - r ( x ) ) Y0:),

x ~ a.

(iii) T h e "input~state" correspondence in the system aT is implemented by the control operator t V r : W r f := u s ( - T), which is continuous by property (ii), See. 2.1. Tile independence of the coefficients of wave equation (2.1) of time leads to the stationary state property of the system. Let I/VT and W T' be control operators of systems aT and (xT' T < T'; introduce the delay operator T T'T' : .~T ~ ,T'T', ( T "r'T' . f ) ( . , t ) =

O, - (T' - T ) ) ,

f(.,t

O<_t
and note t,hat the adjoint operator to it is given by the relation ((TT'T') * f)(.,t)

= f(..t

+ (T' - T ) ) .

O < t < T.

(2.4)

The stationary-state relation has the form W T = w T ' ,'ffT,T'.

(2.5)

The hyperbolicity of problem (2.1)-(2.3) ilnplies the finiteness of the velocity of wave propagation: s u p p u f ( " , 4) C ~ . By the stationary-state property, we have t.V T .~ T, ,~ C_ ~'~'~, 0 < -- 4 _ < T. (2.6) (iv) T h e " i n p u t ~ o u t p u t " correspondence is implemented by tim response operator 1~r : F

~ .T T,

Dora R T -- {.f 9 H ' (Z T) I .f It=0 = 0}, O'ttf [

RT f "-- Ou

y]T'

which is well defined by property (iii), Set:. 2.1. The stationary state property of tile system leads to the relation 1~.T : ( ' T T ' T ' ) * I~ w~ 7 T'T~ (2.7) (7' < T'), whid~ directly tbllows from (2,5). (v) T h e operator L :7-/--~ 7-/, D o t a L = H2(fl),

k:=A+(b,V) determines the evolution of the system n:T. By tile stationary-state property, wave equation (2.1) is equivalent to the relation 0' IV T L t u r Oil ./~T. (2.8) Ot'-' The operator L is non-,selfadjoint. :

3411

2.3. A p p r o x i m a t e controllability.

The set := w "r

=

If

7 T}

is called reachable at the m o m e n t t = {. Obviously, tile family L/{, 0 < { _ T, expands as ~ increases, a n d the inclusion/,/{ C ~{ is valid (see (2.6)). A property of great importance for the BC-rnethod is tile d e n s i t y of reachable sets: clos L/{ = 7-/{, 0 < { < T. (2.9) Similarly to self-adjoint case (with L = A), it is established with the help of the fimdamental HolmgrenJ o h n - T a t a r u uniqueness theorem (see [18]). The proof of (2.9) by using Russell's line of argument, which is presented in [8], is applicable to the case of L = A -t- ( b, V ) with no change. In control theory, property (2.9) is referred to as the a p p r o x i m a t e controllability of the system aT. This means t h a t any given function y E 7-{~ may be approximated arbitrarily precisely (in 7-/metric) by a ~ a v e 'af( 9, {), appropriately choosing a boundary control f. P r o p e r t y (2.9) implies the existence of the so-called wave bases or, more precisely, complete systems of waves. Fix { e (0, T]; let {J~}, j = 1, 2 , . . . , be a system of controls that is complete in s r ' { : clos Lin {.fJ } = .~T,~ (Lin stands for a linear span); then, by (2.9), we have clos Lin {'aI~( 9 , T) } = :He.

(2.10)

W a v e prqjecto'rs 1~ projecting clos N w onto (:los L/e act in the imler space of the system. The relation

P(=GQ

O_<(_
(2.11)

is a useflfl form of relation (2.9). 2.4. L o c a l e x a c t c o n t r o l l a b i l i t y . for sufficiently large times:

An important %ature of the system c~w is that it is exactt2q co'nt'rollable Z4/T = 7-{w ,,

T > ~(},

where T0 is determined by tile geometry of tile dolnail~ (see [4]). On the other hand, being appro:rimately controllable for all times (see (2.9)), tile system is obviously not e,aactly controllable for slnall times: U T # 7-~y ,

T

< T.

