An Optimization Problem of Boundary Type for Cooperative

Intelligent Control and Automation, 2014, 5, 262-271 Published Online November 2014 in SciRes. http://www.scirp.org/journal/ica http://dx.doi.org/10.4236/ica.2014.54028

An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrödinger Operator Ahlam Hasan Qamlo Department of Mathematics, Faculty of Applied sciences, Umm AL-Qura University, Makkah, KSA Email: [email protected] Received 16 September 2014; revised 10 October 2014; accepted 25 October 2014 Copyright © 2014 by author and Scientific Research Publishing Inc. This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

Abstract In this paper, we consider cooperative hyperbolic systems involving Schrödinger operator defined on R n . First we prove the existence and uniqueness of the state for these systems. Then we find the necessary and sufficient conditions of optimal control for such systems of the boundary type. We also find the necessary and sufficient conditions of optimal control for same systems when the observation is on the boundary.

Keywords Hyperbolic Systems, Schrödinger Operator, Boundary Control Problem, Boundary Observation, Cooperative

1. Introduction The optimal control problems of distributed systems involving Schrödinger operator have been widely discussed in many papers. One of the first studies was introduced by Serag [1], which discusses 2 × 2 cooperative systems of elliptic operator. Further research in this area developed the problem by studying different operator types (elliptic, parabolic, or hyperbolic) or higher system degree as in [2]-[6]. Many boundary control problems have been introduced in [7]-[10]. In [3], we discussed distributed control problem for 2 × 2 cooperative hyperbolic systems involving Schrödinger operator. Here, using the theory of [11], we consider the following 2 × 2 cooperative hyperbolic systems involving Schrödinger operator: How to cite this paper: Qamlo, A.H. (2014) An Optimization Problem of Boundary Type for Cooperative Hyperbolic Systems Involving Schrödinger Operator. Intelligent Control and Automation, 5, 262-271. http://dx.doi.org/10.4236/ica.2014.54028

 ∂ 2 y1 ( x ) + ( −∆ + q ) = y1 ay1 + by2 + f1 ( x, t ) ,  2  ∂t  ∂ 2 y2 ( x ) + ( −∆ + q ) y= cy1 + dy2 + f 2 ( x, t ) ,  2 2  ∂t  y , y → 0 as x → ∞,  1 2  y1 ∂y2  ∂= = 0,  ∂ν Σ ∂ν Σ  = y2 ( x, 0 ) y2,0 ( x ) ,  y1 ( x, 0 ) y= 1,0 ( x ) ,  ∂y2 ( x, 0 )  ∂y1 ( x, 0 ) y= = y2,1 ( x ) , 1,1 (x ),  ∂t ∂t

A. H. Qamlo

in Q, in Q, (1)

in R n , in R n .

∂y1 ∂y2 ∈ L2 ( Q ) . , ∂t ∂t where a , b , c and d are given numbers such that b , c > 0 , i.e. the system (1) is called cooperative

(

( )) ,

with y1 , y2 ∈ L2 0, T ;Vq R n

(2)

q ( x ) is a positive function and tending to ∞ at infinity,

(3)

and Q = R × ]0, T [ with boundary Σ = Γ × ]0,T [ . The model of the system (1) is given by: n

 ∂2 y  ∂2 y B (t ) = y ( x ) B ( t ) ( y1 ( x ) , y2= ( x ) )  21 + ( −∆ + q ) y1 − ay1 − by2 , 22 + ( −∆ + q ) y2 − cy1 − dy2  ∂t  ∂t 

( ( −∆ + q ) y1 − ay1 − by2 , ( −∆ + q ) y2 − cy1 − dy2 ) , A ( t ) y ∈ (Vq′ ( R n ) )

since A ( t ) y ( x )=

2

.

We first prove the existence and uniqueness of the state for these systems, then we introduce the optimality conditions of boundary control, we also discuss them when the observation is on the boundary.

