On the optimal control problem for linear Schrödinger equation

Applied Mathematics and Computation 121 (2001) 373±381 www.elsevier.com/locate/amc

On the optimal control problem for linear dinger equation Schro B unyamin Yõldõz *, Murat Subasßõ  niversitesi, Fen-Edebiyat Fakultesi, Matematik Bolumu, 25240 Erzurum, Turkey Atat urk U

Abstract In this paper, two estimates have been proven for the solution of the optimal control problem which is set for the linear Schr odinger equation appearing in quantum mechanics. Acquiring necessary and sucient conditions, the existence and uniqueness of the solution have also been shown. Ó 2001 Elsevier Science Inc. All rights reserved. Keywords: Optimal control problem; Schr odinger equation

1. Introduction Optimal control problems for partial di€erential equations are currently of much interest. One of these problems is determining the quantum mechanical potential. Given simplifying assumptions, this potential is determined on the basis of intuitive concepts [1]. Problems of determining interaction potentials have stimulated the development of scattering theory. The experimental data concerning scattering are processed by minimizing the energy of the system [2]. Optimal control problems of the coecients of di€erential equations do not always have a solution [3]. The inverse problem of determining quantum mechanical potential is not well-posed in the Hadamard sense and this also holds for the variation formulation. In this paper, we obtain two estimates and necessary and sucient conditions for a solution of the inverse problem for the linear Schr odinger equation.

*

Corresponding author. Tel.: +90-442-2331062; fax: +90-442-2330784. E-mail address: [email protected] (B. Yõldõz).

0096-3003/01/$ - see front matter Ó 2001 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 0 ) 0 0 0 1 3 - 8

374

B. Yõldõz, M. Subasßõ / Appl. Math. Comput. 121 (2001) 373±381

2. Formulation of the problem We investigate the existence, uniqueness and smoothness of the solution of the problem Z l 2 2 Ja ˆ …1† jw…x; T ; v† y…x†j dx ‡ akv wkL2 …0;l† ! inf 0

in the class of admissible potential set V ˆ fv: v ˆ v…x†; v 2 L2 …0; l†; 0 6 b0 6 v…x† 6 b1 8x 2 …0; l†g under the conditions i

ow o2 w ‡ a0 2 ot ox

w…x; 0† ˆ u…x†;

v…x†w ˆ f …x; t†;

…x; t† 2 X;

…2†

x 2 …0; l†;

ow…0; t† ow…l; t† ˆ ˆ 0; ox ox

…3†

t 2 …0; T †:

…4†

For this purpose, we shall prove that for the solution of the problem (2)±(4), the following estimates holds:   2 2 2 kwkL2 …0;l† 6 C1 kukL2 …0;l† ‡ kf kL2 …X† and

  2 2 2 kwkW 2;1 …X† 6 C2 kukW 2 …0;l† ‡ kf kW 0;1 …X† : 2

2

2

Also, we will prove that the functional in a closed set and for the arbitrary a > 0 (1)±(4) have unique solution, where l > 0; T x 2 …0; l†, t 2 …0; T †, X ˆ …0; l†  …0; T †. The conditions u 2 W22 …0; l†;

du…0† du…l† ˆ ˆ 0; dx dx

(1) is lower semi-continuous and almost w, the problems > 0 are known numbers and functions u; f ; y satisfy the

f 2 W20;1 …X†;

y 2 W21 …0; l†;

respectively. Also, w 2 L2 …0; l† and a0 > 0; b0 > 0; b1 > 0 are given constants. Let Wpk …X† and Wpk;m …X†; p P 1; k; m P 0 be as de®ned in [4,5] and the symbol 8 signi®es that the given property applies for almost all values of a variable quantity.