(see [2]). In connection with these facts, tile following interlnediate result, which clarifies tile character of controllability fbr a system of tile type (2.1)-(2.3), turns out to be of interest aud importance. Let D be a corot}act set in t2; denote by L/D T the set of restrictions u f ( 9 , T)] D" Theorem

1 (Bardos, 1994). For any T > 0 and D

C

B T,

the rehition

Lt T = L 2 ( D )

(2.12)

is m l i d .

Roughly speaking, this result (see [3]) means the following. If one changes the location of a small compact set D in tile layer ~7" filled by waves, then the set of the restrictions of waves ' J ( -, T)[ D becomes richer as D moves fi'om tile head part of waves (near their fbrward fi'ont F T) toward their tail part (near F). The following remark should be made in addition. Relation (2.12) is established in [3] (with the help of the ulain result of [4]) fbr a wave equation of the form

_(__0 ""

\ Oz k

0:~/ ] a = O.

However, ~ is noted in [4], the presence of lower-order terms does not complicate the proot~. 3-112

2.5. T h e dual s y s t e m .

A system a ~ of the form 'ut,t -- L # v = 0

vtlt:o

'v[t:0 =

in QT;

(2.13)

= 0;

(2.14)

(2.1.5) where tile operator L# := A - ( b , V ) - d i v b is formally adjoint to L, is said to be dual to a system a T. We denote by v ~ = vg(x, t) the solution of problem (2.13)-(2.15); the operators corresponding to the system a~: are denoted by lJ:~, R~, etc. The systems a T and a ~ have common outer and inner spaces: 9v~ = ~ T , 7-/# -- 7-t. The dual system possesses the same properties as the original system: each of the relations (2.5)-(2.12) holds for it. The response operators of the systems ~T and a ~ are connected by a simple relation. In the outer space, we introduce the operator y T : ~]cT..__~ .~T, (YTf)(.,t)

:=f(.,T-t),

O
and the operator b,, multiplying the controls by the flmction b, = b,("/) := (b("/), u(7) }, 7 e F. Integrating by parts in the relation

/

0=

dxdt['u[t(:r,t ) -(L'uZ)(x,t)] v ~ ( x , T - t ) ,

Q'r one can easily obtain the relation R~ = YT(Rr)* yW + b,,

(2.16)

which is wdid on D o m R ~ = Dora R r = {if E H t (Ew)I f It=0= 0}; The operator C T : .F T --~ jzr, C T := ( I I ~ ) * IV T,

here, (fir),

is adjoint to R z in .7=-:.

is called tile conne.cting operator. This term is inspired by the relation (cT.f.g):=T ---- (I,V r f , I,Vr g)~ -- ('uf( 9, f ) , re( 9, T ) ) ~ ,

(2.17)

which links the metrics of tile outer and inner spaces. The fact putting C T in tile tbrcfl'ont of our approach is that the connecting operator may be expressed explicitly and simply in terms of the response operator. which plays the role of the data in tile inverse problem. To state the result, let us introduce the following operators: (i) the response operator R 2w : 5c''r --+ .7-27 corresponding to the system a 2T (to problem (2.1)-(2.3) with doubled final nlonleilt); (ii) the odd-extension operator S T : .pr ___,.T2T,

(srf)(.,t) := { f(''t)" - - . f ( -, 2 T - t),

0_
(iii) the integration operator j'2T : 52T ....+ .~2T, t

(.I2Tf)(.,t) :=

j"d,l.f(.,/I), 0 < t < 2 T . o

34 13

Theorem

2. For any T > 0, the representation

C T.f = I ( s T ) , t{2T j2T sT f, 2

(2.18)

