On the stability of the solution in an optimal control problem for a Schrödinger equation

On the stability of the solution in an optimal control problem

for a Schrödinger equation

Murat Subasßı

a,⇑

, Hülya Durur

b a

Ataturk University, Science Faculty, Department of Mathematics, 25240 Erzurum, Turkey b

Ardahan University, Vocational School of Social Sciences, Ardahan, Turkey

a r t i c l e

i n f o

Keywords: Optimal control Frechet differentiability

Time dependent Schrödinger equation

a b s t r a c t

This research paper deals with the Frechet differentiability and necessary condition for optimality in a problem governed by the coefficient of a Schrödinger equation. In order to obtain a stabile solution by means of strong convergence of any minimizer, we have used a more regular space and considered second adjoint problem for obtaining the gradi-ent of the cost functional. Proving the Lipschitz continuity, we have stated the necessary condition for optimal solution.

Ó 2014 Elsevier Inc. All rights reserved.

1. Introduction and statement of the problem

In this paper, we consider the minimization of the functional

Jað

v

Þ ¼ Z l 0 jwðx; T;

v

Þ  yðxÞj2dx þ

a

k

v

 wk2W1 2ð0;TÞ ð1Þ on the set V ¼

v

ðtÞ :

v

2 H1ð0; TÞ; k

v

ðtÞkH1_ð0;TÞ6

l

n o ð2Þ

subject to the following Schrödinger initial-boundary value problem given on the domainX¼ ð0; lÞ  ð0; TÞ;

i@w @tþ a0

@2w

@x2

v

ðtÞw ¼ 0; ðx; tÞ 2

X

ð3Þ

wðx; 0Þ ¼

u

ðxÞ; x 2 ð0; lÞ; wð0; tÞ ¼ wðl; tÞ ¼ 0; t 2 ð0; TÞ ð4Þ

Equation(3)describes the wave function of a particle having mass of unity and acting in a magnetic field for h ¼ 1. If it is known that particle moves only inside of (0, l) and it has the initial status

u

(x) then the conditions given in(4)are valid. The control function

v

(t) is the potential of outer electric field. The function w(t) is in H1_{(0, T) and plays a vital role in order to}

obtain a unique solution.

The parameter

a

is a positive number used for both obtaining the unique solution and ensuring a balance between the norms of kwðx; TÞ  yðxÞk2 L2ð0;lÞand k

v

 wk 2 W1 2ð0;TÞin numerical investigations. http://dx.doi.org/10.1016/j.amc.2014.10.069 0096-3003/Ó 2014 Elsevier Inc. All rights reserved.

⇑Corresponding author.

E-mail addresses:msubasi@atauni.edu.tr(M. Subasßı),hulyadurur@ardahan.edu.tr(H. Durur).

Contents lists available atScienceDirect

Applied Mathematics and Computation

Similar optimal control problems have been considered in the studies[1–6]with the functional Iað

v

Þ ¼ Z l 0 jwðx; T;

v

Þ  yðxÞj2dx þ

a

k

v

 wk2L2ð0;TÞ ð5Þ on the set ^ V ¼

v

ðtÞ :

v

2 L2ð0; TÞ; k

v

ðtÞkL2ð0;TÞ6

g

n o ð6Þ

subject to the Schrödinger equations like(3)–(4). With these researches the existence and uniqueness for the optimal control problems have been proved. But the stability of the solution could have not been guaranteed by minimizing the functional

(5)on the set(6). Namely, instability often occurs in the manner that there are some minimizing sequences f

v

m_{g  ^V for the}

functional Ia(

v

) such as lim m!1Iað

v

m Þ ¼ Ia¼ inf v2VIað

v

Þ

but these sequences may not converge on the norm of L2(0, T) to the minimum element

v

⁄; lim

m!1k

v

m_{ðtÞ }

_v

_ðtÞk

L2ð0;TÞ9 0

In this study, we propose the minimization of the functional Ja(

v

) on the set V given by(1)–(2)instead of minimization Ia(

v

) on the set ^V. Thanks to the compact embedding of the space H1(0, T) to the space L2(0, T), the weakly minimizing

sequences in H1_{(0, T), converge strongly in L}

2(0, T)[7]. So the desired convergence of limm!1k

v

mðtÞ 

v

ðtÞkL2ð0;TÞ¼ 0 will

be hold with the construction of any minimizing sequence

lim m!1Jað

v

m_{Þ ¼ J} a¼ inf v2VJað

v

Þ for problem(1)–(2).

