pseudo differential operators and neumann problems

PSEUDO DIFFERENTIAL OPERATORS AND NEUMANN PROBLEMS

S. J. Monaquel

Mathematics Department, Faculty of Science, King Abdul Aziz University, Jeddah, Saudi Arabia

Abstract-Our main purpose of this paper is to find the corresponding set of inequalities defining an optimal control of a system governed by Neumann problem for a class of pseudo differential operators with symbols defined in terms of conditionally exponential convex functions also we formulate the boundary control problem for a system governed by Neumann problem.

Index Terms-Pseudodifferential operators, conditionally exponential convex, optimal controls.

1. INTRODUCTION

Consider a class of pseudo differential operators

(

^,

) ( ) (

^,

) ( )

^(1.1)

n

x R

L x D u x =

∫

e ^ζL x ζ u ζ d

where L :RⁿRⁿ → R is a real valued continuous symbol such that Lx, . :Rⁿ → R _is coditionally exponential convex functions.

Definition 1.1

A real valued function L:Rⁿ → R is said to be coditionally exponential convex function if for any x₁,x₂, . . .xn ∈ Rⁿ _and C₁,C₂, . . .Cn ∈ R_, we have

( ) ( ) ( )

, 1

0 (1.2)

n

j k j k j k

j k

L x L x L x x C C

=

⎡ + − + ⎤ ≥

⎣ ⎦

∑

see 2, 4.

Under suitable conditions Lx,D extends from C₀^Rⁿ to a generator of a symmetric Dirichlet

form B,DB with domain DB ⊂L2Rⁿ

and

( ) ( ( ) )

_{( )}

( )

2 0

, , , ,for , .

(1.3)

n

n L R

B u v = L x D u v u v ∈C^∞ R In this paper we are interested in boundary value problems for Lx,D on some open set ⊂ Rⁿ_. We find the corresponding set of inequalities defining an optimal control of a system governed by Neumann problem for Lx,D on 

(

^,

)

ⁱⁿ

(1.4) on

L x D u f Rn

u h

Ω ν Γ

⎧ = ⊂

⎪⎨ ∂ =

⎪ ∂⎩

II. A CLASS OF PSEUDODIFFERENTIAL OPERATORS

Let us Recall some results from 1, 2, 10, see also 4, 5. Let a² :Rⁿ → R be a real valued continuous conditionally exponential convex function, see 2, 11, that is a² is a continuous function such that a²0 ≥ 0 for all t  0 , the function  → e^{−ta2} is exponentially convex.

We define the norm

( ( ) ) ( )

2

2 2

2 2

, 1 , for 0.

(2.1)

n

s a s

R

u =

∫

+a ζ u ζ dζ s ≥ and the sobolev spaces

( ) { ( ) }

2

2

,

2 : , . (2.2)

a s n n

a s

H R = u∈L R u < ∞

The space H^a2,sRⁿ is a real Hilbert space with

the scalar product

( )

, ²,

(

1 ²

( ) )

²

( ) ( )

(2.3)

n

s a s

R

u υ =

∫

+a ζ u ζ υ ζ ζ d and C₀^Rⁿ is a dense subspace of H^a2,sRⁿ.

For a²  ||², the space H^a2,sRⁿ coincides with the usual sobolev space H^2sRⁿ.

It is known that a real valued continuous conditionally exponential convex function a² satisfies the estimate

( ) ( 2)

0≤a2 ζ ≤C 1+ζ , for someC >0 (2.4) and a² ≥ C||^2r for some r ∈ 0, 1, C  0 and all  ∈ Rⁿ_.

Moreover, by 8, 9, we can construct a chain

( ) ( ) ( )

2, 2,

2 (2.5)

a s n n a s n

H R ⊆L R ⊆H⁻ R

In the following we will always suppose that L:Rⁿ Rⁿ → R is a real continuous symbol such that for any fixed x∈ Rⁿ, the function Lx, . :Rⁿ → R is coditionally exponential convex and Lx, has the decomposition

(

,

)

1

( )

2

(

,

)

(2.6)

L x ζ =L ζ +L x ζ

where for a suitable m∈ N, we have

1) ^L1

( )

^{ζ ≤C}

(

^1+a2

( )

^ζ

)

^{, for some C  0}

and  ∈ Rⁿ_;

2) L₂. , ∈ C^mRⁿ and for all  ∈ N₀ⁿ_,

( ) ( ) (

²

( ) )

, 2 , 1

(2.7) m x^βL x _β x a

β ≤ ∂ ζ ≤ϕ + ζ

hold for all  ∈ Rⁿ with some

 ∈ L1Rⁿ;

3) L1 ≥ 20a² for some 0  0 and all

 ∈ Rⁿ, || ≥ R  0;

4)

L1

m α α

ϕ

∑

≤ is small with respect to 0, see

3.

