On Some Fractional-Integro Partial Differential Equations

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International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 52

On Some Fractional-Integro Partial Differential

Equations

Mahmoud M. El-Borai

a

, Abou-Zaid H. El-Banna

b

, Walid H. Ahmed

c a

Department of Mathematics, faculty of science, Alexandria university, Alexandria. Email:m_ m_elbora [email protected]

b_{Department of Mathematics, Faculty of science, Tanta university, Tanta.}

Email:ah [email protected]

c_{Department of engineering mathematics, High institute of engineering, ELsherouk academy, Cairo.}

Email: [email protected].

Abstract-- In this paper, an inverse optimal control problem is introduced with state function governed by fractional partial differential equation. The existence of the control and necessary optimality conditions are proved.

Index Term-- fractional calculus, partial differential equations, Optimal control.

1. INTRODUCTION

There is an increasing interest in the study of dynamic systems of fractional order. Extending derivatives and integrals from integer to non-integer order has a firm and long standing theoretical foundation. Leibniz mentioned this concept in a letter to L‟Hopital over three hundred years ago. Following LHopital‟s and Leibniz‟s first inquisition, fractional calculus was primarily a study reserved to the best minds in mathematics.

Euler [1], Fourier [2] and Laplace [3,4]are among the many that contributed to the development of fractional calculus. Along the history, many found, using their own notation and methodology, definitions that fit the concept of a non -integer order integral or derivative. The most famous of these definitions among mathematicians that have been popularized in the literature of fractional calculus are the ones of Riemann-Liouville and Grunwald-Letnikov. On the other hand, the most intriguing and useful applications of fractional derivatives and integrals in engineering and science have been found in the past one hundred years. In some cases, the mathematical notations evolved in order to be better meet the requirements of physical reality. The best example of this is Caputo fractional derivative, nowadays the most popular fractional operator among engineers and applied scientists, obtained by reformulating the “classical” definition of Riemann-Liouville derivative in order to be possible to solve fractional initial value problems with standard initial conditions (see [22]). Particularly in the last decade of 20th century, numerous applications and physical

manifestations of fractional calculus have been found.

Fractional differentiation is nowadays recognized as a good tool in various different fields: physics, signal processing, fluid mechanics, viscoelasticity biology, electro chemistry,

economics, engineering and control theory (see [23], [24], [25], [26], [27] and [28]).

The fractional calculus of variations was porn in 1996 with the work of Riewe, and nowadays a subject under strong current research. The fractional calculus by considering fractional derivatives into the variation integrals to be extremized. This occurs naturally in many problems of physics and mechanics.

The aim of this paper is studying the existence of the control which maximize the cost functional

( ) ∫ ( ( ) ( ))

Where is constant.

This paper is organized as follows. Section 2 presents some preliminaries on fractional calculus. In section 3 we formulate the fractional partial differential equation which governing the state function. Our main results are stated and proved in sections 4 and 5. In section 6 we introduce an inverse optimal control problem and we study the existence of the optimal control .

2. PRELIM INARIES

In this section, we give some definitions and lemmas which are used further in the paper. For more on the subject we refer the reader to the books ([5], [6], [7], [8]).

Definition 2.1. Let be a continuous on and . Then the expression

( ) _{( )}∫ ( ) ( ) ( )

Is called the Riemann-Liouville integral of order

Definition 2.2. Let . The Riemann-Liouville fractional derivative of order of is defined by

( )

( ) ∫( ) ( ) ( )

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1311701-4949-IJBAS-IJENS @ February 2013 IJENS I J E N S Definition 2.3. Let . The (left) Caputo fractional

derivative of order of is defined by

( )

( )∫ ( )

_{( )}( )

( )

Where ( )

3. PROBLEM FORM ULATION

We consider the following linear fractional integro -differential equation:

( )

( ) ( ) ( )

( ( ) ∫ ( ) ) ( ) ∫ ( ) ( )

( )

( ) ( ) ( )

( ) ( ) ( ) ( ) ( )

For ( ) and under assumption

, -

(, - , -),

where the ( ) is the control input. The control objective is to stabilize the equilibrium ( ) and ( ) is the standard Brownian motion see ([9], [10]).

At the first we assume that the stochastic process

( ) where is constant and hence we generalize our results for some wide class of stochastic process ( ).