But, at the same time, some problems may arise when we choose the problem(1)–(2). For example how the gradient in the space H1_{(0, T) can be computed? How the Lipschitz continuity of this gradient can be proved and how the necessary}

con-dition for obtaining optimal solution can be gotten? Finding stabile solution by means of strong convergence of any mini-mizer requires the usage of a second adjoint problem in the stage of obtaining gradient. In the literature the answers to these types of questions are missing in the space H1_{(0, T). This paper deals with these questions.}

This paper is organized as follows: In Section2, we introduce the definition of generalized solution that we need in the remaining part of the paper. In Section3, we affirm the existence and uniqueness of the optimal solution by means of Goebel’s study[9]. The contribution of this research begins in Section4. In Section4, we give the Theorem 1 about the Frechet differentiability of the functional and we give a second adjoint problem. In Section5, estimating the solution of this adjoint problem, the Lipschitz continuity of the gradient and necessary condition for optimal control is submitted. The gra-dient obtained in the space H1_{(0, T) will lead to further numerical investigations about stabile solutions.}

2. The generalized solution for Schrödinger initial-boundary value problem

Let us define spaces used in this article. The space L2(X) is a Hilbert space consisting of square integrable functions with

its modulus onX. The inner product and norm on this space are defined such as

hw; /iL2ðXÞ¼ Z X wðx; tÞ/ðx; tÞdxdt; kwkL2ðXÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hw; wiL2ðXÞ q :

The space H2_{(0, l) is a Hilbert space consisting of all elements L}

2(0, l) having generalized derivatives of first and second

order from L2(0, l). The inner product and norm on this space are defined such as hw; /iH2_ð0;lÞ¼ Z l 0 w/þ@w @x @/ @xþ @2w @x2 @2/ @x2 ! dx; kwkH2_ð0;lÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hw; wiH2_ð0;lÞ q :

The space Ho2_{ð0; lÞ is a subspace of H}2

(0, l), in which all functions in [0, l] that are equal to zero at 0 and at l are dense. The space H2,1₍

X) is a Hilbert space consisting of all elements L2(X) having generalized derivatives@w_@x,@

2_w

@x2 and

@w @t from

L2(X). The inner product and norm on this space are defined such as hw; /iH2;1_ðXÞ¼ Z X w/þ@w @x @/ @xþ @2w @x2 @2/ @x2þ @w @t @/ @t ! dxdt; kwkH2;1_ðXÞ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi hw; wiH2;1_ðXÞ q : The space H2;1 o

ðXÞ is a subspace of H2,1(X), in which all functions inXthat are equal to zero at 0 and at l are dense. Now, we introduce the definition of the generalized solution to the Schrödinger initial-boundary value problem.

If the initial status

u

(x) is in the space

u

2 H2 o

ð0; lÞ ð7Þ

then the generalized solution of the problem(3)–(4)is the function w 2 Ho2;1_{ðXÞ satisfying the integral equality of} Z X i@w @tþ a0 @2w @x2

v

ðtÞw " # 

g

ðx; tÞdxdt ¼ 0 ð8Þ for8

g

2 L2ðXÞ.

Using Galerkin method[8]it can be proven that the solution in the sense of(8)is exist, unique and satisfies the inequality

kwk2 H2;1 o ðXÞ 6c_0k

u

_k2 H2 o ð0;lÞ: ð9Þ

Here c0is independent from

u

.

3. The existence and uniqueness of the optimal control

To prove the existence of optimal solution for problem(1)–(2)it is enough to show that the conditions of the generalized Weierstrass theorem hold.