2. Some Concepts and Results Here we shall consider some results about the following eigenvalue problem which introduced in [1] and [12]:

( −∆ + q ) φ = λ ( q ) φ in R n  φ ( x ) → 0 as x → ∞, φ  0

(4)

( )

The associated space is Vq R n , with respect to the norm: 12

  2 2 y q=  ∫  ∇y + q y  dx   n   R 

( )

( )

into L2 R n

Since the imbedding of Vq R n 2

( )

(5)

is compact, then the operator

( −∆ + q )

considered as an

n

is positive self-adjoint with compact inverse. Hence its spectrum consists of an infinite seOperator in L R quence of positive eigenvalues, tending to infinity; moreover the smallest one which is called the principal eigenvalue denoted by λ ( q ) is simple and is associated with an eigenfunction which does not change sign in R n . It is characterized by:

λ ( q ) ∫ y dx ≤ 2

R

We have:

n

( ) ( )

∫  ∇y

R

n

2

2 + q y  dx 

( ) ( )

( )

∀y ∈ Vq R n

( ) ( )

Vq R n × Vq R n ⊆ L2 R n × L2 R n ⊆ Vq′ R n × Vq′ R n

263

(6)

A. H. Qamlo

which is continuous and compact.

(

( ))

of measurable function t → f ( t ) which is defined on open

Let us introduce the space L2 0, T ;Vq R n

( 0,T ) and the variable t ∈ ( 0, T ) , T < ∞ denotes the time. ( 0,T ) with Lebesgue measure dt we have the norm:

interval On

12

  2 = f ( t ) L2 0,T ;V R n ( q ( ))  ( 0,∫T ) f ( t ) Vq ( Rn ) dt 

∞

and the scalar product

( f ( t ) , g ( t ) )L (0,T ;V ( R )) = ∫ ( f ( t ) , g ( t ) )V ( R ) dt , 0,T 2

(

( ))

the space L2 0, T ;Vq R n

n

q

(

q

)

n

with the scalar product and the norm above is a Hilbert space.

(

( )) = L (Q ) ,

Analogously, we can define the spaces L2 0, T ; L2 R n

2

with the scalar product:

dt ∫ f ( t ) ⋅ g ( t ) dxdt ( f ( t ) , g ( t )= )L (Q ) ∫ ( f ( t ) , g ( t ) )L ( = R ) 0,T Q 2

2

(

n

)

then we have:

(

( ) ) × L ( 0, T ;V ( R ) ) ⊆ L ( Q ) × L ( Q ) ⊆ L ( 0, T ;V ′ ( R ) ) × L ( 0, T ;V ′ ( R ) )

L2 0, T ;Vq R n

2

2

n

2

2

n

q

2

n

q

q

3. The Existence and Uniqueness for the State of the System (1) We have the bilinear form:

π ( t ; y,ψ= )

1 1 d [∇y1∇ψ 1 + qy1ψ 1 ] dx + ∫ [∇y2∇ψ 2 + qy2ψ 2 ] dx − ∫ y1ψ 2 dx − b R∫n c Rn c n R −

∫

y2ψ 2 dx

Rn

a y1ψ 1dx − ∫ y2ψ 1dx, b R∫n Rn

y y1 , y= ψ = 2,

(ψ 1 ,ψ 2 ) ∈ (Vq ( R n ) )

( ( ))

For all y, ψ ∈ Vq R n

2

(7)

2

.

the function t → π ( t ; y,ψ ) is measurable on

( ( ))

The coerciveness condition of the bilinear form (7) in Vq R n

( 0,T ) .

2

has been proved by Serag [1], by using the

conditions for having the maximum principle for cooperative system (1) which have been obtained by Fleckinger [13], and take the form:

a  λ ( q ) , d  λ ( q ) ,  ( λ ( q ) − a ) ( λ ( q ) − d )  bc

(8)

that means:

(

π ( t ; y, y ) ≥ C y1 Theorem (3.1):

2 q,m

+ y2

(

( )

264

),

( )) ,

Under the hypotheses (2) and (9), if f1 , f 2 ∈ L2 0, T ;Vq′ R n

y2,1 ( x ) ∈ L R n , then there exists a unique solution: = y

2 q,m

C0

(9)

( )

y1,0 ( x ) , y2,0 ( x ) ∈ Vq R n

{ y1 , y2 } ∈ ( L2 ( 0, T ;Vq ( R n ) ) )

and y1,1 ( x ) ,

2

for system (1).