B. Yõldõz, M. Subasßõ / Appl. Math. Comput. 121 (2001) 373±381

375

A solution of the boundary value problem (2)±(4) for each v 2 V is called a function w ˆ w…x; t† from W22;1 …X† which satis®es the conditions (2)±(4) for 8…x; t† 2 X. Theorem. If f 2 L2 …X† and u 2 L2 …0; l†, then the problems (2)±(4) have a solution in the w 2 C 0 …‰0; T Š; L2 …0; l†† and the estimate   kwk2L2 …0;l† 6 C1 kuk2L2 …0;l† ‡ k f k2L2 …X†

8t 2 ‰0; T Š

holds. Proof. Let us take the solution of the problems (2)±(4) as N X

wN …x; t† ˆ

kˆ1

hNk …t†uk …x†;

where the functions uk …x† are eigenfunctions of the Sturm±Liouville problem a0

d2 X ‡ v…x†X ˆ kX ; dx2

X 0 …0† ˆ X 0 …l† ˆ 0

…0 < k1 < k2 <


< kk †:

Therefore according to the Fourier method, we can write wˆ

1 X

hk …t†uk …x†;

…5†

kˆ1

ikk t

hk …t† ˆ uk e Z uk …x† ˆ

0

Z fk …t† ˆ u…x† ˆ

l

l 0

1 X kˆ1

f …x; t† ˆ

Z i

t 0

fk …s†e

u…x†uk …x† dx;

f …x; t†uk …x† dx; uk …x†uk …x†;

1 X kˆ1

fk …t†uk …x†:

ikk …t s†

ds;

…6†

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B. Yõldõz, M. Subasßõ / Appl. Math. Comput. 121 (2001) 373±381

We write the scalar product in L2 as N

N

w ;w


L2 …0;l†

N X

ˆ

hk …t†uk …x†;

kˆ1

62

N X kˆ1

N X

! hm …t†um …x†

mˆ1

2 juk j ‡ 2T

N X kˆ1

Z 0

L2 …0;l† t

2

jfk …s†j ds

by using (5) and (6). If we consider the equality 2 kuk ˆ

1 X kˆ1

2 juk j ;

then we can write the above inequality as Z T

N

N 2

N 2

f …0; t† 2

w

u 6 2 ‡ 2 L …0;l† L L …0;l† 2

2

Hence, we write

N

w

2 wM L …0;l† 2

2 …0;l†

0



6 C1 uN

2 uM L …0;l† 2

Z ‡

T 0

dt:

N

f

2 f M L …0;l† 2

 dt :

f we take limit for N ! M, then the last inequality is true in L2 space. Hence the proof is complete.  Theorem. For the solution belonging to the space W22;1 …X† of the boundary value problem (2)±(4), the following estimate is true:   kwk2W 2;1 …X† 6 C2 kuk2W 2 …0;l† ‡ k f k2W 0;1 …X† : 2

2

2

Proof. For this purpose, let us take wˆ

1 X

hk …t†uk …x†;

kˆ1

hk …t† ˆ uk e

ikk t

Z i

t 0

fk …s†e

ikk …t s†

ds;

where uk is orthonormal in the space L2 …0; l† and orthogonal in W21;0 …0; l† and W22 …0; l†. Since LX ˆ

a0

d2 X ‡ v…x†X ˆ kX ; dx2

B. Yõldõz, M. Subasßõ / Appl. Math. Comput. 121 (2001) 373±381

we have …uk ; um † ˆ k2k dmk ; where

(

dmk

ˆ

1;

k 6ˆ m;

0;

k ˆ m:

Let us consider the partial sum wN …x; t† ˆ

N X

hk …t†uk …x†:

kˆ1

We know that 

N 2

w

uN 2 6 C L L …0;l† 2

2

2 ‡ f N L …0;l†

2

 : …X†

Now, we will obtain a a priori estimate for wNt ; wNx ; wNxx . For this, we write N X

wNt ˆ

kˆ1

h0k …t†uk …x†

N  X

ˆ

kˆ1

N  X

ˆ

kˆ1

ikk uk e

ikk t

ikk uk e

ikk t

Z ifk …t†

kk

ifk …0†e

ikk t

t

fk …s†e

0

Z i

t 0

t

L2

63 …0;l†

It is obvious that 2

N X kˆ1

Z

jfk …0†j 6 C3 and

N 2

Lu

L2 …0;l†

0

T

k2k juk j2 ‡ 3

N X



Z

2

2 jfk …0†j ‡ 3T

kˆ1

jfk …t†j dt ‡

0

0 2 f …t† dt k

2 6 C4 uN W 2 …0;l† : 2

Hence, we write

N 2

w t

L2 …0;l†

6 C5

N 2

u

W 2 …0;l† 2

2 ‡ f N

0;1 W …X† 2

!



