A local inverse problem for Hamilton-Jacobi equation Reconstruction of Riemannian metric

Applied Mathematics

A local inverse problem for

Hamilton{Jacobi equation

Reconstruction of Riemannian metric

I.V. Golubyatnikov

1

, H.H. Hacsalihoglu

2? 1 Novosibirsk State Technical University, Novosibirsk, Russia

e-mail:[email protected]

2 Ankara University, Ankara, Turkey

e-mail:[email protected]

Received: January 23, 2001

Summary.

We study the uniqueness questions for the inverse prob-lem of determining hamiltonian H(xp) = gij_{(x)pipj and a phase}

function w(xt) from the Hamilton-Jacobi equation and the initial-terminal conditions w(xt0), w(xt1). Here the unknown functions

gij_{(x) compose a matrix inverse to that of the metric tensor. We}

obtain uniqueness theorems for special classes of initial-terminal con-ditions in the cases n = 1,n = 2 and in the case of a scalar n n

matrixgij_(x).

Key words:

Inverse problem, hamiltonian, Hamilton system, tra-jectories

Mathematics Subject Classication (1991): 33F28, 35R30, 37J99

1. Introduction

We consider an inverse problem for the Hamilton-Jacobi equation (1) @w_@t +H(xgradxw) = 0

with an unknown hamiltonianH(xp). Herex2GRn,t2t 0t1],

pi =@w=@xi,i= 1::: nand the values of the phase functionw(xt)

? This work was supported by NATO, Grant OUTR.CLG 970357 and by RFBR

are assumed to be known att=t0 and t=t1:

(2) w(xt0) =w0(x) w(xt1) =w1(x):

Similar inverse problems for various classes of dierential equations were studied in 1]. As it was shown in 2], if this hamiltonian has a full involutive collection of integrals, which are assumed to be known, i.e., if corresponding Hamilton system is integrable, then under some additional assumptions on concordance of the initial and terminal data (2) this inverse problem has a unique solutionfw(xt)H(xp)g.

Now, we consider the case when the hamiltonianH(xp) describes some geodesic ow on a domain in a Riemannian manifold. Let (3) H(xp) =gij_(x)pipj:

Here and in the sequel, we always sum over repeated indices.

Since the geodesic ows on Riemannian manifolds usually are not integrable, see 3], no any additional restrictions on the integrals of this hamiltonian will be imposed here. We shall consider the tomo-graphic variant of this inverse problem: the functions gij_{(x) will be}

assumed to be known and constant outside of some setKRn, and

the initial dataw0(x) will be chosen in some special class of functions.

Usually, this will be the class of linear functions.

2. One-dimensional case

For simplicity of exposition we start our considerations with the case of conformal Riemannian metric

(4) H(xp) =2(x) jpj

2

in the one-dimensional spaceR1. Here, the Hamilton-Jacobi equation

(5) @w_@t +2(x) @w

@x



2

= 0 corresponds to the Hamilton system

(6) x_ =Hp = 22(x)

p _p=;Hx =;2 0

x(x)(x)p 2:

It is well-known, see, for example 4], that the hamiltonian 2(x)p2

has constant values on each of the trajectories of this system. Let(x) be such a positive smooth function that for some positive constantsC0,C1 and a nite segment ]

R 1

a1).(x) =C0 for allx = 2]

b1). 0< C1

and let w0(x) = kx for some positive coecient k. The case of a

negative coecient k can be treated in analogous way. For x > + 2C2

0k(t 1

;t

0) and forx <  the system (6) has the form

_

x= 2C2

0p p_= 0:

Hence, the following equality (7) w1(x) = kx ;C 2 0k 2(t 1 ;t 0) p= const

holds outside of the segment + 2C2 0k(t

1 ;t

0)], and the phase

functionw(xt) has the formw(xt) =kx;C 2

0k 2(t

;t

0) outside of

some neighborhood of the domain ] t0t1].

If the projection Q(x) of a trajectory Q of the system (6) from the phase space R2

fxpg onto the space of the variable x begins at

t = t0 in the point x0 =

2 ], then at any point of this trajectory

the following identity holds

(x(t)) @w @x(x(t)t) =C0k (x 0)  @w @x(x0t0):

In the sequel we shall call Q(x) just a projection of the trajectory

Q without mentioning the domain and the range of the projection mapping.