A. H. Qamlo

Proof:

( ( ))

Let ψ → L (ψ ) be a continuous linear form defined on Vq R n = L (ψ )

2

by:

1 1 f1 ( x, t )ψ 1 ( x ) dxdt + ∫ f 2 ( x, t )ψ 2 ( x ) dxdt ∫ bQ cQ ∂ψ ( x, 0 ) ∂ψ ( x, 0 ) 1 1 dx + ∫ y2,1 ( x ) 2 dx, ∀ (ψ 1 ,ψ 2 ) ∈ Vq R n + ∫ y1,1 ( x ) 1 b Rn c Rn ∂t ∂t

( ( )) .

= y then by Lax-Milgram lemma, there exists a unique element

( y1 , y2 ) ∈ (Vq ( R n ) )

(

,ψ ) L (ψ ) , = π ( t ; y= ∀ψ (ψ 1 ,ψ 2 ) ∈ Vq ( R n ) Now, let us multiply both sides of first equation of system (1) by

)

(10)

2

2

such that:

2

(11)

1 ψ 1 ( x ) , and the second equation by: b

1 ψ 2 ( x ) then integration over Q , we have: c 2  1  ∂ y1 ( x ) 1 + ( −∆ + q ) y1 − ay1 − by2 ψ 1dxdt = ∫ f1 ( x, t )ψ 1dxdt  ∫ 2 b Q  ∂t bQ  2  1  ∂ y2 ( x ) 1 + ( −∆ + q ) y2 − cy1 − dy2 ψ 2 dxdt = ∫ f 2 ( x, t )ψ 2 dxdt  ∫ 2 c Q  ∂t cQ 

By applying Green’s formula: ∂y ∂y 1 ∂y1 ( x, 0 ) ∂ψ 1 ( x, 0 ) 1 1 1 a q  dx + ∫ψ 1 1 ν dΣ + ∫ ∇y1∇ψ 1dxdt − ∫ψ 1 1 dΣ + ∫  y1 − y1 − y2 ψ 1dxdt ∂t ∂t ∂t b R∫n bΣ bQ b Σ ∂ν A b b  Q =

1 f1 ( x, t )ψ 1dxdt , b Q∫

∂y ∂y d  1 ∂y2 ( x, 0 ) ∂ψ 2 ( x, 0 ) 1 1 1 q dx + ∫ψ 2 2 ν dΣ + ∫ ∇y2 ∇ψ 2 dxdt − ∫ψ 2 2 dΣ + ∫  y2 − y1 − y2 ψ 1dxdt c R∫n ∂t ∂t cΣ ∂t cQ c Σ ∂ν A c c  Q =

1 f 2 ( x, t )ψ c dxdt , c Q∫

By sum the two equations we get: ∂y ∂y ∂y 1 ∂y1 ( x, 0 ) ∂ψ 1 ( x, 0 ) 1 1 1 ∂y ( x, 0 ) ∂ψ 2 ( x, 0 ) 1 dx + ∫ ψ 1 1 ν d Σ − ∫ ψ 1 1 d Σ + ∫ 2 dx + ∫ ψ 2 2 ν d Σ ∫ b Rn bΣ b Σ ∂ν A c Rn cΣ ∂t ∂t ∂t ∂t ∂t ∂t

∂ψ ( x, 0 ) ∂ψ ( x, 0 ) ∂y 1 1 1 y1,1 ( x ) 1 = − ∫ ψ 2 2 dΣ dx + ∫ y2,1 ( x ) 2 dx , ∫ ∂t ∂t c Σ ∂ν A b Rn c Rn by comparing the previous equation with (7), (10) and (11) we deduce that:

∂y1 ∂y2 = = 0 ∂ν Σ ∂ν Σ ∂y1 ( x, 0 ) ∂y2 ( x, 0 ) = y= y2,1 ( x ) 1,1 ( x ) , ∂t ∂t then the proof is complete.

in R n

4. Formulation of the Control Problem The space L2 ( Σ ) × L2 ( Σ ) is the space of controls. For a control u=

265

( u1 , u2 ) ∈ ( L2 ( Σ ) )

2

, the state

A. H. Qamlo

= y (u )

( y1 ( u ) , y2 ( u ) ) ∈ ( L2 ( 0, T ;Vq ( R n ) ) )

2

of the system is given by the solution of

 ∂ 2 y1 ( u ) + ( −∆ + q ) = y1 ( u ) ay1 ( u ) + by2 ( u ) + f1 in Q,  2  2 ∂t  ∂ y2 ( u ) cy1 ( u ) + dy2 ( u ) + f 2 in Q, 2 (u )  ∂t 2 + ( −∆ + q ) y=   y1 , y2 → 0 as x → ∞,  ∂y ( u ) ∂y2 ( u )  1 = u= u2 , 1,  ∂ν Σ ∂ν Σ  = y2 ( x, 0, u ) y2,0 ( x ) in R n , 1,0 ( x ) ,  y1 ( x, 0, u ) y=  ∂y1 ( x, 0, u ) ∂y2 ( x, 0, u ) = y1,1 ( x ) , = y2,1 ( x ) in R n .  ∂t ∂t 

(

( )) ,

with y1 ( u ) , y2 ( u ) ∈ L2 0, T ;Vq R n

∂y1 ( u ) ∂y2 ( u ) , ∈ L2 ( Q ) . ∂t ∂t

The observation equation is given by z= (u )

= zd For a given

( zd 1 , zd 2 ) ∈ ( L2 ( Q ) )

J ( v ) = y1 ( v ) − zd 1

((

) (

where N ∈ L L2 ( Σ ) , L2 ( Σ ) 2

)

2

2

2 L2 ( Q )

+ y2 ( v ) − z d 2

The control problem then is to find u =

( L ( Σ )) 2

2 L2 ( Q )

+ ( Nv, v ) L2 ( Σ ) 2 .

(

)

(13)

) is hermitian positive definite operator: 2

vex subset of

u ) ) y= ( u ) ( y1 ( u ) , y2 ( u ) ) . ( z1 ( u ) , z2 (=

, the cost function is given by:

( Nu, u )( L ( Σ )) 2

(12)

2

≥γ u

2

( L2 ( Σ ))

{u1 , u2 } ∈ U ad

2

,

γ 0

(14)

such that J ( u ) ≤ J ( v ) , where U ad is a closed con-

.

Since the cost function (14) can be written as (see [11]):

J ( v ) = a ( v, v ) − 2 L ( v ) + y ( 0 ) − z d

2

( L2 (Q ))

2

where a ( v, v ) is a continuous coercive bilinear form and L ( v ) is a continuous linear form on

( L ( Σ )) 2

2

.

Then there exists a unique optimal control u ∈ U ad such that J ( u ) = infJ ( v ) for all v ∈ U ad by using the general theory of Lions [11]. Moreover, we have the following theorem which gives the necessary and sufficient conditions of optimality: Theorem (4.1): 2 Assume that (9) and (14) hold. If the cost function is given by (13), the optimal control u= ( u1 , u2 ) ∈ ( L2 ( Σ ) ) is then characterized by the following equations and inequalities:

 ∂ 2 p1 ( u ) + ( −∆ + q ) p1 ( u ) − ap1 ( u ) − cp= y1 ( u ) − zd 1 in Q,  2 (u ) 2  2 ∂t  ∂ p2 ( u ) y2 ( u ) − zd 2 in Q, 2 (u )  ∂t 2 + ( −∆ + q ) p2 ( u ) − bp1 ( u ) − dp=   p1 , p2 → 0 as x → ∞,  ∂p ( u ) ∂p2 ( u )  1 = 0,= 0,  ∂ν Σ ∂ν Σ  p= in R n , 0 1 ( x, T , u ) 2 ( x, T , u )  p=  ∂p1 ( x, T , u ) ∂p2 ( x, T , u ) in R n . = = 0 ∂t ∂t 