.f E H ~(ET),

is valid. Tile way in which (2.18) is derived does ,lot differ fi'om the one given in [1] for tile one-dimensional case. Representation (2.18) is of the same form as in the self-adjoint case (see [6, 8]); the possibility of the direct generalization to the case L r L* was found by S. Avdonin. Formula (2.18) describes the (continuous) operator C T only oil a dense set; by the unboundedness of R 2T, it is not valid for a r b i t r a r y .f E 9cT. However, Theorem 2 allows us to assert that the connecting

operator C T is uniquely determined by the response operator FteT. 2.6. L e m m a s o n l i n e a l s . We complete this section with auzdliary results, which will be used directly in the inverse problem. These results are derived fi'om the property of controllability described in Secs. 2.3 and 2.4. For convenience, we consider the dual system, not the original one. Fix ~ E (0, T) and introduce tile subspaces

j~-•

:= ~-7" O 5~z'~ and ~T,~ := ~ T @ 7_/~,

which coincide with L., (F x [0, T - (]) and L.~(f~7" \ [~), respectively; decompose the subspace ~ T into the suIn ,]_~T = ,]_~ @ ,]..~T.~. (2.19) Lemma

1. The, lineal (W~)* "]_~TC ,~r is decomposed into the sum

(~)* ~'~ =

(w;f)* ~

+ (w~')**,~~• ,~. .

(2.20)

of" two n(minterse('ting lineals, and ,~T,~ separates the summands:

(H,~)*~'~ns2'~ {0}; (~),~4T,~ =

,~•

Co

rr,~ z

9

(2.21)

Proof. In accordance with a known operator relation, we have KerIW2)*

= ~ +closRanW2

= (see (2.9) for W2)

whence

= ~ + ~T,

(B"~)* acts oil the subspace ~r injeetively; it takes decomposition For arbitrary g E 5cr'~ and y E 74T,,~ , ~• , we have the relations

((W2)* :'J'g)F

= (:'J'~g)~

(2.19) to (2.20).

= (see (2.6) for W 2 ) = 0 ,

which lead to the second relation in (2.21). Let g E (W~)* ~ D ,T'T'~'• . ill this. case, we . have. g = (IV~)*g, y E ~ , Therefore, ['or any h, E ~T,~ we obtain

which iml)lies that y = 0 because W~ ~-v',~ is dense in ~ is t)roved.

3.11.[

and, at the same time, g _L 5c'r'~.

(see (2.9)). Hence, g = ( l ~ ) - 0 = 0. The lemma

C o r o l l a r y . Every element h E (IVy)* 7-IT is uniquely represented in tile form

h = hi + h,; h, 9 ( w D * ue,

h2 9 7 : 'e .

(2.22)

Local exact controllability gives additional information oil the lineals (W~)* 7~~ occurring in the decompositions. L e m m a 2. Let T and ~ 9 (O,T) be sudl that the inclusion-~ ~ C B r holds; let {.f~}, j = 1 , 2 , . . . , be a complete system of controls in jrT,e. Then, the lineal (W~)* ~ e is dosed in ~ T and the representation ( W ~ ) * 7-/e = elosLin {C r

f~}

(2.23)

is valid. Proof. In accordance with Theorem 1, for D = fi~ we have ur

, = a~ w [ ~ F

= see (2.12)) = ~ ,

i.e., the operator G ~ W~ has a closed range coinciding with ~ . The,'efore, the adjoint operator (II~)* G~ acts on ~ isomorphically; hence, the range (W~)* ~/~ is closed. By (2.10), the span Lin { W r .f~} is dense in 7-/~. Consequently, the set

(w2)* Lin { w ~ . f ~ } = Li~ { ( , ~ ) * w ~ . f ~ } = L ~ {C~.f~} is dense in the lineal (I.V~')* 7-/~. Since the latter is closed, we obtain (2.23). The Iemma is I)roved. Decomposition (2.22) plays a central role in the inverse problem. Representation (2.23) provides a t)ossibility of reconstructing the lineal (W~')* 7t ~ in j r ' r having the connecting operator at our disposal. 3. AN AMPLYI'UDE INTEGRAL An operator construction solving the inverse probleln is described. 3 .1 . O p e r a t o r s u m s a n d a n a m p l i t u d e i n t e g r a l . Recall that the cut-off projectors X ~r'e and G ~, wlfich act in ~-T and 7-{, respectively, were introduced in Sec 2.2, (i) and (ii). Choose a partition -- := {~j}, j = 0, 1,... , N : 0 = {0 < {1 < ... < ~x = r of the time interval [0, T]; k,t ,.(=):= max(~j - ~,j-t) be the rank of it. The two sets of projectors

A j X T'a :---- X T''~j - -

xT'(j-I

;

A j a ~ := a (j - G (j-I

in j r r and 7-{ are ,'elated to this partition. Fo,'m the sunl N

A~ :: ~

a ~ a ~ w ~ a ~ x ~,~.