It follows from the previous identity, that the impulsep does not change its sign, hence, the rst equation of the system (6) for this trajectoryQ can be written in the form

dx

(x) = 2C0kdt:

If projection of this trajectory nishes at x1=x1(x0) =x(t1), then

x1 Z x0 dx (x) = 2C0k(t1 ;t 0):

It is easy to see that if

(8) k > ;

2C0C1(t1 ;t

0)



then some trajectories which begin atx0 < have their endpoints at

x1 > . We assume in the sequel that the coecient k satises the

condition (8), and introduce notationsl=; and  =t 1

;t 0.

Determination of the function

(x)

.

Let w1(x) be a smooth

then for allx2]the unknown function (x) is determined from the equality (9) 2(x)  @w1 @x (x)  2 =C2 0k 2

If projection of a trajectory of the system (6) starts at the point

x0

2] and nishes at a pointx

1 > , then  Z x0 dx (x) + x1 Z  dx C0 = 2C0k

and we obtain an explicit expression of this pointx1:

(10) _C1 0k (w1() ;w 1(x0)) + x1 ; C0 = 2C0k:

In particular, forx0 = the projection of this trajectory nishes at

the point x1 =+ 2C 2 0k ; w1() ;w 1() k 

which we denote by x1() =+. Since x1 > , we see that

(11) @w1 @x (x1)  @w1 @x (x0) =k 2

and in this case the condition (8) implies

x1

2(++ 2C 2

0k):

As it was shown above, the initial and terminal data w0(x) =

kx and w1(x) in this inverse problem are not independent. Let us

describe all possible correlations of these functions. c1). For x =2+ 2C

2

0k] the equality (7) holds.

d1). For x 2 ] and a smooth monotonic function w

1(x) the

following inequality holds 0< @w1

@x (x)< C0k

C1



see (9) and the condition b1). e1). Ifx2+], then@w

1=@x=k. In this case the projections

of corresponding trajectories begin atx0 < and nish at x1 > .

f1). If x1

2++ 2C 2

Theorem 1.

If the conditions a1) { f1) and (8) are satised, then the solution of the inverse problem for the Hamilton-Jacobi equation (4) with initial and terminal data w0(x) =kx, w1(x) does exist and

is unique.

Proof. The statement of the theorem follows immediately from the previous arguments. In a similar way we prove the theorem 2 and the theorem 3 below. ut

Numerical experiments with power expansions of analytical func-tionsw0(x) andw1(x) do not give good result because of problem of

concordance of these initial-terminal data.

3. The case of

n

-dimensional scalar matrix

g ij

An analogous result is valid in the higher-dimensional spaces as well. Let S = ] Rn;1

 Rn be a stratum, and a smooth positive

function(x) =(x1x2::: xn) satises the following conditions:

a2).(x) =C0= const for allx = 2S

b2). 0< C1

(x) for all x2S.

c2). Let the unknown function (x) satisfy the estimate

@ @x1(x

1::: xn)< C 2:

Here C0,C1,C2 are positive constants.

Consider the inverse problem (1), (2) in the case when H(xp) =

2(x) jpj

2 and the initial data have the form w 0(x

1::: xn) = kx1

for some positive k. As in the previous section, we describe here such a class of functions (x1::: xn) and such a set of values of

the coecientkthat the projections of trajectories of corresponding Hamilton system with initial points in some neighborhood of the hyperplane x1 =  will nish in the half-space x1 > . Hence, the

projections of these trajectories pass through the stratumS.

In this case the unknown function(x1:::xn) can be determined

from the equality

(12) 2(x1::: xn) jgradxw 1(x 1::: xn) j 2=C2 0k 2

as in the previous section, see (9).

If a projection of some trajectory of the Hamilton system (13) _xi _=Hp i = 2 2(x) pi _pi =;Hxi=;2 0 xi(x)(x) jpj 2

has an initial point in the half-space x1 0

 , then for all points of

this trajectory

(14) (x)jpj=C 0k:

Hence, for all points of this trajectory the equations of the Hamilton system corresponding to the rst coordinate axis have the form

_ x1 = 22(x)p 1 _ p1 = ;2 0 x1(x)(x) jpj 2= ;2 0 x1(x) (x) C 2 0k 2:

In contrast with the one-dimensional case, here the components of the impulse p can change their sign, though the condition (14) is satised in all points of this trajectory.

