American Journal of Theoretical and Applied Statistics 2017; 6(5-1): 66-70 http://www.sciencepublishinggroup.com/j/ajtas doi: 10.11648/j.ajtas.s.2017060501.20 ISSN: 2326-8999 (Print); ISSN: 2326-9006 (Online) Optimal Control of a Class of Parabolic Partial Fractional Differential Equations Mahmoud M. El-borai 1 , Mohamed A. Abdou 2 , Mai Taha Elsayed 2 1 Faculty of Science, Alexandria University, Alexandria, Egypt 2 Faculty of Education, Alexandria University, Alexandria, Egypt Email address: m_m_elborai@yahoo.com (M. M. El-borai), abdella_777@yahoo.com (M. A. Abdou), mai_manal2417@yahoo.com (M. T. Elsayed) To cite this article: Mahmoud Mohamed El-borai, Mohamed A. Abdou, Mai Taha Elsayed. Optimal Control of a Class of Parabolic Partial Fractional Differential Equations. American Journal of Theoretical and Applied Statistics. Special Issue: Statistical Distributions and Modeling in Applied Mathematics. Vol. 6, No. 5-1, 2017, pp. 66-70. doi: 10.11648/j.ajtas.s.2017060501.20 Received: July 28, 2017; Accepted: July 31, 2017; Published: August 9, 2017 Abstract: In this paper, the existence of the solution of the parabolic partial fractional differential equation is studied and the solution bound estimate which are used to prove the existence of the solution of the optimal control problem in a Banach space is also studied, then apply the classical control theory to parabolic partial differential equation in a bounded domain with boundary problem. An expansion formula for fractional derivative, optimal conditions and a new solution scheme is proposed. Keywords: Optimal Control, Fractional Order System, Expansion Formula for Fractional Derivative, Parabolic Partial Differential Equations, Functional Analysis, Interior and Neumann Boundary Controls 1. Introduction In an optimal control problem, one adjusts control in a dynamic system to achieve a goal. The underlying system can have a variety of types of equations such as ordinary differential equations (see [1], [2]), partial differential equations [3], fractional differential equations (see [4], [5] and [6]), stochastic differential equations (see [7], [8], [9] and [10]) or Integra-partial differential equations. Many processes in physics and engineering are described by systems of equations in which derivatives of arbitrary order appear (not necessarily integer). mention problems of describing behavior of viscoelastic diffusion – wave problems. As a matter of fact, if one wants to include memory effects, i.e., the influence of the part on the behavior of the system at present time one may use fractional derivative to describe such an effect. In principle, there are two different approaches to “fractionalization” of the dynamic of a system (see [11], [12] and [13]). In the Fust procedure, integer order derivatives are simply replaced by derivatives of real order. In the second approach, considered to be more fundamental from the physical point of view, Functionalization is made on the level of Hamilton’s principle (see [14], [15] and [16]). This approach, however, leads to differential equations of the process involving both left and right fractional and partial fractional derivatives, thus making the effective solution procedure more difficult. For more results concerning fractional calculus and variational principles with fractional derivatives, (see [17], [18], [19] and [20]). In this paper, considering systems of fractional partial differential equation. denoting A (α, t), B(α, t), (t) and (t) π μ are controls. denoting u(x, t) as the state. The state function u(x, t), satisfy the following partial differential fractional equation: 0 x0 t 2 2 2 D D u(x,t) A( , t) D(x, ) u(x,t) - B( , t) E( , ) = A( , t)f( , ) u u x x xt x β α α β α β α β - ∂ ∂ ∂∂ ∂ (1) For (x, t) ∈ R T = {(x, t): x ∈ δ = [0, 1]; 0<t<T}, A (α, t), B (α, t), E(x, β ), D (x, β ) and F (x, β ) ∈ C 1 [0, 1] with conditions 0 u (0, t) u (1, t) = (t) , = (t) , x x u(x,0) u( ) x π μ ∂ ∂ ∂ ∂ = (2)