Let us accept that we have found a minimizer f

v

m_{g  V for the functional J}

a(

v

), namely the following equality is valid;

lim

m!1Jað

v

m_{Þ ¼ J}

a¼ inf v2VJað

v

Þ:

Since the set V is closed, bounded and convex subset of H1_{(0, T), it is possible to choose a subsequence f}

_v

mk_{g  f}

v

m_g

which weakly converges to a control

v

0

_e

_H1_{(0, T). Again since the set is weak closed, this weak limit is in the set V. On}

the other hand the solution sequence fwmkg of the problem(3)–(4)corresponding to the controls f

v

mk_{g is exist. Similar to}

(9), this sequence holds the inequality

kwmkk

2 Ho2;1_ðXÞ

6_c0k

u

_k2

Ho2_ð0;lÞ:

Besides this sequence weakly converges to an element w02 H o

2;1_{ðXÞ. It can be shown that}

_w

0is generalized solution to the

problem.

Using the above inequality, we can say that

wmð:; TÞ!w0ð:; TÞ; weakly in L2ð0; lÞ

for m ? 1. Since the norm in L2(0, l) is weakly lower semi continuous we can pass the inequality; kw0ðx; TÞ  yðxÞkL2ð0;lÞ6 lim

m!1kwmðx; TÞ  yðxÞkL2ð0;lÞ

:

Taking into account that the sequence f

v

m_{g  V weakly converges in H}1_{(0, T), we conclude that} Ja6J_að

v

0Þ 6 lim

m!1

Jað

v

mÞ ¼ Ja:

Namely the conditions of generalized Weierstrass theorem hold. Therefore the limit

v

0_{is optimal solution to the problem} (1)–(2);

Jað

v

0Þ ¼ Ja¼ J_að

v

Þ:

As to the uniqueness of the solution we refer to the following theorem given by Goebel in[9].

Theorem: Let H be a uniformly convex Banach space and the set V be a closed, bounded and convex subset of H. If

a

> 0, b P 1 are given numbers and the functional J(v) is lower semi continuous and bounded from below on the set V then there is a dense set G of H that the functional

Jað

v

Þ ¼ Jð

v

Þ þ

a

k

v

 wkbH

takes its minimum on the set V for8w 2 G. If b > 1 then minimum is unique. h The following statements are valid in problem(1)–(2);

(a) the set H1_{(0, T) is a uniformly convex Banach space[10],}

(b) the set V is a closed, bounded and convex subset of H1_{(0, T),}

(c) the functional J(

v

) is lower semi continuous and bounded from below on the set V, (d) b = 2.

4. The Frechet differentiability of the functional

In this section we prove the Frechet differentiability of the functional(1)on the set(2). To do this, we first consider the following adjoint boundary value problem

i@/ @t þ a0 @2/ @x2

v

ðtÞ/ ¼ 0; ðx; tÞ 2

X

/ðx; TÞ ¼ 2iðwðx; TÞ  yÞ; /ð0; tÞ ¼ /ðl; tÞ ¼ 0: ð10Þ

As the solution of this problem, we mean the function / ¼ /ðx; tÞ 2 C0ð½0; T; L2ð0; lÞÞ which satisfies the integral equality Z X / i@

c

@tþ a0 @2

_c

@x2

v

ðtÞ

c

! dxdt ¼ Zl 0 i/ðx; 0Þ

c

ðx; 0Þdx  2 Z l 0 ðwðx; TÞ  yÞ

c

ðx; TÞdx ð11Þ for8

c

2 H o

2;1_{ðXÞ. The solution for this problem is exist, unique and satisfies the following inequality} k/ð:; tÞk2L2ð0;lÞ6c1 k

u

k 2 Ho2_ð0;lÞþ kyk 2 L2ð0;lÞ   ;

8

t 2 ½0; T: ð12Þ

Here c1is independent from

u

and y.