266

(15)

(

∂p1 ( u )

( )) ,

with p1 ( u ) , p2 ( u ) ∈ L2 0, T ;Vq R n

∂t

,

∂p2 ( u ) ∂t

( p ( u ) + Nu, v − u )( L (Σ)) 2

2

A. H. Qamlo

∈ L2 ( Q )

≥ 0, = ∀v

( v1 , v2 ) ∈ U ad

(16)

together with (12) , where p ( u ) = ( p1 ( u ) , p2 ( u ) ) is the adjoint state. Proof:

( u1 , u2 ) ∈ ( L2 ( Σ ) )

The optimal control u=

2

is characterized by [11]

J ′ ( u )( v − u ) ≥ 0,

∀v ∈ U ad ,

Which is equivalent to:

( y ( u ) − zd , y ( v ) − y ( u ) )( L (Q )) + ( Nu, v − u )( L (Σ)) 2

2

2

2

≥0

i.e.

( y1 ( u ) − zd1 , y1 ( v ) − y1 ( u ) )L (Q ) + ( y2 ( u ) − zd 2 , y2 ( v ) − y2 ( u ) )L (Q ) + ( Nu, v − u )( L (Σ)) 2

2

2

2

≥0

(17)

this inequality can be written as: T





∫ ( y1 ( u ) − zd1 , y1 ( v ) − y1 ( u ) )L ( R ) + ( y2 ( u ) − zd 2 , y2 ( v ) − y2 ( u ) )L ( R )  dt + ( Nu, v − u )( L (Σ)) 2

n

2

n

2

2

≥0

0

Now, since:   ∂ 2 y1 ( u ) , dt p u + ( −∆ + q ) y1 ( u ) − ay1 ( u ) − by2 ( u )  ( )  1 ∫ 2 2 n ∂t 0 L ( R )

T

= ( p, By ) L2 (Q ) 2

(

)

T  ∂ 2 y2 ( u ) dt. + ∫  p2 ( u ) , + ( −∆ + q ) y2 ( u ) − cy1 ( u ) − dy2 ( u )  2  2 n ∂t 0 L ( R )

where

By ( u ) = B ( y1 ( u ) , y2 ( u ) )

 ∂ 2 y1 ( u )  ∂ 2 y2 ( u ) =  + −∆ + q y u − ay u − by u + ( −∆ + q ) y2 ( u ) − cy1 ( u ) − dy2 ( u )  . , ( ) ( ) ( ) ( ) 1 1 2 2  ∂t 2  ∂t   by using Green formula and (12), we have:  ∂ 2 p1 ( u )  ∫  ∂t 2 + ( −∆ + q ) p1 ( u ) − ap1 ( u ) − cp2 ( u ) , y1 ( u )  2 n dt 0 L ( R )

T

By ) L2 ( Q ) 2 ( p, =

(

)

(

T 2  ∂ p2 ( u ) dt + ∫ + ( −∆ + q ) p2 ( u ) − bp ( u ) − dp2 ( u ) , y2 ( u )   ∂t 2 2 n 0 L ( R )

= B ∗ p, y

)(

L2 ( Q )

)

2

.

then

B∗ p ( u ) = B∗ ( p1 ( u ) , p2 ( u ) )

2 ∂ 2 p2 ( u )  ∂ p1 ( u )  =  + −∆ + − − + ( −∆ + q ) p2 ( u ) − bp1 ( u ) − dp2 ( u )  . q p u ap u cp u , ( ) ( ) ( ) ( ) 1 1 2 2 2 ∂t  ∂t 

and A∗ p ( u )=

( p1 ( u ) , p2 ( u ) )= ( ( −∆ + q ) p1 ( u ) − ap1 ( u ) − cp2 ( u ) , ( −∆ + q ) p2 ( u ) − bp1 ( u ) − dp2 ( u ) )

