Solution of Some Types for Composition Fractional Order Differential Equations Corresponding to Optimal Control Problems

The approximate solution for solving a class of composition fractional order optimal control problems (FOCPs) is suggested and studied in detail. However, the properties of Caputo and Riemann-Liouville derivatives are also given with complete details on Chebyshev approximation function to approximate the solution of fractional differential equation with different approach. Also, the relation between Caputo and Riemann-Liouville of fractional derivative took a big role for simplifying the fractional differential equation that represents the constraints of optimal control problems.The approximate solutions are defined on interval [0, 1] and are compared with the exact solution of order one which is an important condition to support the working method. Finally, illustrative examples are included to confirm the efficiency and accuracy of the proposed method.


Introduction
The idea of fractional derivative dates back to a conversation between two mathematicians: Leibniz and L'Hopital.In 1695, they exchanged about the meaning of a derivative of order 1/2.Their correspondence has been well documented and is stated as the foundation of fractional calculus [1].
A fractional optimal control problem (FOCPs) is an optimal control problem focused on the performance index or the fractional differential equations governing the dynamics of the system or both contain at least one fractional order derivative term.
The formulation and solution of state and variables FOCPs were first established by Agrawal, where the applied fractional variational calculus (FVC) presented a general formulation and solution scheme for FOCPs in the Riemann-Liouville (RL) sense; it was based on variational virtual work coupled with the Lagrange multiplier technique.Since the Caputo fractional derivatives (CFDs) seems more natural and allows incorporating the usual initial conditions, it becomes a popular choice for researchers [2,3].
The Chebyshev polynomials are used to solve composition FOCPs and Chebyshev polynomials  () of degree  are important in approximation theory, since the roots of the Chebyshev polynomials of the first kind, which are also called Chebyshev nodes, are used as nodes in polynomial interpolation.
Agrawal and Baleanu [4] obtained necessary conditions for FOCPs with Riemann-Liouville derivative and then were able to solve the problem numerically.
Elbarbary introduced Chebyshev finite difference approximation for the boundary value problems of integer derivatives [5].In [6], Khader and Hendy studied an efficient numerical scheme for solving fractional optimal control problems.In [7], Akbarian and Keyanpour studied a new approach to the numerical solution of fractional order optimal control Problems.
This paper presents a new approach for computing the approximate optimal control function for optimal control problems with some types of fractional order differential equation such as multi-fractional order differential equation, and composite fractional order differential equations are investigated with first kind Chebyshev approximate polynomial which works very efficiently with fractional orders and optimal control problems.All the discussion and details in addition to the transformations of the steps are given to compute all the useful of the new work.The obtained solutions of the method show that the technique of the approach is very convenient and efficient, and many calculations give high accuracy and may lead to near to exact solution by changing up the order of fractional positive integer numbers.
The multi-fractional differential equations corresponding to optimal control problems and basic theorems have been given with algorithm for multi-composite fractional order optimal control problems.Also, we give illustrative examples for the solution of the approximate systems.
The value of () is observed in detail to explain the activity of solution approximation with mixed boundary conditions.It is important to notice that the calculations are written by using the mathematical software MATHCAD by version (14.0.3.332).
This paper consists of the following sections: in Section 2, some basic definitions and properties of fractional order calculus are introduced (R-L and Caputo fractional derivatives).In Section 3, the shifted Chebyshev polynomials and numerical approximations of CFD and RLFD using Chebyshev polynomials are introduced.In Section 4, we derive the necessary optimality conditions of composition order fractional optimal control problems.In Section 5, we give numerical examples to solve composition FOCPs and show the accuracy of the presented method.Finally, conclusions are presented in Section 6.

Preliminaries
In this section, we give some definitions and properties of fractional derivatives that will be needed later on.