(3.1)

j= I

which is a n operator from jrT tO "]"[. As is easy to verif~y, it is bounded: IIA=rll _< [[IvTI[; ILmA_-_ r c "]-iT. Let us describe the limit attained by stuns (3.1) as the partition is refine(l. Associate with the vector fieM b a function (I) defined on ft \ w in accordance with the formula

'~(:,) := ~

.

I ds I ,

G(~)[0,r(~')} 3-115

where the integral is taken over a segment of the shortest ray that joins x with F; then, define the flmction

z(x)

: = e '~(:~'),

x E ~ \ ca.

The latter flmction deternfines an operator of multiplication ~" : 7 - / ~ 7-/, ~y = e(. )y(. ). It is convenient to introduce a reduced image operator I T : 7-/---+ .T"T given by sT y :=

where ~ is the complete image of the function y (see Sec. 1.4). For the adjoint operator (IT) * 9 7-/ ---+ 9~T, representation (1.3) easily implies

((IT) * .f)(x) = { f3-i/2(~/(x)'v(x))'Z(7(x)'r(*))'O, for the remainingZE x~TE\a.c~

(3.2)

Note also a relation, which is easy to verify:

(IT) * I T = G T,

(3.3)

and recall the definition of the operator y T : (yT.f)( ", t) = f ( . , T - t ) ,

0 < t < T.

T h e o r e m 3. The refinement of the partition leads to the convergence o/:smsls (3.1) to the limit

A r :=

A~ = ~'-~ ( I t ) * y r

lira r(':-)~0

(3.4)

-

in the weak operator topology.

The limit operator is called an arnplihMe integral and is denoted by

/. T

AT =

dG ~ W r dX'r,~.

o

Vv'e onfit the t)roof of Theoreln 3 because it is quite sinlilar to the one i)roposed in [9] for the sclfadjoint case (see also [8]). However, ibi" completeness, in See. 2 we present some heuristic eonsider~tions, which clarii~y the dynmnical nature of the operator A T anti underlie the proof. 3.2. P r o p a g a t i o n o f d i s c o n t i n u i t i e s . Our considerations concern a well-known fact: in hyperbolic system (2.1)-(2.3), discontinuous controls produce discontinuous waves. The discontinuities propagate along the rays fronl F inward ~. Take a smooth control f E j ~ T and fix { E (0, T); let

.f~(.,t)

:= (xT'e.f)( 9,t) = (~(t --(T - e ) ) . f ( - , t ) ,

0 _< t _< T,

t)e a cut-off flmetion of .f to F x [T - (. T]. The control .f~, in general, is discontinuous at t = T - (:

f~(~l,t)

t--T-e+0

=

.f(~t,T-e) -0,

~ ~ r.

(3.a)

t=T--~--O

Consider problem (2.1)-(2.3) with cut-off flmetion ]~ taken as a control. The solution u f-" of it is localized in the Sl)ace-time cylinder QT above the characteristic sm'face

9 T'~:={(x,t) EQT[t=r(x)+(T--~)}. 3-1 l 6

Tile characteristic itself carries a discontinuity of the solution: n e a r ,)~,T,( (and outside a small neighborhood of the set a~ x (0, T)) we have a well-known representation from geometric optics:

v,f~(z,t) = e - t ( z ) / 3 - ~ / 2 ( 7 ( z ) , T ( z ) )

-(T-~)) + . . . ,

x f(3,(z),t - 7 ( z ) ) O ( t - r ( z )