267

(18)

A. H. Qamlo

since the adjoint equation takes the form [11]:

∂2 p (u ) ∂t

2

+ A∗ p ( u ) = y ( u ) − zd

and from theorem (3.1), we have a unique solution

(

( )) ,

p2 ( u ) ∈ L2 0, T ;Vq R n

∂p1 ( u ) ∂t

∂p2 ( u )

,

∂t

( (

p ( u ) ∈ L2 0, T ;Vq ( R n )

))

2

which satisfies

p1 ( u ) ,

∈ L2 ( Q ) .

This proves system (15). Now, we transform (18) by using (15) as follows:

 ∂ 2 p1 ( u )  ∫  ∂t 2 + ( −∆ + q ) p1 ( u ) − ap1 ( u ) − cp2 ( u ) , y1 ( v ) − y1 ( u )  2 n dt 0 L ( R )

T

T 2  ∂ p2 ( u ) + ∫ + ( −∆ + q ) p2 ( u ) − bp1 ( u ) − dp2 ( u ) , y2 ( v ) − y2 ( u )  dt  ∂t 2 2 n 0 L ( R )

+ ( Nu , v − u ) L2 ( Σ ) 2 ≥ 0.

(

)

Using Green formula, we obtain: T



 ∂2





0







 L2 ( R n )

T

∫  p1 ( u ) ,  ∂t 2 + ( −∆ + q )  y1 ( v ) − y1 ( u )  T

+ ∫ −c ( p2 ( u ) , y1 ( v ) − y1 ( u ) ) L2 0

(

Rn

0

0

( Rn )

dt

T    ∂2  dt + ∫  p2 ( u ) ,  2 + ( −∆ + q )  y2 ( v ) − y2 ( u )  dt )  ∂t  0  L2 ( R n )

T

+ ∫ −b ( p1 ( u ) , y2 ( v ) − y2 ( u ) ) L2

dt + ∫ −a ( p1 ( u ) , y1 ( v ) − y1 ( u ) ) L2

T

(R ) n

dt + ∫ −d ( p2 ( u ) , y2 ( v ) − y2 ( u ) ) L2 0

( Rn )

dt + ( Nu , v − u ) L2 ( Σ ) 2 ≥ 0.

(

)

Using (12), we have:

( p ( u ) + Nu, v − u )( L (Σ)) 2

2

≥ 0.

Thus the proof is complete.

5. Formulation of the Problem When the Observation Is on the Boundary The observation equation is given by:

= z (u )

(

u ) , z2 ( u ) ) ( z1 (=

)

= M y (u ) Σ M

((

(( y (u ) ) , ( y (u ) )) 1

) (

M ∈ L L2 ( Σ ) , L2 ( Σ ) 2

)

2

Σ

2

Σ

).

This is interpreted as follows [11]: we take the trace of y ( u ) on Σ , which is particular in

( L ( Σ )) 2

2

. Let

this be denoted by y ( u ) Σ .

zd For a given =

( zd 1 , zd 2 ) ∈ ( L2 ( Σ ) )

2

J ( v ) = y1 ( u ) Σ − zd 1

((

) (

where N ∈ L L2 ( Σ ) , L2 ( Σ ) 2

)

2

, the cost function is given by: 2 L (Σ) 2

+ y2 ( u ) Σ − z d 2

) is defined as in (14).

The control problem then is to find = u

( u1 , u2 ) ∈ U ad

268

2 L2 ( Σ )

+ ( Nv, v ) L2 ( Σ ) 2 .

(

)

(19)

such that J ( u ) ≤ J ( v ) , where U ad is a closed con-

vex subset of

( L ( Σ )) 2

A. H. Qamlo 2

.

Since the cost function (19) can be written as [11]:

J ( v ) = a ( v, v ) − 2 L ( v ) + y ( 0 ) − z d

2

( L2 ( Σ ))

2

,

where a ( v, v ) is a continuous coercive bilinear form and L ( v ) is a continuous linear form on

( L ( Σ )) 2

2

.