Then the operator Lx,D as defined in 1. 1

maps C₀^Rⁿ into the space CRⁿ and the bilinear form associated with Lx,D

( )

^,

(

^,

) ( )

^.

( )

^(2.8)

Rn

B u υ =

∫

L x D u x υ x dx is defined for u, ∈ C₀^Rⁿ.

In the following we suppose that the operator Lx,D is symmetric on C₀^Rⁿ, then Lx,D has a selfadjoint extension on L2Rⁿ with domain H^a2,1Rⁿ. The bilinear form B _extends to a continuous symmetric Dirichlet form with domain H^a2,12Rⁿ, see 4, 5 for the general theory of Dirichlet forms and their properties. In particular, the form B is positive definite on H^a2,12Rⁿ, i.e. Bu,u ≥ 0 , for all u∈ H^a2,12Rⁿ.

Moreover, the form B satisfies Gãrding inequality, see 6

( )

²2 1 _{( )}

2 2

2

0 , 0

, a L Rn (2.9)

B u u ≥γ u −λ u

where 0 is taken from condition (3) and

0  0.

III. FORMULATION OF THE PROBLEM Let ⊂ Rⁿ be an open set with smooth boundary Γ. By 2. 9 , the bilinear form B _{is a} continuous and coercive bilinear form on H^a2,1Rⁿ ⊂L²Rⁿ. Thus, by the Lax Milgram Theorem, see 9, for each f∈ L²Rⁿ we find a weak solution y∈ H^a2,1Rⁿ satisfying Neumann problem relative to the operator L_, defined by 1. 1, and enables us to obtain the state of our system.

Theorem 3.1

If 2. 9 is satisfied then there exists a unique element y∈ H^1,a2Rⁿ satisfying Neumann problem

(

^,

)

^,

(3.1) on ,

L x D y f in u h

Ω

ν Γ

=

∂ =

∂

where

( )

1

cos ,

n

k

k k

u u

v = x n x

∂ = ∂

∂

∑

∂ ^{on Γ,}

cosn,xk  k−th direction cosine of n_, n being the normal at Γ exterior to Rⁿ.

Proof.

Let us choose L to be of the form

( )

( )

^{12, 2}

( )

2

, (3.2)

where , .

Rn

n a

L f dx h d

f L R h H

Γ

ϕ ϕ ϕ Γ

− Γ

= +

∈ ∈

∫ ∫

We note that 3. 2 defines a continuous linear form on H^1,a2Rⁿ, (see 5, 6, 7), from the coerciveness condition 2. 9there exists a unique element y∈ H^1,a2Rⁿ such that

(

^,

) ( )

^(3.3)

B y ϕ =L ϕ

This equation is equivalent to

(

^,

)

^{, in n (3.4)}

L x D y =f R

Multiply both sides by  and apply Green's formula, we get

( )

, (3.5)

n n

R R

L x D y dxϕ = f ϕ dx

∫ ∫

(

^,

)

^(3.6)

Rn

B y y d f dx

Γ

ϕ ϕ Γ ϕ

ν

− ∂ =

∫

∂

∫

Using 3. 3, it follows that

(

^,

)

^{, (3.7)}

Rn

B y f dx h d

Γ

ϕ =

∫

ϕ +

∫

ϕ Γ then,

0 (3.8)

y h d

Γ

ν ϕ Γ

⎛−∂ + ⎞ =

⎜ ∂ ⎟

⎝ ⎠

∫

on , (3.9)

y h Γ

ν

∂ =

∂

now the space L2Rⁿ, being the space of controls, is given.

For a control u the state of the system yu is given by the solution of

( ) ( )

( )

, , in

(3.10) , on

L x D y xu f u Rn

y u h Γ

ν

= +

∂ =

∂

An observation equation Zu  yu is also given, and N ∈ LL2Rⁿ,L2Rⁿ, where N _is Hermitian positive definite, satisfying,

( )

_{( ) ( )}

2 2

, L Rn 2L Rn . (3.11)

Nu u ≥α u

The cost function Ju is the same, and given by

( ) ( )

_{( )}

( )

_{( )}

( ( ) ) ( )

_{( )}

2 2

2

2

2

,

(3.12) ,

n n

n n

d L R L R

d L R

R

J u y u Z Nu u

y u Z dx Nu u

= − +

= ∫ − +

where Zd is a given element in L2Rⁿ.

