Journal of Control Science and Engineering
Volume 2018 (2018), Article ID 3767263, 12 pages
https://doi.org/10.1155/2018/3767263

Review Article

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

Sameer Qasim Hasan and Moataz Abbas Holel

Department of Mathematics, University of Mustansiriyah, Baghdad, Iraq

Correspondence should be addressed to Sameer Qasim Hasan; dr.sameer_kasim@yahoo.com

Received 19 February 2018; Revised 9 May 2018; Accepted 28 June 2018; Published 1 August 2018

Academic Editor: Darong Huang

Copyright © 2018 Sameer Qasim Hasan and Moataz Abbas Holel. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

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 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.

1. 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.

2. Preliminaries

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

2.1. 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 , and is the gamma function.

Definition 2 (see [8]). The left (LCFD) and right (RCFD) Caputo fractional derivatives of a function are defined as

where .

2.2. Some Properties of the Fractional Calculus Are Presented in Detail Which Will Be Needed Later on

(i) The fractional operators are linear [9]:

where is one of or and , are real numbers.

(ii) Let is a positive integer, , and if , ) [9], then

(iii) If , we obtain the following [2]:

Moreover, if is a function such that , we have simpler formulas:

(iv) The relation between the RLFD and the CFD is as follows [2].

For and , 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:

In particular, , and, also from right integral, .

where denotes the smallest integer greater than or equal to and . Recall that for , [10].

3. The Shifted Chebyshev Polynomials

The well-known Chebyshev polynomials are defined on the interval and can be determined by the following recurrence formula [11]:

The Chebyshev polynomials can be expanded in power series as follows:

, and denotes the biggest integral less than or equal to

The Chebyshev polynomials are orthogonal under integration over with the weighting function , with orthogonal condition:

where .

In order to use these polynomials on the interval we define the so-called shifted pseudo-spectral Chebyshev polynomials by introducing the change of variable

The shifted Chebyshev polynomials are defined as follows [12]:

The analytic form of the shifted pseudo-spectral Chebyshev polynomial of degree n is given by the following [10]:

where , and

The orthogonality relation is as follows:

with the weight function for

The function 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 , 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

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

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 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 -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.

Theorem 4 (see [2]). Let be the approximation of the fractional derivative of the function as given by (24). Then it holds

where

3.2. Approximation of the Right Riemann-Liouville Fractional Derivative

Using the right Caputo fractional derivative (RCFD) of a function is defined in (4) when and ,

From relation between the RLFD and the CFD (11), when , we have

Use (37) in (38) to obtain

Let be a sufficiently smooth function in and let be defined as follows:

Substituting (40) in (39), we deduce

Now, we approximate , by a sum of shifted Chebyshev polynomials according to

where

Lemma 5 (see [7]). Let be the polynomial of degree as given by (42); then there exists a polynomial of degree such that

Proof. Let be expanded in a Taylor series at

Then,

The assertion follows, if we choose , with an arbitrary constant From (44), we have

Moreover, can be approximated by means of

We express in (47) by a sum of Chebyshev polynomials and provide the recurrence relation satisfied by the Chebyshev coefficients. Differentiating both sides of (44) with respect to yields

To evaluate in (47), we expand in terms of the shifted Chebyshev polynomials:

where (′) on the summation means that the first term is to be taken with a factor

where On the other hand, we have

By using the relation and from (51), it follows that

Such that
Let

Inserting and 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 , where are the Chebyshev coefficients of

4. The Necessary Optimality Conditions of Composition Order Fractional Optimal Control Problems

In this problem, the constraint has multiorder and composition for the three fractional derivatives , and , where and 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 this type are introduced as follows.

Let and be real numbers and and , and let and be two differentiable functions with domain . We consider a general form of multicomposition fractional optimal control problem:

which subject to the multi derivative-composite fractional dynamic control system,

and the boundary conditions,

where , are fixed real numbers.

Theorem 6. If is a minimizer of (58), (59), and (60), then there exists a function for which satisfies the multi-composite fractional optimality conditions:

(i) The amiltonian system is defined as

(ii) The stationary condition is

Proof. We consider the following multi-composite fractional optimal control problem:

Thus,

Suppose that

Using (8) in (65), we get

Evaluating the Euler equation for of (66),

From (67), we get , as follows:

Now, evaluating Euler's equation for of (65):

Finally, the Euler's equation for of (65) is

From (65) it is clear that , then we get

From (69), (71), and (73) we have that

5. 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 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.

Step 1. Compute the amiltonian function:

Step 2. Derive the necessary optimality conditions from Theorem 6:

Step 3. Find the coupled system.

(i) From substitution (82) in (80), we get

(ii) Find control variable from (82) and substitution in (80) to obtain

Using property (6) for (83) and (84), we get the coupled system:

Step 4a. Solve (85a) by Chebyshev expansion method, and use (48) to approximate.

, where and is defined in (47).

Sum both (86) and (87) to get

, where and is defined in (47).

Step 4b. Solve (85b) by Chebyshev expansion method, and use (24) to approximate .

, where is defined in (25).

Using (89) and (90) in (85a) to get

where and are defined in (25).

Step 5. Find the results at the shifted Gauss-Lobatto nodes , from (22) when , and and , in two cases.

Case 1. Let , and .
, and since , then .  

Now, substituting the value of and , in (88), we get

where , where and are defined in (47).

After calculating (92) at the shifted Gauss-Lobatto nodes and substitution in (88) we get

Now, we calculate (91) and, using the boundary conditions (78), we have

Use (25) to evaluate the computation of and , which is shown in Table 1.