The condition c2) implies that _p1 > ;2C 2 0C 2k 2=C 1: Integrating

along the segment t0t1] we obtain

p1(t1) ;p 1(t0) =p1(t1) ;k >; 2C2 0C 2k 2 C1 :

This rst component of the impulsepis positive if

(15) k < C1 2C2 0C 2 : Since _ x1= 22p 1 >2C 2 1(k ; 2C2 C1 C2 0k 2) it follows that x1(x 0) ;x 1(t 0)>2C 2 1(k ; 2C2 C1 C2 0k 2): If (16) 2C2 1(k ; 2C2 C1 C2 0k 2) >  ;=l andx1

0 is suciently close to, then the projections of corresponding

trajectories of the system (13) pass through the stratum S. This condition (16) implies that

(17) C2 < C 3 1 4C2 0l and (18) C2 1 ; p C4 1 ;4C 2 0C 1C2l 4C2 0C 1C2 < k < C2 1+ p C4 1 ;4C 2 0C 1C2l 4C2 0C 1C2 :

Denote by this interval (18) of admissible values of coecientk. If the conditions (17) and (18) are satised, then the projections of trajectories of the system (13) with the terminalpoints (x1

1::: x

n

1) 2

S have their starting points in the half-space x1

0 < . Hence the

function(x1::: xn) can be determined in the points (x1::: xn) 2

S from the equality (12). The condition b2) for the terminal data

w1(x

1::: xn) is equivalent to the estimate jgradxw 1 j= C0k (x) < C0k C1 

analogous to the condition d1) in the previous section and the con-dition c2) is equivalent to the estimate

@ @x1 1 jgradxw 1(x 1::: xn) j < C2 C0k:

Theorem 2.

If the conditions a2), b2) and c2) are satised, then the inverse problem for the Hamilton-Jacobi equation

@w

@t +2(x1::: xn)

jgradxwj 2= 0

with the initial-terminal data w0(x

1::: xn) = kx1, w 1(x

1::: xn)

and k2 has at most one solution fw(xt)(x)g.

As in the previous section, the existence of solution of this inverse problem is equivalent to the concordance of the initial and termi-nal data w0(x) and w1(x). In the one-dimensional case this

concor-dance follows from the conditions c1) { f1). The higher-dimensional analogies of these conditions should be much more complicated, be-cause two trajectories of the system (13) with starting points at

fx 0p= (k0::: 0) g,fx 0 0p= (k0::: 0) gwithx 0 6 =x0 0 can nish at the points fx 1p g,fx 1p 0 g with p 6=p

0 and the same space

coor-dinates fxig. However, for x

1 <  and forx1 > + 2C2 0k 2 one has w1(x) =kx ;C 2 0k 2, see (7).

4. Two-dimensional case

Analogous result can be obtained for the inverse problem (1), (2) with hamiltonian (3). For convenience of exposition we shall restrict our considerations to the two-dimensional case

H(xp) =g11(x)p2 1+ 2g 12(x)p 1p2+g 22(x)p2 2:

Since here, in this hamiltonian there are three independent unknown functions gij_(x1x2), one pair on the initial-terminal conditions is

not sucient for reconstruction of this hamiltonian, so we assume the results of the following three testing to be known

I. w(1) 0 (x) =a 11x 1+a 12x 2 a2 11+a 2 12=k 2 1 w (1) 1 (x) II. w(2) 0 (x) =a 21x 1+a 22x 2 a2 21+a 2 22=k 2 2 w (2) 1 (x) III. w(3) 0 (x) =a 31x 1+a 32x 2 a2 31+a 2 32=k 2 3 w (3) 1 (x): Let DR

2 be the disk with radiusr centered in the origin and

smooth functions gij_(x1x2) satisfy the following conditions

a3).g11(x) =g22(x) =C

0 = const>0,g 12(x)

0 for all x =2D.

b3). For all points x 2 D and for allX

2+Y2 = 1 and for some

positive constantsC3> C1

C3 > g

11(x)X2+ 2g12(x)XY +g22(x)Y2

C

1>0:

c3). For allx2D,k= 12 and for some positive constantC 2

@gij_(x)

@xk pipj < C2g

ij_(x)pipj:

As in the previous sections, we describe a class of functions gij _and

the set of the values of the coecients k1,k2,k3, for which the

pro-jections of the corresponding trajectories of Hamilton system with the endpoints in D start outside of this disk. In this case the un-known functions gij_(x1x2) can be determined from the system of

linear equations (19) g11 @w (m) 1 @x1  2 + 2g12@w (m) 1 @x1 @w(m) 1 @x2 +g 22 @w (m) 1 @x2  2 =C2 0k 2 m m= 123 at any point (x1x2)

2D. The determinant of this system

is exactly 3 3 Vandermonde's determinant.

