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.

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1 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. m_m_elborai@yahoo.com b Department of Mathematics, Faculty of science, Tanta university, Tanta. ahmedco31@yahoo.com c Department of engineering mathematics, High institute of engineering, ELsherouk academy, Cairo. walid_hamdy81@yahoo.com. 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 noninteger 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 20 th 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. PRELIMINARIES 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 where

2 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 53 Definition 2.3. Let. The (left) Caputo fractional derivative of order of is defined by with boundary ( ) Where ( ) 3. PROBLEM FORMULATION We consider the following linear fractional integrodifferential equation: ( ) where * + ( ) 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 Proof. Differentiating (3.4), we get * where ( ) + Substituting and into and using We obtain the following equation:, - [ ] ( )

3 [ International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No: / ( ) That transforms the system into ] ( ) For the equation to be verified for all, the system for must be satisfied This completes the proof. Remark 3.1. The boundary conditions ( controller in the form ) gives the [ ] 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:.. / / where * + Integrating with respect to from to and using we get ( ) ( ) ( ) 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]). Integrating with respect to from to, we get.. / / 4. ANALYSIS OF PARTIAL DIFFERENTIAL EQUATION OF THE KERNEL We introduce the standard change of variables [13] we have

4 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 55, -.. / /.. / / 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 : ( ) ( )

5 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 56 Theorem 5.1 [19]. for any initial data that ( ) satisfy the compatibility conditions 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 system with boundary condition or is well posed. Thus, we get the following main results. 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

6 International Journal of Basic & Applied Sciences IJBAS-IJENS Vol:13 No:01 57 Substituting and into the inverse transformation as a coordinate transformation from, or a short way of and changing the order of integration, we obtain the writing. following result * + Theorem 6.1. consider the system with the associated functional ( ) where where, ( ) - ( ( )) If is a martingal and hence its expectation equals 0, we can write the system in the form: And ( ( ) where ( ). then we can apply the same approach of the previous sections (see [20], [21]). For ( ) and. Then the control ( ) Minimize the cost functional. 6. APPLICATION (INVERSE OPTIMAL 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 Proof. We define the function such that Using and, we get We point out that, in this section, satisfies the system with a much more complicated boundary condition at, hence, should be understood primarily

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