ds uk …x†:

N Z t X f 0 …t† 2 : kˆ1

T

 ds uk …x†

fk0 …s†e ikk …t s†

By the de®nition of norm in L2 space, we write

N 2

w

ikk …t s†

0

k

377

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B. Yõldõz, M. Subasßõ / Appl. Math. Comput. 121 (2001) 373±381

and

N

w t

2

wM t L2 …0;l†

6 C6

N

u

u

M 2

W 2 …0;l† 2

‡ f N

f

M 2 W

! 0;1 …X† 2

:

Integrating this inequality from 0 to t; …0; l† ! X and if we take limit for N ! M the last inequality belongs to the space L2 …X†: Hence, !

N 2

N 2

N 2

w

u ‡ f : …7† t L …X† 6 C7 2

W 2 …0;l† 2

0;1 W …X† 2

If we apply similar processes for wNxx ; wNx ,

N 2

w

xx L2 …X†

and

N 2

w x

L2

6 C8

!

N 2

2

u 2 ‡ f N W …0;l†

0;1 W …X† 2

2

 

N 2

uN 2 2

f 0;1 6 C ‡ : 9 …X† W …0;l† W …X† 2

2

…8†

…9†

Using by integration (7)±(9), we obtain  

N 2

N 2

w 2;1 6 C10 uN 2 2

f 0;1 ‡ : W …0;l† W …X† W …X† 2

2

2

N

If we take limit for N ! 1, we get w ! w and !

N 2

N 2

N 2

w 2;1 6 lim u ‡ f : W2 …X†

N !1

The proof is complete.

W 2 …0;l† 2

0;1 W …X† 2



3. Smoothness of the problem In this section, we examine the smoothness of the problems (1)±(4). For this purpose we will use the integral principle [6]. Let us consider the problem Ja …v† ˆ J0 …v† ‡ akv where J0 …v† ˆ

Z

l 0

jw…x; T ; v†

wk2~ ! inf; X

…10†

2

y … x†j dx

and V  X~ is a closed and bounded set. Theorem. If the functional J0 …v† is lower semi-continuous in the set V, there is 9G  X~ such that Ja …v† has a minimum in G for 8 w 2 G and 8 a > 0 [6].

B. Yõldõz, M. Subasßõ / Appl. Math. Comput. 121 (2001) 373±381

379

If we take 2

J0 …v† ˆ kw…x; T ; v†

ykL2 …0;l†

Ja …v† ˆ J0 …v† ‡ akv

wk2L2 …0;l† ;

and

then we only need to show that J0 …v† is lower semi-continuous in the set V, by the above theorem. For this purpose, let us calculate J0 …v ‡ Dv† J0 …v† for Dv 2 L1 …0; l†: Hence, we write Z l J0 …v ‡ Dv† J0 …v† ˆ 2 Rejw… x; T ; v† y … x†jDw…x; T † dx 0

2

‡ kDw…x; T †kL2 …0;l† by the formula of the functional where Dw ˆ w…x; t; v ‡ Dv† satis®es the conditions i

oDw o2 Dw ‡ a0 ot ox2

Dw…x; 0† ˆ 0;

w…x; t; v† and Dw

…v ‡ Dv†Dw ˆ Dvw;

…11†

x 2 …0; l†;

…12†

oDw…0; t† oDw…l; t† ˆ ˆ 0: ox ox

…13†

It is obvious that Dw ! 0 for Dv ! 0 and J0 …v† satis®es the Lipschitz condition jJ0 …v ‡ Dv†

J0 …v†j 6 C11 kDvkL1 …0;l† :

Then J0 …v† is lower semi-continuous. Now, we will use the integral principle to show the existence of a solution of the functional Ja …v†. For this purpose let us calculate DJa ˆ Ja …v ‡ Dv†

Ja …v†:

We know that Dv 2 L1 …0; l† and v ‡ Dv 2 V for the arbitrary v 2 V . We can write Z l DJa ˆ Ja …v ‡ Dv† Ja …v† ˆ 2 Rejw… x; T ; v† y … x†jDw…x; T †dx 0 Z l …v… x† w… x††Dv dx ‡ kDw…x; T †k2L2 …0;l† ‡ akDvk2L2 …0;l† ‡ 2a 0

…14†

using the formula Ja …v† where the function Dw satis®es the conditions (11)±(13).