The problem is to find inf,  ∈ Uad, where Uad (the set of admissible controls) is a closed convex susbset of L2Rⁿ. Under this consideration , we have the following theorem.

Theorem 3.2

Assume that 2. 9 holds, the cost function being given by 3. 12, a necessary and sufficient condition for u∈ L2Rⁿ to be an optimal control is that the following equations and inequalities be satisfied

( )

( ) ( ) ( )

in , on ,

(3.13)

in , 0 on

n

n d

Ly u f u R u h

Lp u y u Z R p u ν Γ

ν Γ

= + ∂ =

∂

= − ∂ =

∂

and

( ( ) ) ( )

^{0 (3.14)}

Rn

p u +Nu υ−u dx ≥

∫

for all u, ∈ Uad, where pu is the adjoint state of yu.

Proof

The control v∈ Uad is optimal if and only if

( ) ( )

0, for all _ad (3.15) J u^′ = v −u ≥ υ∈U

That is

( ) ( ) ( )

( )

_{( )}

( )

_{( )}

2

2

,

, 0. (3.16)

n

n

d L R

L R

y u Z y y u

Nu u υ

υ

− −

+ − ≥

If we set

( ) ( ( ) ( ) ( ) ( ) )

_{( )}

( )

_{( )}

( ) ( ( ) ( ) ( ) )

_{( )}

2

2

2

, 0 , 0

, (3.17)

0 , 0

n

n

n

L R

L R

d L R

u y u y y y

Nu u

L Z y y y

π υ υ

υ υ

= − −

+

= − −

The form u, is a continuous bilinear form and L is a continuous linear form on L2Rⁿ, then if we set

( ) ( ) ( ) ( )

_{( )}

2

, 2 _d 0 2 _n (3.18)

L R

J υ =π υ υ − L υ + Z −y since

( ) ( ) ( )

_{( )}

( )

_{( )}

2 2

, y y 0 2L Rn N , L Rn .(3.19)

π υ υ = υ − + υ υ

Then from 3. 11, we have

( )

^{, 2}_{2( )}n , for every ²

( )

(3.20)

n

L R L R

π υ υ ≥α υ υ∈

As in 8, 9, there exists a unique element u _in Uad such that

( )

^inf

( )

^(3.21)

Uad

J u J

υ υ

= ∈

and this element is characterized by

( ) ( )

0, for all _ad. (3.22) J u^′ υ−u ≥ υ∈U

Since L is a canonical isomorphism from H^1,a2Rⁿ into H^−1,a2Rⁿ, we may write

( )

¹

( )

^(3.23)

y u =L⁻ f +u whence

( ) ( ) ( ( ) ( ) ( ) )

_{( )}

( )

_{( )}

2

2

2 , 0

,

(3.24)

n

n

d L R

L R

J u u y u Z y u y

Nu u

υ υ

υ

′ − = ⎡ − − −

⎢⎣

+ − ⎤⎥⎦

But ^y

(

υ−^u

)

−^y

( )

⁰ =^y

( )

υ −^{y u}

( )

^{, (3.25)}

then

( ) ( ) ( ( ) ( ) ( ) )

_{( )}

( )

_{( )}

2

2

2 ,

,

(3.26)

n

n

d L R

L R

J u u y u Z y y u

Nu u

υ υ

υ

′ − = ⎡ − −

⎢⎣

+ − ⎤

⎥⎦

Therefore, after dividing by 2, 3. 15 is equivalent to

( ) ( ) ( )

( )

_{( )}

( )

_{( )}

2

2

,

, 0 (3.27)

n

n

d L R

L R

y u Z y y u

Nu u υ υ

− −

+ − ≥

for the control u∈ L₂Rⁿ the adjoint state

( )

^1,a2

( )

ⁿ

p u ∈H R is defined by

( ) ( ) ( )

, in

(3.28) 0, in

n

Lp u y u Zd R

p u Γ

ν

= −

∂ =

∂

Now, multiplying the first equation in 3. 28 by

y−yu and applying Green's formula, we obtain

( ) ( ) ( )

( )

_{( )}

( )

_{( )}

2 2

, , 0

(3.29)

n L Rn

L R

Lp u y υ −y u + Nu υ−u ≥

and

( ( ) ( ( ) ( ) ) )

_{( )}

( ) ( ( ) ( ) )

( )

( )

_{( )}

2

2

2

, ,

, 0 (3.30)

n

n

L R

L

L R

p u L y y u

p u y y u

Nu u

Γ

υ ν υ υ

−

⎛ ∂ ⎞

+⎜⎝ ∂ − ⎟⎠

+ − ≥

from 3. 1 , we obtain

( ( ) )