(3.6)

where the term vmfishing on ,)~,T,[ is omitted (see, e.g., [19]; in [9], the reader can find a proof due to A. P. Kachalov). At the final moment, the wave u I~ ( . , T) is localized in the subdomain f~r in accordance with (3.6), the wave has a j u m p on its forward front (the surface F~), and the amplitude of the jura t) is determined by the jump of control (3.5) and by the factor e -~ ~-~/e, which is determined by the field b and the divergence of the rays ~-~. In a thin layer fY \ f~--A~ close to F~, one has a representation:

u f~ (x, T) = ~-~ (x)~ -~/'2 ('~(x), v(x)) f ( 7 ( x ) , T

-

T(Z)) + . . .

(3.7)

(the term vanishing on F ~ is onfitted). Now consider snm (3.1); consider a separate term of it on .f E 3,4T; then, by (2.6), we have (G gj - G~J-t)~V T X T'(j-t f = O. Hence, A~ G ~ W r Aj X r'~ f = A~ G ~ W r X r'~j f = A~ G ~ ~/~ (., T).

The wave "u/'~J ( -, T) has a .]ump o n F (j ; the t)rojector A j G '~ selects the part of the wave localized in the layer [b% \ ~
w T

x

= [ (

+..., 0

for tile remaining

:i; z E [~.

]

Sunmling "layer l)y layer" and keeping only the leading terms, we obtain an apt)roximate relation: $ s-'(.z')l~-*/'2(-/(:c),T(:t,))f(-/(:c),T - T(:r)),

:r E ~T \ ~ ,

[

:~' E fl,

0

for the remaining

whose right-hand side coincides with ? - 1 (IT). y T .f (see (3.2)). That is why convergence (3.4) should be expected. The sunnnation of the amplitudes of the jumps of waves propagating in the system a T, which holds in the construction of A T, inspires the t e r m "amplitude integral." Note that geometric optics makes it possible to control the shape of a wave near its forward front. Using representation (:3.7), one can show that for any point z0 E fiT \ ~, tile controls .fl . . . . , .f,~ E 9r r may be found so that the vectors V u A (.c0, T) .... , V'u .t' (xo, T) form a basis in R '~. 3.3. T h e c o n t r o l o p e r a t o r in t h e f o r m o f a n a m p l i t u d e i n t e g r a l . For the dual system ( ~ . Theorem 3 yMds the representation T

"dG~ ~V2 dxT,~ = ? ( I T ) . y T

A~= o

(the change of the t)ower of ? stems from the fact that the field b is replaced by - b in the dual ot)erator L#). For the adjoint operators, we have T

. / d X T , 5 ( ~ . ~ ) . dG ~ -_ y T i T A 0 "Ml7

whence, in accordance with (3.3), we obtain a relation: T

GT ~. = (IT). y T .f__dX T'e (W~)* dG e.

(3.8)

0

Multit)lying by W T froln the right and taking into account the relations

G T F W T = g'G T W T = (see (2.6) for ~ = T) = g'W T, we derive the representation T

~rvvT = (IT)* y T j" dX T'~ (Wf~)* dG ~ W T.

(3.9)

0

Now we apply the representations found above to tile inverse problem. Let 1(- ) = 1 be the unit function on fi; let r := a T z be a cut-off flmction of ~ oil fiT; note that c T = GT g l . Applying both sides of (3.8) to the unit function, taking into account relation (2.11) (controllability!) and tile independence of ( } ~ ) * of the variable of integration, we arrive at T

r

= ( / ' r ) . yT./'dXT,< d[(W~)* P~ 1].

(3.10)

0

Introduce a fanfily of operators: r,e

:= (w2)* pe w r,

0 _< < _< T.

Together with (2.11), this definition shows that (3.9) transforms to the representation T

~VT = g-- t (IT), y T .