Now, we state following theorem about the Frechet differentiability of the functional using the solution of adjoint bound-ary value problem.

Theorem 1. The functional Ja(v) is Frechet differentiable on the set V and its gradient is

J0að

v

Þ ¼ n þ 2

a

ð

v

 wÞ: ð13Þ

Here, the function n(t) is the solution of the adjoint problem

n00 n ¼ Z l 0 Reðw/Þdx n0ð0Þ ¼ n0ðTÞ ¼ 0 ð14Þ

Proof. The increment of the functional Ja(

v

) according to the increment of the control

v

+ dv

e

V is

dJað

v

Þ ¼ Jað

v

þ d

v

Þ  Jað

v

Þ ¼ 2 Z l 0 Re½ðwðx; TÞ  yÞdwðx; TÞdx þ 2

a

Z T 0 ½ð

v

 wÞd

v

þ ð

v

t wtÞd

v

tdt þ kdwð:; TÞk2L2ð0;lÞþ

a

kd

v

k 2 H1_ð0;TÞ: ð15Þ

Here the difference function d

w

= d

w

(x, t) is the solution of the following difference problem according to the increment of the control

v

+ dv

e

V; i@dw @t þ a0 @2dw @x2  ð

v

ðtÞ þ d

v

ðtÞÞdw ¼ d

v

ðtÞw; ðx; tÞ 2

X

ð16Þ dwðx; 0Þ ¼ 0; dwð0; tÞ ¼ dwðl; tÞ ¼ 0:

The solution of the problem(16)satisfies the inequality

kdwð:; tÞk2L2ð0;lÞ6c2kd

v

k

2

L2ð0;TÞ;

8

t 2 ½0; T: ð17Þ

Here c2is independent from dv.

With some manipulations we have the following equation;

2 Z l 0 Re½ðwðx; TÞ  yðxÞÞdwðx; TÞdx ¼  Z X d

v

ðtÞReðw/Þdxdt  Z X d

v

ðtÞReðdw/Þdxdt:

Using the above equality we can rewrite(15)as

dJað

v

Þ ¼  Z X d

v

ðtÞReðw/Þdxdt þ 2

a

Z T 0 ½ð

v

 wÞd

v

þ ð

v

t wtÞd

v

tdt dt þ Rðd

v

Þ ð18Þ where the term R(dv) is

Rðd

v

Þ ¼ kdwð:; TÞk2L2ð0;lÞþ

a

kd

v

k 2 W1 2ð0;TÞ Z X d

v

ðtÞReðdw/Þdxdt ð19Þ

and it satisfies the inequality

jRðd

v

Þj 6 c3kd

v

k2H1_ð0;TÞ ð20Þ

Here c3is independent from dv.

Therefore the increment dJa(

v

) in(18)is rearranged such as

dJað

v

Þ ¼ ZT 0  Z l 0 Reðw/Þdx ( ) d

v

ðtÞdt þ 2

a

h

v

 w; d

v

iH1 ð0;TÞþ oðkd

v

kH1 ð0;TÞÞ ð21Þ

In order to obtain the inner product in the space H1_{(0, T), we purpose a second adjoint problem in the following;} n00 n ¼ Z l 0 Reðw/Þdx n0ð0Þ ¼ n0_{ðTÞ ¼ 0} ð22Þ

With the solution of this problem the statement(21)is written such as

dJað

v

Þ ¼ ZT 0 nðtÞd

v

ðtÞ þ n0_ðtÞd

_v

0_ðtÞ ½ dt þ 2

a

h

v

 w; d

v

iH1_ð0;TÞþ oðkd

v

k_H1_ð0;TÞÞ or dJað

v

Þ ¼ n þ 2h

a

ð

v

 wÞ; d

v

iH1 ð0;TÞþ oðkd

v

kH1 ð0;TÞÞ: ð23Þ

Hence the functional Ja(

v

) is Frechet differentiable on the set V and its gradient is given by

J0

að

v

Þ ¼ n þ 2

a

ð

v

 wÞ: ð24Þ

5. Lipschitz continuity of the gradient and necessary condition for optimal control

In this section, we give a theorem about Lipschitz continuity of the gradient. Then, to state the necessary condition will be possible.