_{( )}

( ( ) )

_{( )}

( )

_{( )}

( ( ) )

_{( )}

( )

_{( )}

2 2

2

2 2

, ,

,

, , 0

(3.31)

n

n

n n

L R L

L R

L R L R

p u f f u p u h h

Nu u

p u u Nu u

υ Γ

υ

υ υ

+ − − + −

+ −

= − + − ≥

that is u∈ Uad,

(

p u

( )

Nu

) (

u dx

)

^{0, for all} Uad ^(3.32) Ω

υ υ

+ − ≥ ∈

∫

which completes the proof.

IV. BOUNDARY CONTROL FOR A SYSTEM GOVERNED BY NEUMANN

PROBLEM

Consider the space H^−12,a2Γ  U (the space of controls), for every control ^{u∈H−12,a2}

( )

^{Γ , the}

state of the system yu is given by the solution of

( ) ( )

in

(4.1) on

Ly u f Rn

u u h u Γ

ν

=

∂ = +

∂

and the observation is given by Zu  yu.

Finally the cost function is given by

( ) ( )

_{( )}

( )

2

2 n , (4.2)

d L R U

J υ = y υ −Z + Nυ υ

where Zd is a given element in L2Rⁿ and, N ∈ LU,U, N is Hermitian, positive definite,

(

^{Nυ υ,}

)

_U ^{≥C υ} _U^{2 , C >0. (4.3)}

We wish to find infJ, ∈ Uad, where Uad

(the set of admissible controls) is a closed convex subset of U_.

Under the given considerations, we have the following theorem.

Theorem 4.1

Assume that 2. 9 holds and the cost function being given by 4. 2. The optimal control u _is characterized by the following system of equations and inequalities

( ) ( ) ( ) ( )

( )

in on

(4.4) in

0 on

n

n d

Ly u f R

y u h u

Lp u y u Z R

p u ν Γ

ν Γ

=

∂ = +

∂

= −

∂ =

∂

and

( ( ) )

_{( )}

( )

2

, , _U 0 (4.5)

p u υ−u L _Γ + Nu υ−u ≥

for all u, ∈ Uad, where pu is the adjoint of the state yu.

Outline of proof.

Using 8, 9, the control u∈ U_ad is optimal if and only if

( ) (

^.

)

0, for all _ad (4.6) J u^′ υ−u ≥ υ∈U

that is

( ) ( ) ( )

( )

_{( )}

( )

2

, , 0

(4.7)

d L Rn U

y u −Z y υ −y u + Nu υ−u ≥

the adjoint state is given by the solution of the adjoint Neumann problem

( ) ( ) ( )

in

(4.8) 0 on

n

Lp u y u Zd R

p u Γ

ν

= −

∂ =

∂

from 4. 7 and 4. 8, we have

( ) ( ) ( )

( )

_{( )}

( )

2

, , 0

(4.9)

n U

L R

Lp u y υ −y u + Nu υ−u ≥ by applying Green's formula, we obtain

( ) ( ( ) ( ) )

( )

_{( )}

( ) ( ( ) ( ) )

( )

( ) ( ) ( )

( )

( )

2

2

2

, ,

,

, 0 (4.10)

L Rn

L

L

U

p u L y y u

p u y y u

p u y y u Nu u

Γ

Γ

υ ν υ ν υ

υ

−

⎛ ∂ ⎞

+⎜⎝ ∂ − ⎟⎠

⎛ ∂ ⎞

−⎜⎝∂ − ⎟⎠

+ − ≥

from 4. 1 and 4. 8, we obtain

( ) ( ) ( )

( )

_{( )}

( ( ) )

_{( )}

( ) ( )

( )

_{( )}

( )

2

2

2

, , 0,

, 0 (4.11)

L Rn

L

L

U

p u Ly Ly u p u h h u

y y u

Nu u

Γ Γ

υ υ υ υ

−

+ + − −

− −

+ − ≥

It follows

( ( ) )

_{( )}

( ( ) )

_{( )}

( )

2 2

, ,

, 0 (4.12)

L Rn L

U

p u f f p u u

Nu u υ Γ

υ

− + −

+ − ≥

which is equivalent to

( ( ) )

_{( )}

( )

2

, , _U 0 (4.13)

p u υ−u L _Γ + Nu υ−u ≥ which completes the proof.

ACKNOWLEDGEMENTS

The author is grateful to prof. Hoda A. Ali for substantial assistance through the paper.

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