/

d,u

dIIT, Q

(3.1,)

o

It is worth noting that tile right-hand side of (3.11), in tkct, contains the values of the flmction e that it takes on the subdomain QT only. As in (3.4), the integrals in representations (3.10) and (3.11) converge ill a weak sense. In the sequel, in solving the inverse problem, we shall calculate integra.ls (3.10) and (3.11) using the inverse data. This will allow us to determine el~r and, thereafter, to recover the control operator W T. 4.

THE

INVERSE P R O B L E M

A procedure of recovering the vector field fl'om the response operator is described. A generalization ~br a wave equation with zeroth- and first-order terms is presented. 4.1. T h e m a i n r e s u l t . Let us state the main result of the present paper. 1-{ecall that the Bardos zones were defined in Set',. 1.5. T h e o r e m 4. Let T aml T' be positive times such that tile zone B r' contains tile layer ~T: then the respo~lse operator R 2T' uuiqu@ determines the ~d(1 b in ~T. The remaining part of the paper is devoted to the proof of Theorem 4. The proof is preseuted ill the form of a procedure yielding b 3-118

4.2.

Recovery

o f t h e l i n e a l s (W~)* ~. Given the operator R 2r', let us find the operator C r' with

the help of representation (2.18). Fix ( E (0,T); in the subspace .T~r',~ C 9vT' choose a complete system of controls {f~}. Thereafter, one can recover the lineal (W~')* 7-/e in 9vT' by means of representation (2.23). The desired lineal (W~)* 7-/~ in bvT may be found with the help of the reduction: in accordance with (2.5), we have

( w ~ ) * ~ e = (Tr'r') * ( w ~ ' ) * ~ e,

o < ~ < T.

4 . 3 . F i n d i n g t h e f u n c t i o n s T. In order to determine r we shall use representation (3.10). In the context of the inverse problem, the operators I T, y T anti X T'~ occurring in the representation are assumed to be given. It remains to find the elements ([{~)* P~ 1, 0 < ( < T. This will be done.in two steps. (i) First, let us find the element (~~rT #) * 1. Introduce the function ~T = ~T(3, t) := T - t, 0 < t < T, in ~ T ; the inclusion >r E Dora (RT) * = Dora (R~)* m a y be easily verifed.

L e m m a 3. The relation

(4.1)

( ~ ) * 1 = y r RT y r z r

is valid. Pwof. Fox"arbitrary g E ANT, we have the following relations: ((14~)*l,g)~r=(1,

VJ~g)~ =

/

dxvg(:r,T)=

~2

/

/_r0,.

0

F

T

= . dr(T-t),

/ f2

T

dx

T

,a , t) = - t)vtt(x o

]

]

"dt(T - t).

o

al[-~-~tj('y,t)-b,,(7)I/J(7, t ) = (~T,R'~.q-b,,g)j:r = s e e (2.16))

/

dx(n# vg)(x, t)

f2

(yT RT y T ~T,g)~:r"

Comparing the begimfing and the end of this chain of identities and taking into account the density of .ANT ill .T"T, we obtain (4.1). Tile lemma is proved. In tile inverse problem, tile operator R r (T < T') is given (see (2.7)). Hence, relation (4.1) determines tile eJeme.nt (14~)* 1. (ii) Having found the lineal (W~)* 7-t~, one can find deconq)osition (2.22):

(W#)*l=h,+h,;

h,

(4.2)

On the other hand, we; have the representation 1 = P<

1+ [pT _ P~]I + [11~ - pT]I,

(4.3)

where the sulmnands belong to tile subsi)aces %/', WT'< ,~• , and 7-g @ ~-~T, respectively. . By (2.9) (for b/~ -W~ ) e r ) , the relations 7-I @,7-/T = 7-/~) d o s Ran I4~ = Ker (W~)* hold; consequently, the third summand ill operator, we obtain a representation:

(4.3)

is annihilated by the operator (IVy)*.