Theorem 3.1. The transformation

( ) ( ) ∫ ( ) ( ) ( )

Transforms the system (3.1), (3.3) into the system

( )

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

( ) ( ) ( )

which is exponentially stable for where

* + and the transformer (namely ( )) satisfies the hyperbolic partial differential equation

( ) ( )

( ( ) ) ( ) ( ) ∫ ( ) ( )

( )

with boundary

( ) ( ) ( ) ( )

∫( ( ) ( )) ( ) ( )

( )

∫( ( ) ) ( )

where ( ) * +

Proof. Differentiating (3.4), we get

( )

∫ ( ) * ( ) ( ) ( )

( ( ) ( )) ( )

∫ ( ) ( ) + ( )

( ) ( ) ( ) ( )

( )

( ) ( ) ( )

∫ ( ) ( ) ( )

where

( ) ( ) ( )

( ) ( )| ( ) ( )|

Substituting ( ) and ( ) into ( ) ( ) and using ( ) ( ) We obtain the following equation:

∫ , ( ) ( ) ( ) ( ) ( )

( )

∫ ( ) ( ) - ( )

[ ( ) ( )

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[

( ) ( ) ( ) ( )

∫( ( ) ( ) ) ( ) ]

( )

For the equation to be verified for all ( ), the system ( ) ( ) for ( ) must be satisfied

This completes the proof.

Remark 3.1. The boundary conditions ( ) gives the controller in the form

( ) ( ) ∫ ( ) ( ) ( )

and

( ) ( ) ( ) ( )

∫ ( ) ( ) ( )

Lemma 3.1. The transformation ( ) is invertible and the inverse transformation is given by

( ) ( ) ∫ ( ) ( ) ( )

Where ( ) denote the kernel of the inverse transformation and satisfies:

( ) ( )

( ( ) ) ( ) ( ) ∫ ( ) ( )

( )

for ( ) With boundary conditions

( ) ( ) ( ) ( ) ( )

( )

∫( ( ) ) ( )

Proof. The proof can be obtained directly by substituting ( ) into ( )-( ) and using ( )-( ) then we can apply the same approach of theorem ( ) (see [11], [12]).

4. ANALYSIS OF PARTIAL DIFFERENTIAL

EQUATION OF THE KERNEL

We introduce the standard change of variables [13]

we have

( ) . / ( ) ( )

That transforms the system ( ) ( ) into

( ) ( (

) ) ( )

(

)

∫ [ (

) ( )]

( )

( ) ( ) ( ) ( ) ( )

∫( ( ) ( ) ) ( ) ( )

( )

∫ . . / / ( )

where ( ) *

( )+

Integrating ( ) with respect to from to and using ( ) we get

( )

( ( ) ) ∫ ( (

) ) ( )

∫ ∫ ( ) (

)

∫ (

) ( )

Integrating ( ) with respect to from to , we get

( ) ( )

∫ . . / /

∫ ∫ (

)

∫ ∫ ∫ ( ) (

)

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1311701-4949-IJBAS-IJENS @ February 2013 IJENS I J E N S

∫ ∫ . .

/ / ( ) ( )

Using ( ), we obtain

( ) ( ) ( )

( ) ( ) ( ) ( )

∫( ( ) ( )) ( ) ( )

Substituting from ( ) into ( ) we obtain

( ) ( ( ) )

∫ ( (

)) ( )

∫ ∫ ( ) (

)

∫ (

)

( ) ( ) ( )

∫( ( ) ( )) ( ) ( )

Integrating the previous equation using the variation of constants formula and substituting in ( ) we get

( ) ( ) , -( ) ( )

where

( )

∫ . . / /

∫ ( )( . / ( ) ( ))

∫ ∫ (

)

∫ ( )∫ (

) ( )

, -( )

∫ ( )∫ . .

/ / ( )

∫ ∫ . .

/ / ( )

∫ ∫ ∫ ( ) (

)

∫ ( )∫ ∫ ( ) (

)

∫ ∫

. /

( . /

. /) ( ) ( )

Lemma 4.1. The sample path ( ) is uniformly ̈ continuous on , - such that

| ( ) ( )| ( )

Where . /

Theorem 4.1. The series

( ) ∑ ( ) ( )

Converges uniformly in and its sum is a solution of ( ) with a bound

| ( )| ( ( )) ( )

Proof. Let ( )

where is defined in ( ). and denote

̅

, -| ( )| ̅ , - | ( )|

̅

( ) , - , -| ( )| ( )

By using (Lemma 4.1.), we can prove that

| ( )| ( )

We estimate now ( ):

| ( )|

( ̅ )( )

( ̅ ̅ ) ̅

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International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 56 ( ̅ ̅ ̅ )( _{) ( )}

Suppose that

( ) ( )

( )

Then, we get

| | ( )

( ) ( )

So, by induction, ( ) is proved.

The uniqueness of the solution can be proved as follows:

Let ̅( ) ̿( ) are two different solutions of ( ) Then ( ) ̅( ) ̿( ) satisfies the integral ( ) in which and are changed to

Using the above result of boundeness we have

| ( )| _{( )}

Following the same estimates as in ( ) we get

| ( )| ( )

( )

Thus, which means that ( ) is a unique solution to ( ). By direct substitution we can check that it is also a unique solution to the system ( ) ( ).