Theorem 2. The gradient J0að

v

Þ holds the following Lipschitz inequality

kJ0að

v

þ d

v

Þ  J

0

að

v

Þk

2

H1_ð0;TÞ6c4kd

v

k2H1_ð0;TÞ ð25Þ

Here c4is independent from dv.

Proof. The increment of the functional J0

að

v

Þ according to the increment of the control

v

+ dv

e

V is

J0

að

v

þ d

v

Þ  J

0

að

v

Þ ¼ ndþ 2

a

ð

v

þ d

v

 wÞ  n þ 2

a

ð

v

 wÞ ¼ dn þ 2

a

d

v

: ð26Þ Here the function dn(t) is the solution of the increment problem

dn00ðtÞ  dnðtÞ ¼ Z l

0

Reðwdd/þ dw/Þdx: ð27Þ

The problem(27)has a solution in H1_{(0, T) and this solution satisfies the inequality} kdnðtÞkH1_ð0;TÞ6 Z l 0 ðjwdkd/j þ jdwk/jÞdx          _L 2ð0;TÞ ð28Þ The function d/ðx; tÞ ¼ /dðx; tÞ  /ðx; tÞ ¼ /ðx; t;

v

þ d

v

Þ  /ðx; t;

v

Þ

that takes place in the right hand side of(28)is the solution of the problem

i@d/ @t þ a0 @2d/ @x2  ð

v

ðtÞ þ d

v

ðtÞÞd/ ¼ d

v

ðtÞ/; ðx; tÞ 2

X

d/ðx; TÞ ¼ 2idwðx; TÞ; d/ð0; tÞ ¼ d/ðl; tÞ ¼ 0: ð29Þ

and this function holds the inequality

kd/ð:; tÞk2L2ð0;lÞ6c5kd

v

k

2

L2ð0;TÞ ð30Þ

The other function

w

dthat takes place in the right hand side of(28)holds the inequality kwdk 2 L2ðXÞ6k

u

k 2 Ho2_ð0;lÞ: ð31Þ

Therefore the inequality(28)has the property

kdnðtÞk2H1_ð0;TÞ62kw_dk2_L 2ðXÞmax_06t6Tðkd/ð:; tÞk 2 L2ð0;lÞÞ þ 2k/k 2 L2ðXÞmax_06t6Tðkdwð:; tÞk 2 L2ð0;lÞÞ:

If the inequalities(12), (17), (30) and (31)about the functions, /, d

w

(.,t), d/(.,t) and

w

dare used then the following

assess-ment is obtained; dnðtÞ k k2W1 2ð0;TÞ 6_c6kd

v

_k2 L2ð0;TÞ: ð32Þ

Here c6is independent from dv.

Taking the norm of(26)in the space H1_{(0, T), we acquire;} kJ0að

v

þ d

v

Þ  J

0

að

v

Þk

2

H1_ð0;TÞ62kdnðtÞk2_H1_ð0;TÞþ 8ð

a

Þ2kd

v

ðtÞk2_H1_ð0;TÞ

and considering(32), we have the following inequality for the gradient J0

að

v

Þ; kJ0að

v

þ d

v

Þ  J 0 að

v

Þk 2 H1_ð0;TÞ6c7kd

v

k2H1_ð0;TÞ:

Here c7is independent from dv.

Hence, it has been demonstrated that the gradient J0

að

v

Þ is continuous on the set V and moreover it holds the Lipschitz

condition with the constant c7> 0. The fact that the functional Ja(

v

) is continuously differentiable and that the set V is convex, in that case according to theorem in[11](p. 28) the following inequality is valid

J0

að

v

Þ;

v



v



 

H1

ð0;TÞP0;

8

v

2 V

if the control

v

⁄

_e

_{V is the minimum point of the functional J}

a(

v

). So the necessary condition for optimal control has been obtained.

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