(1,I:~')* 1 = (IVy)* P~ 1 + (IVy)* [P T - P~]I

Applying this

(4.4)

with summands belonging9 , to (fV~)* ~/~ and (~V~)* wT,a,~• C ,';'r"~• , respectively. In view of the uniqueness ., of tile clecomt)osition of such a tvpe, comparing (4.2) and (4.4) we (:onclude t h a t ht = (I'V#"r) 9 p~ 1. i.e., the required element is found. Making ~ w~ry and again using a decomposition of the type (4.2), one can recover the family of eleinents ( W #r ) 9 pe 1, 0 ~ { ~ T, iIl ,,T"T. Substituting it into (3.10), one finds the flmction e ill fiT. 3419

4.4. R e c o v e r y o f

IevT. Fix .f E JU T and find decomposition ~_.22): C Tf=hq+h2;

hl E(I~Vr

h.ZE5-T'~.

On tile other hand, c T . f = (~l~)* W T f = ( t [ ~ ) *

P~ W T f + ( W ~ ) * [11~ - P~]W Tf,

which is a decomposition of tile same type. This implies hq = ( W ~ ) * P~ W T .f = HT'~ f.

Knowing the family IIT'~f, 0 < ,~ <_ T, and the function ~l~r deternfined above, one can find w ' r f by using (3.11). By continuity, one can extend the formula for W T from j~T to the whole of 5-r. Thus, the control operator is recovered. R e c o v e r y o f t h e field. Relation (2.8) determines the operator L o n the set of smooth waves l t f ( - , f ) E w T j ~ T. As one m a y show, this set is sufficiently rich, so that, m a k i n g . f vary, one can recover the components h i , . . . , bn in w~we equation (2.1) (compare with the concluding remark of Sec. 3.2). Thus, the field bl~ r is recovered and the inverse problem is solved. 4.5.

4.6. G e n e r a l i z a t i o n s .

Let us discuss the possibility of recovering all lower-order terms in the equation ,,tu - ~Xu - ( t,, V'~ } + q',t = 0

(4.5)

from the response operator. W i t h no changes in the method described above, one can recover the operators 1-IT'~ and, thereafter, the operator ~'W 7" (see (3.11)). Relation (2.8) determines the operator ~LE -~ = A + ( b ' , V ) - q'

with coefficients

b' = 1,+ 2eVz-L;

q' = q - (b, c V c -~ ) - z A z - ~ .

(4.6)

In the set of all operators of the same type, the operator ? L ? - l is characterized by the following property: the field U corresponding to this operator has zero t)rojection to the normal rays /?~. We would like to enlphasize that one should not expect a u'niq'tte determination of b and q, since the substitution 'u ---+ X'u (X E C~(f2); y > 0 in f2; X = 1 near F) into the wave equation leads to a dynamical system with the same response operator R T but with another evolution operator xL.~ - t . This implies the ibllowing result. T h e o r e m 5. Let T and T' be times such that the zone B T' contains the layer-~ T ; thou the operator R 2T' determines the coeff&'ients of Eq. (4.5) in ~2"r up to a transformation of the t y p e (4.6). Without difficulty, this theorem is extended to a Rienmnnian manifold with boundary, which has nonempty Bardos zones containing the layers f2T. C o n j e c t u r e s . In the self-adjoint cruse, the B C - m e t h o d yields a result optimal with respect to time: the operator R "2T determines the coefficients in f2T (see [6, 8]). The result of the present paper is far fl'om being optimal, and there are some grounds to hope that it may be improved. We intentionally presented our results so as to einphasize the key point, which is the recovery of the lineals (W~)* 7-L~. 4.7.

If a way of deterinilfing them groin /'~2T (alld not from /~2T') could be fomid, then a result allowing no flu'ther improvement, i.e., the recovery of lower-order terms in the subdomain f2 7" filled by waves, would be immediately obtained. The w a y of establishing such a result could be found if a positive answer to any one of the following questions exists: (i) does the relation

[

,los(WD*

n

= {0}

hold for any 7' > 0"? (ii) Is it true that the subspaces 5-T <-:)elos (W~)* 7-t~ are nontrivial for any T > 0 and their images under the mat)ping W T are dense in 7-[T'~'? The second question is equivalent to the problem on the existence of controls .f E 5 -7, producing waves without tails: supp u f ( 9, T) C f2 T \ ~2e. 3420