Table 1: The values of Caputo fractional derivative when (, and in Case 1) and ( in Case 2 needed later on) are shown.

1
0
1
0
0
0.6722
0.6274
0
0.8154

1
1
1
0
0
0
0
0
0

1
2
1
0
0
0.0387
0.0287
0
0.0572

2
0
1
0.6722
0.6274
-0.6722
-0.6274
0.8154
-0.8154

2
1
1
0
0
0
0
0
0

2
2
1
0.0387
0.0287
-0.0387
-0.0287
0.0572
-0.0572

2
0
2
-0.5777
-0.5210
0.5777
0.5210
-0.7720
0.7720

2
1
2
0
0
0
0
0
0

2
2
2
0.0592
0.0620
-0.0592
-0.0620
0.0396
-0.0396

0.1924
0.1971
0.5185
0.459
0.1402
0.7324

0.3848
0.3942
0.5185
0.459
0.2804
0.7324

when and

The results of Table 1 and boundary conditions from (78) can be substituted in (93); we get

Step 6. Compute approximation solution of and at .

To find and from (23), we get

Now, we substitute , and , in (98), to get

To compute the approximation solution of the control from (76), we get

Using (102), (103), and (100) in (101), we have:

Case 2. Let , and .
, and since , then

Now, we calculate (91) and using the boundary conditions (78), we have

To find and from Table 1 and we get , in the same steps of Case 1.

Now, compute approximation solution of and at , in Case 2:

To find and , from (23), we get

We substitute , and , in (108), to get

To compute the approximation solution of the control , we get

Now, using (110) and (112) in (111), we get

In Table 2, we give the computation of and when values t = 0…1 with different values of orders , for fractional derivative (); see Figures 1(b) and 2(b) and (); see Figures 1(c) and 2(c) for , which converge to exact values with (); see Figures 1(a) and 2(a).

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

0
0
0.0001
0
0
0.0001
0

0.1
-0.283
0.1966
0.078
-2.0660
0.5152
0.78

0.2
-0.416
0.3942
0.0252
-2.9336
0.1920
0.252

0.3
-0.4
0.5928
0.502
-2.9336
0.2078
0.502

0.4
-0.236
0.7925
0.818
-3.0727
0.6427
0.818

0.5
0.078
0.9932
1.196
-2.4852
1.0983
1.196

0.6
0.541
1.1950
1.63
-1.5024
1.5673
1.63

0.7
1.153
1.3978
2.118
-0.1432
2.0459
2.118

0.8
1.915
1.6017
2.658
-0.1432
2.5314
2.658

0.9
2.825
1.8067
3.247
-0.1432
3.0221
3.247

1
3.884
2.0127
3.884
6.0665
3.5170
3.884

(a) Exact solution of for ( and

(b) Approximate solutions of with ( and

(c) Approximate solutions of with (

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

(a) Exact solution of with ( and

(b) Approximate solutions of for ( and

(c) Approximate solutions of with (

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

6. Conclusions

In this paper, we have presented algorithm for solution a class of multi-composite order fractional optimal control problems, in two cases one when and , and the other one when at ( In both cases, the solution is approximated by Chebyshev series. Numerical results for illustrative example show that the algorithm converges from the exact solution when , and we note that the convergent to the exact solution is dependent on increasing of the fractional order of derivative.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

References

K. Oldham and J. Spanier, The fractional calculus theory and applications of differentiation and integration to arbitrary order, vol. 111, Elsevier, 1974.

R. Almeida, S. Pooseh, and D. F. Torres, Computational methods in the fractional calculus of variations, World Scientific Publishing Co Inc, 2015.

C. Tricaud and Y. Q. Chen, “An approximate method for numerically solving fractional order optimal control problems of general form,” Computers & Mathematics with Applications, vol. 59, no. 5, pp. 1644–1655, 2010.

O. P. Agrawal and D. Baleanu, “A hamiltonian formulation and a direct numerical scheme for fractional optimal control problems,” Journal of Vibration and Control, vol. 13, no. 9-10, pp. 1269–1281, 2007.

E. M. Elbarbary and M. El-Kady, “Chebyshev finite difference approximation for the boundary value problems,” Applied Mathematics and Computation, vol. 139, no. 2-3, pp. 513–523, 2003.

M. M. Khader and A. Hendy, “An efficient numerical scheme for solving fractional optimal control problems,” International Journal of Nonlinear Science, vol. 14, pp. 287–296.

T. Akbarian and M. Keyanpour, “A new approach to the numerical solution of fractional order optimal control problems,” Applications and Applied Mathematics. An International Journal, vol. 8, no. 2, pp. 523–534, 2013.

Y. Zhou, Basic Theory of Fractional Differential Equations, World Scientific, Singapore, 2014.

A. B. Malinowska and D. F. Torres, Introduction to the fractional calculus of variations, World Scientific Publishing Co Inc, 2012.

Q. M. Al-Mdallal, M. I. Syam, and M. N. Anwar, “A collocation-shooting method for solving fractional boundary value problems,” Communications in Nonlinear Science and Numerical Simulation, vol. 15, no. 12, pp. 3814–3822, 2010.

N. H. Sweilam and M. M. Khader, “A Chebyshev pseudo-spectral method for solving fractional-order integro-differential equations,” Anziam Journal, vol. 51, no. 4, pp. 464–475, 2010.

B. Obsieger, Numerical Methods III-Approximation of Functions, University-Books, 2013.

C. W. Clenshaw and A. R. Curtis, “A method for numerical integration of an automatic computer,” Numerische Mathematik, vol. 2, no. 1, pp. 197–205, 1960.