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B. Yõldõz, M. Subasßõ / Appl. Math. Comput. 121 (2001) 373±381

It is obvious that Z l Rejw… x; T † 2 0

ˆ

Z

Z

t

0

y … x†j Dw…x; T †dx Z


Re w/ Dv dx dt

l

0

t

0

Z

l

0


Re Dw/ Dv dx dt;

…15†

where the function / ˆ /… x; t† satis®es the conditions i

o/ o2 / ‡ a0 2 ot ox

/… x; T † ˆ

v… x†/ ˆ 0;

2i…w… x; T †

o/…0; t† o/…l; t† ˆ ˆ 0; ox ox

… x; t† 2 X

y … x††;

…16†

x 2 …0; l†;

…17†

t 2 …0; T †:

…18†

If we substitute Eq. (15) in (14), we obtain  Z lZ T Z l
DJa …v† ˆ Re w/ dt Dv dx ‡ 2a … v… x † 0

where Rˆ

Z X

0

0

w… x††Dv dx ‡ R;


Re Dw/ Dv dx dt ‡ kDw…x; T †k2L2 …0;l† ‡ akDvk2L2 …0;l† :

Norm of each term in formula R is to belong to the space L1 …0; l† and their terms are less than kDvkL1 …0;l† . Hence, the Frechet derivative of the functional Ja …v† exists. Hence, by the formula   DJa …v† ˆ Ja …v ‡ Dv† Ja …v† ˆ hJ 0 …v†; Dvi ‡ O kDvkL1 …0;l† gradient of Ja …v† is Z T
0 Re w/ dt ‡ 2a…v… x† J … v† ˆ 0

w… x††:

Now let us give a well-known auxiliary lemma. Lemma. Let the functional J …v† be defined, continuous and differentiable on the convex set V  B. If v 2 V is minimum point of J …v†, then for 8 v 2 V hJ 0 …v †; v

v i P 0

holds [6]. Now, let us accept that the function v ˆ v … x† is a solution of the problems (1)±(4). Then according to the lemma, the following condition holds:

B. Yõldõz, M. Subasßõ / Appl. Math. Comput. 121 (2001) 373±381

Z     Re w / ‡ 2a…v… x† X

 w… x†† …v… x†

381

v … x†† dx dt P 0:

To complete the proof, to show the continuity of J 0 …v† is enough. It is easily seen that Z T 
 0 0 Re wD D/ ‡ Dw/ dt ‡ 2aDv: Ja …v ‡ Dv† Ja …v† ˆ 0

Thus, Z l

0 J …v ‡ Dv† a

0

Z

Ja0 …v† dx 6

l

Dv dx 6 kwD kL2 …X† ‡ 2a 0 Z l ‡2 jDv… x†j dx

Z X

w D/ dx dt ‡ D



D/

L2 …X†

Z X

Dw/ dx dt

‡ kDwkL2 …X† k/kL2 …X†

0

and

Z 0

l

0 J …v ‡ Dv† a

Ja0 …v† dx 6 C12 kDvkL1 …0;l† ‡ C13 kD/kL2 …X† :

…19†

If we consider that kD/kL2 …X† 6 C14 kDvkL1 …0;l† ; then (19) inequality is Ja0 …v ‡ Dv†

Ja0 …v† 6 C15 kDvkL1 …0;l†

and the Lipschitz condition holds. The proof is complete.



References [1] L.I. Schi€, in: Quantum Mechanics, McGraw-Hill, New York, 1955. [2] E. Prugovecki, in: Quantum Mechanics in Hilbert Space, Academic Press, New York, 1971. [3] A.N. Tichonov, V.Y. Arsenin, Solution of Ill-Posed Problems, Willey, London, 1977 (English Translation). [4] J.L. Lions, in: Optimal Control of Systems Governed by Partial Di€erential Equations, Springer, New York, 1971. [5] O.A. Ladyzhenskaya, V.A. Soloninkov, N.N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, Nauka, Moscow, 1967, AMS, Providence, RI, 1968, English Translation. [6] A.D. Iskenderov, R.K. Ta®yev, G.A. Yagubov, Methods of Optimization, Baku, 1994 (in Turkish).