Then using the general theory of Lions [11], there exists a unique optimal control u ∈ U ad such that

J ( u ) = infJ ( v ) for all v ∈ U ad . Moreover, we have the following theorem which gives the necessary and suf-

ficient conditions of optimality: Theorem (5.1):

( u1 , u2 ) ∈ ( L2 ( Σ ) )

Assume that (9) and (14) hold. If the cost function is given by (19), the optimal control u=

2

is then characterized by the following equations and inequalities:

 ∂ 2 p1 ( u ) + ( −∆ + q ) p1 ( u ) − ap1 ( u ) − cp2 ( u ) = 0 in  2  ∂t  ∂ 2 p2 ( u ) + ( −∆ + q ) p2 ( u ) − bp1 ( u ) − dp2 ( u ) = 0 in  2  ∂t  p , p → 0 as x → ∞,  1 2  ∂p2 ( u )  ∂p1 ( u ) =− =− y1 ( u ) Σ zd 1 , y2 ( u ) Σ z d 2 ,  ∂ν ∂ν Σ Σ   p= in p= 0 1 ( x, T , u ) 2 ( x, T , u )   ∂p1 ( x, T , u ) ∂p2 ( x, T , u ) = 0 in  = ∂t ∂t ∂p1 ( u )

( )) ,

(

with p1 ( u ) , p2 ( u ) ∈ L2 0, T ;Vq R n

∂t

,

∂p2 ( u ) ∂t

Q, Q, (20)

Rn , Rn .

∈ L2 ( Q ) together with (16) and (12).

Proof: The optimal control u=

( u1 , u2 ) ∈ ( L2 ( Σ ) )

2

is characterized by [11]:

J ′ ( u )( v − u ) ≥ 0, ∀v ∈ U ad

Which is equivalent to:

( y ( u ) − zd , y ( c ) − y ( u ) )( L (Σ)) + ( Nu, v − u )( L (Σ)) 2

2

2

2

≥0

i.e.

( y1 ( u ) − zd1 , y1 ( v ) − y1 ( u ) )L (Γ) + ( y2 ( v ) − zd 2 , y2 ( v ) − y2 ( u ) )L (Γ) + ( Nu, v − u )( L (Σ)) 2

2

2

2

≥0

(21)

this inequality can be written as: T

∫ ( y1 ( u ) − zd1 , y1 ( v ) − y1 ( u ) )L (Γ ) + ( y2 ( v ) − zd 2 , y2 ( v ) − y2 ( u ) )L (Γ )  dt + ( Nu, v − u )( L (Σ)) 2

2

0

since the adjoint system takes the form [11]:

 ∂2 p (u ) 0 in Q + A∗ p ( u ) =  2 t ∂    ∂p ( u ) = on Σ y ( u ) − zd  ∂ν Σ 

269

2

2

≥0

(22)

A. H. Qamlo

( (

( )))

and from theorem (3.1), we get a unique solution p ( u ) ∈ L2 0, T ;Vq R n

(

( ))

p1 ( u ) , p2 ( u ) ∈ L2 0, T ;Vq R n ,

∂p1 ( u )

,

∂p2 ( u )

∂t ∂t This proves system (20). Now, we transform (22) by using (20) as follows:

2

which satisfies:

∈ L2 ( Q ) .

T  ∂p1 ( u )   ∂p2 ( u )  y v − y u t + , d , y2 ( v ) − y2 ( u )  dt + ( Nu , v − u ) L2 ( Σ ) 2 ≥ 0 ( ) ( )    1 ∫ ∂ν 1 ∫ ( ) ∂ν 0 0  L2 ( Γ )  L2 ( Γ )

T

Using Green formula, we obtain: T ∂y1 ( v ) ∂y1 ( u )  ∂y2 ( v ) ∂y2 ( u )    p u − t + − , d dt + ( Nu , v − u ) L2 ( Σ ) 2 ≥ 0 ( )    p2 ( u ) ,  1 ∫ ∫ ( ) ∂ν ∂ν  L2 ( Γ ) ∂ν ∂ν  L2 ( Γ ) 0 0

T

Using (12), we have: T

T

∫ ( p1 ( u ) , v1 − u1 )L (Γ ) dt + ∫ ( p2 ( u ) , v2 − u2 )L (Γ ) dt + ( Nu, v − u )( L (Σ)) 2

0

2

2

2

≥ 0,

0

which is equivalent to:

( p ( u ) + Nu, v − u )( L (Σ)) 2

2

≥ 0.