Thus, we get the following theorem

Theorem 4.2. The system ( ) ( ) has a unique solution. The bound on the solution is

| ( )| _{( )}

Also, the system ( ) ( ) has a unique solution. The bound on the solution is

| ( )| _{( )}

where is given by ( )

5. MAIN RESULT

In this section we find the unique solution of our system ( ) ( ). Equations ( ) and ( ) establish the equivalence of the norms of and in both ( ) and ( ) From the properties of the damped heat ( ) ( ) with ( ) exponential

stability in both and follows.

Furthermore, it can be proved that if the kernels ( ) ( ) are bounded then the sys tem ( ) ( ) with boundary condition ( ) or ( ) is well posed. Thus, we get the following main results.

Theorem 5.1 [19]. for any initial data ( ) ( ) that satisfy the compatibility conditions

( ) ( ) ∫ ( ) ( ) ( )

System ( ) ( ) ( ) with

Dirichlet boundary control ( ) has a unique classical solution ( ) (( ) ( )) and is exponentially stable at the origin ( )

‖ ( )‖ ( ̅) _‖ _‖ _{( )}

Where is positive constant independent of and is

either or .

Theorem 5.2. For any initial data ( ) ( ) ( ( )) that satisfy the compatibility conditions

( ) ( )

( ) ( ) ( ) ∫ ( ) ( ) ( )

system ( ) ( ) for ( ) with Neumann boundary control ( ) has a unique classical solution ( ) (( ) ( )) and is exponentially stable at the origin ( )

‖ ( )‖ ( ̅ ) _‖ _‖ _{( )}

Theorem 5.3 [14, 17, 18]. The solution of the system ( ) ( ), with , is given by

( )

∑ ( ) ∫ ( ) _{( )}

∫ ( ) ( ) ( )

Where ( ) is a probability density function defined on

( ). The Laplace transform of ( ) is given by ([15], [16]).

∫ _{( )}_{( ) ∑} ( )

( )

The initial condition ( ) can be calculated explicitly from ( ) using the transformation ( )

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1311701-4949-IJBAS-IJENS @ February 2013 IJENS I J E N S Substituting ( ) and ( ) into the inverse transformation

( ) and changing the order of integration, we obtain the following result

( ) ∑ ∫ ( ) _{( )}

* ( ) ∫ ( ) ( ) + ( )

where

( ) ( ( ) ∫ ( ) ( ) ) ( )

( ) ( ) ∫ ( ) ( ) ( )

If ∫ ( ) ( ) ( ) is a martingal and hence its expectation equals 0, we can write the system ( ) ( ) in the form:

( )

( ) ( ) ( ) ( ) ( )

∫ ( ) ( )

( ) ( )

( ) ( ) ( ) ( )

where ( ) ( ( )).

then we can apply the same approach of the previous sections (see [20], [21]).

6. APPLICATION (INVERSE OPTIM AL

CONTROL)

In this section, we show how to solve an inverse optimal control problem. We design a controller that not only

stabilizes ( ) ( ) but also minimizes some

meaningful cost functional. For our result, stated next, we remind the reader that

( ) ( ) ∫ ( ) ( ) ( )

We point out that, in this section, satisfies the system ( ) ( ) with a much more complicated boundary condition at , hence, should be understood primarily

as a coordinate transformation from , or a short way of writing ( ).

Theorem 6.1. consider the system ( ) ( ) with the associated functional

( ) ∫ ( ( ) ( )) ( )

where

( ) ( ) , ( ) ∫ ( ) ( )

̅ ( ) ( ) ( ) ∫ ( )

∫ ( ) - ( )

( ) ( ) ( ) ( ( ̅ ))

∫ ( ) ∫ ( ) ( )

And

(| ( | ∫ ( ) ̅)

( )

For ̅ ( ) ( ̅ ) and .

Then the control

̅ ( ) ( ( ) ∫ ( ) ( ) ) ( )

Minimize the cost functional ( ).

Proof. We define the function ( ) such that

̇( ) ∫ ( ) ( )

( )

Using ( ) and ( ), we get

̇( ) ∫ ( ) ( ) ∫ ( )

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International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 58 ( ) ( ) ̅ ( ) ∫ ( )

∫ ( ) ( )

Then, we can rewrite ( ) in the form

( ) ( ) ( ( ) ( ) ̇) ( )

Substituting now ( ) into the cost functional ( ), we obtain

( ) ∫ ( ( ) ( ( ) ( ) ̇)

( ))

( ( ) ( )) ∫ ( ( ) ( ))

( ( ) ( )) ∫( ( ) ̅ ( ))

( )

Using ( ) and ( ), the Cauchy-Shwartz inequality and Agmons‟s inequality

, - ( ) ( ) √∫ ( )

√∫ ( ) ( )

We can easily prove ( ).

So is a positive-definite functional which makes ( ) a reasonable cost which puts penalty on both states and the control form ( ), we now have

̇

( ) ( ) ( )

( ( ̅ )) ∫ ( ) ∫ ( )

Which prove that the controller ̅ ( ) stabilizes the system ( ) ( ) [ and thus the original system ( ) ( )].setting now ( ) in ( ) completes the proof.

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