4.8. R e m a r k s . The results of this paper were announced at the conferences "Boundary Control and Inverse Problems" (EIMI; St. Petersburg, July-August, 1996) and "Control and Partial Differential Equations" (CIRM-Luminy, 16-20 June, 1997). A generalization of the BC-method to the dynanfical systems governed by nonselfadjoint operators originated in [1]. The first stage of research in the multidiinensional case was implemented by the author together with S. Avdonin. The author is very grateful to him for their collaboration. The author dedicates this paper to O. A. Ladyzhenskaya with love and great respect. Translated by M. Belishev. REFERENCES

1. S. Avdonin and M. Belishev, "Boundary control and the dynamical inverse problem for a nonsel.fadjoint Sturm-Liouville operator," Control Cybernetics, 25, No. 3, 429-440 (1996). 2. S. A. Avdonin, M. I. Belishev, and S. A. Ivanov, "Controllability in a filled domain for tile wave equation with a singular boundary control," Zap. Nauchn. Semin. LOMI, 210, 7-21 (1994). 3. C. Bardos and M. Belishev, "Tile wave shaping problem," in: Progress in Nonlinear Differential Equations and Their Applications, 22 (1995), pp. 41-59. 4. C. Bardos, G. Lebeau, and J. Rauch, "Sharp sufficient conditions tbr tile observation, control, and stabilization of waves from the boundary," SIAM J. Control Optimization, 30, No. 5, 1024-1065 (1992). 5. M. I. Belishev, "An approach to nmltidimensional inverse problems for the wave equation," Dokl. Akad. Nauk SSSR, 297, No. 3, 524-527 (1987). 6. M. I. Bclishev, "Boundary control and tile extension of wave fields," Preprint LOMI P-l-90. 7. hi. I. Belishev, " Boundary control and the dynainical reconstruction of vector fields (the BC-method)," in: P'mceedings of the Co'nfc.rcnce "ContTvl and PDE, ""CIRM-Luminy (1997). 8. M. I. Belishev, "Boundary control in reconstruction of inanifolds and metrics," Inverse P.mbh,.ms, 13, No. 5, ;1-49 (1997). 9. I. M. Belishev and A. P. Kachalov, "An operator integral in the multidimensional spectral inverse probhun," Zap. Na'uchn. Semin. POMI, 215, 9-37 (1994). 10. D. Gromol, W. Klingenberg, and W. Meyer, Riemannsche. Geometric in GTvssen, Springer, Berlin (1968). 11. P. Hartman. "Geodesic paralM coordinates in the large," Am..L Math., 86, No. 4, 7(}5-727 (1964). 12. V. Isakov, "An inverse hyt)erbolic problem with many boundary measurenmnts," Comrn.u;n. PDE, 16, 118;1-1195 (1991). 13. V. Isakov and Z. Sun. "Stability estilnates for hyperbolic inverse problems with local boundary data," Inverse Pwblems, 8, 193-206 (1992). 14. O. A. Ladyzhenskaya, A Mixed P'mblem for the Hyperbolic Equation [in Russian], GITTL, Moscow (1953). 15. O. A. Ladyzhenskaya, Boundary-Value Problems in Mathematical Ph.ysics [in Russian], Nauka, Moscow

(1973). 16. I. Lasiecb~, J.-L. Lions, and R. Triggiani, "Nonhomogeneous boundary-value problems for second-order hyperbolic operators," J. Math,. Pures Appl., 65, 149-192 (1986). 17. S. Nakanmra, "Uniqueness for an inverse hyperbolic problem," Jpn. J. Math., 22, No. 2, 257-261 (1996). 18. D. Tataru, "Unique continuation for solutions to PDE's; between H6rmander theorem and Holmgren's theoreln," Commwn. PDE. 20, 855-884 (1995). 19. B. R. Vainberg, Asymptotic Methods for Equations of Mathematical Ph,ysics [in Russian], Nauka. Moscow (1982).

3,12 [