Fractional Order Calculus
Definition 1 (see [8]).The left (LRLFD) and right (RRLFD) Riemann-Liouville fractional derivatives of a function () are defined as ( where the order of the derivative  satisfies  − 1 <  ≤ , and D() is the gamma function.
For  > 0 and  = [] + 1, the Riemann-Liouville and Caputo fractional derivatives are related by the following formulas: (v) The constant function and power function of Caputo's derivative are as follows: (1)       = 0, where  is a constant.
The shifted Chebyshev polynomials are defined as follows [12]: where The analytic form of the shifted pseudo-spectral Chebyshev polynomial    () of degree n is given by the following [10]: where    (0) = (−1)  , and    () = 1.The orthogonality relation is as follows: with the weight function The function () ∈  2 ([0, ]) can be expressed in terms of shifted Chebyshev polynomials as follows: where the coefficients   are given by 3.1.The Chebyshev-Gauss-Lobatto Points.We choose the Chebyshev-Gauss-Lobatto points associated with the interval [0, ], as follows: These grids can be written as Clenshaw and Curtis [13] introduced an approximation of the function (), as follows: where (  ) on the summation means that the first and last terms are to be taken with a factor (1/2).
Theorem 3 (see [2]).The fractional derivative of order  in the Caputo sense for the function () at the point   is given by Such that where ,  = 0, 1, 2, . . .,  and Proof.The fractional derivative of the approximate formula for the function () in ( 22) is given by Using ( 12) and ( 13), in (18) we have then Therefore, for  = ⌈⌉, ⌈ + 1⌉, . . .,  and by ( 12) and ( 13), in for formula (18), we get Now,  − can be expressed approximately in terms of shifted pseudo-spectral Chebyshev polynomial series, so we have where   is obtained from (21) with () =  − .If only the first ( + 1)-terms from pseudo-spectral shifted Chebyshev polynomials in (23) are considered, the approximate formula for the fractional derivative of the shifted pseudo-spectral Chebyshev polynomials is as follows: From ( 29) and (32), we have From (33), the fractional derivative of order  for the function () at the point   leads to the desired result.
where  −1 =   = 0. On the other hand, we have By using the relation  +1 (V) +  −1 (V) = 2V  (V) and from (51), it follows that Inserting  −1 () −  −1 () and ( − ) −1  () as given (52) and ( 54) into (50) and taking (55) into account, we get The Chebyshev coefficients   of    () as given by (55) can be evaluated by integrating and comparing it with (42): with starting values   =  +1 = 0, where   are the Chebyshev coefficients of   ().In this problem, the constraint has multiorder and composition for the three fractional derivatives , , and , where ,  and  > 0. The multi-order fractional optimal control problems refers to the minimization of an objective functional subject to dynamical constraints on the state and the control which have three fractional derivatives order models.

The Necessary Optimality Conditions of Composition Order Fractional Optimal Control Problems 𝑀
The necessary optimality conditions of this type are introduced as follows.

Illustrative Example
In this section, we consider the following linear-quadratic of multi-composite fractional optimal control problem: which is subject to the multi-composite fractional dynamical system, and the boundary conditions, The exact solution for  =  =  = 1 is given by the following: Now, we develop algorithm for solution (75), (76), and (77).
It is based on the necessary optimality conditions from Theorem 6 and implements the following steps.
The results of Table 1 and boundary conditions from (78) can be substituted in (93); we get To find  0 , Now, we substitute  0 ,  1 , and  2 , in (98), to get To compute the approximation solution of the control () from (76), we get Using ( 102 Now, we calculate (91) and using the boundary conditions (78), we have To find  (0.7) 1,1 and  (0.7) 1,2 from Table 1 and we get ( 1 ) = 0, in the same steps of Case 1.

Figure 1 :u
Figure 1: Display the exact and approximate values of state x(t) with different types of , , .

Figure 2 :
Figure 2: Display the exact and approximate values of control input u(.) with different types of , , .

Table 2 :
The numerical values of the state () and control () with  = 2 which converged to exact values are shown.