Hence, the solvability of the system (19) is equivalent to the pairwise linear independence of the gradients of the functions

w(1) 1 (x 1x2), w (2) 1 (x 1x2), w (3) 1 (x 1x2) in the diskD.

In our case the Hamilton system has the form _ x1= 2g11(x)p 1+ 2g 12(x)p 2 _ x2= 2g12(x)p 1+ 2g 22(x)p 2 _ p1 = ; @g11 @x1 p 2 1+ 2 @g12 @x1p 1p2+ @g22 @x1 p 2 2  

_ p2 = ; @g11 @x2 p 2 1+ 2 @g12 @x2p 1p2+ @g22 @x2 p 2 2 :

Consider the trajectories of this system which correspond to the initial-terminal data w0(x 1x2) =a 1x 1+a 2x 2 w 1(x 1x2) for some a2 1+a 2 2 =k 2. If (20) a1x_ 1+a 2x_ 2 = 2p i(a1g 1i+a 2g 2i)> 2r  k

then the projections of these trajectories which nish at the points of Dstart outside of this disk.

Since p1(t0) =a1 and p2(t0) =a2 it follows that

p1(t) =a1 ;2 t1 Z t0 @gij @x1pipjdt > a 1 ;C 2 t1 Z t0 gij_pipjdt=a 1 ;C 2k 2C2 0 and similarlyp2(t)> a2 ;C 2k 2C2 0:Hence, 2  p1(a1g 11+a 2g 12) +p 2(a1g 12+a 2g 22)  > 2  gij_aiaj;k 2C2 0C 2(g 11a 1+g 12(a 1+a2) +g 22a 2)  > 2C1k 2 ;2C 2 0C 2(g 11a 1+g 12(a 1+a2) +g 22a 2)k 2 > 2C1k 2 ;8C 2 0C 2C3k 3

and the required inequality (20) holds if 2C1k 2 ;8C 0C2C3k 3 > 2rk   i.e., if C2< C 2 1 16rC2 0C 3 and C1 ; p C2 1 ;16rC 2 0C 2C3 8C2 0C 2C3 < k < C1+ p C2 1 ;16rC 2 0C 2C3 8C2 0C 2C3 :

As in the previous section, we denote by this interval of values of the coecientk.

Theorem 3.

If the gradients of the functions w(1) 1 , w (2) 1 , w (3) 1 are

pairwise linearly independent and the solution of the system of linear equations (19) satises the conditions a3), b3) and c3), then the inverse problem for the Hamilton-Jacobi equation

@w @t +g11(x)p2 1+ 2g 12(x)p 1p2+g 22(x)p2 2= 0

with the unknown functionsgij_(x),w(1)(xt),w(2)(xt),w(3)(xt)and

the initial-terminal data I, II, III andk1k2k3

2 has at most one

solution fgij(x),w

(1)(xt), w(2)(xt), w(3)(xt) g.

As in the previous theorem, the existence of solution of this inverse problem is equivalent to concordance of the initial-terminal data I, II, III. In particular, for solvability of our inverse problem each of the functions w(m)

1 (x

1x2),m= 123 should have the form

w(m) 1 (x 1x2) =a m1x 1+a m2x 2 ;C 2 0k 2 m in the domainjam 2x 1 ;am 1x 2 j> rkm.

An analogous result can be obtained for the higher-dimensional spaces as well.

The authors are indebted to M.M.Lavrent'ev and V.P.Golubyatni-kov for helpful discussions.

References

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2. Golubyatnikov, V. P. (1995): Inverse problem for the Hamilton-Jacobi equa-tion, J. of Inverse and Ill-Posed Problems3, 407{410.

3. Taimanov, I. A. (1995): The topology of Riemannian manifolds with inte-grable geodesic ows, Proc. Steklov Inst. Mathematics205, 139{150.

4. Arnol'd, V. I. (1978):Mathematical methods of classical mechanics, Springer-Verlag, New-York { Heidelberg { Berlin.