Thus the proof is complete.

6. Conclusions In this paper, we have some important results. First of all we proved the existence and uniqueness of the state for system (1), which is (2 × 2) cooperative hyperbolic system involving Schrödinger operator defined on R n (Theorem 3.1). Then we found the necessary and sufficient conditions of optimality for system (1), that give the characterization of optimal control (Theorem 4.1). Finally, we also find the necessary and sufficient conditions of optimal control when the observation is on the boundary (Theorem 5.1). Also it is evident that by modifying: -the nature of the control (distributed, boundary), -the nature of the observation (distributed, boundary), -the initial differential system, -the type of equation (elliptic, parabolic and hyperbolic), -the type of system (non-cooperative, cooperative), -the order of equation, many of variations on the above problem are possible to study with the help of Lions formalism.

References [1]

Serag, H.M. (2000) Distributed Control for Cooperative Systems Governed by Schrödinger Operator. Journal of Discrete Mathematical Sciences and Cryptography, 3, 227-234. http://dx.doi.org/10.1080/09720529.2000.10697910

[2]

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[3]

Qamlo, A.H. (2013) Distributed Control for Cooperative Hyperbolic Systems Involving Schrödinger Operator. International Journal of Dynamics and Control, 1, 54-59. http://dx.doi.org/10.1007/s40435-013-0007-z

[4]

Qamlo, A.H. (2013) Optimality Conditions for Parabolic Systems with Variable Coefficients Involving Schrödinger Operators. Journal of King Saud University—Science, 26, 107-112. http://dx.doi.org/10.1016/j.jksus.2013.05.005

[5]

Serag, H.M. (2004) Optimal Control of Systems Involving Schrödinger Operators. International Journal of Control and Intelligent Systems, 32, 154-157. http://dx.doi.org/10.2316/Journal.201.2004.3.201-1319

[6]

Serag, H.M. and Qamlo, A.H. (2005) On Elliptic Systems Involving Schrödinger Operators. The Mediterranean Jour-

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Bahaa, G.M. (2006) Boundary Control for Cooperative Parabolic Systems Governed by Schrodinger Operator. Differential Equations and Control Processes, 1, 79-88.

[8]

Bahaa, G.M. and Qamlo, A.H. (2013) Boundary Control for 2 × 2 Elliptic Systems with Conjugation Conditions. Intelligent Control and Automation, 4, 280-286. http://dx.doi.org/10.4236/ica.2013.43032

[9]

Bahaa, G.M. and Qamlo, A.H. (2013) Boundary Control Problem for Infinite Order Parabolic System with Time Delay and Control Constraints. European Journal of Scientific Research, 104, 392-406.

[10] Serag, H.M. and Qamlo, A.H. (2001) Boundary Control for Non-Cooperative Elliptic Systems. Advances in Modeling & Analysis, 38, 31-42. [11] Lions, J.L. (1971) Optimal Control of Systems Governed by Partial Differential Equations. Springer Verlag, Berlin. http://dx.doi.org/10.1007/978-3-642-65024-6 [12] Fleckinger, J. (1981) Estimates of the Number of Eigenvalues for an Operator of Schrödinger Type. Proceedings Royal Society Edinburgh Section A: Mathematics, 89, 355-361. [13] Fleckinger, J. (1994) Method of Sub-Suber Solutions for Some Elliptic System Defined on Rn. Preprint UMR MIP. Universite Toulouse 3. France.

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